Page 1 ... To my mother, Nancy H. Williams (1938-2002), who taught me how to tell time, and to her ... 1 Introduction: Making Meaning in a Material World .
UNIVERSITY OF CALIFORNIA, SAN DIEGO
M AKING M EANING F ROM A C LOCK : M A T E RI A L A R T I F A C T S IN
AND
CONCEPTUAL BLENDING
T I M E -T E L L I N G I N S T R U C T I O N
A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Cognitive Science by Robert Frederick Williams Committee in charge: Professor Gilles Fauconnier, Co-Chair Professor Edwin Hutchins, Co-Chair Professor Charles Goodwin Professor Ronald Langacker Professor Rafael Núñez 2004
Copyright Robert Frederick Williams, 2004 All rights reserved.
The dissertation of Robert Frederick Williams is approved, and it is acceptable in quality and form for publication on microfilm: ________________________________________________ ________________________________________________ ________________________________________________ ________________________________________________ Co-Chair ________________________________________________ Co-Chair University of California, San Diego 2004
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DEDICATION
To my mother, Nancy H. Williams (1938-2002), who taught me how to tell time, and to her namesake, Rose Nancy Williams (2002-), who smiles at the clock. To Jill, who keeps me on time (or tries to, anyway).
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TABLE OF CONTENTS Signature Page........................................................................................................................ iii Dedication .............................................................................................................................. iv Table of Contents................................................................................................................... v List of Figures, Tables, and Transcripts........................................................................... viii Acknowledgments .................................................................................................................. x Vita ........................................................................................................................................ xii Abstract................................................................................................................................. xiii 1
Introduction: Making Meaning in a Material World ................................................ 1 1.1 1.2 1.3 1.4
Cognitive science and the search for meaning............................................... 1 Meaning as conceptualization .......................................................................... 3 Conceptualization as a dynamic process......................................................... 4 Stabilizing conceptualization ............................................................................ 5 1.4.1 Conceptual sources of stability.................................................................. 5 1.4.2 Social sources of stability ........................................................................... 8 1.4.3 Material sources of stability ....................................................................... 9 1.5 Questions for research .................................................................................... 11 1.6 Chapter organization ....................................................................................... 13 2
Approaching the Study of Meaning-Making in Time-Telling............................... 14 2.1
Conceptual framework.................................................................................... 14 2.1.1 Distributed cognition as a way of analyzing cognitive processes....... 14 2.1.2 Conceptual integration theory as an account of meaning construction ............................................................................................... 19 2.1.3 Cognitive artifacts as material anchors for conceptual blends ........... 22 2.2 Domain for study: time-telling and time-telling instruction ...................... 24 2.3 Methods: Cognitive ethnography and semantic analysis............................ 25 2.3.1 Cognitive ethnography ............................................................................. 25 2.3.2 Semantic analysis ....................................................................................... 27 2.4 Specifics of the study....................................................................................... 28 3
The Cognitive Ecology of Time-Telling .................................................................. 31
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3.1 3.2 3.3
The nature of time-telling problems ............................................................. 31 The evolution of time-telling practices ......................................................... 33 Discussion ......................................................................................................... 39 3.3.1 The movement from natural to artificial to systematic ....................... 41 3.3.2 Changing conceptualizations of time ..................................................... 44 3.4 Time-telling today ............................................................................................ 48 4
Reading the Time: The Analog Clock as a Cognitive Artifact.............................. 49 4.1 4.2
4.3 4.4
4.5
4.6
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The analog clock as a composite artifact...................................................... 49 Image schemas required for clock-reading................................................... 51 4.2.1 The part-whole schema ............................................................................ 51 4.2.2 The center-periphery schema .................................................................. 52 4.2.3 The extension schema .............................................................................. 52 4.2.4 The proximity schema .............................................................................. 53 4.2.5 The container schema............................................................................... 54 4.2.6 The source-path-goal schema.................................................................. 54 Perceiving clock structure: pointing and the three dials............................. 55 Reading absolute times.................................................................................... 58 4.4.1 The form of absolute time expressions ................................................. 59 4.4.2 The order of looking for reading absolute times.................................. 59 4.4.3 Clock-reading strategies for absolute times........................................... 59 Reading relative times...................................................................................... 62 4.5.1 The form of relative time expressions ................................................... 62 4.5.2 The order of looking for reading relative times.................................... 62 4.5.3 Clock-reading strategies for relative times............................................. 63 Discussion ......................................................................................................... 69 4.6.1 Experimental evidence ............................................................................. 71 4.6.2 A cognitive functional system for time-telling...................................... 74 4.6.3 Time-telling as a distributed cognitive activity...................................... 78
The Construction of Meaning in Time-Telling Lessons........................................ 80 5.1 5.2
Two forms of the Artifact Puzzle.................................................................. 80 The development of clock-reading skills...................................................... 81 5.2.1 Foundations of time-telling ..................................................................... 85 5.2.2 The conventional order of instruction................................................... 87 5.3 Episodes of instruction ................................................................................... 89 5.3.1 Reading ‘quarter past’: building the Clock Quarters blend ................. 89 5.3.2 Reading ‘X fifteen’: re-conceptualizing the artifact state..................... 95 5.3.3 Reading ‘quarter till’: adding motion and future spaces ...................... 98
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5.3.4
Reading ‘X forty-five’: re-conceptualizing with the proper hour reading.......................................................................................................103 5.4 Discussion .......................................................................................................106 5.4.1 The same anchors for different blends ................................................106 5.4.2 The linking role of gesture.....................................................................106 5.4.3 Constructing meaning from an artifact................................................107 5.4.4 Guiding conceptualization .....................................................................107 5.4.5 Diagramming the construction of meaning ........................................109 5.4.6 Guided conceptualization and communication ..................................110 5.5 Conclusion ......................................................................................................110 6
Student Time-Telling, Errors, and Conceptual Change ......................................111 6.1 6.2
Components of correct performance .........................................................111 Sources of error..............................................................................................112 6.2.1 Using an inappropriate conceptual model...........................................113 6.2.2 Mis-mapping ............................................................................................117 6.2.3 Using an inappropriate procedure ........................................................118 6.2.4 Making a procedural or memory error.................................................120 6.3 Discussion .......................................................................................................120 6.3.1 The prevalence of hidden errors...........................................................120 6.3.2 The relationship between performance and understanding..............121 6.4 Conclusion ......................................................................................................125 7
Conclusion: Functional Systems, Guided Conceptualization, and the Future of Cognitive Science.......................................................................................................128 7.1 7.2 7.3 7.4 7.5 7.6 7.7
Cognition as a dynamic interactive process ...............................................128 The lifespan of cognitive functional systems.............................................129 The use of speech, gesture, and the material environment in guided conceptualization ...........................................................................................130 The dynamics of learning..............................................................................131 Material anchors in cognition and instruction ...........................................132 Material artifacts, conceptual models, and mental imagery .....................134 The internal/external boundary and the future of cognitive science .....136
Appendix: Figures, Tables, and Transcripts ...................................................................138 References............................................................................................................................214
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LIST OF FIGURES, TABLES, AND TRANSCRIPTS
Figure 2.1: The Regatta conceptual blend .....................................................................139 Figure 2.2: A material anchor for the conceptual blend of standing in line ............140 Figure 3.1: The unequal spacing of reference lines on a T-stick................................141 Figure 3.2: Faces of early mechanical clocks ................................................................142 Figure 4.1: The analog clock as a composite artifact ...................................................143 Figure 4.2: Embodied image schemas important to clock-reading ...........................144 Figure 4.3: A normative procedure for reading absolute time...................................145 Figure 4.4: A normative procedure for reading quarter-hour relative time .............146 Figure 4.5: A normative procedure for reading minute relative time .......................147 Figure 4.6: Time intervals for relative times in English ..............................................148 Table 5.1: The development of clock-reading ability...................................................149 Figure 5.1: The Clock Quarters conceptual blend .......................................................150 Transcript 5.1: ‘Quarter past’ lesson..............................................................................151 Figure 5.2: Building the Clock Quarters conceptual blend ........................................154 Transcript 5.2: ‘X fifteen’ lesson ....................................................................................163 Figure 5.3: Building & running a new conceptual blend for the same clock state..166 Figure 5.4: Reading relative vs. absolute time (long hand on right side of clock)...175 Transcript 5.3: ‘Quarter till’ lesson ................................................................................176 Figure 5.5: Elaborating the conceptual integration network to read ‘quarter till’ ...179 Figure 5.6: Conceptual mappings for reading ‘a quarter till seven’ ...........................191
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Transcript 5.4: ‘X forty-five’ lesson...............................................................................192 Figure 5.7: Constructing an alternate blend with a new hour reading ......................195 Figure 5.8: Reading relative vs. absolute time (long hand on left side of clock) .....204 Table 6.1: Matching components of analog clock-reading .........................................205 Figure 6.1: Different image schemas used to read the hour dial................................206 Figure 6.2: Proximity errors in reading the hour..........................................................207 Figure 6.3: Misuse of the container schema to read the hour for relative time.......209 Table 6.2: Summary of conceptual errors in reading the hour dial ...........................210 Figure 6.4: Changes in the conceptual understanding of ‘quarter past’....................211 Figure 7.1: Adding material anchors to stabilize conceptual elements & relations.212 Figure 7.2: Dropping material anchors as mastery is achieved..................................213
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ACKNOWLEDGMENTS
A work such as this would never see the light of day without the support of many individuals and organizations. My sincere thanks to my advisors, Ed Hutchins and Gilles Fauconnier, for sowing the seeds of this research during my first year of graduate study in cognitive science and for helping me to develop the theoretical ground in which they could take root. Thanks to Seana Coulson for mentoring me through my cognitive science master’s project and to Rafael Núñez for sharing his enthusiasm for the study of gesture as a window into conceptualization. Thanks to Ron Langacker for guiding me through the seminal ideas of cognitive semantics and to Chuck Goodwin for modeling how to study meaning-making in situated activity. I have been blessed to have these people on my dissertation committee. Among the research groups in which I have participated during my time at UCSD, I have a special place in my heart for the Center for Research in Language (thanks to Jeff Elman and Liz Bates for providing both wisdom and coffee), the Conceptual Blending Research Group (thanks to Gilles Fauconnier for attracting so many visiting scholars), the Digital Ethnography Workgroup (thanks to Ed Hutchins and other DEWers for launching this community of cognitive ethnographers), and the Distributed Cognition/Human-Computer Interaction Laboratory (thanks to Jim Hollan for emphasizing the human in human-computer interaction). The members of these groups have created supportive communities that make research like this possible. Special thanks to Aaron Cicourel for sage advice and to Chris Johnson, Amaya Becvar, and Morana Alac for model studies of cognition in the wild. I also thank the remaining faculty of the Department of Cognitive Science for their support and my fellow graduate students for help and commiseration when it was sorely needed. This research could not have been done without the generous participation of four teachers and their classes at two elementary schools. My sincere thanks to the teachers, administrators, parents, and students for giving me consent to observe and record their activities. I especially thank the teachers and students who welcomed my presence, my video camera, and my many questions. I marvel at what you do every day. In addition to supportive communities and willing participants, research also depends on financial support. I thank the Center for Research in Language for a training grant early in my period of graduate study and the Harris family for a Hepps Fellowship that freed me from teaching obligations when I was doing fieldwork. The writing of the dissertation was supported by a dissertation fellowship from the Spencer Foundation for Research Related to Education, to whom I will always be grateful. The views expressed in this dissertation are, of course, solely my own, as are any errors that may have crept in, so while all of these groups are to be thanked, I alone am to be blamed for anything that appears here.
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My deepest thanks go to my family and to my wife’s family for supporting my mid-life foray back into postgraduate education. A special note of love and remembrance goes to my mother, Nancy, who was unyielding in her support of my education but who, alas, did not live to see me complete the Ph.D. No sooner had she left us than my wife and I discovered a new life was on the way. The birth of our daughter, Rose, provided that all-important final impetus to finish the degree and return to the world of work (well, paid work, anyway). Thanks to my sister-in-law, Heidi, for assuming childcare duties while I finished my research and for providing special elixirs (her own secret recipes) to ensure successful completion. I especially recommend the ‘Excitement and Enthusiasm Enhancer’ and the ‘Essence of Doctoral Thesis Acceptance’. Finally, none of this would matter without the love and support of my wife, Jill, who amazes me with her willingness to put up with my curious mind and ceaseless wanderings. No matter where we go, as long as she is with me, I am home. And lastly, I thank you, the reader, for gazing upon the marks I have made here and doing the hard work of constructing meaning from them. I hope that you find pleasure in the work and that you experience the joy of a fresh insight somewhere along the way.
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VITA
1985
B.S. in Mechanical Engineering B.A. in German Purdue University, West Lafayette, Indiana
1985 – 1991
Systems Engineer IBM Corporation, Denver, Colorado
1990 – 1993
Graduate Student, School of Education University of Colorado at Denver
1992
M.A. in Secondary Education University of Colorado at Denver
1993
Colorado Teaching License, Secondary English and German
1993 – 1998
Teacher, Green Mountain High School Jefferson County School District, Golden, Colorado
1996 – 1997
Visiting Lecturer, Sprachenzentrum Universität Passau, Germany
1998 – 1999
Teaching Assistant, Linguistics Language Program University of California, San Diego
1999 – 2003
Teaching/Research Assistant, Cognitive Science University of California, San Diego
2000
M.S. in Cognitive Science University of California, San Diego
2004
Ph.D. in Cognitive Science University of California, San Diego
2004
Assistant Professor, Education Lawrence University, Appleton, Wisconsin
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ABSTRACT OF THE DISSERTATION
Making Meaning from a Clock: Material Artifacts and Conceptual Blending in Time-Telling Instruction by Robert Frederick Williams Doctor of Philosophy in Cognitive Science University of California, San Diego, 2004 Professor Gilles Fauconnier, Co-Chair Professor Edwin Hutchins, Co-Chair
How do conceptual processes interact with the material world in cognition? Integrating distributed cognition with cognitive semantics, this dissertation investigates relations between material structures and conceptual operations in the everyday activity of time-telling. Telling time is crucial to modern life but cannot be mastered without instruction. Where did our time-telling artifacts and practices come from? How do we read the time from an analog clock face? How does each new generation learn to perform this activity? To address these questions, the author gathered data on the history of time-telling artifacts and practices, analyzed the conceptual mappings involved in constructing time readings, conducted a cognitive ethnographic study of time-telling instruction in elementary classes (grades 1-3), and analyzed student errors when solving time problems. Episodes of interaction were transcribed for speech, gesture, and manipulations of artifacts by combining conversation analysis conventions with annotated images from the video data. Conceptual integration theory was then used to analyze the construction of meaning step-by-step in the unfolding discourse. The historical analysis shows how our modern ways of measuring and conceptualizing time were shaped by the selective pressures of an evolving cognitive ecology. The analysis of clock-reading details
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differences in the order of looking, imposition of image-schematic structure, and mapping from conceptual models when constructing absolute (‘three thirty’) versus relative (‘half past three’) time readings. In examining how functional systems for time-telling get perpetuated into new generations, the cognitive ethnography uncovers the process of ‘guided conceptualization’: a teacher controls the sequence of action while using artifacts, gestures, and speech to guide learners’ conceptual operations, building and elaborating the blended spaces used to generate time readings. In this process, gestures play a key role in setting up material anchors for conceptual elements. Finally, the analyses uncover sources of student errors, which often remain hidden, and show how conceptual changes can occur after a history of apparently successful performance. By relating actions in the world to the conceptual operations involved in meaning-making, the dissertation explores phenomena that span the internal/external boundary, taking a step toward a more complete and coherent account of human cognition.
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INTRODUCTION: MAKING MEANING IN A MATERIAL WO RL D
1.1
Cognitive science and the search for meaning
The third cognitive revolution is underway. The first cognitive revolution, the one that originated the field of cognitive science, took place a half-century ago. That revolution overthrew behaviorism as the dominant approach to the study of mind and opened the inner workings of the stimulus-response “black box” to scrutiny. The new fields of artificial intelligence and information-processing psychology recast stimulus and response as “input” and “output” and then used abstract formal systems to model the processes linking one to the other. The first thirty years of cognitive science, now referred to as “classical cognitive science,” were dominated by this computational view of mind. Thinking, we were told, is nothing more than symbol processing, governed by syntax and syntactic operations. Meaning is simply the way that symbols map onto the world (or possible worlds). History, culture, context, and affect can be sorted out later, after the fundamental computational architecture of cognition is understood (Gardner 1985). The second cognitive revolution came with the renaissance of connectionism in the mid-1980s (Rumelhart and McClelland 1986). Connectionist (or parallel distributed processing—PDP) models of cognitive processes preserved the notion of inputs and outputs but undermined the dichotomy between symbols and rules. In a connectionist network, data structures and the operations that act on them are not clearly distinguishable. Both are subsumed by the strengths of connections between simple processing units. In learning to relate inputs to outputs, a single network can produce both rule-like mappings and list-like exceptions. Some connectionist networks, called “neural networks,” model the behavior of particular sub-regions of the brain in an effort to produce biologically plausible models of brain functions. The growth of connectionism in cognitive science has gone hand-in-hand with advancements in neuroscience and with the development of statistical, probabilistic, and dynamical system models of cognitive processes. Together, these approaches have prompted increased attention to structure that is inherent in the input and to the learning processes that adapt the cognitive system to make use of that structure. While providing more subtle, flexible, and adaptive models of cognitive processes than the computational models of classical cognitive science, connectionist and statistical models have perpetuated the computational spirit of cognitive science by focusing on how the brain (or an area within the brain) processes information. Even so, by breaking the hegemony of symbol-processing, these approaches have reduced
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the syntactic bias that has long hampered the study of meaning-making in cognitive science. The third cognitive revolution began alongside the second one, also reacting against the current of mainstream cognitive science. Whereas the second had focused on the nature of computational processes in the brain and had questioned the biological plausibility of the dominant symbol-processing account, the third questioned the whole disembodied view of cognition as a computational process acting on representations inside the head. This movement, which came to be known as “embodied cognition,” emphasized the importance of the body and embodied engagement with the world (see Johnson 1987; Lakoff and Johnson 1999 for philosophical accounts). In artificial intelligence, the embodied cognition movement found expression in the new field of cognitive robotics, centered on the argument that it is better to use the world as its own model (Brooks 1995). In cognitive psychology, it surfaced in the ecological approach to visual perception, which emphasized direct engagement of the visual perceptual system with invariants arising from the experience of a body moving through its physical environment (Gibson 1979). Most important for our purposes was the emergence of an embodied “cognitive linguistics,” a radical departure from the formal linguistics that had been an integral part of classical cognitive science. The revolution began with the publication of Lakoff and Johnson’s seminal book Metaphors We Live By (1980), which described how abstract thought is grounded, via conceptual metaphor, in embodied experience. Several years later, Fauconnier published Mental Spaces (Fauconnier 1994, orig.: 1985), in which he introduced simple cognitive structures— mental spaces and connectors across spaces—that (together with a few general principles) solve key problems of reference in linguistics and the philosophy of language. Shortly thereafter, Langacker published his two-volume Foundations of Cognitive Grammar (1987; 1991), which eschewed the traditional division between syntax and semantics, subsuming them both under the notion of “assemblies of symbolic structures” that pair forms with meanings (in the sense described in section 1.2). These and related works, such as Talmy’s groundbreaking studies of force dynamics (1988) and fictive motion (1996) in language and thought, laid the groundwork for the field of cognitive semantics. Cognitive semantics, more than any other area of cognitive science, directly addresses questions about the nature of meaning and how meaning is constructed. In stark contrast to the computational view of cognition as consisting of formal operations on strings of inherently meaningless symbols (meaning being relegated to the way the symbols map onto the world), cognitive semantics treats meaning-making as the central engine of human cognition. The construction of meaning depends upon embodied experience in the world, mental simulation, and extension into abstract domains via analogy, conceptual metaphor, and conceptual blending. These capacities are the focus of cognitive semantic research. Work in cognitive science today provides mounting evidence that cognition is fundamentally embodied and embedded, situated and dynamic. Cognition depends in crucial ways upon embodied engagement with the world and with other cognitive
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beings; so therefore do the meanings we construct. In order to explore how this works in a familiar domain of everyday cognitive activity, we must first be clear what we mean by “meaning.”
1.2 Meaning as conceptualization What is “meaning”? In this dissertation, when we talk about meaning, we are not talking about Platonic ideals that are disembodied, abstract, and non-localizable. Nor are we talking about logical truth conditions, i.e. what the world would have to be like for a sentence to be “true,” nor even about mapping from a logical formula to some constructed world, model, or other formal object. In short, we are not taking the extensional view that the meaning of an expression is some entity in the world that it refers to, nor that it is some possible set of entities that an expression denotes. In keeping with cognitive semantics, our view of meaning is intensional. Meaning is constructed in the mental experience of a cognizer with a brain in a body engaged with a world that includes other cognizers. Our interactions with others and the world play crucial roles in the meanings we construct (and are able to construct), as do our capacities for the conceptual operations described throughout this dissertation. By itself, the world has structure and process but no meanings. Meanings arise in the embodied engagement of the brain-bearing organism with the world and, derivatively, from mental simulation based on such engagement (with subsequent elaboration). The brain is a physical organ of the body, a kind of adaptive controller for embodied activity (Clark 1997); as such, it co-evolved with the body to cope with the exigencies of life in the physical world. Because of this, “all facets of the mental worlds we construct derive ultimately from our embodied experience as physical creatures in the real world” (Langacker 1999). This basic philosophical underpinning of cognitive linguistics is referred to as “experiential realism,” and it underlies the view of meaning that carries throughout this dissertation (for a more detailed exposition of experiential realism, see Lakoff 1987; Lakoff and Johnson 1999). What we are talking about, then, is meaning as conceptualization, as the embodied experience of a conceptualizer building, connecting, and integrating mental spaces structured by conceptual models and anchored by perceptual experience and potential action. Conceptualization is a process. It plays out through time, much of it below the threshold of conscious awareness. Conceptual structures derive initially from early sensorimotor experience interacting with the world (Mandler 1991; 2000; 2004). Conceptualization seems to involve reactivation of neural assemblies in sensorimotor areas (Glenberg 1997; Barsalou 1999). Conceptualization elicits and may be affected by affective responses, although this relation remains largely unexplored. New conceptual structures can arise via conceptual operations like analogy, conceptual metaphor, conceptual blending, and the elaboration of mental simulations. A central goal of cognitive semantics is to spell out these processes in detail.
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If meaning is in the mental experience and associated visceral, emotional, and pre-motor responses of a conceptualizer, then how do we account for the apparent meaningfulness of objects in the world? Are books, letters, newspapers, road signs, photographs, paintings, nods, points, and even spoken sentences simply devoid of meaning? In the absence of a conceptualizer with the appropriate cognitive capacities to construct meaning based on them (i.e. to “interpret” them), the answer must be yes. But this is a less than satisfactory answer, for clearly some things are more interpretable than others; many are even intentionally designed for interpretation. The verb “interpret” suggests that there is some kind of meaning in the object, waiting to be uncovered. The notion that there may be correct interpretations or misinterpretations furthers this view. Clearly, the nature of objects in the world plays some kind of role in the creation or determination of meaning, and the nature of that role is a topic of this dissertation. For the moment, let us be content with the idea that meaning is constructed by a conceptualizer engaged in some activity, and that the surroundings or circumstances contain at best some kind of “meaning potential” (Langacker 2001) that will tend to evoke certain conceptual content and/or a certain construal of that content. This meaning potential can be exploited to a greater or lesser degree by the conceptualizer in the course of activity.
1.3 Conceptualization as a dynamic process I have argued that meaning arises in the activity of the brain in a body engaged with the world. Cognitive scientists talk about states of the brain as if they were discrete entities, but in living beings, brain activity is continuous and ever-changing. Talking about a state of the brain is merely a kind of shorthand for referring to a snapshot of brain activity at a given instant in time. Conceptualization is therefore continuousy in motion. According to Langacker (2001), “conceptual structure is dynamic: it emerges and develops through processing time, this temporal dimension being inherent and essential to its characterization” [emphasis in original]. In examining dynamicity, Langacker is interested in the way conceptualizations are built up over processing time in discourse—a natural effect of the serial order of speech, shifting attention, and the availability of reference points. But dynamicity is also relevant in another way: if conceptualization plays out in processing time (where activity lasts from milliseconds to multiple seconds), then there should be limits to the possible range and complexity of mental simulations. In contrast to “possible worlds,” for example, Fauconnier characterizes mental spaces as “partial structures” that “proliferate when we think and talk” (Fauconnier 1997). Coulson (2001) has suggested that mental spaces are “partitions in working memory.” The limits of working memory are well known (Miller 1956) and should restrict the complexity and number of mental spaces that could be active at any given moment. From an evolutionary point of view, while a capacity for limited mental simulation could provide adaptive benefits by enabling an organism to anticipate actions or outcomes, sustained mental simulation would be maladaptive if it diverted attention from
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dangers in the immediate environment, such as a predator lurking nearby. The persistent daydreamer quickly gets eaten. Perhaps for this reason, mental simulation is fleeting: sufficient for the construction of meaning in ongoing discourse but seemingly insufficient for maintaining complex ideas and insights over time. It seems that the restricted capacity and temporality of conceptualization should place limits on human cognitive achievement. And yet, humans do achieve. How do we account for the complexity of modern life? Given our limited conceptual apparatus, how are we able to learn complex concepts, to carry out complicated computational procedures, and to sustain our species’ cognitive achievements over time, even build on them, across multiple generations?
1.4 Stabilizing conceptualization To understand human cognitive achievement, we need to understand what it is that provides structure and stability to conceptualization. Let us consider three possible sources of stabilizing structure: conceptual, material, and social.
1.4.1 Conceptual sources of stability One of the most well established facts in cognitive psychology is that the amount of information that can be held in working memory is much higher when the information is organized into meaningful chunks (Miller 1956). For example, a random string of more than about seven digits (14927417761271941) is extremely difficult to maintain, but if the digits are recognized as a sequence of meaningful dates (1492, 7/4/1776, 12/7/1941), then they can be recalled with ease (in this example, especially by an American who was alive during World War II). Similarly, strings of random letters (SXCBNJKDWQMD) are harder to maintain than words (CATMANDOGBIG). Some advantage appears even with pronounceable pseudowords (TIGDATMOGNAB), while the greatest advantage comes from a complete meaningful string (BANTHEBIGDOG). What differentiates the meaningful chunks from the random sequences? To begin with, the meaningful chunks are organized by some kind of recognizable pattern. This pattern is what cognitive psychologists call a “schema.” A schema for a date, such as MM/DD/YYYY, is a patterned arrangement of elements (MM, DD, and YYYY) with each element being a number within a particular range (compare 10/12/1992 to 78/52/6498). The schema provides both relational structure and constraints: the first digit in MM, for example, must be a 0 (blank) or a 1 because MM ranges from 1 to 12. A schema for a word, such as (C)V(C), i.e. consonant-vowel-consonant with one or both consonants being optional, can organize a string of letters into a pronounceable pseudo-word (TIG). Meaningfulness can also be enhanced if the information organized by the schema associates with other knowledge, as when a date relates to a significant historical event (07041776) or when a pronounceable string is a familiar word (DOG). Further
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enhancement occurs when a chunk of information organized by a schema can be related to other chunks by a higher-level schema, as when a word occurs as part of a sentence (BANTHEBIGDOG). Schemas can be nested, one within the other, providing organization at multiple levels. A sub-schema is simply a schema nested within another schema operating at a larger scale. Schemas give meaningful structure not only to entities, but also to events. Recurring patterns of event structure give rise to schemas called “scripts,” which organize expectations about participants, the roles they play, and the sequence of actions that occur. A classic example is the restaurant script (Schank and Abelson 1977), which has roles for customer and waiter, and a sequence of activities like ordering, eating, and paying, each of which may be organized by its own schema (sub-schema). Schemas may substitute for one another within scripts, providing variations for such things as eating in a cafeteria vs. dining in a restaurant, or paying by credit card vs. paying cash. The impact of such scripts on our behavior and on our expectations for the behavior of others tends to go unnoticed until we encounter a situation where the script is disrupted, as often happens the first time a traveler enters a restaurant in a foreign country. Something akin to schemas or scripts has been posited in the various subfields of cognitive science. In artificial intelligence, knowledge structures with related slots, possible fillers, and default values are called “frames” (Minsky 1995). The term “frame” has also been used in linguistics to refer to schemas that organize semantic knowledge (Fillmore 1982). A connectionist take on schemas (Rumelhart, Smolensky et al. 1986) emphasizes how schemas can be flexibly instantiated in each new set of circumstances. In contemporary cognitive science, the terms “frame” and “schema” tend to be used interchangeably (for a review of frames and frame-based models in cognitive science, see chapters 2 and 3 of Coulson 2001). In this dissertation, the term “conceptual model” will refer to what has been called a “schema,” “script,” or “frame”; this is akin to what Lakoff (1987) calls an “idealized cognitive model” or ICM. The term “cultural model” (D'Andrade 1989) will be used to emphasize that a particular conceptual model is intersubjectively shared by members of a social group. The term “schema” or “image schema” will be reserved for embodied image schemas (Johnson 1987; Lakoff 1987), pre-conceptual patterns for basic experiences like containment or motion along a path. Embodied image schemas, further described in chapter 4, are believed to form the foundation of the conceptual system (see Mandler 1991 for one account of how image schemas are formed from perceptual schemas; see Barsalou 1999 for a contrasting view that dispenses with the perceptual/conceptual distinction). Image schemas and conceptual models structure the mental spaces used to construct meaning and create expectations that guide behavior. Shared image-schematic structure also plays a key role in conceptual blending (discussed in chapter 2) by providing a basis for linking input spaces in a conceptual integration network. Because humans have the same body configurations and inhabit the same world, embodied image schemas are universal to all normally developing human beings. Conceptual models, on the other hand, depend upon the activities of the
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cultural group and the meanings that group members create for and with one another. Embodied image schemas provide a minimal basis for human intersubjectivity, but shared conceptual models (cultural models) provide the basis for communication and mutual understanding. They enable us to distinguish winks from twitches (Geertz 1973). They create expectations that can make silence or inaction as meaningful as noisy activity. Without intersubjectively shared conceptual models, communication would be little more than stimulus and response, and the richness of human cultural life would be an impossibility. Conceptual models relate to other conceptual models, forming webs of mutual support and constraint. Yet this does not preclude the possibility that some conceptual models may contradict others, especially when applied to particular situations, so that the same content might be construed in alternate, mutually exclusive ways. Indeed, this may be the source of many misunderstandings. While conceptual models provide coherence and stability to conceptualizations, conflicting models can create incoherence or paradox. These conflicts provide an opportunity for originality or insight by creating exactly the conditions under which conceptual blending is most challenging and potentially most fruitful. To see an example of the power of conceptual models in providing stability to conceptualization, consider the history of research into human reasoning based on the Wason card-selection task (as discussed in Hutchins in press). In the original task (Wason 1966), subjects were shown four cards and asked which cards would need to be turned over to test the validity of a rule. For example, a subject shown four cards displaying A, 3, 2, and K is asked which cards should be flipped to test the rule “If a card has a vowel on one side, it has an even number on the other.” In this case, subjects most commonly choose the cards displaying A (a vowel) and 2 (an even number), instances of the two categories mentioned in the rule. Flipping over the A card is correct: an odd number here disproves the rule. But flipping over the 2 card is not: either a vowel or a consonant could occur without violating the rule. The other correct response is to flip over the 3 card (i.e. not an even number): if this card has a vowel on the other side, it also disproves the rule. The answer the subjects choose correctly (the A card) is an example of modus ponens reasoning: if p, then q; p; therefore q. The answer that many subjects miss (the 3 card) is an example of modus tollens reasoning: if p, then q; not q; therefore not p. In general, the research shows that subjects have little difficulty with modus ponens, but they often err with modus tollens, which requires additional mental manipulation: shifting attention to q, then to the inverse of q, then reasoning back to the inverse of p. Johnson-Laird, Legrenzi, and Legrenzi (1972) later discovered that subjects’ performance on modus tollens improves when tested with more realistic content like postage stamps and letters. In a review of this research, Johnson-Laird and Wason (1977) attribute this improvement to familiarity with the materials. In a clever study, D’Andrade (1989) recast the logic problems into two types. The first, exemplified by the rule “If Roger is a musician, then Roger is a Bavarian,” has familiar content, but the link between p and q is arbitrary. The second, exemplified by the rule “If this is a garnet, then it is a semi-precious stone,” also has familiar content, but the link between p and q is
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supported by a culturally coherent conceptual model. D’Andrade found that subjects were able to do modus tollens correctly for the coherent problems but not for the arbitrary ones. D’Andrade argued that without a culturally coherent conceptual model linking p and q, subjects lost the contingency between p and q when they made the perspective shift necessary for modus tollens. Here we see an example of how a conceptual model stabilizes the representation of conceptual relations to support reasoning, and how the lack of such a model impairs cognitive performance.
1.4.2 Social sources of stability Where do intersubjectively shared conceptual models, i.e. cultural models, come from? Cultural models are intimately tied to cultural practices, the activities engaged in by members of a social group. In some cases, a group member develops a conceptual model through directly interacting with others in the course of an activity. This depends upon the individual assuming a role in the participation framework of the activity and then performing that role with resulting success or failure. Because the activity is patterned (reflecting in part the extant conceptual models of other participants), the behaviors of others will prompt and reinforce appropriate behaviors in the novice (or, alternatively, discourage or punish inappropriate behaviors). As the novice aligns his behavior to the group, performs successfully, and comes to understand the performance, he develops conceptual models about the way the activity works and how to participate in it. Participation frameworks are relevant to everyday social activities like turn-taking in conversation as well as to specialized group activities with differentiated but interrelated roles, such as team navigation aboard naval ships (Hutchins 1995). In many cases, the novice performer receives direct instruction or assistance from others, helping him to perform beyond his current level of competence until he develops the necessary skills and conceptual models to perform unaided. He may even repeat verbal instructions to himself while performing the activity, using them as a resource to guide his behavior while relevant conceptual models are still being formed (see chapter 7 of Hutchins 1995 for one such account). If, as suggested here, our own embodied experience forms the foundations of our conceptual system and our participation in cultural activities extends it, then we must confront the problem that it is simply not possible to participate directly in every activity of significance to the cultural group. There must be other ways of developing cultural models. D’Andrade’s subjects, for example, were still able to use a cultural model to reason about garnets and semi-precious stones even when they had no direct experience with garnets to draw upon. How is this possible? One answer is that members of a cultural group can develop useful conceptual models simply by observing and interpreting the behaviors of others engaged in activity. Another, more important answer, and the one relevant to D’Andrade’s example, is that members of a cultural group form conceptual models as they encounter representations of the actions and thoughts of others in the stories that group members tell one another (or act out, or write, or portray in still or moving images,
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and so on). In this way, individuals can learn from happenings that took place at other times and/or in other places, or which in fact never took place at all. The richness of human communication generates a great variety of mental experience beyond that which any individual could generate alone through direct physical interaction with the world. Indeed, the ways in which language prompts and guides (but does not uniquely determine) conceptualizations are a primary focus of cognitive linguistic analysis. The social organization of activity and the richness of human communication support the formation of conceptual models and greatly expand the repertoire of conceptual models an individual has available to structure and stabilize conceptualizations. In addition, they make possible an increasing diversification of activity, a distribution of physical and cognitive labor, while simultaneously helping to ensure that the most important and culturally relevant conceptual models will be intersubjectively shared. And yet, social organization is relatively invisible, and communication ephemeral. While social interaction shapes conceptualization in episodes of activity, this support vanishes as soon as the interaction is over. Even when social interaction helps to form conceptual models, these, too, are relatively invisible and disappear when members of a cultural group die out. To look for further sources of stability in the dynamic process of conceptualization, we need to examine the residue of cognitive activity, the marks it leaves upon the world.
1.4.3 Material sources of stability As humans evolved, their survival depended upon close attention to the environment, including changes in weather or climate, the availability of plant and animal sources of food, the movements of predators and presence of other dangers, and the activities of other hominids. Human activity left its own traces in the environment: paths through the landscape, waste products, and remnants of food, tools, and shelters. While the earliest hominids followed food sources and relied upon simple tools of stick and stone, Homo sapiens eventually built settlements and civilizations, modifying the environment in more extreme ways and building myriad structures and tools to support their activities. The patterns inherent in these physical structures and tools reflect important aspects of conceptual models and social organization, but unlike conceptual and social structures, these physical structures are durable parts of the world. They are ready-at-hand to support human activity, including human conceptual activity. Looking at the physical environments humans occupy, we immediately see cultural values, social distinctions, and the nature of activities reflected there. Consider the church spires that rose above medieval European towns; compare these to the commercial skyscrapers that tower over modern American cities. Human life in these two settings is dominated by different values and different sorts of activities. Likewise, but on a smaller scale, consider a lecture hall, a café, a restroom, or a parking garage. Each is divided into distinct spaces, demarcated in some way, and set in a particular arrangement with respect to one another. Each contains barriers,
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access points, and mechanisms for controlling access. The structures of these spaces afford and constrain particular sorts of activities. The objects within these spaces trigger certain activities and the conceptualizations that accompany them. The persistence of these cues helps to maintain or restart appropriate conceptual processes throughout the course of activity. It is not just permanent structures like buildings that have this effect, but also temporary arrangements of objects in space (Kirsh 1995). The arrangement of objects on the desk in front of me as I type tells a story about my activities. Near at hand are books and articles I am referring to. Within reach are a stack of notepaper and a cup containing pens, pencils, and markers. To one side are piles of papers and folders, each pile related to one of the projects I am involved in. To the other side are some photos and treasured objects that remind me of loved ones. In the drawer of the cabinet behind me, out of view, are articles in folders arranged in alphabetical order by author’s last name. Next to that is a bookcase with shelves of related books also arranged in order by author’s last name. If I focus my attention to the computer screen in front of me, I am still confronted by a spatial structure that reflects an enormous number of conceptual groupings and distinctions. These and many other aspects of my environment serve as resources that help me to structure my work activity and the thought processes that accompany it. In addition to structuring their environments for action, humans also create specialized objects or tools to support particular activities, including cognitive activities. Sometimes we make opportunistic use of available structure, even our own bodies, to perform activities like counting or measuring (e.g. holding up fingers while adding or subtracting, or pouring “two fingers” of whiskey). Often we create short-lived structures as temporary supports for cognitive activity, such as tying a string around a finger as a reminder to perform some future action, sketching a map in the sand while planning a route, or writing a grocery list to guide our shopping at the store. And sometimes humans create durable, long-lasting tools, such as a measuring cup, a scale, or a compass, which we use to help us accomplish cognitive activities that recur again and again. We also create durable, portable representational artifacts like maps and charts that can be brought into coordination with other similar artifacts to support complex systems of cognitive activity like navigating around the world (Latour 1986). Some cognitive artifacts, like the Rosetta stone, even outlast the civilization in which they were used. The artifacts that humans create and use clearly support our cognitive activity by serving as external memories and computational tools. But do they also support our construction of meaning? We might expect material artifacts to serve as inputs to conceptual processes, even to trigger conceptual operations, but does the material world play a supportive, stabilizing role in the dynamic process of conceptualization? Hutchins (in press) claims that it does. Hutchins argues that material structures are used to anchor conceptual models during cognitive activity. In Hutchins’ examples, a structured configuration of material elements is used to anchor conceptual elements and relations (a conceptual model) so that the material elements become proxies for the conceptual elements. This makes it possible to act on the
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material structure to carry out some type of reasoning or computation. Hutchins’ claim is that these “material anchors” stabilize the representation of conceptual relations, just as the familiar cultural models stabilized the relations between the premises for the modus tollens task in D’Andrade’s tests of logical reasoning. Material anchors like these would be particularly helpful for complex, multi-step meaningmaking or computational operations.
1.5 Questions for research The factors described here—conceptual, social, and material—are clearly interrelated and likely to interact in complex ways. Together, they make up the cultural component of human cognition and produce what Tomasello (1999) calls the “cultural ratchet effect,” the ever-increasing complexity of human cognitive life. Understanding the dynamics of these interactions is a long-term goal of cognitive scientific research. To begin to explore how conceptual models, social interaction, and material artifacts support conceptual operations, we can start by considering a test case: a familiar everyday cognitive activity that depends upon a material artifact and that is complex enough that it cannot be mastered without instruction and practice. For this dissertation, I have selected the domain of daily time-telling. The reasons for this choice are elaborated in chapter 2 (section 2.2). The fundamental idea is that telling time involves a person interacting with an artifact (a clock), thus providing an opportunity to explore relations between conceptual models and material artifacts in the performance of a cognitive activity. In addition, learning to tell time requires instruction, which provides an opportunity to observe, record, and analyze social interactions and concomitant manipulations of material structure involved in establishing time-telling practices in a new generation of learners. The overarching questions guiding the research are the following: a. How do conceptual models and material artifacts interact in cognitive processes (here construed as involving embodied engagement with the world)? b. How is meaning constructed in situated activity? The general research program thus has a twofold focus: (1) understanding how cognitive activities are accomplished in real-world contexts; and (2) understanding the meanings that participants construct as they accomplish those activities and how it is that they construct those meanings. The present study focuses on the use of a cognitive artifact—a clock—to perform what might seem to be a simple cognitive activity: telling time or reasoning about time relations. As we will see, clock-reading turns out to be fairly complex. Examining the interplay of material structures and conceptual processes in time-telling should help us to address two important related questions:
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c. How does conceptualization render the material world meaningful? d. What role does the material world play in supporting meaning-making? Question (c) gets at the heart of the human experience: living in a world of meaning. Question (d) addresses the hypothesized role of the material world in stabilizing or supporting conceptualization, and thus acting as more than mere input to an internal symbolic process. Adapting these general research questions to the study of time-telling produces the following specific research question: 1. How do conceptual structures and material structures interact in the cognitive process of telling time from an analog clock? If we consider the time-teller and clock as together composing a functional time-telling system, this raises the question of how such a system comes into being. The question of origins relates both to the historical development of time-telling and to the perpetuation of time-telling in each new generation. With that in mind, we also take up the following questions: 2. Where did today’s time-telling practices, artifacts, and conceptual models come from? Why do they have the form that they do? 3. How do a teacher’s actions, gestures, and speech during instruction shape the construction of meaning by learners and support the establishment of functional time-telling systems? Any analysis of time-telling as a cognitive activity should also account for ways that the system breaks down or changes over time: 4. How can we account for learners’ errors in time-telling? 5. How can we account for differences in learners’ conceptual understanding and for changes in conceptual understanding that occur during performance? By taking a single domain of cognitive activity as a test case, we have the opportunity to examine a range of questions about the interrelation of conceptualization, material structure, and social interaction in real-world cognitive activity. In order to be plausible, our analysis needs to address all of these questions in a single, coherent framework that accounts for the empirical data in this and other studies.
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1.6 Chapter organization Chapter 2 lays the groundwork for what follows. It describes the conceptual frameworks used in the study, namely distributed cognition and conceptual integration theory. It also presents the reasons for choosing time-telling as a test domain and details the methods employed in the investigation, specifically cognitive ethnography and semantic analysis. The subsequent four chapters present the findings of the study. Chapter 3 discusses the evolution of time-telling systems and how these were shaped by a changing cognitive ecology, while providing the historical context for the artifacts and practices used to tell time today. Chapter 4 considers the analog clock as a cognitive artifact and examines in detail how time readings in both absolute form (“ten thirty”) and relative form (“half past ten”) are constructed from the clock face. Chapter 5 analyzes time-telling instruction, showing how a teacher’s use of artifacts, gestures, and speech shape learners’ construction of meaning. Chapter 6 uses the account developed in the preceding chapters to examine sources of error in student time-telling and the kinds of conceptual changes that can occur even after students appear to be performing correctly. Finally, Chapter 7 brings together the themes developed in the preceding chapters and discusses their implications for the future of cognitive science.
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2
APPROACHIN G THE STUDY OF M EANING MAKING IN TIME-TELLING
2.1 Conceptual framework Any research study approaches its subject from the perspective of a conceptual framework. The present study uses two theoretical frameworks, one rooted primarily in cognitive anthropology and the other in cognitive linguistics, but both expanding beyond their fields of origin to encompass broader areas of cognitive science. The cognitive anthropological framework is distributed cognition, the study of how cognitive processes are distributed across the individual and aspects of the material environment, across multiple individuals in interaction, and across time. The cognitive linguistic framework is conceptual integration theory, an outgrowth of mental space theory, which describes how meaning is constructed in networks of mental spaces and the principles that govern these meaning-making processes. In this dissertation, I attempt to integrate these two theoretical frameworks to provide a more comprehensive understanding of cognition and meaning construction in situated activity.
2.1.1 Distributed cognition as a way of analyzing cognitive processes There is no evidence that the brains of humans living in modern high-tech societies differ in any way from the brains of humans living in tribal societies with primitive technology. Indeed, there is no evidence that our brains differ at all from those of our ancestors who lived 50,000 years ago. We are Stone Age people living Space Age lives. How is this possible? While cognitive science has long focused its study on processes internal to the individual brain, it’s clear that any reasonable understanding of cognition will demand that we account for processes that go beyond the individual and that account for the complexity of the world we live in today. One approach that aims to do just that goes by the name of “distributed cognition.” The fundamental argument of distributed cognition (Hutchins 1995; 1996; 2001) is that to understand processes like memory, learning, reasoning, and problem solving, we need to expand our unit of analysis beyond the individual mind to include the social and material contexts of cognitive activity. Distributed cognition takes as its unit of analysis entire cognitive functional systems. A local functional system consists of an individual interacting with one or more artifacts to perform
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some cognitive activity. A global functional system consists of multiple local functional systems in interaction; this makes social organization and patterns of communication defining elements of the global functional system’s cognitive architecture. The global functional system may have cognitive properties that are very different from those of the individual participant. Even the local functional system consisting of task performer and cognitive artifact may have cognitive properties that differ from those of the individual whose skull has been the traditional boundary for cognitive scientific inquiry. From the perspective of distributed cognition, cognitive processes are distributed in three important ways: (1) across internal and external representational media; (2) across members of a social group; and (3) through time, in that the products of earlier cognitive processes change the nature of later ones. The classic study of distributed cognition is Hutchins’ detailed cognitive ethnography of team navigation aboard a navy ship (Hutchins 1995). A navigation team, at its most basic, consists of five members: a plotter, a bearing recorder, a fathometer reader, and two pelorus operators (left and right). Each team member interacts with particular tools—each tool being the product of a history of cognitive activity—to instantiate a local cognitive functional system. Information flows from one local functional system to another within the global functional system as representational states are propagated across various representational media. For example, a pelorus operator uses a telescopic sighting device called an “alidade” to bring a hairline, a visible landmark (e.g. a lighthouse), and a gyrocompass card with compass readings into alignment in order to produce a representation, in the form of a three-digit number (e.g. 242 degrees), of the direction of the ship’s position from the landmark. Through a phone circuit, the pelorus operator tells the reading (transforming it into a series of speech sounds) to the bearing recorder, who records the bearing (transforming it into written digital form) in a log book, a form of external memory. The plotter reads the number from the log book, transforms it into the position of a locking arm on a special protractor called a “hoey,” brings the hoey into alignment with the grid on a navigation chart (also a product of a long history of cognitive activity), and then draws a line on the chart, producing a line of position. The entire navigation team, their technology, their organization, and their communication make up the global functional system that converts a relative direction of the ship’s position into a line on a navigation chart. Repeating this process and others like it brings information from different sources into the same representational space—the chart—and layers constraints upon one another to perform a computation—in this case, fixing the ship’s position. In this example, we see cognition “as computation realized through the creation, transformation, and propagation of representational states” (Hutchins 1995). A representation of the ship’s relationship to the landmark is propagated from the alidade to the pelorus operator to the bearing recorder to the bearing log to the plotter to the hoey to the chart. The representation is transformed as it flows from one medium to another, including into and out of the working memories of human beings and into the states of various physical media. A distributed cognition
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analysis tracks the flow of information across media both inside and outside the individual as it explicates how the cognitive work of the system gets accomplished. Unlike studies of situated action, distributed cognition does not stop at the boundary of the skull. Its systems-level approach begins with observable phenomena and works inward, refining “a functional specification for the human cognitive system” (Hutchins 1995). Taking a systems-level view of cognition as a distributed process makes it possible to preserve the notion of cognition as computation and to compare different functional systems that address the same computational problems. As an example of this, Hutchins compares Micronesian navigation to Western navigation (chapter 2 of Hutchins 1995). On the face of it, the two systems seem entirely different. Westerners use compasses, charts, chronometers, and other specialized measuring and calculating tools to navigate on the open ocean. Micronesians use none of these, yet they still manage to cross vast expanses of open ocean in canoes and successfully make landfall on tiny, remote islands. Micronesian navigators do make use of cognitive artifacts, including, for example, the paths of stars across the night sky (linear constellations), and real or imaginary reference islands (“etak” islands), whose position from the canoe is referenced to temporal and spatial landmarks (sunrise, noon, sunset, and the linear constellations) at various points in the voyage. Using Marr’s three levels of analysis for information-processing systems—computation, representation/algorithm, and implementation (Marr 1982)—Hutchins demonstrates that while Westerners and Micronesians use completely different procedures and representations implemented in profoundly different ways, their systems of navigation are nonetheless computationally equivalent. They use the same constraints to answer the same basic questions. This equivalence can only be seen if the unit of analysis of the cognitive process includes the entire socio-technical functional system. One cognitive artifact employed by the Micronesians was an inspiration for the present study. A Micronesian navigator gazing at the horizon at dawn mentally superimposes a pattern of stars onto the visible scene, using the position of the sunrise to anchor the star pattern in imagination. He then uses this combined physical/mental cognitive artifact as an analog computer to figure out the present position of the etak island relative to his canoe. Hutchins presents this as an example of “situated seeing.” The pattern of stars in the night sky becomes a computational tool through the formation and application of particular conceptual models. Near the equator, stars rise more vertically out of the horizon. Stars that rise at the same point can be mentally connected to form linear constellations that cross the night sky. These are referenced to their most salient stars or groupings. The linear constellations stand in consistent spatial relation to one another, supporting the formation of a conceptual model linking the entire night sky into a sidereal compass. Considerable experience interacting with this highly structured physical-mental object makes it possible for the Micronesian navigator to instantiate
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the entire structure in imagination.1 Once this level of familiarity has been achieved, then merely anchoring one corner of the compass can fix the entire compass in the navigator’s mind. The navigator can use something as simple as the sunrise to superimpose the sidereal compass onto the horizon even when no stars can be seen, thereby instantiating a powerful and useful cognitive artifact. To what extent does such situated seeing permeate everyday cognitive activity? How do such combined material-conceptual cognitive artifacts get established? These are important topics of research. The distributed cognition approach has been fruitfully applied to the study of teams of people interacting with technology to perform complex activities, including team ship navigation (Hutchins 1995), jet piloting (Hutchins 1996; Holder 1999), air traffic control (Halverson 1995), and even fishing (Hazlehurst 1994). A distributed cognition analysis tracks such things as the flow of information through the system, the transformation of representations, and the layering of constraints to produce computational outputs. But what does distributed cognition have to say about the construction of meaning? In their pioneering paper “Constructing Meaning from Space, Gesture, and Speech” (1997), Hutchins and Palen analyze the communications of the cockpit crew during a training exercise in a high-fidelity flight simulator, focusing on the second officer’s explanation of an apparent problem in the fuel system. In their analysis, they explicitly reject the view that gestures and the setting primarily provide context for the interpretation of speech. Instead, they show how space (i.e. structure in the setting), gesture, and speech are combined in the construction of complex multilayered representations. In the second officer’s explanation, the gestures are superimposed on the spatial layout of the fuel panel (a meaningful artifact to the members of the flight crew) and the speech is superimposed on the gesture. The gestures provide indexical reference to parts of the panel, enactments of actions on the panel, and representations of events in the fuel system, while the accompanying verbalizations signal how these actions are to be interpreted (i.e. as relating to the fuel panel or the fuel system it represents), place the actions in a temporal framework (as relating to the past or present), indicate conceptual relations (like logical disjunction), and express the second officer’s epistemic relation to the actions (i.e. his beliefs or stance toward what is being portrayed). What is important is that none of the layers—space, gesture, or speech—is complete or coherent by itself. The second officer’s explanation is a communicative performance interweaving visual and auditory elements into a complex representational object. “Granting primacy to any one of the layers of the object,” Hutchins and Palen argue, “destroys the whole” (p. 39). A similar coupling of environmental structure with gestures and speech will be evident in the time-
1
I encourage the reader to try this. Imagine the night sky and attempt to point to the locations of particular stars or constellations. Try to draw the night sky from memory on a piece of paper. Personally, I can produce only a few constellations and can locate very few correctly with respect to one another.
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telling instruction analyzed in chapter 5.2 Where the dissertation extends Hutchins and Palens’ work is by relating these manipulations of the environment, gestures, and speech to the conceptual operations involved in the construction of meaning. For this we need a theory of meaning construction, the subject of section 2.1.2. Distributed cognition’s other foray into the subject of meaning occurs in chapter 7 of Cognition in the Wild (Hutchins 1995). In this chapter, Hutchins describes a novice attempting to use a written procedure to guide his performance of a task. In examining how the novice uses both the written procedure and the task world to construct meanings in an internal semantic medium, Hutchins assumes “that the meanings that appear in this medium are image-like and that they are the same kinds of structures that would result from actual performance of the task” (p. 299). He then goes on to say: It is difficult to place the meaning of the step cleanly inside or outside the person, because some component of the meaning may be established by a kind of situated seeing in which the meaning of the step exists only in that active process of superimposing internal structure on the experience of the external world. That is, at some point in the development of the task performer’s knowledge the step may not have a meaning in the absence of the world onto which it can be read. Perhaps the meaning of a step can reside cleanly inside a person only when the person has developed an internal image of the external world that includes those aspects onto which the mediating structure can be superimposed. (p. 300) Here we see an instance in which the external world supports the construction of meanings that could otherwise not be achieved. Once experience has led to the formation of relevant conceptual models and the ability to image familiar structure, similar meanings can be constructed without the direct support of the world. Not surprisingly, distributed cognition sees the construction of meaning as a distributed cognitive process, depending in crucial ways upon interaction with the material world and with other meaning-makers, and building on the products of past cognitive activity. This does not discount the fact that meaning-making depends upon individual capacities for conceptualization and that it is the human participants in cognitive activity who experience meaningfulness:
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Like the example analyzed in Hutchins and Palen (1997), the episodes of time-telling instruction analyzed in chapter 5 are explanatory discourse. To what extent the tight coupling of space, gesture, and speech varies in other forms of discourse remains to be investigated.
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Meaning, attributed by intentional beings—indeed, by the brains of those intentional beings (in bodies, in environments, in cultures)— can be crucially public, distributed, and indispensably dependent upon objects and situations, but it is still the people, not the tools or the spaces, who find things meaningful. (Turner 2001) The conceptualizer and the world are interrelated. The world has structure, but it takes the conceptualizing activity of a mind (in a body in the world) to turn that structure into meaning. What meanings the conceptualizer can produce depend upon the world he inhabits. The process of constructing meaning can change the internal structure of the conceptualizer, and the conceptualizer can change the structure of the world. Both of these make new meanings possible. Distributed cognition can take us part of the way toward understanding this process. What we still need is a theory of the conceptual operations involved in constructing meaning.
2.1.2 Conceptual integration theory as an account of meaning construction This, in fact, is the power of the imagination, which, combining the memory of gold with that of the mountain, can compose the idea of a golden mountain. - Umberto Eco, The Name of the Rose (p. 188) If we wish to examine how meaning is constructed by participants in the functional systems described by distributed cognition, then we need a theory of conceptualization. The semantic analyses in the current study are grounded in conceptual integration theory (Fauconnier and Turner 1994; 1998; 2002), a cognitive linguistic account of how meaning is constructed in networks of interconnected mental spaces, and of the principles that govern these operations. Conceptual integration theory is commonly referred to as “conceptual blending,” focusing attention on the blended spaces whose emergent properties have been the principal focus of study. The simplest conceptual integration network begins with elements in a mental space and uses a conceptual model (a “frame”) to interrelate them, creating a structured conceptualization in a blended space. An example of such a “simplex network” is construing an interaction between two people as an argument. The argument frame assigns the participants to roles in particular relation to one another, creating an interpretation of the nature of the observed activity. Framing gives structure to the blended space and brings in related knowledge; the dynamic nature of conceptualization leads to further elaboration or “running the blend” to generate inferences. Slightly more complex is the “frame-compatible network,” in which an already framed space receives additional structure from a second frame that is
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recruited without displacing the first, as when one discovers that two people who are business associates are also lovers. The lovers frame does not directly clash with the business frame, but it does lead to additional inferences that would not have been generated by the business frame alone. More interesting are the conceptual integration networks in which two or more inputs are already framed by conceptual models; disparity between these models complicates the construction of a blended space. The blending process is simplest when both inputs are framed by the same conceptual model, producing overlap in their basic image-schematic structures. Such “mirror networks” were the first to be identified in what became the study of conceptual blending. A standard example discussed in Fauconnier (1997) and Fauconnier and Turner (1994; 1998; 2002) is the Regatta blend. In this example, a magazine article compares the progress of a modern catamaran, Great America II, sailing from San Francisco to Boston in 1993 to that of the clipper ship Northern Light, which sailed the same route in record time in 1853. In the article, the catamaran crew is said to be “barely maintaining a 4.5 day lead over the ghost of the clipper Northern Light.” What would be required to understand this statement? The answer is a conceptualization structured by the conceptual model of a race. The notion of a race is not present in either of the inputs but results from the construction of a blended space, as depicted in Figure 2.1. One input to the blend is the current journey of the catamaran Great America II in 1993. The other is the journey of the clipper ship Northern Light in 1853. The two spaces share image-schematic structure, namely a path of motion from a source (San Francisco) to a goal (Boston), as well as a means of motion, i.e. sailing on the water. There are obvious counterparts: the routes, the ships, the directions of travel, and so on. There are differences, too: the specific types of ships (catamaran vs. clipper), the crews, the times of the journeys, the weather and sailing conditions, etc. The blended space is constructed via partial projection from each of the inputs. The routes, differing in their detail, are nevertheless fused into one, compressing Analogy into Identity. The ships are projected separately but now become two elements within the same mental space. The time and weather conditions project from the 1993 input, while those from the 1853 input are left out. Once brought together in the same blended space, the two ships sailing the same route at the same time form a recognizable pattern, which activates the conceptual model of a race. This, then, makes possible the generation of an inference like “maintaining a 4.5 day lead.” This example illustrates the basic ideas of conceptual blending: cross-space mappings that connect counterparts; selective projection from the inputs to the blended space; emergent structure arising in the blended space through composition, completion, and elaboration; and the generation of new inferences not possible in either of the inputs. Like the navigation chart discussed in the section on distributed cognition (section 2.1.1), the blended space brings diverse elements into the same representational space—a mental space of a tightly integrated scene at human scale—where can then be subjected to further cognitive operations. In mirror networks (as in simplex and frame-compatible networks), the conceptual models that frame the inputs do not clash with one another. Clashes
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between elements can be resolved by projecting the element from one space only (e.g. the sailing conditions of 1993) or by projecting the elements as separate entities (e.g. the ships from both time periods). In more complex integration networks, the inputs are structured by conceptual models that do clash with one another. Resolving these clashes can likewise involve projecting the conceptual model from only one input and not the other (a “single-scope network”), or it can involve projecting the conceptual models from both inputs (a “double-scope network”) but in a way that adapts them to one another in the blended space. As one might expect, double-scope networks yield the greatest originality. An instance of a single-scope integration network is the Boxing CEOs blend discussed in Fauconnier and Turner (2002). In this example, the chief executive officers of two competing corporations are depicted as boxers fighting inside a boxing ring. One input comes from the domain of business, where two companies compete with each other in the marketplace. The other input comes from the domain of boxing, where two athletes punch each other in a ring until one knocks the other out or is declared the winner. Abstract competition is understood via conceptual metaphor as physical struggle; this provides shared image-schematic structure with the boxing input, a basis for the conceptual blend. In business, chief executive officers are metonymically linked to their companies; this provides counterparts to the boxers in the boxing input. Still, how to blend the two inputs is not at all obvious. In most ways, the conceptual models associated with business and boxing clash. Businessmen do not pummel one another. Boxers do not (within the domain of boxing) sell products and services and try to win market share. In the Boxing CEOs blend, the conceptual model of a boxing match, taken from the boxing input, is used to frame the blended space. Specific elements, like business suits in place of boxing shorts, connect the blended space to the business input. Inferences generated in the blended space are transferred back to the business input, which is the target of the discourse. Notice that in this example, the rhetorical purpose of the blended space is to comment upon the business relationship, not upon the sport of boxing. Business relationships are abstract and diffuse. The conceptual model of a boxing match produces a compressed, tightly integrated scene that can be used to generate inferences about business: one CEO, for example, knocks out the other, driving him out of business. “Borrowing compression” from one domain to support reasoning in another is a common conceptual strategy, and it is found in many integration networks. Maintaining connections to the inputs is also essential in order to ensure that inferences transfer to whichever space is the current target of discourse. The construction of meaning involves operations across the conceptual integration network, not solely within the blended space. Clashes are resolved differently in double-scope networks. In these cases, the inputs are framed by different conceptual models, and the blended space includes partial frame structure from each input, plus emergent structure of its own. Doublescope integrations are fundamentally creative. Indeed, Fauconnier and Turner have argued that the capacity for double-scope integration is a distinguishing feature of
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cognitively modern human beings (Fauconnier and Turner 2002, chapter 9). An example of a double-scope integration discussed by Fauconnier and Turner (2002) is the Computer Desktop Interface, popularized by the Apple Macintosh computer. In this interface, we have inputs coming from two very different domains: office activities and computer operations. The office domain includes actions like typing documents, placing documents in folders, and disposing of them in a trashcan. The computer domain includes operations like entering data and using programs to operate on data. The interface combines elements of both to create a hybrid computer-office workspace. Sometimes the frame structures align reasonably well: dragging a file icon over a folder icon “moves” the file into that folder. In reality, the data is not physically moved, but pointers are adjusted so that it is associated with the new folder—all this being invisible to the user. (Here we ignore the phantom icon that is dragged to the folder before the move is actually executed, an accommodation to clashing frame structure that is discussed by Fauconnier and Turner.) Many times, the frame structures clash with one another: dragging a file icon over a folder icon from a different storage device (disk, CD, networked computer, etc.) “moves” the file into that folder but also leaves it in the original folder—a violation of the physics of real offices. In this case, a copy of the data is made and placed in a new physical location while the original data remains unaltered. Two more frame clashes involve the trashcan icon. First, unlike in a real office, the trashcan sits on the desktop. Second, moving the diskette icon over (into) the trashcan icon does not dispose of the diskette or its contents; instead, it ejects the diskette from the diskette drive. New users find this nerve-wracking, fearing they will lose the data on the diskette by moving it into the trashcan, the standard action for disposing of files on the computer. Despite these clashes, the blended computer-office workspace has been very successful. Inexperienced users are able to take advantage of their conceptual models for office activities to learn to operate the computer more quickly. Over time, the clashes become less apparent as interacting with the computer interface becomes familiar and supported by its own conceptual models. In their book, The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities (2002), Fauconnier and Turner explore the kinds of relations that connect counterparts in different input spaces (Identity, Analogy, Representation, etc.), the ways in which relations get compressed to human scale in the blended space (compressing Analogy into Identity, shortening Cause-Effect chains, tightening metonymies, scaling or syncopating Time and Space, etc.), and the competing pressures that govern conceptual integration (e.g. maximizing integration while maintaining topology and connections to the inputs). These topics go beyond the scope of this chapter; relevant concepts will be introduced as needed for the analysis of specific data.
2.1.3 Cognitive artifacts as material anchors for conceptual blends In recent work (Hutchins in press), Hutchins uses conceptual integration theory to analyze cognitive artifacts whose spatial configurations support reasoning about
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temporal relations. Examples include a Japanese hand calendar, which uses the structure of the hand to support naming the day of the week that corresponds to a given date; the method of loci, which uses the landmarks along a familiar path to aid recall of topics in a particular order; the medieval compass rose, which can be used to compute the time of high tide in a port; and several others, including the analog clock, which is a primary focus of the present study. These cognitive artifacts are specialized analog computing tools realized through the marriage of material and conceptual structure. A straightforward example is the cultural practice of queuing or standing in line, depicted in Figure 2.2 (adapted from Hutchins’ original analysis). Here the relevant material structure is an arrangement of physical bodies in space, one behind the other, oriented in a particular direction. The top left circle in Figure 2.2 represents the perceptual space of a person viewing this arrangement of bodies. The square behind the circle shows that this mental space is anchored by structure in the environment. Onto this perception the viewer imposes a particular imageschematic structure: a path of motion oriented in the direction the bodies are facing. This provides the basis for relating the arrangement of bodies to the ‘first come, first served’ cultural model of service, shown in the top right circle. Counterpart mappings link the bodies in the perceptual space to the slots in the conceptual model (first to arrive, second to arrive, etc.). In this simplex network, conceptual integration produces the blended space shown at the bottom: the interpretation of the perceptual scene as people waiting in line for service. Once constructed, the blended space supports the generation of specific inferences, such as that the person in position 2 will receive service after the person in position 1 but before the person in position 3, or that three people will be served before the person in position 4, and so on. The construction of the blended space in this conceptual integration network is an essential step for forming this interpretation of the state of the world. Notice that the world is more than mere input to this conceptual process. Hutchins argues that the material structure, here the configuration of bodies in space, anchors the conceptual blend, stabilizing and maintaining the set of conceptual relations during subsequent reasoning or computation. This might be simply generating an inference, such as who is to receive service next (for the standing-in-line example), or it might be some type of physical operation used to compute a result, such as finding the day of the week that corresponds to the 8th of August (for the Japanese hand calendar example). When a physical structure acts as a material anchor for a conceptual blend, the material elements become proxies for conceptual elements, engaging the perceptual and motor systems in the performance of the cognitive activity at hand. At the very least, the stabilization provided by the material anchor supports the generation of inferences that might otherwise falter due to the complexity or instability of the conceptual relations involved. Hutchins’ analysis was the starting point for the present study. If material structures do act as anchors for conceptual blends, and if this process is important to human cognitive achievement, then how do these crucial blends get set up and used in situated activity? For that matter, how does each new generation learn to
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instantiate and elaborate these blends, maintaining and even advancing the cognitive sophistication of our species? I set out to investigate these questions by studying a domain of everyday cognitive activity that utterly depends upon interacting with material cognitive artifacts, namely telling time from a clock.
2.2 Domain for study: time-telling and time-telling instruction The present study investigates interactions between material and conceptual structures in the domain of time-telling. Why time-telling? Time-telling was chosen for several reasons: (1) it is a fundamental everyday cognitive activity that is familiar to the reader and that cannot be dismissed as irrelevant or peculiar to some specialized domain; (2) it clearly depends upon interactions between conceptual models and the material environment; and (3) it is amenable to ethnographic study, particularly with respect to questions about how time-telling functional systems get established. Telling time is a ubiquitous human cognitive activity. It is central to both everyday life and work in specialized domains. Telling time with reasonable accuracy enables group members to coordinate their activities with one another, distributing physical and cognitive labor. This distribution is a defining characteristic of the complex industrial (and now high-technology) society we live in. Improvements in the precision and accuracy of time-keeping have gone hand-in-hand with the development of industry and science and the rise of the modern world. Although the passage of time is part of the human experience, time itself is inherently abstract. It cannot be directly sensed or manipulated. It is therefore understood analogically, in the relations among events, or metaphorically, in terms of space and motion. With the standardization of time measurement, time has even come to be understood as a valuable but limited commodity which, like money, must be carefully spent and not wasted. Because time is inherently abstract, telling time has always depended on perceiving some aspect of the material world that changes state in a regular way. Early time-telling depended upon regularities in the motion of the sun and the stars and the changing length and direction of shadows. Current time-telling depends upon specialized artifacts that recreate and regularize these natural rhythms. The most important of these historically, and the one that has been most widely used, is the analog clock, whose clock face has become a near-universal symbol for time. In a sense, all of these material structures from shadows to clocks “keep time,” but they do not “tell time.” In each case, telling time depends upon bringing perceptual and conceptual processes into engagement with these material structures in order to instantiate local functional time-telling systems. The modern analog clock, the artifact most focused upon in this study, is the material outcome of an extended history of artifact development within an evolving cognitive ecology (sketched in the next chapter). Reading the time displayed on such a clock seems nearly effortless to a competent adult, but learning to do so requires
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specific instruction and years of practice. This makes time-telling amenable to ethnographic study. Clock-reading abilities develop piecemeal in a process that extends over time. Teachers use overt behaviors to teach learners how to read times from a clock. Learners who have yet to master the task make errors that provide clues to underlying conceptual operations. Each of these factors provides an opportunity for insight into how conceptual and material structures interact in cognitive activity and how these are brought together to form a local functional system. In the United States (and many other countries), time-telling instruction has become a standard element of an important social institution: elementary school. The topic of time-telling appears throughout the mathematics curriculum in the primary grades (kindergarten through 3rd grade), providing an opportunity to observe and record time-telling instruction in a public setting. Such recordings make it possible to do detailed analyses of the construction of meaning in specific episodes of clock-reading instruction and to explore the roles that artifacts, gestures, and speech play in these meaning-making processes. Together with these recordings, observations of student time-telling can inform the analysis of how a competent clock-reader instantiates a functional system for telling time and how this process goes awry, providing further insight into the interplay between material structures and conceptual processes.
2.3 Methods: Cognitive ethnography and semantic analysis The conceptual framework guiding this research couples distributed cognition with conceptual integration theory. Accordingly, the method of investigation couples cognitive ethnography with semantic analysis. By drawing on cognitive linguistic theories of meaning construction, the semantic analysis aims for a deeper level of detail than that typically found in studies of situated action, which have tended to focus strictly on observable behaviors. By studying how meaning is constructed step-by-step in episodes of real activity, the cognitive ethnography aims for an understanding of process that eludes most cognitive linguistic analyses, which have tended to focus on static examples or generic descriptions of activity.
2.3.1 Cognitive ethnography Like other forms of ethnography, cognitive ethnography is based on participant observation within a community of practice. It includes observations of naturally occurring activity, collection and analysis of artifacts, interviews with informants, and the like, while aiming for an explication of the meanings constructed by members within that community. But cognitive ethnography also goes beyond this. It aims for an understanding of process: how members accomplish cognitive activities in actual settings and how they construct meanings in real episodes of activity. Rather than simply explicating the social, material, and conceptual resources available, cognitive ethnography seeks to understand how members bring these resources to
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bear on real-life cognitive problems, how they coordinate these resources, and what complex properties emerge from these interactions. It does so not by providing generic descriptions of cultural activity (i.e. what typically or usually happens), but by finding the rich detail in what actually does happen moment by moment in real instances of activity. In this way, cognitive ethnography supports the analysis of cognition as a distributed process: how information flows through the system, how representations are constructed in various representational media, how computations are carried out, how the person, group, or world adapt or are adapted to preserve informational states and useful procedures, and so on. These kinds of questions can only be answered by close, detailed analysis of real-world activity in natural settings. At the same time, the extended experience of participant observation that has long been the hallmark of ethnographic research still plays an important role in cognitive ethnography, especially in providing warrants for interpretations of video data. Because it aims at fine-grained analysis of what actually happens, cognitive ethnography takes advantage of the latest digital technology to record and analyze episodes of activity. Digital video makes it possible to record activity in sharp detail and to play it back repeatedly and at different speeds (fast, slow, even frame-byframe) in order to examine the intricate details of interaction and to detect patterns that play out at different timescales. Computational tools provide means to rerepresent the data in various forms and to conduct further analyses to answer specific theoretical questions. Digital media also support the sharing of data among different researchers, supporting the advancement of science as a distributed enterprise. As a method of investigation, cognitive ethnography offers several potential benefits to cognitive science (Hutchins in press). Cognitive ethnography addresses important questions that are difficult to investigate with other methods. Chief among these are how cognition is embedded in the material and social environment, how cognitive processes are organized, and how such processes actually play out. By illuminating functional constellations that span traditional disciplinary boundaries, cognitive ethnography can make visible a new range of phenomena and re-shape our understanding of what the human mind actually does. Cognitive ethnography can also work hand in hand with other methods of investigation in cognitive science. It can support experimental studies by generating hypotheses, providing stimulus materials, and addressing concerns about ecological validity. It can aid computational simulation and formalization by deepening our understanding of the phenomena to be simulated. And it can inform the design of new technologies by investigating systems of activity before, during, and after implementation. In the present study, cognitive ethnography is used to provide a deeper understanding of a phenomenon that is not well understood, namely how conceptual processes interact with the material world in cognitive activity. It is also used to investigate whether the conceptual mappings hypothesized by cognitive linguists are apparent in real-life instances of meaning construction. If they are, cognitive ethnography should help to illuminate the processes involved, including the roles played by speech, gesture, and manipulation of the material environment.
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2.3.2 Semantic analysis Conceptual blending analyses usually begin with a sample phenomenon—a piece of language, an image, a mathematical idea, a social ritual—whose interpretation appears to involve multiple conceptual domains. The choice of sample may be driven by some perceived novelty in the phenomenon or by its potential relevance to a current issue in the development of conceptual integration theory. Analysis of a conceptual blend is basically a process of constructing the conceptual integration network behind the blend: diagramming the inputs, crossspace mappings, compressions, and so on that would be involved in producing or interpreting the sample under analysis. The network diagrammed in such a way is necessarily idealized; the actual integration network constructed by a specific person at any given moment will depend on many factors, including (among others) past history, the present setting and situation, and the current focus of discourse. A speaker and listener in discourse, or two people reading the same headline or viewing the same advertisement, might construct different integration networks, leading to different interpretations or even misunderstanding. To the extent that the networks constructed by different participants align, we say that they have achieved intersubjectivity. For the analyst, diagramming the normative conceptual integration network associated with a particular sample includes the following general steps. First, identify the inputs. These are the mental spaces structured by different conceptual domains. Next, look for shared image-schematic structure across the inputs which could provide a basis for integration. In blending diagrams, this shared structure is shown in a separate “generic space” centered above the two inputs; often, as in this dissertation, the generic space is omitted when not a focus of the analysis (see Fauconnier and Turner 2002 for a discussion of the cognitive properties of the generic space and the role it plays in conceptual integration networks). Next, identify counterparts in the input spaces and label the cross-space mappings (“vital relations”) that connect them. Consider how these relations are compressed in the blended space, turning one relation into another (e.g. Analogy into Identity). Examine selective projection from the inputs to the blended space, how the blend is composed, patterns that emerge there, and whether there is recruitment of additional frame structure (i.e. activation of additional conceptual models). Compare the frame structure of the blend to that of the inputs to identify the type of integration network under study. Identify emergent properties of the blend and inferences that are generated there. Consider the rhetorical purpose of the blend and which space in the integration network is the target of discourse, identifying how inferences transfer from one space to another (e.g. from the blend to one of the inputs). Analyze the properties of the conceptual integration network in terms of the governing and optimality principles that have been identified by previous studies of conceptual integration. Explore the ways in which the blended space contains an integrated scene at human scale and how the blend gets elaborated in further discourse, and so on. Which of these various steps are employed in any particular analysis depends
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upon the particular goals of the study. For our purposes, we would add to the list: identify any material anchors that are present and what roles they play. By marrying cognitive ethnography to semantic analysis, we move beyond the analysis of static examples and into the analysis of the dynamic construction of meaning in situated activity. Digital video recordings and multimodal transcription (showing speech, gestures, and manipulations of artifacts) make it possible for us to examine how these conceptual integration networks are constructed step by step in the discourse. This moves us from a single conceptual integration diagram to a series of diagrams representing the state of the network at different points in the discourse. By diagramming the construction of meaning one step at a time, we open the way to an analysis of how each speech act, gesture, and manipulation of the material environment affects the ongoing conceptualization.
2.4 Specifics of the study The analyses included in this dissertation are based on cognitive ethnographic research conducted in two elementary schools during the 2002-03 school year. Both schools are located in the greater San Diego area. One is a private church school in the inner city serving a predominantly working class, multi-ethnic population. The other is an independent private school in an affluent community near the University of California serving a mostly white population. The study included the 1st, 2nd, and 3rd-grade classes at the church school and the 2nd-grade class at the independent school, and each class had 20-22 students. During the instructional portion of the math lesson, the teacher typically stood at the front of the room using the whiteboard, overhead projector, or other teaching tools to deliver instruction while the students sat at their desks listening and marking their workbooks or worksheets. After the instructional presentation, the teacher would move among the students as they worked the problems, checking the students’ work and providing assistance as needed. In the 1st-grade class, which was part of a combined 1st- and 2nd-grade group with a single teacher, the students moved to a rug to receive mathematics instruction from the teacher, who either sat in a chair or stood at a feltboard or whiteboard in front of the students. After the instructional portion of the lesson, the students returned to their seats to complete the assignment in their workbooks. Because time-telling is part of the mathematics curriculum, data were gathered in these classes through participant observation during the portion of the school day devoted to mathematics instruction. The primary investigator, a licensed schoolteacher with a masters degree in education, observed the instructional portion of the lesson from the back of the room and then joined the teacher in moving among the students, checking work and offering assistance. This allowed the investigator to establish a role for himself in the classroom and to build rapport with students, while also providing the opportunity to observe individual students’ work and to ask questions in a way that was natural to the instructional interaction. When lessons focused on time-telling, the instructional presentation was recorded on digital
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video, including any student performances that occurred during the instructional portion of the lesson. When the students worked individually at their desks, the investigator circulated among them making fieldnotes that were elaborated as soon as the lesson was over. After the lesson, teaching materials and student worksheets were collected and copied for subsequent analysis. This primary data collection was supplemented by interviews with teachers and with individual 3rd-grade students reading times from digital and analog displays, solving verbally posed time problems (e.g. “you’re meeting your friend at 10:30; how many minutes is it until then?”), and explaining their solutions. These individual meetings were a way of supplementing the brief interactions that occurred during the classroom lessons. They provided the opportunity to follow up on observations, informally test hypotheses about the students’ reasoning, and collect additional samples of student errors. Although these interviews were loosely structured, an attempt was made to have students read a variety of times (hour, half-hour, 15minute, 5-minute, and 1-minute) and to move from solving simpler problems (e.g. reading hour- and half-hour times) to solving problems of greater difficulty (e.g. reading 1-minute times or computing the minutes remaining till some future time). Students were free to interact with a clock face with movable hands, to gesture, or to write on scratch paper during their solutions and explanations. These sessions were recorded on 8-mm analog video for later review, but only episodes from naturally occurring classroom lessons were subjected to step-by-step analysis of the construction of meaning in the discourse. The digital video recordings of classroom lessons were first roughly transcribed, and then specific portions of the direct instruction were subjected to more detailed transcription. The rough transcriptions are segmented to show lesson structure and timing. Within each segment, notes indicate the speech content, observable gestures or artifact manipulations, and any student errors that occurred. From this rough transcription, it was apparent that each lesson contained brief episodes of direct instruction by the teacher about some specific aspect of time-telling, lasting typically 30 seconds to 2 minutes, interspersed with periods of student practice. These episodes of direct instruction were subjected to more detailed transcription, producing multimodal transcripts showing speech in conversation analysis format (with timed pauses, emphasis, vowel-lengthening, etc.) and showing gestures and artifact manipulations in still images from the video recordings annotated with colorcoded arrows and labels. Each annotated image is linked to boxed speech that cooccurred with the gesture or action. The multimodal transcripts provide the basis for the detailed semantic analyses in chapter 5. These were constructed by working through the transcript line by line and diagramming the construction of meaning at each step in the process, following the conventions of conceptual integration diagrams with additional modifications to accommodate the particulars of this study. The groundwork for the ethnographic study included research into the origins of Western time-telling in order to trace the development of time-telling artifacts and practices, explore the pressures that shaped them into the form we know today, and
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examine concomitant changes in the ways that Westerners have conceptualized time. This historical survey is the subject of chapter 3.
3
THE COGNITIVE ECOLOGY OF TIMETELLING
Consider the ordinary analog clock with two hands. Our folk theory of timetelling says that this clock consists of two superimposed dials, one showing hours and the other showing minutes. The short hand points to the hour while the long hand points to the minute. In fact, a two-handed clock is rather more complicated than our folk theory suggests. It actually consists, for example, of not two but three superimposed dials—an artifact of its history—and each dial is read in a different way. Different ways of reading and combining the dials are reflected in different forms of time expressions. In order to unravel the complexities of the clock as a cognitive artifact, it will be helpful to begin by examining the origins of the clock. In this chapter we look at the historical factors that have shaped the daily time-telling artifacts and practices—and, indeed, the very ways of conceptualizing time—that we know today.
3.1 The nature of time-telling problems The problems in time-telling boil down to trying to find the answers to two questions: (1) “When?” and (2) “How long?” (Brearley 1919). Just as answering the question “Where?” involves fixing a location in space, answering the question “When?” involves fixing a point in time. It seems to us that we always occupy some absolute position in space-time—after all, we are here, now—but in fact the determination of a location in space-time is always relative. Locations in space are fixed relative to objects; points in time are fixed relative to events. On the scale of everyday life, I say that something is located “on my desk” or will take place “after lunch.” On a larger scale, positions are fixed relative to prominent features of the landscape, such as mountains, coastlines, or human-made structures, or according to some standard system of measurement, such as latitude-longitude, anchored by the Earth. Similarly, points in time can be fixed relative to prominent natural events, such as sunrise, or according to some standard system of measurement anchored by reference events. Changes in time-telling systems over the centuries have involved this sort of shift from natural reference events to a system of measurement with few natural anchors, as we shall see. In everyday life, the problem of fixing a point in time has two primary forms. The first is fixing the current moment in time, i.e. finding one’s position relative to the flow of events. In the history of time-telling, this type of problem has been
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solved with the aid of shadow sticks, sundials, and clocks of various kinds. The second is ascertaining when some anticipated future moment in time has arrived, especially when that moment is far enough removed from the present to make tracking of the intervening moments impractical. In other words, how can it be possible to devote attention to other matters and then be made aware when a certain point in time has been reached? The solution to this problem has typically involved devices that make some kind of attention-getting noise at the end of an event of fixed duration, such as water clocks that dump pebbles onto a metal platter or mechanical or electronic clocks that sound alarms. It was actually one form of solution to this problem that led to the modern clock. Related to the problem of finding a future moment in time is the problem of measuring durations of time, or answering the “How long?” question. In one form of the problem, we wish to measure the duration of an event. Events are bounded—they have a beginning, middle, and end—and so one way to solve this problem is to make two point measurements, one at the start of the event and one at the end, and then to determine the difference. Notice that this approach works only if we have some way of relating the two point measurements to one another—in other words, only if they can be brought together within some system of time measurement that encompasses both point measurements (see Latour 1986 for a similar point about the use of maps in navigation). Historically, this problem has been solved by using artifacts that incorporate some type of scale, such as sundials, graduated water clocks, or candles with equally spaced markings along their length. Tools of this sort support a second way of solving the “How long?” problem. Rather than finding the difference between the start point and end point, one can track the passage of time by counting the intervening moments. This method requires two things: a measurable standard interval and some form of counting system. The counting solution is implemented in modern count-up timers such as stopwatches. Another version of the “How long?” problem is the reverse of the one above. In this version, rather than measuring the duration of an independently-occurring event, we wish to measure out a particular interval of time which will be used to structure the event, to set the temporal boundaries for its beginning and end. One way to do this is by reference to a second, repeatable event of fixed duration. This is the method described by Geertz for timing the Balinese cockfight (Geertz 1973), where the length of a cockfight is set by the time it takes a coconut to sink to the bottom of a barrel of water. The reference event can be built right into the structure of an artifact, as in a sandglass, where the flow of sand from the upper bulb into the lower bulb is an event of fixed, repeatable duration. The amount of sand can be specifically chosen to reference a standard interval of time within a larger system, as in the “hour glass,” a sandglass used to measure out the duration of an hour. A variable form of this type of artifact is the modern kitchen timer or “egg timer,” which, in contrast to the stopwatch, is a count-down timer. This kind of artifact incorporates a scale so that it can be set to measure out different standard intervals, and, like the artifacts mentioned above for finding a future point in time, it also
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incorporates some type of alarm for signaling that the end of the pre-set time interval has been reached. All of the problems described above can be addressed in some way by the use of the modern clock. The clock has become such a powerful tool both by combining elements of its predecessors and by functioning within a broader, now globally standardized, system of time measurement. How this came to be will be sketched in the next section. Once this system of time measurement was in place, what remained, and what has been most essential to the advancement of science, was the problem of extending this system far beyond ordinary human scale: of measuring enormous stretches of time and fixing points in time vastly removed from the present (as in geology or astrophysics), or of measuring minute stretches of time or fixing points in time with extreme precision (as in atomic physics). These advances, depending largely on the use of different reference events and the development of finer-grained methods of measurement, go beyond the scope of this work. For our purposes, we will focus on the kind of time-telling involved in everyday human activity.
3.2 The evolution of time-telling practices Humans, like other living things that evolved on this planet, have a rudimentary time sense deriving from our own biological rhythms, which are intimately tied to the rhythms in our natural environment such as alternating periods of light and darkness. These biological rhythms alone are insufficient to solving the kinds of time problems posed in section 1.1. We also experience the passage of time through events unfolding on many different scales: the life cycle from birth to maturity to death, the changing of the seasons, the course of a single day, and the moment-to-moment flow of experience. Time seems to pass quickly when we are enthusiastically engaged and slowly when we are bored or tired, making it impossible for us to measure time accurately without reference to some regularly occurring, independent event. The events that became the basis for the earliest human systems of tracking time were those that took place overhead: the celestial motion of the sun and stars and the changing phases of the moon. Because humans are diurnal creatures (active during the day and asleep at night), the most salient reference event for early time-telling was the apparent motion of the sun. To the observer, the movement of the sun seems imperceptibly slow, but its path of motion is defined by the successive positions it occupies as it rises above the horizon, crosses the sky, and sinks beneath the opposite horizon. The sun’s apparent motion produces three distinct reference points: sunrise, apex, and sunset. The exact location of the sunrise and sunset and the height of the apex all vary with the seasons, but certain things are invariant. Among these are the apparent speed and direction of the sun’s motion, the change from ascent to descent at the apex, and the location of the apex at the midpoint of the sun’s path. The sunrise and sunset define the start and end of the day, and the apex (noon) divides the day into two equal parts, an early part during which the sun
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ascends (morning) and a later part during which it descends (afternoon). Once this path of motion is recognized and these reference points are established, a sense of proximity allows one to establish two additional reference points. As the sun ascends, it moves from being closer to the horizon to being closer to the apex. At the exact moment of this transition, the sun is equidistant between the two: this is mid-ascent (mid-morning). The analogous moment during the sun’s descent is middescent (mid-afternoon). Thus we have three major reference points—sunrise, noon, sunset—and two minor reference points—mid-morning and mid-afternoon— which divide the day into two to four equal parts. The duration of these parts varies from season to season as the days grow longer and shorter. Finally, a sense of the sun’s current position along its path from sunrise to apex and then from apex to sunset provides a rough way to locate oneself within the period of daylight and to anticipate the coming darkness. The apparent motion of the sun produces distinct effects on the earth as well: among the most salient (especially in the sunny latitudes where the time-telling practices described here originated) are changes in the appearance of shadows. As the sun moves across the sky, shadows shift direction. They also grow shorter and then longer again. The earliest systems for tracking time during daylight hours made use of these regularities. 4000 years ago, the Egyptians used a shadow stick—a simple stake driven into the ground—to make a cognitive artifact for tracking the progress of the day by following the changing direction and length of the stick’s shadow. Later (approximately 3500 years ago), the Egyptians developed a specialized artifact to use the length of shadows to divide the period of daylight into parts. This artifact is now called a “T-stick.” It consists of a stick with a vertical T at one end. The T casts a shadow that crosses the stick at some point. The width of the T’s crossbar ensures that this shadow continues to cross the stick as the shadow changes direction (nearness to the equator keeps the sun’s path close to vertical, making this possible). At sunrise, the T-stick is placed on a flat surface with its T-end toward the sun. As the sun rises, the crossbar shadow moves along the length of the stick toward the T. At noon, the stick is reversed, and the crossbar shadow moves along the stick away from the T. The exciting innovation comes in the form of lines drawn across the stick. As the crossbar shadow moves toward or away from the T, it intersects these lines at regular intervals, creating a series of temporal landmarks. The lines are not equally spaced as on a ruler; instead, their spacing increases with distance from the T (Figure 3.1). This ensures that as the sun rises or sets, the intersections of the crossbar shadow with the reference lines occur at equal time intervals. The T-stick uses six reference lines to divide the day into twelve equal parts. Aside from whatever other significance the number twelve may have had within Egyptian culture, it happens to be a very practical number mathematically. Twelve has many factors: it can be divided by two, three, four, or six, making it possible to form many sub-groupings of parts of the day without having remainders. The practice of dividing the day into twelve parts was later maintained by the Greeks, Romans, and Arabs, and, as we shall see, became a key structuring element for the system of time measurement we use today. It is important to emphasize that
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while the Egyptians divided the day into twelve parts, these were not the hours we know now. They were temporary hours: equal parts whose length changes from day to day. Temporary hours are long in the summer, when the period of daylight is longest, and short in the winter, when the period of daylight is shortest. The number of the temporary hour (e.g. the sixth hour) marks the current position along the path from sunrise to sunset. While the T-stick relies on the changing length of shadows, the next major innovation in time-telling, the sundial, relies on the changing direction of shadows, which sweep an arc from west to north to east in the northern hemisphere. Sundials were used in Egypt about 3300 years ago, in Greece about 2500 years ago, in Rome about 2300 years ago, and in Arabia about 2200 years ago, as they spread through the great civilizations in that part of the ancient world. Sundials continued to be the primary means for telling time in Europe up to the start of the modern era. Unlike T-sticks, sundials do not have to be manipulated; they can be permanently fixed in position. As the sun travels across the sky, the shadow cast by an indicator (gnomon) sweeps across a scale, dividing the day into twelve hours. These are still temporary hours, longer in summer and shorter in winter. They are also strictly local: the scale on a sundial is particular to latitude, while the occurrence of sunrise, noon, and sunset are particular to longitude. Because of wobbles in the earth’s rotation and the changing speed of the earth as it follows its elliptical path around the sun, the accuracy of sundials varies up to a quarter-hour in either direction at different times of the year, but this variability posed little problem for the time-telling needs of the day. Parallel to the sundial and serving a slightly different need were time-telling artifacts that depended upon the flow of a substance into or out of a container. The most common such devices were sand glasses and water clocks. Some form of water clock (clepsydra) was used in Egypt 3100 years ago, in Greece 2500 years ago, in Rome 2300 years ago, and in Arabia 2200 years ago, proliferating through the ancient world alongside sundials. Water clocks were particularly useful where the sun was not visible: indoors, on cloudy days, and at night. The scales on water clocks were calibrated to the scales on local sundials, so that water clocks also divided the day into twelve hours (temporary and local). This provided the basis for another innovation: by analogy to the day, a water clock could be used to divide the night into twelve parts. The nighttime division could be accomplished using a daytime scale from the opposite season, one whose endpoints aligned with the current sunset and sunrise. Through this means, the twelve hours for daytime were wedded to twelve hours for nighttime, producing a total of twenty-four hours in the period from one sunrise to the next. This was the origin of our twenty-four hour “day” (notice how the word that refers to the most salient part of the cycle, the period of daylight, is extended to refer to the entire 24-hour period; this occurred in many languages). Of course, at this point the daytime and nighttime hours were not the same length. Daytime hours were long in summer and short in winter, while nighttime hours were the opposite. Only at the equinoxes were the daytime and nighttime hours equal. Nevertheless, the pairing of sundial and water clock provided
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a complete time-telling system. Water clocks were not without their problems, though. Like T-sticks, water clocks require manipulation: they have to be filled at regular intervals. And like sundials, water clocks are not very accurate: changes in water pressure and the erosion or blockage of openings disrupt the rate of flow. A significant advantage of water clocks—one that was crucial to the development of the mechanical clock—was that water clocks could include some form of alarm to signal the arrival of a future moment in time. Plato’s garden had a water clock that filled over the course of the night until it tipped a bucket of pebbles that rained down on a metal platter and woke his students in the morning. In Europe in the Middle Ages, water clocks were rigged to ring small bells at night, alerting monks when it was time to ring the monastery bells for the call to prayer. It was this last, bell-ringing function that first gave rise to the mechanical clock (Barnett 1998; Landes 2000). The mechanical clock appeared in Europe about 650 years ago. It was a faceless bell-ringing device that had replaced the water-flow system by an assemblage of gears, weights, and regulators. This history is apparent in the very word clock, which comes from the Middle English clokke, deriving from the Medieval Latin clocca, meaning “bell” (cf. German Glocke, Dutch klok, and French cloche). Although the clock was first built as a bell-ringer, quickly thereafter a sundiallike face was added to the clock. Instead of using a moving shadow from a gnomon as an indicator, the clock used a single metal rod driven by a mechanical gear assembly. The clock face was divided into twelve segments, one for each of the traditional twelve hours, with intermediate marks at the half-hours and quarterhours, reflecting the markings common to sundials of the time. Rounding the hemicircle of the sundial to a full circle satisfied the basic mechanical need of returning the indicator to its starting point, but by doing so, it also eliminated the empty space formerly taken up by night. This was not a problem, though, as the nighttime hours could simply be measured out by a second revolution of the indicator around the dial. Notice the power of this maneuver: for the first time, daytime and nighttime hours were brought together in a single device. But this unity also produced a clash: daytime and nighttime hours were not supposed to be the same length—daytime hours should be long when nighttime hours were short, and vice versa. For centuries, this problem was solved by using conversion tables to convert clock time to sundial time. Sundial time was thought to be “real time” because it was based on nature, and sundials continued to provide the standard long after clocks came into common use. This was helped by the fact that clocks were notoriously inaccurate and needed constant resetting, using sundials as the calibration standard. Eventually, with improvements in the accuracy of clocks and with the rise of commerce and industry, clock time began to dominate secular life. In medieval Europe, labor in the burgeoning industry of textile manufacturing was governed by clock hours. Meanwhile, religious life continued to be dictated by the prayer hours determined under the old system of time-keeping. The two systems existed in tension, side by side, for many years. By the mid-eighteenth century, secular life took the lead, and sundials were being converted to clock time. Temporary hours became a thing of the past, and equal hours became the new standard.
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Up to this point in our historical survey, we have considered only the division of the day into hours. What about the further subdivision of time into smaller parts? Like the shadow on a sundial, early mechanical clocks had only a single indicator: a rod that rotated around the dial in what is now called a “clockwise” direction, the same direction that shadows move in the northern hemisphere. This rod often had an arrowhead or other shape that made its pointing function explicit. The dial was divided into twelve segments labeled with Roman numerals, with XII (the start and endpoint of the cycle) at the top.3 Typically, between each pair of adjacent hour numbers was a symbol, such as a diamond or fleur-de-lis, marking the half hour, and often there was an additional marking for the quarter hour, as in Figure 3.2(a). The division into quarter-hours was a carry-over from the sundial, and since early clocks were even less accurate than sundials, this division sufficed for many years. Further subdivisions first appeared on a separate dial. These dials could be nested, as in Figure 3.2(b), where a dial indicating quarters and eighths of an hour is nested within the hour dial, making it possible to read the time as “eleven and one-quarter hours.” Each indicator is made long enough to point to the correct dial. When the invention of the pendulum clock by Huygens in 1657 made clocks more accurate, it soon became prudent to divide the hour into still smaller parts called “minutes” (literally “small parts”).4 These, too, were displayed on a separate dial incorporated into the clock face in a variety of ways. Eventually, it became conventional to arrange the minute dial, with its greater number of divisions, around the outside of the hour dial, as in Figure 3.2(c). Here the shorter hand indicates the number of hours while the longer hand indicates the number of minutes. The two dials are read separately, in one or the other sequence, to construct a time reading that includes both hour and minute components.5 Lining up the origins of the scales on these two dials, so that both start and end at the top as in Figure 3.2(c), has the fortuitous effect of bringing the five-minute marks into direct alignment with the hour marks. The dials still remain separate, though, because the intervening divisions fail to line up: there are four divisions between each pair of hour labels and five divisions between each pair of five-minute labels. A way around this clash can be found in Figure 3.2. 3
Some clocks had twenty-four segments, with a XII at the top for noon and another XII at the bottom for midnight; this was especially common when the hour dial was nested with other dials indicating days, months, and so on, in an astronomical clock. A driving force for such astronomical clocks was the power of capturing the motions of celestial bodies, long associated with gods, in the mechanics of human-made machines. Isaac Newton famously compared the workings of the universe to a giant clock; this mechanistic view of the universe lasted until Einstein.
4
Just as for the compass rose, the division into parts has roots in the Babylonian sexagesimal system, which used the highly factorable base 60 to avoid dealing with fractions, which were beyond their mathematics. The division of the day into 12 parts is likely to have been rooted in the Babylonian system as well (12 x 5 = 60).
5
A dial for seconds, literally “second small parts,” was added to the clock face later. It first appeared in the form of a separate miniature dial, befitting its distinct function as a counting device for timing a patient’s pulse or measuring the speed of a racehorse
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Comparing the inner dial in Figure 3.2(b) with the outer dial in Figure 3.2(c) makes it clear that the minute dial can also serve the secondary function of marking the fractions of the hour: 15 marks the end of the first quarter-hour, 30 the second, 45 the third, and 60 the fourth. This makes the intervening tick marks on the hour dial superfluous. Dropping these marks opens the way to combining the two dials into a single dial structure: the modern clock face. The modern clock face has major tick marks indicating both the hours and five-minute intervals, numeric labels for the hours, and intervening tick marks for the minutes. Even though the dials have combined, the indicators must still be kept separate and distinguishable; for this reason the modern clock face preserves the convention that the short hand indicates the hours while the long hand indicates the minutes. By combining the dials into a single structure with two indicators, the modern clock supports a shift in cognitive strategy from sequentially reading the separate dials to recognizing distinct patterns of hand configurations on the combined dial (a strategy which, we will see in subsequent chapters, seems to be used primarily for naming hour and half-hour times). This simplification and standardization also makes it easier to become so familiar with the structure of the clock face that some of its physical elements, such as the intervening tick marks or numeric labels, can be omitted. When this is the case, an experienced time-teller can still read the time so long as the length of the indicators can be distinguished and the orientation of the clock face is clear. We have arrived at the modern clock, our principal tool for reading the local time, but there is yet another important development to be considered: the establishment of time zones. Even when the clock overtook the sundial as the measure of “real time,” it continued to be pegged to a single natural invariant: the apex of the sun marking noon. This relationship appears in the very structure of the clock face: the hour hand points straight up at noon, indicating the position of the sun. So does the minute hand, due to the alignment of the two scales, creating a distinctive clock state that marks the restarting of the hour count. Because the time of solar noon varies along an east-west axis, towns at different longitudes all had different local times.6 With the building of railroads, differences between local times made the scheduling of trains both confusing and dangerous. The railroads responded by using the clock times of major cities, such as London, as reference times for train travel within entire regions. As with sundials and clocks, the two systems of time-keeping co-existed and clashed for a while, until small towns finally relinquished their last hold on the sun and adopted the city times as standard. Eventually, the entire earth was divided into twenty-four longitudinal time zones, all pegged to solar noon at the reference location of Greenwich, England. 6
Indeed, comparing the observed solar time to the local time at a known location is a standard way of finding longitude, especially at sea. The problem is that one needs some way of determining the exact time at the reference location. Before the advent of radio time signals, the reference time had to be precisely and accurately kept by a timepiece that could withstand the pitching, rolling, and harsh weather conditions of a ship at sea. For a fascinating history of the development of the marine chronometer, see Sobel (1995).
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The story does not end here. The system of time measurement is still in flux, shaped by the pressures of human activity. Some countries or regions use times that differ from those of the standard time zones by a half-hour, quarter-hour, or some other interval. Boundaries between time zones are negotiated and adjusted to keep growing communities in the same zone as neighboring metropolitan areas. The location of the International Date Line, the 24-hour jump that restarts the system of time zones as one circumnavigates the globe, continues to be debated. Some places adopt Daylight Saving Time, resetting their clocks seasonally to keep their schedules in closer alignment with sunrise, while others stay on Standard Time throughout the year. Atomic clocks bring new precision to time measurement. Vibrations of the cesium atom become the new standard for measuring a second. A leap second is added at the start of a new year to bring Greenwich Mean Time back into alignment with solar noon. Like other complex systems, time-telling continues to evolve.
3.3 Discussion The history of time-telling is the history of a changing cognitive ecology. In this history we see three temporal frameworks that developed in parallel, what Barnett (1998) calls natural time, God’s time, and secular time. The earliest societies were founded on agriculture and governed by natural time: by the rising and setting of the sun. Human activity in rural societies had no need for precise time-keeping and derived little value from time-telling innovations, creating little selective pressure for their development. The rise of cities led to the establishment of temporal reference points for the coordination of activities. In ancient Rome, midday was publicly announced. By the 3rd century, the 3rd and 9th hours were also announced. These hours—the 3rd, 6th, and 9th—became the required hours for Christian prayer. Thus began the development of the second temporal framework: God’s time. The system of prayer hours was codified in the 6th century when St. Benedict established the Canonical Hours of lauds, prime, terce, sext, nones, vespers, compline, and matin, which corresponded to just before daybreak, just after daybreak, third hour, sixth hour, ninth hour, eleventh hour, after sunset, and at night. (Later historical developments shifted nones to midday; this was the origin of our word “noon.”) These hours were, of course, still the temporary hours of the sundial, and so they varied from day to day. In the monasteries, the Canonical Hours were strictly observed. For most monks, this meant listening for the ringing of bells that signaled the call to prayer, but for at least one monk in each monastery, it meant carefully tracking the passage of time to know when to ring the bells. This was accomplished through a variety of techniques, including reading a sundial during the day, tracking the movement of the stars at night, and, importantly, using a water clock to measure out intervals and ring miniature alarm bells when other methods were not possible. The Benedictine order grew, establishing monasteries throughout much of Europe, and the Canonical Hours were adopted by other orders of monks. This spread the framework of God’s time across the continent. At this point, climate
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began to play an important role. In northern Europe, cloudy skies obscured the sun and stars for days at a time, creating a selective pressure for devices like water clocks and their descendants, mechanical clocks, which could function even when the temperature dropped below freezing. Meanwhile, the ringing of monastery bells became a familiar pattern that could be relied upon by people in adjacent towns, spreading the temporal framework of God’s time into secular life. This framework took on even greater importance with the publication of the Book of Hours, a prayer book keyed to the Canonical Hours that became the best-selling book in Europe throughout the 15th and 16th centuries. These two temporal frameworks, natural time and God’s time, continued side-by-side for many centuries with little tension, due both to their compatibility and to the separation in their spheres of influence: God’s time rang out in the towns, while natural time governed the rhythms of life in the country. Tension did exist between God’s time and the third temporal framework: secular time. As we have seen, the need to track hours for prayer created a pressure for the development of the mechanical clock. But the emergence of the clock created a new invariant: the clock hour. Unlike all previous hours, the clock hour never changed: it was the same, day or night, at any season of the year. In the system of temporary hours, this was a nuisance because clock time had to be converted to solar “real time.” Yet it also created an entirely new possibility: the establishment of a standard unit of time. What was needed for clock time to become dominant was a cognitive ecology in which a standard time unit would confer a selective advantage. That ecology was created with the rise of industry. The largest industry in medieval Europe was the manufacturing of textiles, and here the clock hour provided a useful way to measure labor. Once adopted by the textile industry, clock-based secular time took on increasing power. The temporary hours of God’s time and the clock hours of secular time came into direct conflict in the cities and towns. Annoyed by the confusing clamor of bells, King Charles V of France decreed in 1370 that all bells in Paris shall ring in accordance with the clock in the Royal Palace, which measured out twenty-four equal hours. During the Renaissance, an equal-hour sundial was created to fill the need for a calibration standard for the mechanical clock, marking a shift in dominance. By the time of the Enlightenment, commercial life reigned in the cities and towns and clock hours had become the established standard. Throughout this period, the dominance of secular time was helped along by improvements in clocks that made them more accurate, reliable, and affordable to the general public. Once the third temporal framework, secular time, began to dominate, it too was subjected to further selective pressures. One was a pressure toward the coordination of times in different locations, intensified by the ecology of rapid transportation. Time zones unified times within regions and aligned times across regions. Today, no matter how far east or west one travels in the developed world, the number of minutes past the hour remains the same; only the hour changes. Pressure toward accuracy in time-keeping came from the cognitive ecology of navigation and the pressing need to find the longitude at sea, while pressure toward precision came from medicine, sports, and science, adding seconds to the minutes and hours
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measured by the clock. Finally, economic pressures led to the development of inexpensive, reliable watches that made precise, accurate time-keeping possible—and eventually, quite necessary—for the general working public. As we can see, the cognitive ecology of time-telling we inhabit today is the outcome of a contingent historical process. At each step of the way, the practices of time-telling and their associated artifacts were shaped by the cultural activities of particular social groups and their shifting power relations. These activities created selective pressures that favored certain innovations, spreading them through larger populations. In some cases, divergent practices co-existed harmoniously within different realms of activity. In others, practices clashed until selective pressures favored one and led to the extinction of the other. While we can speak of the evolution of time-telling, we must remember that unlike biological evolution, this historical process reflects both local adaptation and conscious design (Hutchins 1995, chapter 8). In the case of finding the longitude at sea, for example, the problem was well known and well defined for centuries, and the solution resulted from a long, iterative process of intentional innovation and testing. Nevertheless, it is useful to speak in terms of a cognitive ecology because of the interrelationship and interdependence of social structure, cultural activities, specific tools and practices, and associated meanings. We have seen, on the one hand, how cultural activities shaped innovations in time-telling, as when the religious practice of praying at particular times created a need that drove the invention of the mechanical clock. Conversely, we have also seen how such innovations in time-telling changed cultural activities, as when the appearance of fixed-length clock hours resulted in a new way of measuring labor. To this we add changes in social structure and the relative power of different institutions that came to favor the secular system over the religious one. The result of this historical process is today’s cognitive ecology of time—a myriad of schedules, tremendous coordinations of activity made possible by, and utterly dependent upon, precise, accurate time-keeping. Two aspects of this historical evolution merit further attention. One is a particular trend in time-telling artifacts and practices. The other is associated changes in how Westerners have conceptualized time. Let us look at each of these in turn.
3.3.1 The movement from natural to artificial to systematic Up to this point we have considered how time-telling artifacts and practices developed within the broader ecology of social organization, valued cultural activities, and the meanings associated with those activities. As we consider this history, we are struck by a particular trend in time-telling artifacts and practices: a movement from the natural to the artificial to systematization and standardization. The earliest time-telling simply made opportunistic use of an invariant in the environment: the movement of the sun and corresponding changes in shadows on the ground. The intersection of a shadow with a natural feature of the terrain could serve as a temporal landmark, but this landmark was strictly local. Borrowing terms
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used by Goodwin (1994), we can describe the next step in the evolution of timetelling artifacts as modifying the environment to highlight the regularity, perhaps by clearing away surrounding brush or by placing rocks or marks where shadows intercept natural features. Note the shift here from opportunistic use to greater intentionality. The temporal landmarks now stand out more distinctly, making them available for use as points of reference in communication and as structuring resources for activity. The next step is to bring these local temporal landmarks together in some relatable way (cf. Latour 1986) by imposing some type of consistent coding scheme (Goodwin 1994). In ancient Egypt, this scheme took the form of the twelve temporary hours, made manifest by the carefully spaced marks on the T-stick, which persisted in the structure of the sundial and ultimately shaped the structure of the clock. While time-telling became more intentional, regular, and consistent, it remained bound to the local environment and directly dependent upon a natural invariant for its function. The next step in the evolution of time-telling involved freeing the cognitive artifact from its direct dependence on the natural invariant; in other words, beginning to separate the artifact from the environment. We see this in the development of water clocks, which transposed the scale derived for shadow motion onto a vessel that could be used indoors, on cloudy days, or at night. The system of time-keeping was still based on the apparent motion of the sun, but once the water clock was set, the sun was not required for its operation. The relevant regularities of sun and shadow motion were captured in the system of water flow built into the artifact. Now the sun cycle of a summer day could be used to measure time on a winter night. The mechanical clock took this separation from the environment one step further. Because it measured out fixed-length hours regardless of day, night, or season, the mechanical clock eliminated the use of sunrise and sunset as cognitive reference points for time measurement. Previously, sunrise and sunset were the boundaries separating the temporary hours of the day (of one length) from the temporary hours of night (of a different length). Fixing the length of the hours detached the timescale from these boundary conditions and left it anchored to a single natural invariant: the apex of the sun at noon. Because this was the only fixed correspondence between clock time and sun-based time, noon became the new boundary condition. After being the sixth hour of the day (and later the ninth hour), noon eventually became the zero hour, the starting point for counting the 12-hour cycle. This pushed the start of the second 12-hour cycle to the middle of the night (midnight, the analogy to midday). Accompanying this was a disruption of the calendar day. Each calendar day, a single day/night cycle, had begun with sunrise, the proverbial start of a new day. Because noon was the last link between nature and the system of daily time measurement and had become the boundary for restarting the hour count, noon would seem to be the logical boundary between one calendar day and the next. This would have the unfortunate effect of splitting the daylight period across two calendar days. Because humans are active during the day and asleep at night, it is less problematic to split the nighttime hours, and so noon’s
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opposite, midnight, became the boundary between one calendar day and the next. In the system of clock time, all of the daylight hours still occur during a single calendar day, but that calendar day is now a half-night/day/half-night cycle. And we still have two cycles of twelve hours per calendar day, but these are no longer daytime hours and nighttime hours; they are now designated as ante-meridian (before noon) or post-meridian (after noon). The development of the mechanical clock pushed time-telling away from nature but toward standardization. By fixing the length of an hour, the clock created a standard unit of time, and, as we have seen, this took on great cultural importance. But the clock also did something more: it made time the same across different latitudes. As one moves farther from the equator, the difference in length between day and night becomes more extreme. The farther north one goes, the longer the summer days and the shorter the winter days. Sundials used sunrise and sunset as the boundary conditions for the twelve-hour day, so if one traveled north in the summer, the day-hours grew longer, making the sundial time different at different locations. Only at noon did the sundial times at different latitudes align. By fixing the length of hours and making sunrise and sunset irrelevant, the clock made time the same no matter how far north or south one traveled. Of course, time did still vary as one traveled east or west, across different longitudes, because the earth’s rotation makes noon different at each east-west location. The clock was already an artifact that could be used to tell time at any location, so the longitudinal solution was to break the fixed correspondence between noon on the clock and the apex of the sun’s path and instead aligning the times on all clocks within each longitudinal region. This moved the system of time-telling even further from nature, leaving solar noon at Greenwich, England, as the only formal link between the two (other alignments between clock noon and solar noon which occur at a single longitude within each time zone are just happenstance, byproducts of the way the system is set up). Just as counting the hours from noon created a boundary problem for the calendar day, dividing the earth into time zones created a boundary problem for restarting the twenty-four hour cycle: a longitude where the time on one side would be twenty-four hours different from the time on the other side. The solution, analogous to locating the boundary of the calendar day in the middle of the night, was to locate this boundary, called the International Date Line, in the middle of the Pacific Ocean. By this point, time-telling had become thoroughly artificial but quite systematic. In the evolution of time-telling artifacts and practices sketched here, we see the system of time-telling emerging through opportunistic use of local invariants in the natural environment, separating itself further from the environment with each new innovation, and becoming increasingly systematized and standardized as time-telling in different locations was brought into coordination. This process reflects a shift in the cognitive ecology from personal time-telling needs, such as when to tend certain plants or animals; to local time-telling needs, such as the coordination of activities among individuals in a social group (e.g. meetings for prayer, trade, or governance); to regional and ultimately global time-telling needs, such as the coordination of
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representations of time across multiple groups in multiple locations (e.g. for transportation or broadcasting). Here we see how the pressures of human activity drove the system from the natural to the artificial to the systematic, ultimately resulting in a worldwide system of time-telling in which the relationship of the parts to one another has superseded any connection to the natural invariants that first gave rise to time-telling as a cognitive activity. We see vestiges of this history in the modern clock: a 12-hour scale with two cycles per day, reflecting the twelve temporary hours first used to divide the period of daylight; the clock hands pointing straight up at noon, reflecting the original correspondence with the sun being directly overhead; and the clock hands moving in what is now called a clockwise direction, reflecting the way the shadow of the gnomon moved across sundials in the northern hemisphere, where the time-telling systems described here were developed. Although these aspects of the analog clock reflect historical contingencies that trace back to invariants in the natural environment, enough generations have passed since the clock was developed that its structures now seem rather arbitrary. In this section, we have considered a trend in the evolution of cognitive artifacts for telling time; now it is time to consider how our ways of conceptualizing time have co-evolved with and through these changes in time-telling practices.
3.3.2 Changing conceptualizations of time The history of time-telling is also the history of changing conceptualizations of time. We experience time in terms of change or motion; consequently, our conceptualizations of time tend to be grounded by a conceptual metaphor in which time is understood as unidirectional motion through space (Lakoff and Johnson 1980; 1999). Either we approach future events, or if we construe ourselves as stationary, future events approach us. We can see this in the close parallels between the problems of time-telling and position-fixing described in section 3.1. Because humans around the world have the same embodied experience of motion and change, this way of conceptualizing time seems to be universal. But how we conceptualize units of time—days, hours, and so on—is not. These conceptualizations are structured by culturally-specific conceptual models that have been shaped by the evolving cognitive ecology of time-telling, including interactions with particular forms of artifacts. The earliest time-telling artifact, the shadow stick, already reflects an important conceptual leap: the emergence of the repeating, cyclic day as a conceptual entity. This conceptual accomplishment relates to what Fauconnier and Turner (2002) call the Cyclic Day blend, a conceptual blend built on the analogy of one day to the next. The analysis here will follow the general outline of Fauconnier and Turner’s analysis but will pull it apart into a series of intermediate stages based on our sketch of timetelling history. As Fauconnier and Turner observe, each input to the conceptual blend is a single day: the sun rises above the horizon, ascends, reaches its apex, descends, and sinks below the opposite horizon. Each time this happens, it is a new event. Although there are differences between these events—the appearance of the
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sky, the amount of light that reaches the earth, the angle of the sun at apex, the duration from sunrise to sunset—there are clearly many similarities. The brain is exquisitely sensitive to repeating patterns, and our biology, like that of many other animal species, evolved in ways that respond to these changing patterns of light (the circadian rhythms being the most obvious example). But we, as human beings, make an important, additional conceptual leap. We see the new sunrise as analogous to previous sunrises, the new apex as analogous to previous apexes, and the new sunset as analogous to previous sunsets, and so on. From the perspective of conceptual blending, we connect elements in the input space of a new day to elements in the input spaces of previous days, using Analogy connectors to join sunrise to sunrise, apex to apex, and so on. We use other Analogy connectors to relate the direction of the sunrise (east), its arcing path across the sky, and the direction of the sunset (west), picking out the similarities in the sun’s path despite variations in the exact locations of sunrise and sunset and the altitude of the sun at midday (tracking these variations was a later conceptual development). In the conceptual blend, the Analogy connections are compressed into Identity, producing the conceptualization of Day as a single entity: an event, made up of ordered parts, which repeats again and again. This level of conceptual understanding supports the intentional construction of a cognitive artifact, the shadow stick, to provide a material anchor for the conceptual division of the Day, first into halves, and then into smaller parts. This artifact then makes it possible to detect how those parts change with the seasons, growing and shrinking in a recurring pattern. Where Analogy is compressed into Identity, Difference is often compressed into Change. The difference in the amount of daylight from one day to the next in the input spaces gets compressed in the blended space into a change in the duration of the single entity, Day. Day grows longer with the approach of the warm season and shorter with the approach of the cold season. Adding markings to the artifact, as in the T-stick and the sundial, provides a means to further subdivide the day, eventually into twelve parts. This produces a new conceptual entity, which I will call the Day-Hour. The Day-Hour is conceptually quite different from the hour we know today. It has no fixed duration from day to day and no meaning outside of the period of daylight. The Day-Hour, a form of what I referred to earlier as a temporary hour, is simply a division. In the blended space, it is exactly one-twelfth of Day. Projected back to any input space, such as the current day, the Day-Hour is exactly one-twelfth of the current period of daylight. But the Day-Hour is not just any continuous duration that amounts to onetwelfth of the period of daylight; it is an ordered segment of the day whose beginning and end depends on the reference points of sunrise (start of the 1st Day-Hour), apex (end of the 6th and start of the 7th Day-Hour), and sunset (end of the 12th Day-Hour). Any given moment of the day occurs during a particular Day-Hour (such as the 3rd Day-Hour), and the temporal moment at which one Day-Hour ends and the next begins corresponds to the spatial event of the shadow of the gnomon intersecting a particular mark on the artifact. Each such intersection is a significant cognitive reference point: the demarcation of the change to the next Day-Hour. Under this
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system, it is possible to peg the beginning (or end) of an event to one of these reference points and thereby to coordinate social gatherings and activity. The system of Day-Hours thus provides a series of temporal landmarks. What it does not provide is a useful way to measure durations of time. Day-Hours change in length from day to day, providing no standard metric. The measuring out of fixed intervals remains a cognitively separate task accomplished through separate means, namely reference to some repeatable event of fixed duration such as the sinking of a coconut or the flow of sand or water through a hole in a vessel. Just as days can be blended into Day, the same compression—Analogy into Identity—can blend the endless succession of nights into a single conceptual entity: Night. At this stage of development, Day and Night are two distinct conceptual entities, each with its own event structure. Day is defined by the event structure of the sun moving along its path; Night is defined by a different set of celestial events: the movements of stars, star patterns (which, via conceptual blending, become constellations), and the moon. Each daylight-event is followed by a darkness-event, which is followed by a different daylight-event, then a different darkness-event, producing an infinite string of distinct experiences. In terms of the conceptual blends that compress analogous days into Day and analogous nights into Night, there are only two events—Day and Night—that alternate back and forth, each occurring when the other does not. In mythology, this alternation is typically explained by the actions of different gods, with, for example, one resting while the other is active. What concerns us is that at this point, Day and Night are still two distinct conceptual event-entities and not yet part of the same entity. As we saw with Day, the blended space for Night compresses Difference into Change, making Night a conceptual entity that also grows and shrinks with the changing seasons. In fact, when Day grows, Night shrinks, and vice versa. In mythology, this relation can be understood benignly as a form of turn-taking or aggressively as some kind of ongoing struggle between gods; in either case, Day and Night remain distinct. The move toward uniting Day and Night comes with the next development in time-telling: the use of sundial scales on water clocks. When calibrated to the sundial, a water clock can take the place of a sundial on a cloudy day. But a water clock can also be used at night. The use of a sundial-scale water clock at night divides Night into twelve parts, creating the Night-Hour as a counterpart to the DayHour. Like Day-Hours, Night-Hours are temporary hours that change from night to night. When Day-Hours are long, Night-Hours are short, and vice versa; only at the fall and spring equinoxes are Day-Hours and Night-Hours are the same. In terms of reference points, sunset marks the start of the 1st Night-Hour while sunrise marks the end of the 12th Night-Hour; there is no reference event for midnight. In order to read the water clock at night, the reader must consult a sundial scale from the opposite season. This opposition supports a further inference: that Day and Night are not merely alternates, but opposites. Opposition is a more powerful conceptual relation. Conjoining the scales for Day and Night makes something else conspicuous: although individually, the durations of Day and Night vary considerably, the duration of both together is invariant (the imprecision of early
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time-keeping obscuring the minuscule changes as days grow longer or shorter). This union of Day and Night forms a new conceptual entity: a fixed-duration Day/Night cycle that I will call a DAY (written in all capital letters), in keeping with the practice in many languages of referring to the entire cycle by its most salient part. Finally, we have arrived at what Fauconnier and Turner call the Cyclic Day. Getting here has not only depended upon the important cognitive ability known as conceptual blending; it has also depended in crucial ways upon the construction and interaction with particular sorts of material artifacts. The transfer of sundial scales to water clocks and the use of water clocks at night brought Day and Night into the same conceptual space and made the counterpart relations of Day-Hours and Night-Hours manifest. The resulting conceptual entity, a DAY, with its total of twenty-four hours, is still unlike the calendar day we know today. To begin with, it extends from sunrise to sunrise. More importantly, it has a dynamically varying internal structure: within the fixed bounds of a DAY, the boundary between Day and Night oscillates back and forth, like an accordion, stretching and compressing the twelve temporary DayHours and Night-Hours on either side. This oscillation is halted by the invention of the mechanical clock and the appearance of the fixed-length clock hour, although it will take centuries to complete the conceptual shift making clock time more “real” than solar time. Day-Hours and Night-Hours are replaced by a single Hour, so that a DAY is divided into twenty-four equal parts that never change. The starting of the hour count at noon and the subsequent shifting of the boundary between one DAY and the next to midnight completes the series of changes that lead to the conceptual day we know today: a half-night/day/half-night periodic cycle, extending from midnight to midnight and consisting of twenty-four hours divided into two groups of twelve by the midpoint of the day, which is noon. Sunrise and sunset remain boundary conditions between day and night, now simply the period of daylight and the period of darkness, but they are no longer part of the system of time-keeping. For this reason, we have to check an almanac to find out what time the sunrise or sunset will occur on any given day. Time zones disrupt the relation between noon and the apex of the sun, and the practice of adopting Daylight Saving Time disrupts this even further, although we still think of noon as the time when the sun is directly overhead. The shift to clock-hours creates the standard hour, which continues to be used to measure labor today. As a standard metric, the hour has been further subdivided into 60 minutes, each of which is divided into 60 seconds (the same subdivisions used for degrees of longitude and latitude on the earth’s surface). Assigning numeric values to hours, minutes, and seconds leads to a new form of time-telling artifact, the digital clock, in which the current time is displayed as a series of numbers that change according to the regular rhythm dictated by the system of time measurement. Finally, turning time into a metric has other important conceptual consequences, as Lewis Mumford observed: “When one thinks of time, not as a sequence of experiences, but as a collection of hours, minutes, and seconds, the habits of adding time and saving time come into existence” (Mumford 1934). Thus we see the roots
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of the familiar conceptual metaphor Time is Money, analyzed by Lakoff & Johnson (1980; 1999). Time becomes a limited resource, a valuable commodity like money which must be spent carefully and not wasted. This conceptual metaphor structures much of modern life. In this history I have attempted to show how our modern time concepts of day, hour, and minute are historical accomplishments that emerged from the dynamic interplay of cultural practices, material artifacts, and conceptual models (Day, DayHours, a DAY, and so on). It is, in many respects, a chicken-and-egg story. The conceptual shifts and blends that led to our modern concepts of time were driven by the artifacts constructed to solve local time-telling problems and the relations these artifacts made manifest. The emergent changes in ways of conceptualizing time had consequences for cultural practices, which, in turn, drove further refinements in time-telling artifacts, and so on, in a developing cognitive ecology.
3.4 Time-telling today The result of this historical process is the time-telling system we know today: a system of time zones, two forms of material artifacts—analog clocks or watches with two to three rotating indicators, and their descendant, digital clocks with numeric readouts—and associated conceptual models, particularly the standard division of the day into hours, minutes, and seconds. This sets the stage for an analysis of clock-reading, the subject of the next chapter.
4
READING THE TIME: THE ANALOG CLOCK AS A
C O G N I T I V E A R T I FA C T
Our look at the history of time-telling has brought us to the time-telling artifacts, practices, and conceptual models we know today. Today, in the course of everyday activity, ordinary people read the time by consulting the display on a clock or watch. This display takes one of two forms: either analog—a circular dial with two or three rods serving as indicators—or digital—a series of numbers separated by colons. The digital clock derives from the analog clock but represents the current time as it is conventionally written in numeric form. This supports rapid naming of the time—one simply names each number in sequence—but does not provide much support for conceptualizing time relationships. In this and subsequent chapters, we focus primarily on the analog clock, specifically a clock that displays hours and minutes but not seconds. The conventional form of this clock has a face with numbers, tick marks, and two hands of different lengths.7 Here we examine what is involved in reading the current time from such a display.
4.1 The analog clock as a composite artifact The analog clock appears as a single dial with two indicators, one longer than the other. The indicators rotate in what is called a clockwise direction, but they move at imperceptible speeds, so in any given act of reading the time, they appear in fixed positions. The problem of time-telling is thus reduced to one of reading the current structure of the clock face, namely the positions of the indicators with respect to the dial. Our folk theory of clocks tells us that the clock face (which I will often refer to simply as “the clock”) is really two superimposed dials, one for hours 7Our
conventional way of describing the analog clock provides a marvelous illustration of what cognitive linguists call the “invariance hypothesis” (I thank Ron Langacker for reminding me of this). Here we have a conceptual metaphor in which the clock is conceptualized in terms of body parts, a common source domain for metaphorical mappings. The clock display is associated with the face, the most expressive part of the body and the locus of attention in interpersonal communication, while the indicators are associated with hands, the body parts used in pointing. This produces the unnatural configuration of hands on a face. The invariance hypothesis explains why this does not bother us: conceptual mappings preserve the topology (image-schematic structure) of the target domain, which in this case is the clock display, and import as much imageschematic structure from the source domain as is consistent with that preservation. For more about the invariance hypothesis, see Lakoff (1990) and Turner (1993).
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and one for minutes, but our history of time-telling has taught us that the twohanded clock actually consists of three superimposed dials—one for hours, one for minutes, and one for quarter-hours—as depicted in Figure 4.1. Two things become immediately apparent. The first is that reading the clock depends upon the perceptual separation of layers of structure, so that the number labels are associated with the hours, the tick marks between the numbers with the minutes, and so on. Some structures, such as the major tick marks at the number labels, are associated with more than one dial—in this case, with the hours on the hour dial and the fiveminute marks on the minute dial. The major tick marks at 3, 6, 9, and 12 are associated with all three dials: standing for the third, sixth, ninth, and twelfth hour; fifteen, thirty, forty-five, and sixty minutes; and the end of the first, second, third, and fourth quarter-hour. Of course, it is not necessary to decompose the clock display into three separate faces in order to read the time; it is merely necessary to attend to the relevant structures on the clock face and to construe them in the appropriate way at each step in the process of time-reading. How this is done will be the main subject of this chapter. The second thing that is immediately apparent in Figure 4.1 is that not all of the scales have explicit numeric labels. The hour scale has labels, but the labels for the minute scale and quarter-hour scale need to be filled in from imagination. The resulting adjacencies create curious correspondences: the label ‘3’ comes to be associated with “three,” “fifteen,” or “one-quarter”; the label ‘6’ with “six,” “thirty,” or “one-half,” and so on. Because the scales on the dial form a complete circle, in accordance with the cyclic nature of time measurement, the label ‘12’ has many associations: (1) in terms of order, it stands for the number between 11 and 1 in the repeating cycle; (2) in terms of quantity, it stands for twelve, zero, sixty, no quarters, and four quarters (or a whole); (3) in terms of the scales, it marks the origin and upper bound; (4) in terms of the indicators, it marks the starting point (source) and endpoint (goal) of motion around the dial; (5) in terms of the solar day, it stands for midday (noon) and midnight; and (6) in terms of the system of time measurement, it marks the boundary between one sixty-minute hour and the next (on the minute dial), the boundary between the first twelve-hour scale (ante-meridian) and the second twelve-hour scale (post-meridian) in a single calendar day, and the boundary between one calendar day and the next. So in addition to perceiving the relevant structures and construing them in the appropriate way at each step of the process, time-reading also involves filling in structures from imagination, associating vocal or sub-vocal labels (“ten”) with specific locations (the second major tick mark clockwise from the top), or making unconventional associations of quantity (15 or ¼) with symbol (‘3’)—which, if reflected upon, can be reconciled logically by considering the symbol to count groups of five (three groups of five is fifteen) or twelfths of an hour (three twelfths is one fourth or one quarter) as demarcated by the major tick marks. Finally, it should be noted that any single reading of the time from an analog clock involves exactly two of the three dials, and these must be dials that use different indicators. Thus it is possible to read the time using either the hour dial and the minute dial or the hour dial and the quarter-hour dial. The order in which
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these dials are read depends on how the time is expressed. One can read the hour dial and minute dial (“ten thirty”), the minute dial and hour dial (“thirty minutes past ten”), or quarter-hour dial and hour dial (“half past ten”), although it is unconventional in American English to read the hour dial and quarter-hour dial (“ten and a half”). In order to understand how these different time readings are constructed, we begin by introducing several embodied image schemas that are fundamental to time-telling and that lay the groundwork for the analysis that follows.
4.2 Image schemas required for clock-reading Embodied image schemas, introduced in chapter 1, are recurrent patterns in perceptual-motor experience. They derive from our bodily interaction with the physical world and, as a result, have a generally spatial or force-dynamic character. The simple patterns of embodied image schemas give structure to the conceptual models through which we make sense of the world. Image schemas thus tie perception to conception. Conceptual models structure mental spaces, while shared image-schematic structure provides the basis for linking mental spaces into conceptual integration networks and forming blended spaces with emergent structure. In the case of time-telling, the clock face presents nothing more than an arrangement of material elements. Interpreting that arrangement involves first imposing image-schematic structure onto the arrangement of elements and then making connections to the conceptual models involved in time-reading. Image schemas have been analyzed in some detail by cognitive linguists (see especially Johnson 1987; Lakoff 1987; Lakoff and Johnson 1999). Of these, the ones that are relevant to clock-reading include the part-whole, center-periphery, extension, proximity, container, and source-path-goal schemas (Figure 4.2) The first three— part-whole, center-periphery, and extension—are relevant to perceiving clock structure (the three dials), while the last three—proximity, container, and sourcepath-goal—are relevant to particular ways of reading the clock (interpreting the states of these dials) and to the errors that arise, a subject we explore in chapter 6. Here, we briefly introduce each image schema, and then we examine the roles these image schemas play in clock-reading.
4.2.1 The part-whole schema The part-whole schema (Figure 4.2(a)) is a spatial-structural image schema with three elements: a whole (something perceived as a unity or Gestalt), the parts that make up the whole, and the configuration of those parts, i.e. the way the parts stand in relation to one another. The whole is made up of the parts, and the parts compose the whole. The configuration links the parts into a whole; without the proper configuration, the parts may exist, but the whole does not (consider the difference between a pile of automobile parts and an engine). The part-whole schema derives directly from our embodied experience: we experience our own
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bodies and the bodies of others as wholes made up of parts configured in a particular way. Distinguishing the part-whole structure of objects is important to how we interact with the physical environment, e.g. grasping the handle of a cup. The partwhole schema is a structuring element in many conceptual models, making it possible, for example, to see social groups as families or ideas as theories, wholes made up of parts in particular configurations. We have already seen the part-whole schema at play in time-telling: in conceiving of the day as made up of hours, minutes, and seconds. In this chapter we will consider the role of the part-whole schema in perceiving relevant structures on the clock face.
4.2.2 The center-periphery schema The three structural elements of the center-periphery schema (Figure 4.2(b)) are simply an entity, a center, and a periphery. Again, our embodied experience provides a direct basis for the schema: our bodies have centers, where the main organs are and where the main functions of life are carried out, and peripheries, the exterior portions and extended limbs through which we interact with the world. As the body example implies, the center is essential and important, the periphery less so. The center is also the source of impetus or animus that drives the periphery. Extended into abstract domains via conceptual metaphor, the center-periphery schema gives us the understanding that what is important, essential, or the source of emanation is “central.” The clock face, being circular, has a distinct spatial center, with the numeric scale and tick marks around the periphery.
4.2.3 The extension schema The extension schema (Figure 4.2(c)) extends or continues a trajectory of motion. A moving object continues moving in the same direction unless obstructed or diverted by some opposing object or force. A body at rest, when acted upon by another object or force, begins moving in a specific direction. Just before motion has started or just after it has been blocked or diverted, there is a sense of where the motion will go or would have gone had it been able to continue. This anticipated trajectory, often represented by a dotted arrow in diagrams, is an essential component of many force-dynamic image schemas, such as compulsion, blockage, diversion, and so on (discussed in Johnson 1987). The extension schema is not limited to cases of actual or potential motion. It can also be applied to objects where no motion is present or anticipated, through the common cognitive mechanism of fictive motion (Talmy 1996). In fictive motion, the object itself, whether perceived or imagined, does not move. What does move is the locus of visual or mental attention. To see why this happens, extend your arm in front of you and look at your thumb. The fovea, the only part of the visual field in sharp focus, is about the size of your thumbnail. When you look at a larger object, your eyes move about, bringing different parts of the object into the fovea. This is visual scanning, and it has a mental counterpart, mental scanning, that occurs when
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an object is “viewed” in imagination. When an object has extension in one dimension, in other words is long and narrow, visual or mental scanning will tend to move along the length of the object in a consistent direction, e.g. either left-to-right or right-to-left if the object is horizontal. The direction of scanning has the effect of construing one end of the object as the start and the other as the end (through imposition of the source-path-goal schema, discussed in section 4.2.6). For example, in the sentence “The fence runs from the house to the barn,” the fence is construed as starting at the house and ending at the barn, but in the sentence “The fence runs from the barn to the house,” the start and end are reversed.8 In either case, once scanning begins to proceed in a particular direction, the extension schema can be used to continue the trajectory of the scan beyond the end of the object and off into space in a particular direction. This turns out to be incredibly important for human interaction, especially for following another’s gaze and for pointing (discussed in section 4.3).
4.2.4 The proximity schema The proximity image schema (Figure 4.2(d)) relates to closeness and potential interaction. Sensitivity to proximity is essential for survival. The approach of another is potentially threatening and generates arousal, heightened awareness, and readiness to respond. We observe that things which are close together are more likely to interact, especially when one or both are animate agents. In the proximity schema, one object, either a body in motion or with the potential for motion, is construed as a trajector, while another, often stationary, is construed as a landmark. The trajector draws near to the landmark or else simply occupies a position near it. The landmark may be imbued with a center-periphery image-schematic structure, so that motion toward the landmark is motion toward its center, heightening the sense of possible interaction—proximity may thus be imbued with a force-dynamic character. An object between two others may be closer to the center of one, and thus more peripheral to the other, or it may be equidistant between the two, equalizing the sense of potential interaction. An object moving along a path (section 4.2.6) is close to the source near the start of motion and close to the goal near the end of motion. In clock-reading, the proximity schema is related to perceiving an indicator as pointing toward one location or number versus another and as approaching or leaving a salient reference point.
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A derivative of fictive motion is fictive change, as in “The scar extended from his ankle to his knee” where the scar is already present and unchanging. In this case, the fictive change (extending) comes from the construal of the scar as lengthening itself during the scan up the leg. This is a kind of fictive motion in which one end of the scar is construed as fixed while the other end, the one coinciding with the locus of attention, is construed as moving along with the scan, thereby lengthening the scar as it moves. I believe that this type of fictive change, where the endpoint moves with the scan, has its roots in the familiar experience of visually tracking a moving object. Fictive change is discussed in Langacker (1999).
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4.2.5 The container schema The container schema (Figure 4.2(e)) has three basic parts: an interior, an exterior, and a boundary separating the two. We experience our bodies as containers with contents. We put things into our bodies and excrete things out. We also experience ourselves being inside a container every time we go into a room or building. We frequently put objects into containers or take them out. While these containers have actual physical boundaries, the container schema can also be used to construe a portion of space as a container, a bounded region with an interior and exterior, even when no physical boundary exists. Thus, an airplane can be said to be entering American air space. Through conceptual metaphor, the container schema provides inferential structure to many abstract domains, as when someone is in trouble and wants to get out of it. In time-telling, the container schema is applied to portions of the clock face to create bounded regions with particular interpretations.
4.2.6 The source-path-goal schema The source-path-goal schema (Figure 4.2(f)) derives from the basic event structure of an object moving from one spatial location to another. Its important elements are: a trajector (the moving object), a source (the origin or starting point of the motion), a goal (the destination or endpoint of the motion), a path (the sequence of contiguous locations occupied by the trajector in moving from source to goal), a current location (where the trajector is along the path), and a direction of motion (away from the source and toward the goal). The source-path-goal schema has its origins in our embodied experience of moving from one place to another and of seeing other people, animals, and objects do so. We move toward destinations for particular reasons and so come to associate purposes with destinations. This conflation of experience gives rise to a conceptual metaphor whereby the sourcepath-goal schema structures our understanding of abstract progress toward a goal: purposes are destinations, progress is motion toward a destination, things that impede progress either slow that motion (or block or deflect it) or move one back to a previous location on the path, while things that aid progress either speed motion or move one to a location closer to the destination. It is particularly important that motion involves time (recall the discussion at the end of chapter 3). The trajector begins moving from the source at some moment in time, occupies a sequence of locations along the path during the intervening moments, and arrives at the goal at some later moment in time. This is a bounded process occurring during a bounded interval of time; in other words, it is an event. Through conceptual metaphor, the source-path-goal schema can be used to structure conceptualizations of many different kinds of processes or events, where the trajector corresponds to the current moment in time, the beginning of the process corresponds to an initial state (source), the unfolding of the process or event corresponds to a series of intermediate states (locations along a path), and the end of the event corresponds to a final state (goal). In time-telling, different ways of reading
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the clock are different paths from a source (the clock state) to a goal (the current time). These different paths involve different impositions of image-schematic structure onto the state of the clock face, as we will see in sections 4.4 and 4.5. Finally, we have already discussed (in section 4.2.3) the importance of fictive motion, in which a source-path-goal image schema, and thus motion, is imposed upon a static scene. In these cases, the trajector corresponds to the locus of visual or mental attention, the source and goal to the starting point and endpoints of the scan, and the path to the sequence of locations scanned.
4.3 Perceiving clock structure: pointing and the three dials Now that we have introduced these embodied image schemas, we can examine how they apply to the perception of the clock face. The clock face is circular. Its two indicators are anchored at the geometric center, which coincides with the axis of rotation and the source of the driving force that turns the indicators around the clock face. Around the periphery are the marks and labels that support interpretation of the clock state. The first step to interpreting a clock state is to understand that the indicators are pointing. Let us first consider pointing as a human behavior and then look at how pointing can be attributed to an object. In humans, the basic image-schematic structure of pointing is provided by a combination of the center-periphery schema, the extension schema, and the source-path-goal schema. It also involves the kind of visual scanning we discussed in looking at fictive motion. In order to interpret a point, we need to understand that the pointing finger is not itself the focus of attention (a common error made by infants and pets). Rather we are to scan away from the body center or origo (center-periphery schema) along the extended finger (sourcepath-goal schema) and beyond the end of the finger in the same direction (extension schema) until we encounter some likely referent (goal). The referent may be encountered immediately in direct contact with the end of the finger (a touch-point) or the scan may have to continue the trajectory for some distance until a referent is found. The greater the distance from pointer to referent, the harder it is to extend the trajectory accurately, resulting in a wider search space for a possible referent. If the addressee has difficulty locating the referent, the pointer will typically move toward it, narrowing the search space, and may (if possible) even close the gap between the pointing finger and intended referent until they come into direct contact. For a point to function successfully as a communicative act, both the pointer and the addressee must share the conceptual model of pointing. The conceptual model of pointing minimally includes the following basic elements: a pointer, who uses a part of his or her body (or a surrogate or extension, such as a held object) to direct the visual attention of the addressee toward some intended referent; an addressee, who performs a visual scan and searches for the referent; and a particular form of visual scan, which originates at the pointer’s body, extends beyond it, and ends at the referent. The form of this scan is primarily structured by the source-path-goal
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schema, where the center-periphery schema locates the source in the pointer’s body, the extended body part defines the path, the extension schema extends the trajectory of the scan beyond the end of the body part, and the expectation of a goal defines the search for a referent. Once the conceptual model of pointing is in place, it can be applied to other extended body parts—a foot, the lips, an elbow—always scanning from origo beyond the end of the body part and on into space along the same trajectory toward an anticipated goal. In much the same way, we can interpret pointing done with the eyes, whether intentional or unintentional. We follow the direction of gaze from the eyes to a likely referent. When close enough to see the eyes clearly, we can even respond to the convergence of the eyes by following their combined trajectories to a point of intersection (the focal point), helping us to narrow the search space for possible referents to an approximate distance in front of the looker. What happens when the function of pointing is attributed to an object rather than a person? Take, for example, a sign in the shape of an arrow. In order for the arrow to function as a pointer, the viewer must interpret the arrow according to the conceptual model of pointing. As we have seen, this model includes an attribution of intentionality—that there was a human being, the pointer, who had a conceptual model for pointing and who placed the arrow there to serve a communicative function, namely to point toward some intended referent. Because there is no body present, the center-periphery schema cannot be used to locate the origo and orient the source-path-goal schema in a particular direction. This limitation is overcome by the shape of the arrow. When a physical arrow is fired from a bow, it moves through space in the direction of the arrowhead, not the tail. A symbolic arrow has the same orientation, with the arrowhead pointing toward the goal. This has become so highly conventionalized that the conceptual link to the motion of an actual physical arrow is no longer required. Even the symbolic arrow can be degraded considerably, so long as it is possible to discern a triangle or point at one end (the arrowhead), which then allows the extension and source-path-goal schemas to be applied. In most early mechanical clocks, the indicators were in fact shaped like arrows, making the intended direction of the point explicit. In many clocks since then, including the watch I wear on my wrist, the indicators are simply rods, yet they still serve a pointing function. How does this work? The answer lies in the fact that the center-periphery schema can still be applied. Even though a clock face does not contain a person (although some novelty watches do include a cartoon figure who points), the clock does have a clear geometric center, which we have already noted is the anchor point, source of impetus, and center of rotation for the two indicators. If we simply locate the source of our source-path-goal schema at this center, direct the path of the visual scan along the indicator, and extend that path beyond the end of the indicator, we will intersect some point on the scale around the clock’s perimeter, which is the goal, i.e. the intended referent. Because the clock face has two indicators that serve pointing functions, it is necessary to distinguish between them. As we saw in chapter 3, it became conventional to distinguish them on the basis of
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length. The shorter hand serves one pointing function, while the longer hand serves another. So far in our discussion, clock-reading involves a conceptual model for pointing (structured by three image schemas and roles for the pointer and addressee) and a perceptual distinction between the lengths of two indicators that serve pointing functions. The next step is to figure out what the indicators are pointing to. It is at this point that the part-whole schema becomes relevant. Consider once again the three dials of the clock depicted in Figure 4.1. Each of these dials is a conceptual whole made up of parts in a particular configuration. Earlier in this chapter we described the clock face as three superimposed dials—three layers of structure, at least two of which must be perceived in order to read the clock. In fact, the clock face has no layers at all; it has only a single arrangement of structural elements. Some of these elements—such as the major tick marks—participate in more than one dial, while others—such as the minor tick marks—participate in only one. The perception of layers is a conceptual achievement, and it results from different impositions of the part-whole schema. One imposition of the schema creates the hour dial, connecting the short indicator in the center with the numbers and major tick marks around the periphery. Another imposition of the schema creates the minute dial, connecting the long indicator in the center with the major and minor tick marks around the periphery. The third imposition of the schema creates the quarter-hour dial, again connecting the long indicator in the center with only four of the major tick marks around the periphery. To some extent, these impositions of the part-whole schema are aided by Gestalt perception, specifically the tendency to perceive similar elements arranged in a regular pattern as a single entity. Thus, the tick marks are easily perceived as a group. In some cases, though, Gestalt perception can get in the way: in the quarter-hour dial, for example, four of the major tick marks need to be extracted from the others. These tick marks can be made to stand out through the imposition of an additional conceptual model that joins them in a group (i.e. as parts of a whole); this conceptual model is a circle divided into canonical quarters (as demonstrated in chapter 5). In all of the dials, Gestalt effects help only around the periphery; the part-whole schema provides the only connection of these peripheral structures to a single indicator in the center.9 The resulting conceptual unities have both part-whole and center-periphery schematic structure. Both play a role in the pointing function described earlier: the center-periphery structure anchors the image-schematic structure of pointing while the part-whole structure provides a class of possible referents. Before leaving the topic of perceiving the structure of the clock face, it is important to take note of one more conceptual model: the number line. The number line is one of the central conceptual blends of mathematics (Lakoff and Núñez 2000). It combines the arithmetic understanding of quantity with the geometric 9
In fact, there may be a slight motivation toward particular pairings of peripheral structure with central indicators: the shorter indicator points to the numbers, which are closer to the center of the clock, while the longer indicator points to the tick marks, which lie further away (see chapter 3 for historical reasons for this arrangement).
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understanding of distance. Conceiving of numbers as points on a line makes it possible to use spatial reasoning to carry out arithmetic operations. On a clock face, the numbers around the perimeter derive from the ancient system of counting the temporary hours of daylight (1st hour, 2nd hour, 3rd hour, and so on). The clock differs from earlier artifacts not only in measuring out a standard interval of time, the clock-hour, but also in having equally spaced tick marks that go all the way around the dial. This means that the numbers on the clock face can be interpreted as a number line wrapped around the perimeter of a circle, so that equal arcs of distance, at any orientation, always correspond to equal intervals of time. It is not necessary to recognize this in order to read the current time; the image-schematic structures that support perception of the dials, the conceptual model of pointing, and additional image-schemas and mappings to be described later in this chapter make it possible to do that. What adding the conceptual model of the number line does is enable spatial reasoning about time relations using the structure of the clock face. This becomes important when the activity shifts from simply reading the current time to solving some type of time problem, such as determining how much time has elapsed since the start of an event or how much time remains until the end of an event or the start of some future event. These problems involve reasoning about relations between the current time and some past or future reference time. Such problems can be solved through a combination of time-reading and arithmetic operations or, via the conceptual model of the number line, through spatial reasoning on the clock face itself. On a digital clock, only the former is possible. Despite this important difference, we will restrict our focus in this chapter to the task of reading the current time from an analog clock display.
4.4 Reading absolute times When the time is exactly on the hour, there is only one conventional way to express it: as the number of the hour followed by the phrase “o’clock.” In all other situations, there are two forms of time expressions: absolute (“five thirty”) and relative (“half past five”). Both refer to the same point in time, but each relates to a different set of conceptual operations. Absolute time is generally regarded as more formal and precise and is the preferred form of time expression in official announcements. It is also the only form of time with a standard numeric written format (compare ‘8:15’ with “a quarter past eight”). It is also the only conventional form for expressing times on the 24-hour schedule used for official time-keeping throughout Europe and by the American military. So, for example, the military time corresponding to 3:42 p.m. would be announced as “fifteen forty-two” and not as “forty-two past fifteen” or “eighteen till sixteen.” Until recently, times were commonly expressed in relative form in casual conversation and often rounded to the nearest five-minutes, reflecting the reduced effort needed to read times at the major tick marks on analog clocks (as discussed below). Several decades ago, the marriage of quartz crystal timekeeping to liquid crystal displays revolutionized the watchmaking industry and led to the rapid
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proliferation of cheap, accurate digital wristwatches. Since that time, absolute time expressions have become increasingly common in everyday speech, reflecting the simple number-naming strategy for reading a digital display.10 In this section, we look at absolute time: examining the form of absolute time expressions, the order of looking at the dials in the clock face, and the image-schematic structures and strategies involved in reading the time. We then do the same for relative times in section 4.5.
4.4.1 The form of absolute time expressions Absolute time expressions express a point in time as a number of hours followed by a number of minutes (or “o’clock” if there are no minutes to express). An example is “twelve forty-three,” which means ‘twelve hours and forty-three minutes’. The units are usually omitted but may be included for clarification or emphasis, in which case the conjunction “and” may be added. Represented schematically, the full form of an absolute time expression is ‘ hours and minutes’, while the more common abbreviated form is simply ‘ ’, where ‘H’ and ‘M’ denote the numbers of hours and minutes, respectively.
4.4.2 The order of looking for reading absolute times As implied by the form of absolute time expressions, reading the absolute time from an analog clock involves first reading the hours and then reading the minutes. This generates a particular order of looking: first, look at the structures on the clock face that are relevant to reading the hours (the components of the hour dial); then look at the structures relevant to reading the minutes (the components of the minute dial). The quarter-hour dial is not used. Each of these acts of looking is a particular form of situated seeing in which image-schematic structure is imposed upon the clock face. To complicate matters further, the strategies for reading the hour and minutes, and the image-schematic structures associated with these strategies, are not the same. Failing to realize this difference produces clock-reading errors.
4.4.3 Clock-reading strategies for absolute times In order to construct an absolute time reading, the clock-reader must first scan the clock to locate the two indicators and to distinguish them based on length (Figure 4.3). Once this is done, the clock-reader can set about reading the current hour. Reading the current hour begins with attending to the structures relevant to the hour dial, namely the short indicator at the center and the circle of numbers and major tick marks around the periphery. Assuming that these are linked by the image10
With a digital wristwatch, the practice of matching the precision of time-reporting to the demands of the social situation can force extra cognitive effort: rounding the displayed time appropriately. Decades ago, if someone had asked a passerby for the time, a response like “ten fifty-three” would have seemed absurdly precise. Today, it hardly raises an eyebrow.
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schematic structures described in section 4.3 and that the conceptual model for pointing is applied to the indicator, the indicator points either to a location that coincides with one of the numbers and its associated major tick mark or to a location that lies somewhere between two numbers. If the indicator points directly to one of the numbers, then the task is simple: name the number that the indicator points to. But if the indicator points to a location between two numbers, the task is considerably more complex. A familiar strategy from the common experience of finding the referent of a point would be to impose the proximity schema and name the number that the trajectory of the point comes nearest. This, in fact, turns out to be the wrong strategy for reading the hours and is a frequent source of error (discussed in detail in chapter 6). The hours can be read correctly by ignoring proximity and instead imposing a container schema. Conceptualizing the space between adjacent numbers as a bounded region or container produces twelve possible containers that the indicator could be located in. The region containing the indicator corresponds to the current hour, and its name is given by the lower boundary, i.e. the prior number in the counting cycle (e.g. the hour is “three” if the indicator is in the region bounded by 3 and 4). This way of reading the hours is depicted in Figure 4.3. In actually reading the current time, it is not necessary to conceptualize all twelve of the bounded regions for hours; it is only necessary to conceptualize the container around the short indicator at its current location. Once the number of hours has been named, the next step is to name the number of minutes. In this case, attention shifts to the long indicator and the associated structures of the minute dial, specifically the ring of major and minor tick marks around the periphery. Once again, the indicator is construed as pointing to a location around this ring. In this case, invoking a container schema is not the preferred strategy. Instead, it is customary to round to the nearest minute, so invoking the proximity schema produces a particular referent: the major or minor tick mark nearest the pointed-to location. This tick mark corresponds to the number of minutes, but such marks are typically unlabeled. Correctly naming the number of minutes depends upon a second strategy: either producing the associated label from memory (as described below) or counting to the indicated tick mark from some recognized landmark. A typical counting strategy (especially when learning to read a clock, as in chapter 5) is to count by fives using the major tick marks, and then to increment (or decrement) the count by ones using the minor tick marks until the tick mark pointed to by the indicator is reached. In order to count correctly, it must be recognized that the starting point for the count is the major tick mark at the top of the clock (coinciding with the hour label ‘12’) and that the counting should proceed in a clockwise direction. The count can be shortened by recognizing the time at a major tick mark (i.e. providing the associated numeric label) and then counting up or down from that reference point. The other strategy of filling in the label from memory is supported by two possible associations. One is to associate a number of minutes with each hour label, so that the label ‘5’ is associated with 25 minutes (the number of the label times five). Another is to associate the names of numbers of minutes with their corresponding
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spatial patterns on the clock face. Whenever the long indicator points to the tick mark for twenty-five minutes, for example, the indicator has a distinctive orientation on the clock dial. This recurring pattern comes to be associated with a particular name (“twenty-five”). This strategy of pattern naming is supported by the fact that the discrimination set for major tick marks is just 12 patterns (“o’clock,” “five,” “ten,” “fifteen,” etc., up to “fifty-five” and then “o’clock” again).11 Once the pattern names for major tick marks are learned, nearby minor tick marks can be found through quick-counting (or quick mental addition or subtraction) since a minor tick mark is never more than two away from a major one. The brain is highly adept at pattern recognition, so pattern naming can become reliable for minor tick marks as well, in which case the discrimination set is 60 possible patterns. As clock readers become proficient, counting comes to be relegated to the status of backup strategy (Siegler and McGilly 1989). In similar fashion, pattern naming can be used to name whole absolute times, obviating the need for sequential reading of the hours and minutes. In this case, the distinctive configurations of short and long indicators (when correctly oriented) can come to be recognized and named. The possible combinations of hour hand positions and minute hand positions would be enormous if the hands moved independently, but of course they do not. The hands are geared together, meaning that the vast majority of potential combinations are “built out” of the artifact. In fact, any position of the minute hand can co-occur with only twelve possible positions of the hour hand (each within one of the bounded regions corresponding to a particular hour). Rounding to the nearest minute produces a total of 720 possible hand configurations. For just the major tick marks, the total of possible configurations is reduced to 144. For just the quarter-hour times, the total is 48. For the common reference times of the hour and half-hour, the set is 24, while for the hour times alone it is 12. Because of the size of the discrimination sets, the order of learning (discussed in chapter 5), and the frequent use of the hour and half-hour times as landmarks for scheduling daily activities, the rapid recognition and naming of entire hand configurations is likely to start with the hour and half-hour times and then possibly to extend to the quarter-hour and five-minute times, depending on the experience and proficiency of the clock-reader. In fact, proficient clock readers are likely to rely upon a variety of strategies for reading absolute times, including wholeclock pattern naming for landmark times, sequential hour reading and minute pattern-naming for five-minute times, and sequential hour reading and minute counting from a nearby reference point as a fallback strategy. Even when pattern naming becomes a reliable strategy for reading many absolute times, imposing image11
The distinctive pattern name of “o’clock” is typically the first learned. It means literally “on the clock,” a relic from the days when mechanical clock times had to be distinguished from the standard sundial times. This label is used for times on the hour, so we say “four o’clock” rather than “four zero” or “four hours and zero minutes” (but note the military practice of stating such times as “four hundred hours” or “sixteen hundred hours,” explicitly acknowledging the two zeroes in the minutes position and the omission of the colon as a separator: 400 or 1600).
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schematic structure onto the clock face may still provide advantages for reasoning about relations between times. This takes us into the topic of relative times, the subject of the next section.
4.5 Reading relative times Relative times differ from absolute times in that they involve expressing the current time not as a specific numeric value but instead as a relationship to some reference time, as in “ten past one” or “quarter till three.” In English the conventional reference times are either the previous hour or upcoming hour. In other languages, such as German or Dutch, the half-hour may be used as a reference time, resulting in expressions like “five till half-eight” for 7:25. In Czech, the quarter-hour may be used. Relative times differ from absolute times in practically every way: in the forms of the time expressions, in the order of looking at the dials in the clock face, and in the image-schematic structures employed in constructing time readings.
4.5.1 The form of relative time expressions Relative time expressions take the standard form ‘ ’, paralleling the form of the spatial expression ‘ ’. In English, the interval is expressed in quarter-hours12 or minutes, the relationship is either past (after) or till (to), and the reference point is either the previous hour or the upcoming hour. Considered this way, there are essentially two forms of relative time expressions in English: ‘ ’ and ‘ ’. Where either form could be applied, the quarter-hour form takes precedence: “half past” is common, but “thirty past” is not. At times other than the canonical quarters, the minutes form is used.
4.5.2 The order of looking for reading relative times As can be seen from the form of the relative time expressions, the order of looking for reading relative time is the opposite of that for reading absolute time: look first at the long indicator and then at the short indicator. When looking at the long indicator, several decisions must be made: (1) whether to round to the nearest 12
Why not use thirds of an hour? One reason comes from the history of time-telling described in chapter 3: Lines marking the quarter-hours were part of the sundial scales that pre-dated the clock. These were carried over onto the early mechanical clocks, which had only a single indicator and a dial for hours. Another reason relates to the act of division itself. The simplest possible division is into two parts. Successive division into two parts produces halves, quarters, eighths, and so on (like the scale on an American ruler). Once something has been divided into halves, a division into thirds is no longer afforded.
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major tick mark; (2) whether to read the time interval past the previous hour or till the next hour, and (3) whether to read the time interval in quarter-hours or minutes. Relative times are often rounded to the nearest major tick mark, reducing the discrimination set of time intervals considerably while still producing a reading within a couple of minutes of the exact time, accurate enough for most everyday time-telling. When the time is very close to the reference hour (e.g. three minutes till four), the exact interval is likely to be named. Whether the time is read as past or till is generally based on proximity: we read the time relative to the nearest hour. This keeps the time intervals to a half-hour or less. In terms of the clock, this practice divides the clock face down the middle. When the long indicator is on the right side, times are read as past, and when it is on the left, they are read as till. How the boundary itself is treated differs between languages, with, for example, English favoring past while German favors till.13 Finally, whether to read quarter-hours or minutes depends upon the actual location of the long indicator on the dial. When the long indicator points to one of the three major tick marks that are part of the quarter-hour dial (but not straight up, which is “o’clock”—the reference time), then a reading of the time interval in quarter-hours is facilitated and generally preferred (compare “half past seven” to “thirty past seven,” “quarter past seven” to “fifteen past seven,” and “quarter till eight” to “fifteen till eight”). When the long indicator points more than a couple of minor tick marks away from one of these three locations, rounding to the quarter hour produces a reading that is too imprecise for modern sensibilities, given the nature of our lifestyles and the accurate time-keeping of contemporary clocks and watches. At these in-between locations, reading the time interval in minutes is preferred. Once the quarter-hour or minute interval has been named and the relationship to the reference time (past or till) has been expressed, attention shifts to the short indicator and the reading of the appropriate reference hour. Reading the reference hour for relative time differs from reading the current hour for absolute time because times during the second half of the hour are referenced to the upcoming hour rather than the previous one. This difference demands a different strategy, to be discussed below.
4.5.3 Clock-reading strategies for relative times Just as for absolute time, constructing a relative time reading begins with scanning the clock to distinguishing the two hands. Figure 4.4 depicts the subsequent process for constructing a quarter-hour relative time, while Figure 4.5 does the same for a minute relative time. In both cases, once the long hand has been 13
This may be an oversimplification. Germans use the expression “halb acht” (half eight) to refer to 7:30, but they also use the expressions “ein Viertel acht” (one quarter eight) for 7:15 and “drei Viertel acht” for 7:45, as if the expanse of the clock face has been construed as a container being filled (the container schema) and the progress of the filling is being tracked by a source-path-goal schema with the time always referenced to the goal or full state.
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picked out, the clock reader can set about reading the time interval, but in order to do so, the reader must first decide whether to read the time past or the time till. In principle, the time could be read in either direction, but in general, the time is read to the nearest reference point—the nearest hour—unless one of the reference points has greater importance or relevance to the situation that prompted the time reading. This might happen if the upcoming hour marked the start or end of some important event, in which case saying “It’s forty minutes to one” would be called for. Generally, proximity reigns, and whether to read past or till is decided simply by noting whether the long hand is on the left or right side of the clock face. Conceptually, this means dividing the clock face along a vertical line of symmetry, producing the familiar left/right symmetry of our own and others’ bodies and of most objects we interact with. The left and right sides are related through the partwhole schema. By construing these sides as containers (imposing the container schema), we can see which side contains the long hand. The orientation of the long hand, made salient when the conceptual model of pointing is applied, provides another cue. The practice of referencing relative times to the nearest hour also makes it possible to use the position of the short hand to decide between past and till simply by imposing the proximity schema and seeing which hour number the hand is nearest. Because the order of looking makes the long hand the initial focus of attention and because its position on the left or right of the clock face is much easier to discern, the position of the long hand is likely to be the salient cue. If the situation calls for breaking with the convention of referencing the time to the nearest hour, then the choice of past or till will depend upon the imposition of a sourcepath-goal schema. In this case, the position of the long hand is construed as a location along a path of motion from the top of the clock around clockwise back to the top. The time is then referenced to the more important hour, producing a ‘past’ reading if that hour is the source and a ‘till’ reading if it is the goal of motion along this path. Once the choice of past or till has been made, the clock reader proceeds to read the time interval. The time interval can be read using two different kinds of conceptualizations: state-based and motion-based. These two possibilities result from the nature of the clock itself. The indicators on the clock do move, supporting a motion-based conceptualization in terms of the source-path-goal image schema. But the indicators also move so slowly (even imperceptibly) that their positions on the clock face appear fixed, supporting a state-based conceptualization in terms of the shape and orientation of bounded regions of space (similar to using the container schema to read the hour for absolute times). Let us examine this difference for situations in which the time interval is read in quarter-hours. In terms of its motion from one reference time to the next, the long indicator traces a path from the top of the dial around the dial in a clockwise direction until it reaches the top again. In this case, the source and the goal are the same, but the indicator moves away from the source to the right and approaches the goal from the left (from the point of view of the observer facing the clock). When the long indicator points due right, it has traveled one fourth of the way along the path from source to goal. This is quarter
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past the reference hour (the source). When the indicator points straight down, it has traveled halfway from source to goal. This is half past the reference hour (still the source). When the indicator points due left, it has traveled three quarters of the way along the path from source to goal and has one quarter of the path remaining; this is quarter till the reference hour (now the goal). Let us now consider the same time intervals in terms of a state-based conceptualization, i.e. a conceptualization that does not involve motion. When the long indicator points due right, it can be conceptualized as defining a boundary for an enclosed region of space—a container. The other boundaries of this region are an imaginary line from the center of the clock to the top of the dial—this is the reference position of the indicator—and the arc of the dial that connects the top to the right, enclosing the space. Filling this container produces a conceptual object with a familiar shape, namely a quarter-circle, which activates another conceptual model. This quarter-circle is also canonically oriented: it is the upper right quadrant of a circle divided by drawing a line down the center and a second line across the middle. The reference circle, the background against which this profiled object is understood, is the circular dial of the clock face. The circular dial can itself be conceptualized, via perceptual filling and the use of the part-whole schema, as a solid circle made up of four parts, with the profiled quarter-circle being the upper right part in the four-part configuration. This conceptual quarter-circle is to the right of the line of symmetry dividing past and till and is therefore “a quarter past.” We can proceed in analogous fashion when the indicator points due left: filling the conceptual region bounded by the indicator, reference line, and arc of the dial to create a conceptual object that is another quarter-circle, this time forming the upper left quarter of a divided circle. Because the quarter-circle is to the left of the line of symmetry, it is read as “a quarter till.” When the long indicator points straight down, the region bounded by the indicator, reference line, and arc of the dial can be filled to produce a different but related conceptual object: a half-circle. In this case, the bounded regions of space are similar, forcing a choice of whether to fill the right side or left side. Convention has us fill the region on the right side, producing a halfcircle object which, via the part-whole schema, constitutes the right half of the clock circle and is thus read as “half past.” This convention might simply reflect priority: past-times precede till-times, so the time continues to be read as past until it crosses over the boundary into till.14 In each of the three cases described here, the shape, orientation, and location of the bounded object-region on the clock face provides sufficient structure to name the interval as “quarter past,” “quarter till,” or “half past” without any conceptualization of motion. 14
It is certainly possible, although unconventional, to conceive of the time as half till the next hour, or even as three quarters past the previous hour or three quarters till the next hour. This is simply a matter of filling different regions and making them the profiled conceptual objects. Several factors tend to suppress these conceptualizations. Two have already been mentioned: the conventional dividing line between past and till (based on proximity) and the priority of past times. A third factor is the perceptual tendency to perceive angles less than 180 degrees and to fill space accordingly.
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The same strategy works for reading the time intervals in minutes at the major tick marks. Filling the interior of the angle between the long indicator and the reference line produces the set of distinctive shapes shown in Figure 4.6. Here the names of the shapes are not motivated by the conceptual model of dividing a circle into parts. Instead, the names come from the number of tick marks enclosed by the filled space between the long indicator and the reference line; these represent the number of minutes past or till the reference hour. There are only eight additional patterns to be distinguished, a small discrimination set for the brain, which is highly adept at pattern recognition. There are also some underlying relationships that support correct naming of the patterns. Consider first the patterns on the right side of Figure 4.6. The positions of the long indicator that correspond to 5 past, 10 past, 20 past, and 25 past are the same positions that in absolute time corresponded to 5, 10, 20, and 25 minutes, respectively. While this alignment supports naming of the patterns on the right side of the clock face, it provides little help on the left side: 5 till, 10 till, 20 till, and 25 till line up with the absolute times 55, 50, 40, and 35 minutes, respectively. Each pair does sum to 60 since there are sixty minutes in an hour, but making use of that relationship imposes an additional cognitive load and the potential for an addition or subtraction error. Comparing the patterns on the left of Figure 4.6 to those on the right suggests a simpler strategy: perceptual mirroring.15 Here we make direct use of the line of symmetry down the middle of the clock—a conceptual structure that was not used for absolute time because it was simply irrelevant. This line of symmetry is used to divide past and till, but it can also be used to support correct naming of the patterns on the left side of the dial. Each such pattern is the mirror image of its counterpart on the right, so it can be named with the same number and the substitution of the word “till.” This memory strategy is suggested by the arrows in the figure. If the name of a pattern cannot be recalled, it is possible to use a counting strategy to determine the number of minutes in the time interval. The basic idea is to count the tick marks from the reference (the top of the clock face) to the tick mark pointed to by the long indicator. For major tick marks, we count by fives, and for minor tick marks, we count by ones. Here again the line of symmetry plays a role. For times past the reference hour, we count the tick marks from the top of the clock to the indicator in a clockwise direction, moving down the right side of the dial. For times till the reference hour, we reverse direction and count the tick marks from the top to the indicator in a counterclockwise direction, moving down the left side of the 15
Perceptual mirroring seems to be a natural part of Gestalt perception: filling in symmetry. If the specific value of the numeric labels is ignored, the clock face is highly symmetric. It is actually symmetric around many axes, but the vertical axis is the most salient for perceiving symmetry (aligning the left-right visual field and matching the familiar symmetry of bodies and many other objects we interact with frequently). It is conceivable that one could engage in a mental operation like folding one side of the clock over the other—indeed, this is a way of demonstrating symmetry for real-world objects—but this operation seems unnecessary, given the power of perceptual mirroring.
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dial. This seems counter-intuitive because it involves moving in the opposite direction of clock-hand motion and thus in opposition to the indicator that is approaching the reference. This is especially troublesome when this sense of approach is reinforced by a source-path-goal schema. So why not start at the position of the indicator and count clockwise to the top? This works fine if the indicator is at one of the major tick marks, in which case we can count by fives, but if the indicator is at a minor tick mark (say 10:37), then we are put in the awkward position of counting by ones first (1, 2, 3) and then having to increment by fives (8, 13, 18, 23—which is sequential mental addition, not reciting a familiar sequence) or else having to finish the entire count by ones, which is tedious. If instead we suppress the source-path-goal schema (the sense of motion), treat the indicator as a static object in a fixed position, and then count down the left side of the clock, we reap the benefits of counting by fives first and then finishing the count by ones: 5, 10, 15, 20, 21, 22, 23. When the indicator does point to a minor tick mark, the strategies of pattern naming and counting can be combined: we can use pattern naming to identify the nearest major tick mark and then increment or decrement the count to the minor tick mark. In practice, this means counting up or down by just one or two since the nearest major tick mark is never more than two marks away. Finally, consider one more strategy discussed for absolute times: making associations between numbers of minutes and the numeric hour labels (associating “twenty-five” with ‘5’, for example). Such a strategy proves difficult with time intervals. For times past the hour, it is easy to make the correct associations because they are the same associations made for absolute time: ‘5’ associates with both 25 and 25 past. But for times till the hour, the associations are confusing: ‘7’ is associated with 35 and 25 till, ‘8’ with 40 and 20 till, and so on. The duplication of numbers—4 with 20, 8 with 40 and 20—creates a large potential for recall errors. Dehaene (1997) discusses a similar problem for learning the multiplication tables. Like the multiplication tables, the associations can be learned but only with considerable practice and persistent errors. The one exception that would seem easy to recall is the association of ‘10’ with 10 till, but even here the previous association with 50 and the habit of suppressing the name of the visible number while trying to name the number of minutes could actually slow the response. For times till the hour, ignoring the numeric hour labels and relying instead on pattern naming aided by perceptual mirroring seems a more reliable strategy. Each of the strategies outlined here, involving memory associations, different image-schematic structures, various conceptual models (quarter-circles, counting), and so on, can be used independently, but they can also be combined. In practice, different strategies are likely to be used for different intervals, perhaps memory association for one and perceptual mirroring for another, and different strategies might even be used to name the same interval on different occasions (this was observed in children in Siegler and McGilly 1989). Part of developing a rich conceptual understanding of time-telling involves using multiple strategies that rely upon different ways of conceptualizing the clock face and then discovering the
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relationships among these conceptualizations. This topic will be explored further in chapter 6. Once the past or till relationship has been noted and the time interval has been read in quarter-hours or minutes—both of which can occur simultaneously in pattern naming—it is then time to read the reference hour. To do this, the clockreader shifts attention to the short indicator. This indicator points to a location between two major tick marks and their associated number labels (recall that the indicator points directly at a number only when the time is “o’clock” and a relative time reading is unnecessary). For absolute times, the region bounded by two numbers was conceptualized as a container and all positions of the short indicator within this container corresponded to the hour named by the prior number in the sequence (“three” when the indicator was between 3 and 4). This container-based conceptualization does not work for relative time. Relative time is expressed as an interval past or till a reference hour, so only the reference hours—the positions of the short indicator at the o’clock times—actually matter. To read the reference hours correctly, we need to make use of a source-path-goal image schema. The current position pointed to by the indicator corresponds to an intermediate location along a path from source to goal. The source is the location pointed to by the indicator at the previous o’clock time: this is the prior number (counterclockwise and previous in the counting sequence) and its associated tick mark. The goal coincides with the location that will be pointed to by the indicator at the next o’clock time: this is the upcoming number (clockwise and next in the counting sequence) and its associated tick mark. In a situation where the short indicator points between the 3 and the 4, for example, the source is the location at 3 while the goal is the location at 4. For time past the reference hour, we name the source, and for time till the reference hour, we name the goal. This directs attention back to the previous number for times past and up to the next number for times till. If we follow the convention of reporting times as past when the long hand is on the right side of the clock and till when it is on the left side, the reference hour—source or goal—will also turn out to be the nearest number. This correlation makes it seem appropriate to use the proximity schema to read the reference hour, in other words simply taking the location pointed to by the short indicator and then rounding up or down to the nearest hour number (just as we did for the long indicator, rounding up or down to the nearest minute). Such a strategy was in fact used to read early mechanical clocks that had a single indicator: the pointed-to location was simply rounded to the nearest quarter-hour mark. Unfortunately, this strategy does not work reliably for a modern two-handed clock. As we have seen, the reading of the hour on a two-handed clock is governed by the container schema (for absolute time) or the source-path-goal schema (for relative time). Misuse of the proximity schema to read the hour turns out to be a frequent source of error, a topic we return to in chapter 6. Proximity can be used appropriately with the short indicator only as a basis for selecting whether to read the time as past or till before turning attention to the long indicator and reading the appropriate interval. Doing this involves an extra shift of attention (short
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hand—long hand—short hand) but also eliminates the need to impose a conceptual dividing line on the minute and quarter-hour dials.
4.6 Discussion In our analysis of clock-reading, we have described several different clockreading strategies with varied image-schematic structure (proximity, container, or source-path-goal), reliance on pattern recognition or other memory associations, and, in some cases, recruitment of additional conceptual models (for dividing a circle or counting). We have also related these strategies to different linguistic forms for expressing the time: a single form for hour times (“twelve o’clock”); a choice of two forms for most other times, either absolute (“seven forty”) or relative-minutes (“twenty to eight”); and a choice of three forms for half- and quarter-hour times: absolute (“two fifteen”), relative-quarter-hours (“quarter past two”), and relativeminutes (“fifteen past two”)—although the relative-minutes form is comparatively uncommon. The analysis presented an orderly alignment of linguistic form, order of looking, and clock-reading strategies. While this alignment produces efficient timetelling, it is certainly not the only way to proceed. It is quite possible to mix different strategies, vary the order of looking, and convert a time read in one form into a different form for verbal expression, so long as the net result is an appropriate and correct output. Digital displays support rapid reading of the absolute time by conforming to the left-to-right reading convention and providing the hour and minutes numbers to be directly named. But with digital displays, reading the relative time can involve additional mental operations, such as adding up to 60.16 Analog clocks provide spatial structure that can be used to construct a time reading. This demands some imposition of appropriate image-schematic structure to construct an absolute time reading, but it also provides spatial-perceptual support for reading intervals for relative time readings and for relating these intervals to the appropriate reference 16
Why adding up and not subtracting? Consider reading the relative time from a digital display of 7:42. Compare the following orders of terms: (1) 42 plus what is 60? (2) What plus 42 is 60? (3) 60 minus what is 42? (4) 60 minus 42 is what? The questions are logically equivalent but not cognitively equivalent. (1) and (2) involve building up rather than taking away. (1) and (3) express the unknown in intermediate position as the interval between given and total. (4) ends with the information being sought. Only (1) starts with the given information; all of the others require that this information be held in working memory. Starting with the information already in mind and working up from there, as in (1), is a likely default strategy.
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hour. In addition, the analog clock provides familiar spatial patterns that trigger rapid naming, especially for landmark times. While the form of the display may spur one kind of time-telling (e.g. exact absolute time for digital displays), the pragmatics of the communicative situation may call for another (e.g. approximate relative time or the time relative to some upcoming event), creating a momentary mismatch and the possible mixing of strategies or the conversion of time from one form to another. A person who has come to rely on a diminished set of clock-reading strategies—either because they were the only ones learned or because they have become ingrained through habit—may proceed less efficiently, depending upon how the strategies match both the form of the display and the demands of the social situation. An interesting prediction arising from the account here is that times in the second half of the hour should be harder to tell than times in the first half of the hour, especially for inexperienced time-tellers. Why should this be so? Consider the times in the first half of the hour being read from an analog clock. There are two ways to read these times—absolute and relative—and the two forms are related by a simple transformation: simply reverse the terms and add or delete “past” (or “after”). “One ten” is “ten past one.” “Twenty after seven” is “seven twenty.” The associations for absolute and relative times reinforce one another; the direction of increase (five, ten, fifteen,…) matches the direction of motion of the clock hands; and the hour, although the current hour for absolute times and the prior reference hour for relative times, remains the same. Now consider the times in the second half of the hour. “One fifty” is “ten till two.” “Twenty to eight” is “seven forty.” Several things are happening here. One is the misalignment of the number associations (50 with 10, 20 with 40), providing disruption of associations rather than reinforcement. Another is the changing direction of increase: the numbers for till times go up as you move in a counterclockwise direction, opposite the direction of increase for past times and opposite the motion of the clock hands. A third factor is the change in the hour, from the current hour for absolute time to the upcoming reference hour for relative time. Finally, even when using absolute times exclusively, the numbers in the second half of the clock are larger than in the first half; which may make them take longer to process (Dehaene 1997). For digital clocks, the difference for absolute times from the first to the second half of the hour is likely to be minimal since the numbers can simply be read and do not have to be recalled. It should also take only slightly longer to name relative times in the first half of the hour than absolute times, owing to the need to reverse the normal reading order to read minutes first and then to add “past.” More difficulty is likely to arise for relative times in the second half of the hour, where the interval from the displayed number of minutes to sixty has to be computed (or recalled from frequent experience) and the displayed number of the hour has to be incremented by one. Because timetelling in everyday life involves making choices about looking, strategies, and reporting, the mismatches and complications in the second half of the hour should be more disruptive for time-tellers who use both forms of reporting rather than one
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or the other exclusively, and it should be more pronounced for those with less clockreading practice.
4.6.1 Experimental evidence Is there any experimental evidence to support the account of clock-reading given here? Studies of children’s clock-reading will be discussed in chapter 5, but it is worth mentioning now that a study of 8- and 9-year-old readers of analog clocks conducted by Siegler and McGilly (1989) found that 5-minute times were indeed easier to read in the first half of the hour than in the second half; in fact, whether the time was in the first half accounted for two-thirds of the variance in both median solution times and percent errors. Recent data from Bock, Irwin, and Davidson (in press) show a less marked difference for adults (American undergraduates): naming absolute times from an analog clock did take longer during the second half of the hour, while naming relative times took longest in the vicinity of the half hour, a slowing that could reflect both the magnitude of the time intervals and proximity to the conceptual dividing line between past and till. For digital clocks, absolute times were named with the same speed throughout, while relative times took slightly longer to name in the second half of the hour. The same study also found that while American undergraduates reading analog clocks had a strong preference for absolute expressions, the number of relative expressions increased when the minute hand was in the upper left quadrant of the clock. Bock et al. argue that this shift to relative expressions is caused by the impending change in the current hour (the hour used in absolute time-telling), hypothesizing that the increased activation of this hour interferes with the naming patterns. From the perspective of the image-schematic structure we have been discussing, it also seems possible that the combination of approach and proximity encourages greater use of relative expressions, with the shift perhaps triggered by the prominence of “quarter till” when the long hand crosses into the upper left quadrant. After the hour has changed, proximity might still encourage relative expressions for the first few minutes until a major tick mark is reached and absolute expressions again come to the fore. A recent study of Dutch speakers reading times from a digital clock (Meeuwissen, Roelofs et al. 2004) found that relative expressions referring to the past hour were produced faster than those referring to the half-hour (a Dutch reference point) or the upcoming hour, both of which require numerically transforming the minutes and incrementing the hour by one. Meeuwissen et al. also found that the time to produce relative expressions increased as the time interval changed from 0 to 5 to 10 minutes; this effect was not found in the Bock et al. data for American English speakers, although relative times appeared to take slightly longer to name at 20-25 minutes till the hour. The recent work by Bock et al. referred to above follows up on more extensive response-time and eye-tracking studies by Bock, Irwin, Davidson, and Levelt (2003), who examined clock-reading by American and Dutch undergraduates. Bock et al.’s focus was on the relationship between visual perception and linguistic production. They used clock-reading as a test domain because it has a fixed set of visual stimuli
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and highly formatted linguistic responses. Before conducting their experiments, Bock et al. did a normative study eliciting names of five-minute times from analog and digital displays. The analog displays were circles with long and short indicators but no numbers or tick marks, while the digital displays were four conjoined boxes containing single digits (including a leading zero for hours less than 10) and no colon separator. Why more prototypical displays were not used is unclear. What Bock et al. found is that American undergraduates at a Midwestern university favored absolute time expressions (94% of responses to digital displays, 89% to analog), while their Dutch counterparts favored relative time expressions (79% to digital displays, 92% to analog). Bock et al. speculate that this bias could reflect a greater prevalence of digital clocks and watches in the United States or differences in how time-telling is taught in the two countries. It might be the case that older Americans would use more relative expressions, like the Dutch, while younger Americans have been more affected by the digital invasion. In both groups, digital displays elicited more absolute expressions than analog displays did, possibly reflecting both the ease of reading absolute time from a digital display and the spatial-perceptual grounding that an analog display gives for relative time-telling. In both languages, landmark times (hour, half-hour, and quarter-hour) were named consistently, while times close to reference-hour changes (the hour boundary for absolute times and the half-hour past/till boundary for relative times) were named most variably. The general picture is a bias for particular forms of time reporting in each group skewed by the affordances of the time displays. Bock et al. then conducted two clock-reading experiments, one with American English speakers and Dutch speakers and one with only American English speakers. In both cases, the experimenters pre-assigned the form of time-expression to be used—absolute or relative—and then collected eye-tracking and response-time data while participants read analog and digital displays (this time without boxes and with a colon separator). Some participants saw the displays for only 100 ms—too short for eye movements—while others saw them for 3000 ms—long enough for successive fixations to different parts of the display. (The second experiment differed by having participants change either time expressions or displays halfway through the session.) Bock et al. found that responses were faster to digital displays in all cases, noting that the hour and minute information is explicit and located in dedicated regions. This advantage was even greater for absolute times, reflecting the speed of the simple leftto-right number-reading strategy, while for relative times, participants adjusted the strategy to read minutes first (as evidenced by the eye-tracking data). For digital displays, till times should take longer than past times due to the additional mental manipulation involved in computing the time interval and incrementing the hour count by one, but we cannot evaluate this because past and till times were not separated in the analysis. It is therefore also unclear how digital till times compared to their analog counterparts. For analog displays, relative times were read more quickly than absolute times in most cases (the exception coming from the Americans at the long display time, perhaps reflecting the American bias toward absolute expressions and the associated practice effects).
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In terms of where participants looked when reading the times, for digital displays participants tended to maintain fixation on the center (where the prestimulus fixation point had been) or shift close by, to the hour-ones or the minutetens. For analog displays participants tended to maintain fixation on the center during rapid presentation and to fixate on the hour and minute hands during longer presentation. With longer presentation on both types of displays, participants fixated successively on hours and minutes in the order that coordinated with the form of the time expression. So for relative time-telling, the fixation to minutes (minute hand or minute-tens) tended to precede the fixation to hours, especially for digital displays. For absolute time-telling, the fixation to hours (hour hand or hour-ones) tended to precede the fixation to minutes, especially for analog displays. This confirms the order of looking described in our analysis. But in each experiment, one American English speaker persisted in looking at the hour hand before the minute hand on analog clocks even when using relative time expressions. In one case, this could simply be perseveration: the form of expression was changed from absolute to relative halfway through the experiment. In both cases, it could reflect the practice effects of Americans’ preference for absolute time expressions or perhaps a check of hour hand proximity to inform the choice of a past or till reading. It’s also important to note that during rapid presentation, when participants remained fixated center and had no time for subsequent eye movements, response rates were slightly faster with little drop in performance (only a 6% increase in dysfluencies or errors). This means that participants were able to tell the time successfully after a single glimpse of the clock, perhaps helped by the fact that clock displays are so familiar and thus easy to hold in mind (no masking stimulus was used in the experiments). In Bock et al.’s analysis of “seeing-for-speaking,” they argue that the first glimpse provides a fast, parallel apprehension of the scene that is used to plan subsequent eye movements; this is followed by a slow, incremental process of linguistic formulation, during which the eyes move successively to each referent about 900 ms before it is named in the utterance. It seems that for clock-reading, the subsequent eye movements are not necessary to perform the task but may provide additional support. Bock et al. offer the hypothesis that “the eyes help to enforce the linearization and separation of information placed in serial order” (p. 683), claiming that fixation on an element both activates associated information in memory and de-activates what was previously active. This seems likely to be true, but successive fixations on elements also use the world as a working memory, rich with detail and affording the direct apprehension and verification of information and thus a reduction in error. Moreover, this is likely to be necessary when the objects are less familiar and the world is needed to anchor the conceptualizations involved in performing the task. Had the short-duration displays been shown to inexperienced clock-readers, I expect that they would have had a much more drastic effect on performance. For our purposes, the data support a relationship between the order of looking and the form of the time expression but offer no information about the imposition of image-schematic structure involved in conceptualizing the time. In Bock et al.’s
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words, “we can say virtually nothing about the perception of the clock displays” (p. 683). The only hint comes from the fact that with analog displays, the first gaze tended to be longer for absolute than for relative time, perhaps reflecting the effort involved in suppressing proximity and instead relying upon a container schema to read the hour. For digital displays, the first gaze tended to be longer for relative times, likely reflecting the additional work involved in computing the appropriate interval for till times. While the fixations of the eyes and the timing of responses provide important clues as to what type of information is used when, and how complex the processing might be, they provide little insight into the underlying conceptual processes. Two additional comments about the Bock et al. study are merited. One is that the form of the utterance was fixed in advance, not spontaneously produced, in order to ensure an equal number of each type of utterance for the experimental comparisons. This eliminated some of the decision-making that goes on during a normal instance of time-reading and reporting, when the choice of where to look, what strategies to use, and how to report the time all impact one another. The second comment is that the only task in the study was naming the time, not reading the time relative to some past or future event, as is commonly the case for everyday clock-reading. In time-telling in real contexts, the specific goal of the clock-reading is likely to impact the nexus of decisions named above.
4.6.2 A cognitive functional system for time-telling Now that we have explored the ways in which a person can read the clock and report the current time, let’s take a step back and consider what constitutes the timetelling system. The problem of telling time can only be solved by a distributed cognitive system. Like the navigation systems analyzed by Hutchins (1995), timetelling systems address the basic question “Where am I?”, but instead of locating the ego (self) with respect to an arrangement of landmarks, they locate the ego with respect to a sequence of events. Spatially, we remain in one location on the earth’s surface unless we move by our own impetus (or are moved by some external force) to another location. We are also free to move in various directions. Temporally, regardless of impetus, we are moved at a steady rate in a single direction in ordinary human experience. The question “Where am I?” becomes “When am I?” in the onedimensional flow of time. The standard system of time measurement developed by humans (described in chapter 3) makes it possible to transform the question and ask instead, “What time is it?” This question defines a computational problem; that problem is solved by a computational system made up of clock-reader and clock in interaction, and the computational system produces a time reading (in absolute or relative form) as its output. Recall from chapter 2 that when a person and artifact together constitute a computational system, Hutchins calls this a “local functional system.” Multiple local functional systems interacting with one another to solve a computational problem, as in a socially organized activity like team ship navigation, constitute a “global
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functional system.” Both local and global functional systems are distributed cognitive systems. The local functional system for time-telling consists of a clock and a clock-reader. The clock does two things: (1) it changes state in a regular way that coordinates with the system of time measurement, and (2) it makes available, through the structure of the clock face, a spatial arrangement of physical elements that constrain and support a time-telling process. This constraint is essential because it causes the time-telling system to produce certain outputs (time readings) and not others. But the support is equally crucial because it anchors the conceptual relations involved in reading the time. To see the clock face as mere input to an internal symbolic operation is to miss the critical supporting role of the material environment and therefore to misrepresent the nature of the cognitive process. We have seen how the clock-reader imposes image-schematic structure onto the clock face and then uses the clock face to anchor key conceptual relations while carrying out the sequential operations involved in reading the time. The fact that a highly experienced clock reader can name the time after only a brief glimpse of the clock does not belie the distributed nature of the time-telling system. Instead, it indicates two potential outcomes of frequent repetition over an extended period: (1) the ability to maintain a highly familiar image in imagination, without sustained perceptual input, and to use this image to support conceptual operations; and (2) a shift toward a pattern-naming strategy that is actually less cognitive because it relies on automated processes of pattern recognition and memory association. The same computational problem, “What time is it?”, can be solved by different functional systems using different operations. Given the two types of clock displays and two standard forms for reporting times, we have examined four general types of functional systems for telling the current time: (1) taking an analog clock display as input and generating an absolute time reading as output; (2) taking an analog clock display as input and generating a relative time reading as output; (3) taking a digital clock display as input and generating an absolute time reading as output; and (4) taking a digital clock display as input and generating a relative time reading as output. These functional systems are clearly related, not only in the overlap in their constituents but also in their historical development. The first type of functional system for timetelling used an analog display (T-stick, sundial, water clock, mechanical clock, etc.) as input and generated a relative time as output, i.e. a reading with respect to the reference hours. Increases in the accuracy and precision of time-keeping led to the second type of functional system: using the analog display as input and producing a specific absolute time as output. The third type of functional system, with a digital display input and absolute time output, derived directly from this second one. It eliminated the analog display and replaced it with a self-updating standard written form of the absolute time, which is the numeric output that would have resulted from an analog clock-reading operation. This change in display reduces the complexity of operations for absolute time-reading to simple number naming. For the fourth type of functional system, using a digital display to generate a relative time reading, the form of the artifact forces a mental transformation to convert the
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displayed numbers (absolute time) into a relative time. It does not support direct construction of a relative time reading. Not only can the same computational problem (“What time is it?”) be solved by different functional systems, but the same combinations of artifact and person can compose functional systems that solve different computational problems. In timetelling, this occurs when the problem changes from “What time is it?” to “How long has it been since X?” or “How long will it be till Y?” These are questions about the relation between the ego (self) and some past or future event. The answers to these questions are time intervals, expressed as some quantity of hours and minutes.17 Once again, even though the functional systems of digital-clock-and-reader and analog-clock-and-reader produce the same outputs and are therefore computationally equivalent, the relations of person to artifact and the roles played by each in the system are quite different. In the functional system with a digital display, the clockreader is likely to read the current point in time and then mentally compute the relation to another point in time. If those times lie on opposite sides of an hour boundary or cross noon or midnight, the mental computation becomes much more complicated, involving base-60 or modulo-12 manipulations. In the functional system with an analog display, the clock-reader can use the spatial properties of the dial to read time intervals that cross such boundaries: an arc of a certain length is always the same time interval, no matter where it begins and whether it crosses the hour boundary at the top of the clock. Notice that changing the question from “What time?” to “How long?” changes the affordances of the artifacts (the artifactperson-task relations), and this affects the efficiency of each functional system.18 Digital clocks afford rapid reading of the current absolute time while analog clocks afford spatial perception of temporal intervals. The relevance of these affordances depends upon the goal of clock-reading in the current activity. Rarely does one read the clock with no purpose in mind (except, perhaps, in a classroom exercise or experiment). When the computational problem does change, such as to “How long is it until Y?”, various functional systems may again be instantiated, including one that takes the analog clock display and the reference time of Y as inputs and generates a time interval as an output, and another that takes the digital clock display and the 17
For longer intervals of time, the answer might come in the form of some quantity of days, weeks, months, or years. Answering the “How long?” question at larger scales involves a different functional system based on a different cognitive artifact: a calendar.
18
In using the term “affordance,” I here refer to the relation between the object and the person with respect to some form of activity. A flat rock, for example, affords sitting to a person with the appropriate body configuration and for whom sitting has become a relevant activity. The term “affordance” may seem to refer to a property of an object, as when Norman (1993) talks about how particularly shaped door handles afford pushing or pulling, but it should be remembered that such an affordance is not an intrinsic property of the object per se but rather a relation between the object and a person engaged in a particular sort of activity. This is closer to the sense intended by Gibson (1979).
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reference time of Y as inputs and generates a time interval as an output. Because of the affordances of the displays, the first functional system might involve spatial reasoning while the second might involve numerical computation, although other strategies and mixing of strategies are possible (and apparent in Friedman and Laycock 1989), e.g. using numeric computation to relate the analog time to the time of Y, or using mental imagery of a clock face to relate the digital time to the time of Y. Again we see that that although the functional systems are computationally equivalent, the cognitive problems posed for the individual within each system are quite different. This highlights an important tradeoff in the construction of cognitive artifacts: a specialized artifact may support a functional system for accomplishing a particular computation with minimal cognitive effort on the part of the user (sometimes little more than direct perception), but such an artifact tends to be ill-suited to supporting functional systems aimed at other kinds of computations. A less-specialized artifact can support a wider variety of functional systems for carrying out different kinds of computations, but it can also make greater cognitive demands on the individual to make each system functional. When compared to the analog clock, the digital clock supports a more robust functional system for the task of reading the current absolute time because it enables the individual performer to rely on a simpler cognitive strategy, direct number reading. But the digital clock supports a less robust functional system when the task switches to computing a time interval because it forces mental manipulation of numbers instead of direct examination of anchored spatial relationships. Designing the artifact to reduce the cognitive effort of the individual may make for a more robust (although less flexible) functional system, but it also has implications for conceptual understanding. Constructing an absolute time reading from an analog clock face is more effortful and error-prone than simple number-reading because it requires a series of conceptual operations, but a byproduct of those operations may be a deeper understanding of the conceptual relationships among the elements of the system of time measurement. In other words, it is certainly easier to read ‘2:54’ as “two fiftyfour” than to construct that reading from an analog clock face, but the problem remains: what does “two fifty-four” actually mean? Constructing the reading from an analog clock face might afford deeper insight into at least some of the relationships that make “two fifty-four” meaningful. In this discussion of functional systems for time-telling, we must be careful not to think of these functional systems as algorithms or machines held in storage and waiting to be rolled out and applied whenever a relevant computational problem presents itself. While functional systems do depend upon the coordination of relatively stable structures like material artifacts and conceptual models, they are nevertheless flexibly constructed in each instance of activity according to the demands and resources of the particular situation. One instantiation takes an analog clock state as input and generates a verbal report of the relative time as output; another takes an analog clock state as input and generates a written numeric form of the absolute time as output; yet another takes an analog clock state and a reference time for some future event as inputs and produces a verbal report of a time interval
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as output; and so on. Similar conceptualizations, such as imposing a container schema to bound a region of space around the hour hand, do show up in more than one system, but each system is flexibly assembled to meet the demands of the task at hand. This is why it is so difficult to write a fixed algorithm that exactly characterizes multiple real instances of time-telling. It is indeed possible to construct an algorithm that takes the clock face as input and produces a time-reading as output, even one that might serve as a normative description for the sequence of operations most adult clock-readers engage in when reading the time in a time-telling experiment (as I did in Figures 4.3-4.5), but actual performances of time-telling in real activity are bound to deviate from this norm in some way. Actual performances are improvised in the moment, assembled on the fly to meet the demands of the situation while making use of the constraints and affordances of the setting. Different states of the same display afford different types of time readings, while different social situations create different expectations about the form (even the exactness) of the output. In a particular instance of time-reading, the functional system actually assembled may be more or less efficient than the normative description, perhaps driven initially by salient aspects of the display or habits of attention from past clock-reading experiences and then impacted by situational characteristics that force transformation of the output (as in Bock et al. (2003), where that one American clock reader persisted in using the absolute-time order of looking when the experiment demanded relative time outputs). In other situations, anticipation of the ultimate purpose or function of the output could impact the shape of the functional system from the very start of activity. In examining children’s clock-reading (discussed further in chapter 5), Siegler & McGilly found that all of the children in the study used multiple strategies to read times from an analog display and that each individual child used different strategies to read some of the same exact clock times in testing sessions less than three days apart. Improvisational assembly, flexibility, and variability are key characteristics that distinguish a cognitive functional system from a standard machine or computer program.
4.6.3 Time-telling as a distributed cognitive activity We have argued that time-telling is a distributed cognitive activity because a reading of the time results from the interaction of person and artifact and, indeed, is impossible if one or the other is removed. But as a cognitive activity, time-telling is distributed in other important ways as well (following Hutchins 2001). It is distributed across time, in that the functional system for time-telling incorporates products from prior generations. Both the forms of the artifacts (digital and analog clocks) and the conceptual models of time measurement we use today are the residua of past cognition, the outcomes of a contingent historical process described in chapter 3. A local functional system for time-telling also works only because time measurement is socially distributed. We have already seen how the global system of time measurement depends upon socially-agreed-upon standards (such as time zones) and upon the calibration of clocks to measurements established by others
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(such as Greenwich Mean Time). Rarely is the clock-reader the same person who constructed the clock. The same person can read many clocks, and the same clock can be read by many people. Each of these acts of clock-reading instantiates a local functional system for telling the time. The conceptual models involved in timetelling are intersubjectively shared—they are cultural models—and the linguistic forms used to name the time are, like other linguistic constructions, shared by members of a language community. For our purposes, it has been convenient to ignore the temporal and social distribution of time-telling as a cognitive activity and instead to concentrate on the material distribution: the interaction of clock and clock-reader in a local functional system. How does such a functional system come into being? We have already explored the historic roots of the system, but the fact remains that people have a limited lifespan. For time-telling to persist across generations, each new generation must learn how to instantiate a functional system for time-telling—in other words, how to read a clock. As we have seen, clock-reading is quite complex, and when performed by an expert, essentially invisible. Children are likely to learn that clocks are important by noticing how frequently adults orient to them. They are also likely to learn that clocks tell us what time it is as they engage in activities with caregivers who monitor the time. What they cannot learn simply by co-participating with experienced clock-readers and watching them tell time is exactly how to go about constructing time readings from an analog clock. Instead, the continuation of timetelling as a cognitive activity (and, indeed, a crucial one in our society) depends upon another uniquely human activity: direct, active instruction. This is the subject of the next chapter.
5
THE CONSTRUCTION OF MEANING IN TIME-TELLING LESSONS
When we confront a text in an unfamiliar language, we think that there must be meaning within the text and if only we had the key, we could unlock the text and get the meaning out. But this is an illusion. A text is only a pattern of marks. Those who read a text extract nothing from the page; rather they construct meaning in their own minds using their own powers of imagination, prompted by the familiar patterns of marks that trigger associations and conceptual operations. This meaningmaking activity works because others, including those who made the marks, can do it too. No one is born knowing how to read; reading takes instruction and practice. So it is with time-telling as well. The functional system for time-telling—clock and clock-reader—does not come into being on its own. It has to be set up, and setting up the system depends upon the actions of experienced clock-readers guiding novices through the conceptual operations involved in interpreting clock states. Guided meaning-making is profoundly important to maintaining the cognitive sophistication of the human species. How it unfolds in situated activity is therefore an important topic of study. In this chapter, we examine how the actions of teachers guide the construction of meaning by learners learning to read times in the formats described in chapter 4.
5.1 Two forms of the Artifact Puzzle On a remote island lives a culture that has survived for many centuries with no contact with the outside world. One day, a member of this culture finds a strange object on the shore: a small, round, shiny object with a clear top. Beneath the top are two black rods of different lengths, joined at the center of the object, and around the edge are many black marks and unusual-looking symbols equidistant from one another. What to make of this curious object? It does not appear to be anything natural, so it is must have been constructed by someone—it must be an artifact. That thought alone awakens a sense of intentionality: why was the object constructed, to what purpose? And how does it accomplish that purpose? The finder of this watch is confronting the Artifact Puzzle, which typically occurs in one of two forms. In the current situation, the artifact puzzle appears in the form of an archaeological problem: taking what appears to be a human-made object and trying to discern where it came from, what it was for, and how it was used—its origin, purpose, and function. When the object is a cognitive artifact, i.e.
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some type of representational or computational tool, then the archaeological problem also becomes one of trying to figure out what conceptual models (typically cultural models) were needed to use the artifact, what connections (or “mappings”) were made between the artifact and the conceptual models, how these were manipulated in some sequence of actions, and how these actions enabled the user of the artifact to accomplish specific goals during the course of an activity. Consider again our finder of the watch. If, by chance, the watch is still running, then the finder might notice certain regularities, both in how the watch changes state and how those changes of state coincide with natural events. For example, the artifact changes appearance in a steady rhythm and the apparent movement of the shorter rod from one symbol to another coincides with the apparent movement of the longer rod all the way around the circle. The artifact also appears to be in the same state whenever the sun is overhead. These regularities offer clues to the device’s possible purpose. The finder might make opportunistic use of these regularities, perhaps relating them to the timing of certain events. More likely, though, the object would remain little more than a curiosity, an adornment for the body or home, perhaps valued for its uniqueness. It would be exceedingly difficult for our amateur archaeologist to make much sense or use of the object within the activities of that cultural group. When it comes to clocks and watches, children in our own culture also face a form of the Artifact Puzzle: figuring out what these objects are for and how to use them. This form is similar to the archaeological problem, but it has one important difference: the context in which it occurs. This form of the Artifact Puzzle occurs within the cultural group that uses the artifact; it is therefore a learning problem. Cultural activity provides three important resources to help solve this problem. The first is a relevant cognitive ecology—a web of related activities and meanings in which the clock plays an important role. The second is scaffolds of various kinds, particularly other artifacts that incorporate or annotate the clock face in some way. The third is direct, active instruction from more experienced members of the cultural group. Children are taught to read clocks by their parents, older siblings, and moreexperienced peers. In industrialized countries, clock-reading instruction is a standard part of the mathematics curriculum in primary schools. Clocks and basic time concepts are introduced in pre-school or kindergarten, and instruction and practice in time-telling continue at least through the 3rd grade. Together, the cognitive ecology, scaffolds, and direct instruction help children become proficient time-tellers, in a learning process that typically spans several years.
5.2 The development of clock-reading skills When do children learn to tell time? How do their skills develop? Despite the importance of time-telling, there have been few studies of how time-telling skills develop. Springer (1952) explored what 4- to 6-year-old children know about telling time prior to any school instruction. Case, Sandieson, and Dennis (1986) studied
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adults’ clock-reading strategies and then designed, tested, and compared two curricula for teaching mentally retarded teenagers to tell time. The most comprehensive study of time-telling abilities in different age groups was by Friedman and Laycock (1989), who conducted experiments with students in 1st through 5th grades (6 through 11 years old) to determine how well the children could read times from analog and digital displays, transform times by adding 30 minutes, relate times to activities, and judge the order of times throughout the day. Siegler and McGilly (1989) studied a particular age group, 2nd- and 3rd-graders (8- and 9-year-olds), and examined the strategies these children used to read times from an analog clock. The clock times used in these studies varied considerably. Springer’s study included only hour, half-hour, and quarter-past times. Case et al.’s study covered the full range of clock settings but grouped them as hour, 5-minute, and 1-minute (i.e. did not separate out the half-hour or quarter-hour times). Friedman and Laycock used times at :00, :30, and :43 for the clock-reading task and at :30, :50, and :23 for the plus-30 transformation task. Siegler and McGilly, though covering the narrowest age group, focused on the broadest range of times, including hour, half-hour, quarter-hour, other 5-minute, and 1-minute times. None of the studies distinguished between absolute and relative time reporting. Clearly, there are gaps in our knowledge of how time-telling skills develop, but a tentative picture is beginning to emerge. Table 5.1 shows the proportion of children who could read various times correctly in the age groups covered by these studies. In terms of reading the time from an analog clock, few 4-year-olds can do more than recognize and name hour times that are related to salient activities like mealtimes or bedtime. About one-third to one-half of 5-year-olds can read hour times, while about 10% can read half-hour times, and fewer can read times at the quarter hour. At 6 years of age (entering 1st grade), three-quarters of students read hour times correctly, about one-third read half-hour times, and about one-sixth read times at the quarter hour. By 2nd grade (78 years of age), readings of hour and half-hour times are nearly always correct, and by 3rd grade (8-9 years), students read 5-minute times with 80% accuracy. From 2nd to 3rd grade, the reading of 1-minute times improves from about one-third to twothirds correct. By 4th grade (9-10 years), students read one-minute times with nearly 80% accuracy, showing very slight improvement by the end of 5th grade. In general, children find it easier to read the time than to set the clock. While performance with analog clocks varies greatly according to the displayed time, for digital clocks, there is little difference between reading hour, half-hour, and other times. Children can read all of these by 2nd grade, which probably reflects the development of number-reading skills more than the development of time-telling. In each of these studies, researchers collected some information about the strategies children used, giving us some idea of how the children went about reading the times. For digital displays, the strategy for children is the same as for adults: direct reading. Although some mental manipulation would be required for relative time-telling, these studies did not report on children’s use of relative time expressions for digital displays. For analog displays, the strategies described in the studies—immediate recognition or some form of counting— focus on how the
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minute hand is read; the reading of the hour hand is rarely discussed. Hour times (o’clock times) are directly named, and so, typically, are half-hour times, although they may be counted if not recognized. 5-minute times are either directly named or counted by fives from the hour, half-hour, or some other recognized 5-minute mark. By 3rd grade, children can directly name most (about three-quarters) of the 5-minute times. Counting is presumably more common with younger children and more likely to start at the hour than at some intermediate mark. 1-minute times are usually counted by ones from an adjacent 5-minute mark. If the minute hand is closer to the upcoming 5-minute mark and this mark is readily named, then children may count down to the 1-minute time. Usually, children count up, counting by fives to the 5minute mark just prior to the minute hand’s location (if this mark has not been immediately recognized) and then finishing the count by ones. Other forms of counting do occur, such as counting by ones across 5-minute marks, but these are less common. Direct naming is also used for 1-minute times—by rounding the time to the nearest 5-minute mark. Do proficient time-tellers ever reach the point where they name 1-minute times directly? Case et al. found in their small sample that adult time-tellers relied on direct naming for 5-minute times but still mostly incremented or decremented from the nearest 5-minute mark to read 1-minute times.19 Case et al. point out that the direction of incrementing or decrementing reverses when reading till times, making it more difficult to read 1-minute times on the left side of the clock, a claim that is supported by their response-time data.20 This reversal also occurs if children count by fives from the top of the clock to name till times, although this was not discussed in any of the studies. Finally, it should be noted that all of the children in the Siegler and McGilly study used at least three different strategies, i.e. three different permutations of direct naming and counting from different starting points, and that individual children used different strategies to name the same time on trials less than three days apart. Friedman and Laycock’s study also included a transformation task in which children were asked to say what time a clock would display in 30 minutes (the children were reminded that 30 minutes is the same as half an hour). This change in task profoundly altered the success rates for digital displays, putting them on a more equal footing with analog displays. The times used in the study were 30 minutes, 23 19
Other studies of adult time-tellers have examined mental imaging of clocks (Paivio 1978), processing of times in word, digital, and analog formats (Gookasian and Park 1980), and where the eyes look when reading clocks (Bock 2003), but all of these have relied exclusively on 5-minute stimuli.
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Case et al. mention that the “minutes to” strategy was the most common strategy for reading values on the left-hand side of the clock by the undergraduate and graduate students in their study. Less than two decades later, Bock et al. (2003) found that undergraduate students were much more likely to read absolute times on the left side of the clock, although the proportion of relative (till) responses did increase during the last quarter-hour (Bock et al., in press). This is evidence of a generational shift in time-telling strategies. Whether this shift is related to the prevalence of digital watches, to changes in time-telling curricula, or to other factors remains uncertain.
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minutes, and 50 minutes past the hour, requiring the children to complete the hour, add to a 1-minute time, and add across an hour boundary, respectively. With a digital clock, adding 30 minutes involves mental addition—adding 3 to the tens place (or counting up)—and noting whether the sum exceeds 60 and by how much. The process can be speeded up for the 30-minute times if it is recognized that adding 30 completes the hour. With an analog clock, the problem can be solved in the same way—reading the current time and then performing a mental addition—but with the added burden of maintaining the current numeric time in working memory. On the other hand, an analog clock makes it possible to solve the problem using a spatialperceptual strategy: adding 30 minutes is tantamount to mentally rotating the long hand clockwise by 180 degrees, while paying attention to whether the hour boundary has been crossed. This spatial-perceptual strategy involves creating a conceptual blend: the imagined future state of the long hand is superimposed onto the current state, with the present and future long hands forming a line across the clock; the visible clock face anchors this conceptual blend. How do children perform on the plus-30 transformation task? For 30-minute times, the success rates for digital and analog displays are identical and lead the other times by one to two years: a majority correct by 2nd grade, three-fourths correct by 3rd grade, and nearly all correct by 4th grade. Analyses of the methods used by children show that one-third (digital) to one-half (analog) recognize the completion of the hour and that there is a developmental trend away from counting toward addition, imagined movement of the minute hand, and complete-the-hour strategies. For 23minute times, success rates for digital and analog displays are also nearly identical (the digital slightly higher), reaching a majority correct by 4th grade—three years after most children can read digital times correctly. Children rely primarily on addition for both types of displays but with a greater proportion of counting for analog displays. For 50-minute times, the success rate for digital displays is the same as for 23-minute times, reaching a majority correct by 4th grade. But the success rate for analog displays runs a full year ahead, reaching a majority correct by 3rd grade. Where does this analog advantage come from? Most children still rely on addition for the digital displays, but for analog displays, the majority switch from adding to counting on the clock face. Taken together, the results for 23-minute and 50-minute times suggest that children make use of the spatial-perceptual strategy, even across the hour boundary, when the minute hand points to a major tick mark, but they have trouble using the strategy when the minute hand falls between major tick marks. For digital displays, the implications are less clear. Children at each grade level appear equally able or unable to transform a 1-minute time within the hour (:23) as to transform a decade time across the hour (:50). More research is needed to determine how these factors combine, i.e. whether transforming a decade time within the hour (e.g. :20) is even easier or whether transforming a 1-minute time across the hour boundary (e.g. :53) is even harder. When children were asked to transform decade times by subtracting 30 minutes, the results were the same as for adding 30 minutes, showing the same advantage for analog displays. It remains to be determined whether the analog advantage would hold up for other 5-minute intervals, but the high incidence
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of counting on the clock face suggests that it would; in fact, for non-decade intervals (e.g. 25 minutes) the analog advantage might be even greater because addition or subtraction with digital displays would involve operating on both the tens and the ones.21 Finally, it should be noted that while numerical methods accounted for about three-quarters of the responses with digital displays, up to a quarter of the responses did involve some kind of clock face imagery. Because children learn to read the digital clock at an earlier age and because digital watches and clocks are now so widespread, the use of clock face imagery to solve time problems might be less common today than when this study was published nearly twenty years ago.
5.2.1 Foundations of time-telling Learning to tell time is about more than reading the clock. It takes little more than number-reading skills to name the time displayed on a digital clock, but in order to make sense of this time, to interpret what it means, children need additional conceptual knowledge. Much of this is developed in kindergarten and 1st grade classes. Part of this knowledge is an emerging time sense, an awareness of the duration of activities and when they typically take place. Children also need to develop the conceptual model for a cyclic day (discussed in chapter 3), a cycle of day and night that repeats over and over. They need to develop a conceptual model for the parts of the day—morning, afternoon, and evening—and their sequential relations, and to understand that noon marks the middle of the day while midnight marks the middle of the night. Teachers help children understand these divisions of the day by relating them to the kinds of activities children engage in: getting up, eating lunch, taking a nap, going to bed, and so on. Introducing the conventions of the time-telling system—hours, minutes, and seconds—brings additional complications. Children need to develop a sense for the duration of an hour, a minute, and a second, and this comes from becoming aware of the passage of conventional units of time and how these relate to one’s embodied experience of what an hour, minute, or second feels like (how quickly or slowly it goes by, although this subjective feeling varies greatly depending on whether children are attending to the time or are engaged in other activities). Around 4-5 years old, children learn to relate certain hour times with activities, e.g. going to kindergarten at eight o’clock. The system of numbering the hours builds on children’s knowledge of the conventional number sequence (but not number quantity—the hour named “8 o’clock” is no bigger than the hour named “7 o’clock,” it simply follows it in the sequence). Children need to learn the convention that the hours are numbered from 1 to 12 and that the numbering starts over at noon and midnight. (Since there is no “zero” hour, the numbering appears to start at one o’clock—one hour after noon 21
Transformations involving adding or subtracting non-5-minute intervals, such as 23 or 48 minutes, would be more likely to force assembly of multi-step operations, either overt actions on the clock face with pointing as a placeholder or use of paper and pencil to carry out a series of arithmetic operations. These go beyond the scope of the immediate discussion.
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and midnight.) The restarting of the hour count also introduces the use of “a.m.” and “p.m.” to distinguish the hours that occur before noon from those that occur after noon. This adds complexity to putting hour times in the right order: 10 a.m. occurs before 1 p.m. even though 10 comes after 1 in the conventional number sequence.22 As with other systems of measurement that children learn (length, weight, volume, temperature, etc.), the system of time measurement includes equivalencies that children must simply memorize: 1 day equals 24 hours, 1 hour equals 60 minutes, and 1 minute equals 60 seconds. Learning to read an analog clock may actually support these memorized associations by tying them to spatialperceptual experiences: it takes 60 tick marks for the minute hand to go all the way around the clock, and when it goes all the way around the clock once, that is one hour.23 Actually reading an analog clock depends upon a variety of other conceptual knowledge. Children need to be able to distinguish length—long and short—in order to distinguish the hour and minute hands on a clock, and then, of course, they need to remember which hand goes with which dial. Children need to impose source-path-goal image-schematic structure onto the clock face to understand that the motion of each hand “starts” at the top (the 12) and proceeds in a particular direction (clockwise), completing a cycle when it reaches the top again—even though the motion of the hands is actually continuous, neither starting nor stopping at any point, and is so slow that the hands appear stationary. In order to read the hour and minute dials, children need an understanding of the number line (also practiced in 1st and 2nd grade), and for the minute dial, they also have to supply the missing labels (recall that the perceptual problems involved in reading the clock dials were explored in chapter 4). By the time most children start school, they have already learned to count by ones, but children are also expected to be reasonably proficient at counting by fives before they are taught to read 5-minute times from the clock. Learning to read 5-minute times then facilitates learning to read 1-minute times by shortening the length of the count (i.e. enabling children to count up or down from a major tick mark rather than counting by ones from the top of the clock). In order to learn to read quarter-hour times, children first learn to divide a circle into halves and quarters, a prototypical case of what will eventually become a broader conceptual understanding of geometric fractions (but see chapter 6 for discussion about how a more dynamic understanding of quarter-hour times develops). The understanding of half- and quarter-hours lays the foundation for understanding other time intervals (such as 20 minutes) as shapes bounded by the long hand or arcs of distance around the clock circle. Children need to learn that the same-length arc, no matter where it 22
Friedman and Laycock (1989) found that children know the order of daily activities at least by 1st grade but most cannot order digital clock times until 2nd grade. Children begin to be able to order mixed sets of times and activities in 1st grade (i.e. better than chance) but continue to increase in accuracy until 3rd grade.
23
Siegler and McGilly (1989) speculate that analog clock-reading contributes to the conflation of timepassed and distance-traveled observed in Piagetian studies of children’s time understanding.
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starts, always represents the same time interval; this understanding helps to facilitate relating the current time to the reference time for some past or future event. We have also seen that reading different kinds of times—hour, half-hour, quarter-hour, 5-minute, 1-minute—calls upon different kinds of strategies—direct retrieval, counting by fives, counting by ones, incrementing/decrementing, etc.—so learning to identify the kind of time to be read before actually reading it may be another important step in learning to read the time from an analog clock. Finally, language phenomena affect clock-reading as well. We know that there are two different conventional linguistic formats—absolute and relative—for reporting times and that these different formats align with different clock-reading strategies. That means that even when children can classify the clock face display in terms of the kind of time to be read, the choice of how to go about reading the time can still be affected by anticipating the form of the output, the particular expression to be used. Or, conversely, the expression might be chosen later based on how well it fits the clock-reading strategies already employed. Learning to navigate these conditions and choices turns discrete cases of time-telling into expert time-telling ability. Finally, once children can read and name times correctly, they still need to learn the conventions for reading and writing times: that a colon is written between the number of hours and the number of minutes but not named when the time is read aloud; that a zero is added before single-digit minutes and pronounced “oh”; that o’clock times are written with ‘:00’ and that ‘:00’ is pronounced “o’clock”; etc. As children become proficient at reading the clock, they are confronted by more complex kinds of time problems: relating the current time to other times, reading time intervals, answering questions about the start, end, and duration of events that span multiple hours, and so on. The algorithms children learn in school for doing place-value arithmetic (addition with carrying, subtraction with borrowing) seem to work fine when no boundaries are crossed—when the minute totals do not exceed 60 and the hour totals do not exceed 12—but for larger spans of time, the algorithms break down or produce errors unless children learn to handle the base-60 and modulo-12 manipulations. Where boundaries are encountered, problems can often be broken into two sub-problems, one for either side of the boundary, whose results are combined. Boundary-crossing time problems demand either a series of pointing and counting actions on the clock face or some kind of multi-step paperand-pencil strategy assembled on an ad hoc basis. As we saw earlier, performing even simple transformations like adding 30 minutes can lag successful clock-reading by two or more years.
5.2.2 The conventional order of instruction How do the learning problems described above translate into a sequence of instruction? It is difficult to tell to what extent the order of development of timetelling skills described above reflects the developmental abilities of children, inherent characteristics of the clock-reading task, or simply a conventional structure to the curriculum. Clearly, all of these play some role. It seems reasonable to teach hour
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times first, and then half-hour times, and then other 5-minute times, saving the 1minute times for last. Whether absolute or relative time-telling is taught first seems more a matter of convention. The current trend in American schools is to focus on absolute time, which corresponds to the form of time displayed on digital clocks, before teaching relative time, which is useful for exploring time relationships, although the expressions “half past” and “quarter past/till” continue to be introduced as alternative labels for the :30, :15, and :45 times. In other countries, relative time may be taught first, as it seemed to be for earlier generations of Americans. Teaching one or the other first, rather than both at once, reduces the potential for confusion, although ultimately children are still expected to be familiar with both. In the two elementary schools included in the present study, time-telling instruction begins in kindergarten with relating parts of the day and activities, introducing the conventional units of time, introducing the clock face, and learning to name hour times, which means simply reading the number pointed to by the hour hand. (Only when other times are mixed in do children also need to focus on the position of the minute hand.) In 1st grade, children continue to read hour times and relate them to activities. They also learn to read half-hour times and distinguish them from the hour times by paying attention to whether the long hand points at the 12 or the 6. To name the half-hour times correctly, children are taught to look from the hour hand back to the previous number. They are also taught that another way to name half-hour times is “half past X,” because the long hand has gone halfway around the clock and because the short hand is halfway between X and Y. When children can count by fives, the 15-minute times are introduced (X:15 and X:45), followed by the other 5-minute times. Quarter-past and quarter-till are also typically introduced around this time, but children prefer to use X:15 and X:45, probably because the absolute forms are used for other clock times. Once children are somewhat accustomed to reading 5-minute times, the 1-minute times may be introduced by the end of 1st grade; otherwise, they are introduced early in 2nd grade. Reading 1-minute times involves counting by ones from a 5-minute mark or, if the 5minute marks are not easily recognized, starting at the top of the dial and counting by fives to the 5-minute mark and then by ones. 2nd grade contains few entirely new time-telling topics but much clock-reading practice. In 3rd grade, reading the time till the upcoming hour is introduced and related to the time past the previous hour, shifting the focus to relative time. 3rd-graders also confront more time problems that involve reasoning about relationships between the displayed time and another time, including times that cross hour boundaries. Late in 3rd grade, children see problems in which they have to read the time from analog displays with major tick marks but no numbers, like the displays found on many analog watches. From this point on, time-telling instruction consists mainly of practice and remediation.
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5.3 Episodes of instruction From this brief discussion we begin to see why time-telling presents such a complex learning problem—more complex than just perceiving the hour, minute, and quarter-hour dials embedded in the clock face—and we begin to appreciate why it takes so long for children to learn. Learning to tell time with an analog clock, in particular, depends upon a variety of pre-existing or co-developing conceptual models that need to be brought to bear upon the clock state in order to render it meaningful. How then do children learn to make meaning from a clock? They learn to do so through guided conceptualization. In clock-reading instruction, teachers use gestures and speech to annotate the clock face while guiding students through the process of time-telling. In these episodes, we see how teachers use speech to activate relevant conceptual models, use gestures and co-gesture speech to guide mappings between these models and the structures of the clock face while building blended spaces, and use the physical structure of the clock (and other artifacts) to anchor these conceptual blends used to read the time. In what follows, I illustrate this claim with specific episodes of clock-reading instruction that show the interplay of conceptual models with the clock face and highlight the processes that guide meaning construction.
5.3.1 Reading ‘quarter past’: building the Clock Quarters blend The first episode comes from a 1st-grade lesson that occurred about midway through the school year. At this point in the year, the students are adept at recognizing hour and half-hour times. In this lesson, the teacher introduces quarter past. At the start of the lesson, she puts a solid circle on a felt board and the students call out “whole” and “circle.” She then covers this circle with two vertically-oriented half-circles and the students call out “half.” Next, she puts four quarter-circles over the half-circles and the students call out “one fourth,” “two fourths,” etc., as each is added. The students are giving responses from the previous day’s lesson, in which they learned to divide a circle into parts; this is why the majority call out “whole” rather than “circle” when the first object is put on the board. Now the actual instruction begins. The teacher says “and another way that we call one fourth,” pulling the upper right quarter-circle (the one that will be the focus of the lesson) slightly apart from the others, “we sometimes call it a quarter.” She says “like money,” opening her hand palm up, and then “a quarter,” making a small circle with her index finger and thumb, resembling a coin. She says “we need how many quarters to make a dollar?” and the students respond “four.” She says “four quarters make a dollar,” raising four fingers. Then she drops three fingers, keeping her index finger up, and says “so one quarter is one fourth, right?” She repeats “a quarter” as she reaches to the felt board, grasps the quarter-circle and says “it’s a quarter of the circle,” and then puts it back in place with the others. Now she steps away from the felt board and seats herself in a chair facing the students. This entire interaction takes about a minute and a half and completes the first part of the
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lesson. The teacher has reactivated the students’ conceptual knowledge of dividing a circle into halves and fourths, anchored by the objects on the felt board, and introduced a new label, “one quarter,” whose more abstract meaning of one fourth is derived through analogy with another conceptual domain, money, in which the term refers to a specific coin, which, as it turns out, happens to be one fourth of a dollar.24 The teacher attaches this new label to the quarter-circle object previously called “one fourth,” now referring to it as “a quarter of the circle.” Now that this essential vocabulary has been introduced, the teacher is ready to begin the clock-reading portion of the lesson, the first part of which is transcribed in Transcript 5.1.25 In this portion of the lesson, lasting only thirty-three seconds, the teacher constructs a key conceptual blend. One of the inputs to this blend is the conceptual model of dividing a circle into parts, which the teacher has just brought to mind with the first part of the lesson. The other input is the clock face anchored by the teaching clock the teacher holds in her hands. Analogy connectors link the parts of the circle to the parts of the clock face, as shown in Figure 5.1. Through a common compression—Analogy into Identity—these linked parts are fused in the blend, producing the emergent structure of the clock face divided into canonical quarters. This emergent structure has an important consequence: it provides the conceptual underpinning for a new way to name times, as “quarter past,” “half past,” or “quarter till.” Note that up to this point, the quarters have not been related to hours or minutes or to either of the clock hands; the students are merely learning to see them when they look at the clock face—a form of situated seeing that will support reading relative time in terms of quarter-hours. The blend is important, although fairly unremarkable. What is interesting for our purposes is how the blend gets constructed. We will explore this process by 24
Note the level of abstraction here. The quarter-circle is a one-fourth part of the physical circle, but the quarter coin is not a one-fourth part of the physical dollar (a dollar torn into four parts). Rather it has a value of 25 cents, which happens to be one fourth of the value of a dollar, which is 100 cents. Up to this point in the students’ experience, the word “quarter” was the name of a type of coin; now it is taking on the more general meaning of one fourth of something (the motivation for the name of the coin). Even more abstractly, that something does not have to be a tangible object; it could be something intangible like monetary value.
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The transcript follows the conventions used by Goodwin (see, e.g., Goodwin 2003), showing speech in conversation analysis format and gestures with annotated still images from the video. The numbers in parentheses (0.5) indicate the lengths of pauses in seconds. A bracket ([) is used to join overlapped speech, a hyphen (-) for speech that is abruptly cut off, and an equal sign (=) for speech that is connected with what follows. Tildes (~) connect rushed speech, while a colon (:) indicates lengthening of a vowel or consonant. A degree sign (°) marks quiet speech, while bold italics indicate emphasis in pitch and volume (in some cases plain italics show slight emphasis). A question mark (?) indicates rising intonation, a period (.) falling intonation, and a comma (,) a risingfalling contour. (Single parentheses) surround uncertain transcriptions, while (( double parentheses )) enclose comments from the transcriber. [Boxed speech] co-occurs with the gesture stroke or action in the accompanying image. In the images, red arrows mark gestures, while blue arrows mark manipulations of objects. A dashed arrow indicates continuation of a gesture or action in a pause following the end of speech.
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working through the transcript line by line, examining each step in the construction of meaning. First, an observation is in order. While natural conversational discourse contains many false starts, overlaps, breakdowns and repairs, the instructional discourse transcribed here exhibits a more ordered structure. The teacher, addressing a group of 6- and 7-year-olds, speaks a single phrase or clause at a time, pausing for a half second or longer after each utterance. Each phrase or clause ends with an emphasized word (or word pair); these tend to be open-class words referencing key conceptual content. From an informational point of view, the effect is to introduce one new piece of information at a time and then to pause to allow that piece of information to be processed. The gestures that accompany the speech are slow and deliberate (more controlled than in ordinary discourse), avoiding extraneous motions that might distract from the focus. In this episode, the students’ contributions tend to be of two types: (1) filling in key words when prompted by the teacher (note the teacher’s vowel lengthening and hesitation at the end of line 20, followed by the students’ filling in the word “nine” [22]); and (2) providing short responses (“yeah” [15-16, 26]) following the teacher’s checks for understanding (“right?” [14, 25]). When a student does provide a correct piece of information, the teacher repeats it before moving on. These characteristics will be apparent in other episodes of instruction we examine. Because the student contributions are minimal and subject to teacher repetition, we will focus on the teacher speech and accompanying actions as we analyze the construction of the blend. We proceed as follows: taking each line of teacher speech and action and diagramming the inputs, cross-space mappings, and blends involved in the construction of meaning. This is shown in Figure 5.2(a-i). In 5.2(a), the teacher picks up the teaching clock while saying “if I take my clock.” It is important to note that the artifact the teacher calls “my clock” is not, in fact, a clock—it does not keep time. It is an artifact specially constructed for clock-reading instruction, made to look like a clock face but with no driving mechanism, with hands that are geared so that moving the long hand causes the short hand to move in a clock-like way. The word “clock” activates the students’ knowledge of clocks and what they are for (a diminished subset of the adult understanding of clocks), and, through the perceptual similarity of the teaching clock to a real clock, associates this knowledge with the artifact the teacher is picking up. (In this and subsequent diagrams, a mental space that is anchored by an object in the world will have a box behind it: here the input labeled ‘teaching clock’ is a perceptual space anchored by the physical object the teacher is holding; the blended space that results from the conceptual integration is anchored by this same object.) In Figure 5.2(b), the teacher holds her right hand in an L-shape, points at the top of the clock, and traces a circle around the perimeter of the clock while saying “it’s the same circle shape.” Here the speech recalls the conceptual idea of a circle, activated just moments before by the sight of the circle on the felt board, while calling attention to the shape of the teaching clock. The gesture traces this shape on the clock face. Tracing, as Goodwin (2003) points out, has both an indexical and an iconic component. In this case, it indexes the portion of the clock face being
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pointed to while iconically drawing a circle, in effect superimposing a conceptual circle onto the clock face. In the blended space, Analogy is again compressed into Identity, so that the teaching clock is perceived as a kind of clock-circle. The circular band of the clock is profiled (brought into the foreground and made the locus of attention), while the rest of the clock face remains in the background. Notice how this meaning emerges from the interplay of speech (“circle shape”), gesture (trace), and artifact (what the trace couples with). The gesture adds other information: while a “circle shape” has no beginning or end, the gesture imposes a starting point at the top of the clock, a particular direction of motion around the clock, and an ending point at the top again. This is the familiar source-path-goal schema that is essential for time-telling, and it anticipates what will come next in the lesson (reading ‘quarter past’ times) after the clock-quarters blend is constructed. Nothing about this imageschematic structure is apparent in the artifact the teacher is holding or in what she says; the structure is apparent only in the gesture. Of course, this could be accidental—after all, a trace has to start and end somewhere—but several characteristics suggest otherwise. In other data, references to the circular shape of the clock take other forms: a two-handed gesture superimposed on the clock face (curved hands with palms facing toward one another), starting the trace on the clock nearest the current hand position (e.g. near the 8), tracing in a counterclockwise direction, continuing the trace past the point of origin (making an additional partial loop or multiple loops, generally obscuring the start and end points), making the circular motion at high speed, and making the gesture somewhat away from the clock face (near the clock but not carefully superimposed over it). The slow, deliberate nature of this gesture, the clear starting point at the top of the clock (not closest to the hand’s initial position), the exact following of the clock band with a constant speed of motion, the clear end point at the top after completing a single loop and no more, all suggest that the form of the gesture was intentional, bearing source-path-goal information in addition to shape information. In the next sequence of actions (Figure 5.2(c-d)), the teacher adjusts the state of the artifact before performing another trace. She raises the short hand to the 12, bringing it into alignment with the long hand (which is already pointing straight downward) as she says “and I divide it,” and then she traces over the hands in a single, deliberate motion from the top of the clock to the bottom, back to the top, and back to the bottom again (5.2(d)) as she says “up and down here,” pauses, and then says “divide it into halves, right?” Several things are noteworthy here. By bringing the short hand straight up when the long hand points straight down, the teacher has brought the teaching clock hands into positions that would be impossible on an actual clock. In this case, the hands are being used opportunistically to create a divider, a material anchor for the conceptual line dividing the clock-circle into two parts. This construal is prompted by the accompanying speech: “and I divide it.” The teacher reinforces this conceptualization of a dividing line with another trace,
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down-up-down (saying “up and down here”).26 The repeated back-and-forth nature of this gesture serves to highlight the dividing line without imposing the kind of specific source-path-goal structure evident in tracing the circle. The net effect is to cancel the motion of the two clock hands and to combine them into a single static entity: a dividing line. The next move is to call attention to the shapes defined by the circle and dividing line: the half-circles on the left and right side of the clock face. As the teacher performs the last gesture stroke (downward) she says “divide it into halves, right?” This profiles the bounded regions on either side of the dividing line, filling them and turning them into conceptual objects. A part-whole schema joins them together as half clock-circles, and the artifact anchors their configuration. Notice that up to this point the roles of the emphasized words at the end of each utterance: “clock,” “shape,” “divide it,” “here” (profiling the location of the conceptual division indicated by the gesture), “halves”. With the exception of “here,” these words profile conceptual elements derived from the divided circle input and applied to the clock-circle blended object. In Figure 5.2(e-f) the teacher adds another dividing line to the clock-circle. In 5.2(e) she announces her intention with “if I wanted to divide it into quarters,” pausing and intoning the word “quarters” with emphasis as she picks up a pointing stick from her desk. The phrase “divide it into quarters,” encourages a further conceptual operation rooted in the ‘divided circle’ input and carried over to the blended clock-circle, namely the addition of a horizontal dividing line (as indicated by the dashed line in the upper left input in Figure 5.2(e)). The teacher’s next action makes this manifest. She swings the pointing stick in front of the teaching clock, aligning it horizontally across the middle of the clock face. Once the pointing stick is in place, she says ‘we go from the nine to the three, right?” This is a fictive motion description, in that “we” are not actually going anywhere; we are merely shifting our locus of attention. The description is accompanied by an eye point since both of the teacher’s hands are now occupied. The teacher says “we go from the,” lengthening the final vowel and pausing, as she does when she wants the students to fill in a piece of information. When a student starts to say the word “nine,” the teacher saccades to the 9 and also says the word “nine,” lengthening the vowel as she scans smoothly along the stick (with a slight head turn) to the 3. She completes her statement with “to the three” as a student also exclaims “three!” This eye point is, in fact, another trace. Via the extension schema (as discussed in chapter 4), the teacher’s gaze contacts the pointing stick at the 9 and then draws a line along the top of the pointing stick to the 3 (Figure 5.2(f)). This trace establishes a conceptual dividing line across the clock-circle, anchored by the pointing stick. For both the vertical and horizontal dividing lines, the teacher has made opportunistic use of material structure 26
Notice that the teacher says “up and down” but the gesture stroke goes down first. “Up and down” is an idiom that profiles bidirectional motion along a vertical axis; as an idiom, it is applied to the entire motion event, not composed according to the particulars of that motion, as in “up and then down.” Similar idioms are “back and forth” and “to and fro”, which profile bidirectional motion along the horizontal axis.
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to anchor the conceptual entities. In the first case, the problem was to turn the two hands into a single entity; here the problem is to delimit a portion of the pointing stick as relevant. The parts of the stick that protrude past either side of the teaching clock are irrelevant to the division of the circle. The eye-trace and speech together establish a segment of the stick as the material anchor for the conceptual entity. Naming 9 and 3 has the added advantage of setting up landmarks that will be useful for reading ‘quarter past’ and ‘quarter till’. An additional observation is now in order. While a circle made up of four parts is a static entity, dividing a circle into four parts is a multi-step process. In classes I observed, whenever a teacher or student drew lines to divide a circle, the first line was drawn from top to bottom, dividing the circle into halves, and the second line was drawn from left to right, further subdividing the circle into quarters. Of course, it does not have to be done this way: the lines could be produced at different orientations, but this would be less prototypical. A vertical line produces the familiar left-right symmetry that is part of our embodied experience. Once this line is drawn, it constrains the orientation of the second line, which must be horizontal. Of course, the lines could be drawn in the opposite directions (bottom-to-top and right-to-left), but this would oppose the familiar direction of reading and writing that is also part of our embodied experience. And perhaps it is just conventional to divide objects this way. These are subtle constraints, but the pattern of dividing circles (and squares and other shapes) is recurrent and familiar, making the orientation of left/right halves and top-left/bottom-left/top-right/bottom-right quarters canonical. In the lesson we are analyzing here, the first conceptual line was drawn (traced over the clock face) from top to bottom (and then back up and down again), while the second was drawn (traced) from left to right (from the students’ perspective), leading to a canonical division of the clock-circle into quarters. The teacher’s next utterance, “from nine to three we have four parts” (Figure 5.2(g)), shifts attention to the bounded regions defined by the dividing lines (similar to what happened in Figure 5.2(d)), conceptually filling the four areas and bringing them into profile as distinct objects, parts of a whole (the part-whole schema). In Figure 5.2(h) the teacher emphasizes that these are equal parts, and as she says “four equal parts” she glances at each in turn, a series of saccades starting at the top-left part (from the student view), moving to the bottom-left, bottom-right, and then topright part. This is a subtle sequence of eye points accompanying the speech. Finally, in Figure 5.2(i), the teacher adds “on our clock,” bringing the clock component of the blended clock-circle back into profile and refocusing attention on the clock input. This is important because the clock is actually the target of the discourse, and the purpose of the blend is to generate inferences that can be transferred back to the clock input. Ultimately, the students are to see the demarcation of quarter-hours on the clock face, not to reason about circles. In what immediately follows in the lesson, when the teacher shows the students how to read a quarter past six, she refers to the upper right quadrant as “a quarter of the clock.” In the construction of this blend, the artifact that the teacher is holding serves as a material anchor for both the clock blend (teaching clock seen as clock) and the
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subsequent clock-circle blend. Opportunistically, the clock hands and pointing stick serve as material anchors for the conceptual dividing lines separating the clock-circle into four quarters. The mapping of elements of the conceptual model of dividing a circle into quarters onto the clock face is accomplished through a series of traces (including one eye-trace), iconic-indexical gestures whose iconicity superimposes a conceptual element over its material anchor and whose indexicality demarcates the portion of material structure that is to anchor the conceptual element. Co-gesture speech provides the conceptual construal, while additional speech activates relevant models and brings particular elements of the models into profile.
5.3.2 Reading ‘X fifteen’: re-conceptualizing the artifact state The next episode occurs about one minute later in the same lesson, after students have practiced as a group naming several ‘quarter past’ times set by the teacher. In these thirty-one seconds of interaction, the teacher guides students through an entirely different way of conceptualizing the same clock state in order to read the relative time of ‘quarter past three’ as the absolute time ‘three fifteen’. The episode is transcribed in Transcript 5.2. As before, we will work through the transcript step by step, diagramming the construction of meaning (Figure 5.3(a-i)). The episode begins with the teacher holding the teaching clock while the students name the time in unison (what teachers call “choral response”) as “a quarter past three.” The reading of this time is depicted in Figure 5.3(a). Here we see that the quarter-hour is read first, by focusing attention on the position of the long hand and using the Clock Quarters blend developed earlier, and then the hour is read by focusing on the short hand and imposing particular image-schematic structure. Because the students have read several ‘quarter past’ times in a row, thereby building an association between the position of the long hand and the linguistic frame ‘quarter past __’, their attention is likely to be focused on reading the hour, which has changed each time the teacher reset the teaching clock. For reasons to be discussed in chapter 6, the students are likely to be reading the hour using an incorrect image schema, but so long as there is no distortion of the output, the teacher does not attend to the hour reading. At this point, the teacher says “now another way that we say it” (Figure 5.3(b)), prompting the construction of a mental space for a new conceptualization. As she says this, she moves the long hand back to the 12. Why? As will become apparent in what follows, she is setting the artifact to a source state that will lead to a particular goal state, namely the same state of the teaching clock the students have just seen but anchoring a different conceptualization. This is a use of the sourcepath-goal schema to structure a conceptual process that will be realized through a series of physical actions. After a 1-second pause, the teacher completes her statement with “is we count by fives,” emphasizing the word “fives.” This adds to the new space a conceptual model that has become familiar to the students through practice: counting by fives. Counting involves bringing a sequence of verbal labels, which have been learned as a kind of rhythmic chant, into coordination with a series
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of objects through a sequence of actions, such as sequential touching or pointing. (When the objects are not physically present, fingers may be used as proxies, raised or touched in sequence.) When counting by fives, the chant is “five, ten, fifteen, …” and the objects are sets composed of five elements. Two things are important in counting: coordinating the verbalizations with the actions-on-objects and ensuring that each object is touched only once while no object is omitted. For young children, this takes some practice. Experienced counters accomplish this by imposing a source-path-goal image schema on the collection of objects to be counted: one object is selected as the starting point, a path is followed through the collection that touches each and every object only once, and a final object is reached as the end point, producing the final count. If the objects are moveable, a different strategy may be used, one that simplifies the tracking by making intelligent use of space (Kirsh 1995): sliding or shifting each object as it is touched from one space to another, thereby distinguishing the already-counted group from the to-be-counted group. This strategy builds and runs a conceptual blend in which the separate groups of objects, in distinct regions of space, act as material anchors for the conceptual categories. In the case of the clock face, the numbers and tick marks are not moveable, so sequential touching or pointing is used for counting the minutes and a source-path-goal schema guides the counting action. Returning to Figure 5.3(c), “we count by fives” activates the conceptual model for counting and prompts the string of verbal labels to be used: ‘five, ten, fifteen,’ and so on. The next couple of diagrams (Figure 5.3(d-e)) show how the conceptual model of counting is mapped, element by element, onto the teaching clock. Continuing, the teacher says “when we move this, from number to number.” As she says “when we move this” (Figure 5.3(d)), her hand is held in an L-shape with her index finger touching the long hand. The word “this” prompts a search for an object; following the point leads directly to the referent in contact with the teacher’s finger: the long hand of the teaching clock. “Move” describes an action, cuing how the long hand is to be mapped into the model for counting: moving the long hand is the action that will bring the series of ‘five, ten, fifteen …’ labels into coordination with a series of objects. “From number to number” (Figure 5.3(e)) names the series of objects: the series of numbers on the clock face. The ‘from/to’ construction and the simultaneous gesture—the same L-shaped point touching the tip of the long hand (now over the 12) and then bouncing to the 1, 2, 3, and 4—provides the starting point and sequence of touches that will be used in the counting. While the speech provides the general form of the path, the gesture establishes the starting point and direction; the goal is left unspecified. Just as held objects are commonly used as proxies for pointing, the long hand will serve as a proxy for the index finger for counting, “touching” each number in turn (because the pointing model is already associated with the long hand, this seems perfectly natural). While the teacher completes the last part of the gesture she appends a key piece of information: “there’s five minutes between each number” (Figure 5.3(f)). This statement introduces another conceptual domain, the system of time measurement, and, together with the gesture, maps onto the teaching clock an interval of five minutes
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between each adjoining pair of numbers (notice that no attention has been drawn to the minor tick marks, which will not be focused on until the students start learning 1-minute times). Recall that these numbers have just been linked to the counting model, so the interval of five minutes gets mapped to the counting model as well. The structures line up: when we count by fives, each counted object is a set composed of five elements; now we understand those elements to be minutes. In the ten seconds of interaction depicted in Figure 5.3(b-f), the conceptual domains needed to count five-minute intervals on the clock face have been activated and the conceptual elements have been linked to material elements on the clock face. All that remains is to do the actual counting. In Figure 5.3(g), the teacher says “so if we were going to count by fives it would be,” setting up a mental space for the blend in which all of the linked components will be integrated in the action of counting. Then she enacts the count; in other words, she actually “runs the blend” she has so carefully constructed. Figure 5.3(h) shows how this unfolds: (1) the teacher looks at the long hand, moves it to the 1, and glances up at the students as she says “five”; (2) then she looks at the long hand, moves it to the 2, and glances up at the students as she says “ten”; (3) finally, she looks at the long hand, moves it to the 3, and (without glancing up) says “fifteen.” In each of these moves, another five-minute interval is mapped onto the clock face, producing a total of fifteen minutes from the 12 to the 3. Notice that once the teacher has set the artifact to the source state (Figure 5.3(b)), the conceptual mappings needed for the blend are all made through gestures and accompanying speech; only after all of the mappings have been set up does the teacher actually grasp the long hand and run the blend as she changes the clock state, step-by-step, from the source to the goal state. Once the goal state is reached, she releases the long hand and resumes gesturing. When the count is completed, the clock reaches the same state it was in at the start of this episode (Figure 5.3(a)) when the students read the time as “a quarter past three.” But now, via the new conceptual blend the teacher has just constructed and enacted with the teaching clock, this same state is conceptualized differently. In Figure 5.3(i) the teacher changes gesture hands as she names the time in two different ways: first, with her right hand she releases the long hand and grasps the bottom corner of the clock as she says the word “quarter” in “so this is quarter past three”—a reference to the time the students read 20 seconds earlier; next, with her left hand, she touches the clock at the 3 (where the long hand is pointing) on the word “it” as she says “but it is” and then grasps the teaching clock with her left hand at the 3 and lifts it with both hands as she emphasizes the word “also”; finally, she holds the clock up in front of her as she announces “three fifteen.” In doing this she equates the relative time reading with the absolute time reading: “this” refers to the state of the clock; “is quarter past three” refers to the previous reading as conceptualized via the Clock Quarters blend; “but” signals a disjunction (in this case, an alternative reading); “it” refers again to the clock state; “is also” sets up the equation to the new conceptualization; and “three fifteen” re-conceptualizes the clock state in terms of the conceptual model of counting by fives as applied to the
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teaching clock and linked to the system of time measurement. No explicit mention is made of the difference in the order of looking at the long and short hands or in how the hour is to be read, although these are also important to reading the times correctly. Also implicit in the initial re-positioning of the long hand, the number-tonumber gesture, and the actual counting action—but never explicitly mentioned—is the fact that the starting point for the counting action is the top of the clock, where the number 12 is located. This is also the zero point for the measurement of the time interval in minutes. Miscoordinating the counting labels with this starting point turns out to be a source of error in children’s clock-reading, as described in chapter 6. For the remainder of the lesson, the teacher shows two more settings of the teaching clock, calling on the class to read them as both ‘X fifteen’ and ‘quarter past X’, and then continues resetting the teaching clock five more times and calling on individual students to read each time both ways. She also has each student write the ‘X fifteen’ time in digital form (with a colon) on the whiteboard. In the short lesson examined in part here and in the previous section, the teacher has guided the students through two different ways of conceptualizing the same clock state. This is depicted in Figure 5.4. The image in the lower left shows the teacher holding the teaching clock as the students read “a quarter past three.” The image in the lower right shows the teacher holding the teaching clock twentytwo seconds later as she announces the time as “three fifteen.” The two images are nearly identical, but the meanings the students are to construct are entirely different. Where do the differences in meaning come from? They depend upon different sequences of actions on the artifact, both mental and physical (a different order of attending to the long and short hands, the action of counting), different impositions of image-schematic structure (container vs. source-path-goal), the activation of different conceptual models (dividing a circle, counting by fives, units of time measurement), different mappings that link conceptual elements to material anchors (the long hand as the boundary of a region vs. the long hand as the index of a count), and, most importantly, the different conceptual blends that result. The same state of the clock anchors different conceptual blends, but each of these blends depends upon that state of the clock to sustain a set of conceptual relations for reading the time, whether in absolute or relative form.
5.3.3 Reading ‘quarter till’: adding motion and future spaces The next episode comes from a lesson that occurred in the same 1st grade classroom one day later. In this lesson, the teacher uses the Clock Quarters blend from the day before to teach the students how to read ‘quarter till’. She begins by putting the felt quarter-circles up on the felt board and pointing at the upper right quarter-circle, asking the students if they remember what that is called. A student responds, “one fourth.” The teacher asks for the other name, and another student says, “a quarter.” The teacher repeats, “a quarter,” and then moves to her chair and sits down. She picks up the teaching clock, which currently displays ten o’clock, and
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says that when the big hand moved from 12 to 3 (moving the hand), it was? Ten fifteen. Or? Quarter past ten. Now she begins the portion of the lesson transcribed in Transcript 5.3. In this fifty seconds of interaction, the teacher reactivates the Clock Quarters blend, modifies the clock state by moving the long hand to the 9, helps the students see the upper left clock-quarter, and then announces the relative time reading of “a quarter till seven.” At first glance, the lesson seems to be a simple elaboration of the conceptual blend constructed the day before, but it turns out to be much more complicated than that. The teacher actually constructs a more complex conceptual integration network relating current and future time spaces. This shift reflects a change in the underlying image-schematic structure, a topic we will return to shortly. For the moment, let us proceed as before, diagramming the construction of meaning step by step, as shown in Figure 5.5(a-l). In Figure 5.5(a), the teacher attempts to re-establish the Clock Quarters blend from the day before, reminding students to “remember the quarters” as she places the pointing stick across the clock face, but the current state of the teaching clock disrupts this blend. Some of the analogies are still present: the circular band of the clock face re-anchors the conceptual circle while the pointing stick re-anchors the horizontal dividing line, but the anchor for the vertical dividing line is missing. Instead, the clock hands are arrayed differently, creating a disanalogy between the conceptual model of the divided circle and the perception of the clock face. Attempting to blend the two inputs by compressing Analogy into Identity merely compresses Disanalogy into Difference, an undesirable effect. To remedy this, the teacher pauses and adjusts the clock hands (Figure 5.5(b)), simultaneously raising the long hand to the 12 and lowering the short hand to the 6. This arrays the hands oppositely from the day before, but this difference does not matter because the two hands again combine to form a single divider, a material anchor for the conceptual dividing line down the center of the clock-circle. Now the teacher repositions the pointing stick across the clock face to anchor the horizontal dividing line. As she does so, she announces the reference points on the clock face that anchor the ends of the dividing lines: the 12 and the 6, the 9 and the 3. With the disanalogy undone and replaced by analogy, the compression follows easily, yielding the Clock Quarters blend from the previous day. Now the teacher is ready to initiate the instruction that will lead to an understanding of ‘quarter till’. In Figure 5.5(c), she begins by building a space for a hypothetical situation—“now what if the big hand”—and touches the big hand as she names it, bringing it into profile. She continues in Figure 5.5(d), saying “moved all the way around to the nine” while moving the big hand slowly and steadily around the clock, in a clockwise direction, releasing it when it points at the 9. Several features encourage a construal of this motion in terms of the movement of an actual clock hand, as shown by the additional input space in Figure 5.5(d): the opportunistic starting position at 12 (six o’clock), which happened to result from aligning the clock hands to form a divider; the clockwise motion when a counterclockwise path would be more direct; the vocal emphasis on the words “all the way around”; the steady
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speed in which the hand is moved and the pause while the teacher waits for the hand to reach its destination before saying “to the nine”; the phrasing ‘the big hand moved” rather than “I moved the big hand,” removing the teacher as the agent of the movement; and the construal of this actual physical movement as a hypothetical. In the perceptual space of the teaching clock, the movement is real, but in the blended space in which the teaching clock is conceived of as an actual clock, the movement is hypothetical: forty-five minutes of actual clock motion has been compressed into a mere five seconds. In Figure 5.5(e), the teacher repeats the statement “moved all the way around to the nine” while setting down the pointing stick and grasping the clock at the 9 exactly on the word “nine.” After lifting the clock and sliding her left hand underneath, which frees her right hand, the teacher asks “how far is it” (releasing her right hand from the clock) “until it gets up here?” (tracing an arc from the 9 to the 12), and then she repeats the trace in diminished form during the subsequent pause (Figure 5.5(f)). Significantly, the teacher asks how far it is rather than how long it would be, framing the question in terms of distance (space) rather than duration (time). Time does enter the picture with the word “until,” which sets up a future space in which the hand reaches its goal or destination, a destination glossed as “here” just when the teacher’s pointing finger reaches the tick mark at the top of the clock above the 12. Interestingly, the actual physical state of the teaching clock anchors the present blended space, which is the hypothetical current clock time, while the gesture traces the path to the future state, the clock time when the big hand reaches the 12. Blending the hypothetical present (long hand pointing at 9) with the anticipated future (long hand pointing at 12), as shown in Figure 5.5(f), outlines a particular structure in the Clock Quarters blend, namely the top-left clock quarter. Of course, the series of conceptual operations since the Clock Quarters blend was re-established at the start of the lesson, including two additional nested blends, is likely to have obscured the original mapping. The teacher responds to this problem by switching anchors (Figure 5.5(g)). While still holding the teaching clock with her left hand, she leans over and uses the extended pinkie of her right hand to point to the quartercircles on the felt board. Saying “from here,” she touches a point at the far left; pausing, she traces a path along the edge of the top-left quarter-circle; and saying “to here,” she touches a point at the top. Emphasis on each word “here” co-occurs with the touch-points. This clear imposition of source-path-goal motional structure onto the felt quarter-circles invites a reinterpretation of the similar gesture that occurred moments before on the face of the teaching clock. The previous link between the circle-quarters and the clock face supports a re-apprehension of the Clock Quarters blend, this time anchored by the felt quarter-circles, while the teacher’s from/to construction and simultaneous gesture over the top-left quarter-circle provide an explicit answer to her “how far?” question. At the moment when the teacher sits back and calls on a student to answer this question (Figure 5.5(h)), there are two artifacts visible in the environment that can be used to anchor the complex quartercircle/clock/present/future blended space: the felt quarter-circles and the teaching
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clock. The teacher calls on a student who provides the desired answer: “One quarter or one fourth.” Returning to gesturing over the teaching clock (Figure 5.5(i)), the teacher repeats the portion of the answer that she sought—“it’s a quarter, right?”—while again tracing a path from the 9 along the arc of the clock-circle to the 12. She then immediately repeats the gesture, slightly diminished, while saying “it’s a quarter of the clock” (Figure 5.5(j)), again bringing the clock component back into profile because it is the clock-quarter, not the quarter-circle, that is the focus. Now that she has taught the students to see the top-left clock quarter, bordered on one side by the present position of the long hand (pointing at the 9) and on the other side by the future position of the long hand (pointing at the 12 at the top of the next hour), the teacher proceeds to teach the students how to use this particular form of situated seeing to name the currently displayed time (Figure 5.5(k)). She says “so we say it’s a quarter till,” emphasizing “quarter till” and lengthening the word “till,” then pausing as she typically does when she wants the students to fill in the information. She again traces an arc from the 9 to the 12 on the word “quarter,” holds briefly at the top on the word “till,” and then swings her hand down to the 7, pointing there while she waits for a student response. As soon as a student starts to say “seven,” the teacher says “seven.” In tracing the arc from 9 to 12, her gesture marks the path of motion of the long hand from its materially-anchored present position to its imagined future position. When her point shifts down to the 7, it indicates where the short hand will be once the long hand has reached the 12; by following directly after the trace, it references the position of the short hand in the imagined future state. In Figure 5.5(l), the teacher repeats the “quarter till seven” reading with the same gestures, but with slightly different timing, this time enunciating the word “seven” as her point reaches the 7. What is striking about this lesson, when compared to the ‘quarter past’ lesson of the day before, is the ubiquity of motion and the introduction of different time spaces into the discourse. Rather than simply pointing the long hand at the 9 and teaching the students to see the upper-left quarter-circle shape and associate it with the label “quarter till,” the teacher has the students follow the motion of the long hand (simulating clock motion) “all the way around” from the 12 to the 9, and then she repeatedly traces the continuation of that motion to the 12. Why so much motion? The answer has to do with the change in the reference hour. Recall that absolute time can be read without conceptualizing motion, using a container schema for the hour and a pointing model for the minutes. ‘Quarter past’ times can also be read without conceptualizing motion, by recognizing the shape of the upper-right clock-quarter, labeling it “quarter past,” and then reading the hour with the container schema, just as for absolute times. This produces the correct output even though it is conceptually incorrect. Relative times are referenced to the start and end points of motion—the previous and upcoming hour—and not the current hour, as in absolute time. For a relative time like ‘quarter past’, the students should use the source-pathgoal image schema and look back to the previous hour as the source or starting point of motion. Because the students can continue to produce the correct time name
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without learning this new way of conceptualizing the hour reading, there is no compelling reason for the teacher to teach it. A compelling reason does emerge when students are taught to read ‘quarter till’: the reference shifts to the upcoming hour. Now it is imperative that the students read the hour differently, as the goal or destination of motion, so that they will look ahead to the upcoming hour and name the time correctly. For this reason, the teacher brings in the source-path-goal schema, enacting the motion of the long hand from its source (the 12) along its path (around the clock in a clockwise direction) to its current position (at the 9) and then tracing the portion of the path remaining to the goal (the 12 again). The students are being taught to see that this remaining portion is one quarter of the path; they are not being taught to see a bounded region of space that constitutes a quarter-circle shape. Notice that when the teacher switches anchors and gestures over the felt board, she traces a path along the outside edge of the top-left quarter-circle rather than tracing its full outline or simply pointing at it, and this trace is accompanied by the path-of-motion statement “from here to here.” The future space gets introduced precisely because it is necessary to reference the goal state to read the hour correctly, even though the teacher focuses on the ‘quarter till’ part of the reading and provides no instruction for reading the hour. (But see below, in the 6:45 lesson, where the teacher does focus on the hour, anticipating errors due to proximity. One can actually produce ‘quarter till’ time readings without conceptualizing motion by naming the upper-left quarter-circle shape and then using proximity to read the hour, but see chapter 6 for why this might be a problem.) There is no reason to expect that the teacher has spent time reflecting on these issues—she is simply trying to teach the students how to read this relative time—but we do see evidence of these conceptual distinctions appearing in her gestures and speech exactly when they become critical to producing the correct output. The complete, correct conceptual models for reading the time as “a quarter till seven” are shown in Figure 5.6. Here we see that a source-path-goal image schema for motion has been added to the quarter-circle model used to read ‘quarter past’. Because the long hand and the short hand are geared together, the source-path-goal motion of the long hand when it makes a full clockwise revolution from 12 to 12 correlates exactly with the source-path-goal motion of the short hand when it moves from one hour number to the next. This is reflected in the conceptual model for reading the reference hour, as shown in Figure 5.6. When the long hand is projected ahead to its goal state at the 12, the short hand is projected ahead to its goal state at the next hour number, in this case the 7. Because source-path-goal image schemas characterize motion along a path from one point to another and because motion unfolds through time, past and future spaces enter into the instructional discourse. The teacher invokes a future space to teach the students to read ‘quarter till’ times. She avoided invoking a past space to teach them to read ‘quarter past’, relying instead upon the static imagery of containers and shapes; this may have implications for the students’ conceptual understanding.
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5.3.4 Reading ‘X forty-five’: re-conceptualizing with the proper hour reading Following the teaching of “quarter till seven” transcribed above, and before any student practice reading ‘quarter till’ times, the teacher moves immediately to teaching the students how to read the same time in absolute form as ‘X forty-five’. This portion of the lesson is transcribed in Transcript 5.4 (the first line is taken from the end of the previous transcript). Diagrams of the construction of meaning are shown in Figure 5.7(a-i). Similarities to the ‘X fifteen’ lesson (section 5.3.2) are immediately apparent. In Figure 5.7(a), the teacher sets up a new space (“the other way is if”) and activates the conceptual model for counting by fives (“we count by fives”). In accordance with this model, this again sets up an expectation that a sequence of verbal labels, in this case “five, ten, fifteen,” etc., will be brought into coordination with a series of objects through a sequence of actions. As the teacher says this, she moves the long hand back to the 12, setting the artifact to the source state that will lead to the goal state, just as she did in the ‘X fifteen’ lesson the day before. Notice that it would have been faster to move the long hand clockwise to the 12, but the teacher wants to arrive at a goal state that is identical to the current state of the teaching clock, so she winds the hand back counterclockwise to the 12. In Figure 5.7(b-c) the teacher again sets up the mappings from the counting model to the teaching clock, but this time she is less explicit. In Figure 5.7(b), she reestablishes the mapping to the long hand she is holding (“with the big hand”), making it clear that the action of moving the long hand will again be used to coordinate the verbal labels with the to-be-counted objects, just as it was the day before. In Figure 5.7(c), still holding the long hand, she says “let’s count by fives to the nine,” emphasizing the word “nine.” Taken alone, this statement would be nonsensical: one cannot count by fives to nine. But in the context of the teaching clock, “the nine” refers to the numeral 9 on the clock face, and this 9, through metonymy, actually stands for the co-located major tick mark that is part of the minute dial. The teacher’s complete statement “with the big hand let’s count by fives to the nine” invokes a source-path-goal schema but leaves much implicit. Only the goal is stated explicitly (“to the 9”); the source (from the 12) and specific path, both direction (clockwise) and manner (jumping from number to number), are left implicit. These can reasonably be assumed to be filled in from the previous day’s lesson and other prior experience. Also left implicit are the mappings linking the objects-to-be-counted to the series of numbers on the clock face and linking both the counting model and teaching clock to the system of time measurement (recall Figure 5.3(e-f)). In Figure 5.7(d), the teacher and students execute the count together, coordinating their behavior with one another as they run the counting blend anchored by the teaching clock. Both the teacher and the students chant the verbal labels in unison and in constant rhythm while the teacher moves the long hand from number to number around the clock face. Again, as before, the long hand is used as a finger-proxy, touching each number-object in sequence as an index of the count.
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At the same time, for those whose current conceptual integration networks include an ‘actual clock’ input (recall Figure 5.2(a)) and who, through a conceptual blend, see the teaching clock as if it were an actual clock, the effect of this counting action is a series of time compressions: time jumping ahead by five minute intervals, pausing just long enough for the minutes to named. The teacher has done nothing to call attention to the ‘actual clock’ input, although the sight of a clock face (when recognized as a clock face) can be expected to activate this input. If enough attention is focused on the counting that the teaching artifact fails to be seen as a clock while the counting unfolds, then this clock blend can re-emerge later when the time is being read; indeed, the teacher may even call attention to it, as in Figure 5.2(i) (“on our clock”). When spaces are linked in a conceptual integration network, focus shifts from space to space as meaning is constructed and inferences get transferred (Fauconnier and Turner 2002). Meaning construction involves the entire network, not just the blended space. The teacher’s next move is to interrupt the rhythm of the verbal chant when the long hand reaches the goal of pointing at the 9. The chant (lines 8-19 and Figure 5.7(d)) follows a steady rhythm with repeated cycles of rising and falling intonation. There are four steady beats (STRESSED/unstressed/STRESSED/unstressed) per rising-falling unit of intonation. Each intonation unit contains two counting labels, one during the rising part (the first STRESSED/unstressed pattern) and one during the falling part (the second STRESSED/unstressed pattern). In this chant, the label “forty-five” would fall in the first half of an intonation unit, on the rising part, and would immediately be followed by “fifty” on the falling part. Although the teacher continues to move the long hand in steady rhythm to the 9, she disrupts the rhythm of the verbal chant, pausing before saying “forty-five,” shifting the stress to “five” (disrupting the STRESSED/unstressed pattern), and lengthening the vowel in “five” beyond the point where a rhythmic continuation would be possible. She also releases the long hand, grasps the side of the teaching clock by the 9, and lifts the teaching clock as she says “forty-five.” In Figure 5.7(f), the teacher then gives a full reading of the time displayed on the teaching clock she is holding up in front of the students. The clock is, of course, in the same state it was in twenty seconds earlier when the teacher read the time as “a quarter till seven” (Figure 5.5(l)), but now she announces the time as “six forty-five,” in accordance with the counting blend she and the students have just run. In naming the time, she inserts a reading of the hour in front of the minute reading she has just derived; the conceptual input for reading the hour, structured by a container schema, is added in Figure 5.7(f). At this point in the lesson, the teacher’s instruction suddenly shifts to the correct reading of the hour (Figure 5.7(g)). With the index finger of her right hand, the teacher traces along the short hand, away from the center of the clock face, extending the trace (a conceptual line) beyond the end of the short hand to the dial around the perimeter—an enactment of the conceptual model of pointing discussed in chapter 4. As she does so, she bends the extension of the trace toward the 7, so that her finger comes to rest on the major tick there. This gesture is accompanied by the statement “it’s not seven.” What’s going on here? A
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negation sets up a mental space for some state of affairs but marks that space as counterfactual. In this case, the teacher’s gesture enacts the very event that her speech marks as counterfactual, the use of a proximity model to read the hour, as shown by the leftmost input space in Figure 5.7(g) and the subsequent counterfactual blend. This statement is immediately followed by a reason that contains another negation: “’cause it’s not on the seven yet.” Although this statement contains a negation, it actually invokes the correct conceptual model, the container-based model shown in the rightmost input space in Figure 5.7(g). In the subsequent blend, what is counterfactual is the location of the short hand in the hour container for the 7; the short hand does not enter this bounded region of space until it is on the 7 (a future state profiled by the word “yet”—the future space is not shown separately in Figure 5.7(g)). In the factual state of affairs, the short hand is between the 6 and the 7, i.e. “not on the seven yet.” In the first case (“not seven”), the negation cancels the mapping from the proximity model to the clock face; it says that the proximity model does not apply and should be removed as a conceptual input. In the second case, (“not on the seven yet”), the negation marks as counterfactual in the present the upcoming future state in which the short hand enters the 7-container. It says that the container model and mapping onto the short hand are correct, but that the present location of the short hand does not fall within the conceptual container associated with the 7. Notice that ‘not being on the seven yet’ is a condition that is compatible with the container interpretation but incompatible with proximity, where being closer to the 7 is all that counts. In Figure 5.7(h), the teacher announces both of the alternate readings, the absolute and the relative, gesturing only for the hour portion of each reading. She says “it’s six forty-five,” and on the word “six” she traces a line from the center of the clock straight down to the 6, defining the salient boundary of the 6-container. After the word “or,” she says “quarter till seven,” and on the word “seven” she drops her point directly onto the 7, the goal state for the hour hand in the sourcepath-goal ‘quarter-till’ reading. She elaborates this second reading in Figure 5.7(i) when she adds “it’s almost seven,” sweeping her point from the long hand toward the 12 on the word “almost” (indicating the motion of the long hand toward the goal) and then swinging her point down to the tick mark below the 7 on the word “seven” (the goal state of the short hand once the long hand reaches its goal). Throughout this brief explication (twelve seconds) of how to read the hour correctly, we see that key elements of the conceptual models—both the correct and incorrect models—show up in the gestures over the clock face. The accompanying speech provides time readings (“six forty five”; “seven forty-five”; “quarter till seven”), describes states of affairs (“not on the seven yet”; “almost seven”), and fills in logical relations (“not”; “or”; “cause”) that mark spaces as counterfactual or specify the relation of one space to another. The contrast between the conceptual models and mappings for the two different readings of the same clock state is shown in Figure 5.8. Here again we note the different sequences of attending to the long and short hands for the relative and absolute readings. We also see that both of the long hand readings involve mapping
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independently derived conceptual models (dividing a circle and counting) onto the clock face, while the short hand readings depend only on imposing different image schemas (source-path-goal vs. container). Both of the long-hand readings also involve blending the same image schema, source-path-goal, with the conceptual model mapped onto the clock face, but the role of the image schema differs in each case. For the relative reading, the source-path-goal schema defines a path of motion around the circle’s perimeter and the tip of the long hand follows this path. For the absolute reading, the source-path-goal schema defines a path of motion for the act of counting and the pointing finger (or its proxy) follows this path. While both types of readings depend upon the position of the long hand, the relative model emphasizes the overall geometry of the space while the absolute model emphasizes the quantity of objects present in a particular sub-region. Each of these conceptualizations is anchored by the same clock state, as shown by the images at the bottom of Figure 5.8 taken from the moments in the lesson when each reading was announced.
5.4 Discussion 5.4.1 The same anchors for different blends The diagrams and images in Figures 5.4 and 5.8 show how the same physical state of the clock face anchors different conceptual blends based on different conceptual inputs during absolute and relative time reading. Even though the clock state is the same, different conceptual models can recruit different sub-structures on the clock face to anchor conceptual elements. For instance, the quarter-hour relative time readings use the circular band of the clock to anchor the conceptual clock-circle in the Clock Quarters blend, while the absolute time readings ignore this band and instead use the sequence of numbers as anchors for the objects to be counted. Different conceptual models can also recruit the same sub-structures as anchors but use them to anchor different conceptual elements. For instance, the numbers adjacent to the short hand anchor the source and goal of motion in the relative reading of the hour, but these same numbers anchor the boundaries of a container in the absolute reading of the hour. In both cases, the short hand (or, via the pointing model, the location it points to) anchors a conceptual ‘trajector’ that is either somewhere along a path from source to goal or somewhere inside a container. Multiple blends are possible with the same anchor: the situated seeing involved in using a cognitive artifact to perform a computational task is essentially the serial instantiation of relevant blends.
5.4.2 The linking role of gesture How do children learn to make these blends? These lesson examples show the critical role that gestures play in establishing the conceptual-material mappings, both in highlighting sub-structures that are to anchor particular elements and in enacting
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the conceptual elements that are to be anchored by these structures. Often, these two functions—the indexical and the iconic—are accomplished by the same gesture, a trace. Of course, without sufficient context, gestures are ambiguous or indeterminate. Co-gesture speech provides key information about how the gestures over the clock face are to be construed. Naming conceptual models, conceptual elements, and material structures brings them into profile at key moments during the gesture. Deictic references (“here,” “this”) skew the interpretation of a pointing gesture as indexing a location or an object. Particular linguistic constructions (“from __ to __”) add image-schematic structure (source-path-goal) to a gesture being made over an object. In addition, we have seen how speech builds mental spaces and marks them as past/present/future or as counterfactual, distinctions which are essential to the construction of appropriate meanings, and of how speech provides logical connectives that relate one space to another in the network.
5.4.3 Constructing meaning from an artifact Let’s take a step back and consider this picture in its entirety. In time-telling, we are talking about deriving meaning from the state of a material artifact. These time-telling lessons make clear that the idea of “deriving” is misleading: nothing is being taken from the clock; on the contrary, much is being added. We do not derive meaning; we construct meaning by adding structuring elements (image schemas and mappings from conceptual models) to the perceptual space anchored by the clock, we compress relations to form blended spaces (likewise anchored), and we generate inferences in the blended space which can then be transferred to other spaces in the evolving conceptual integration network as the discourse develops. The material structure of the clock face provides important and necessary constraint for these meaning-making operations. Once the conceptual elements have been mapped to their anchors, it is the physical relationships of the anchors that determine the conceptual relationships that matter for telling the time: e.g., whether a trajector is inside a particular container, or how much of the path it has left to travel before reaching its goal. Manipulating these physical relationships manipulates the conceptual relationships they anchor, so setting the clock an hour ahead for Daylight Saving Time moves the short hand out of one conceptual container and into another, changing the current hour.
5.4.4 Guiding conceptualization In order to guide learners through the process of setting up these blends and using them to generate the appropriate inferences, teachers do several important things. First, teachers control states of the artifact. They set the artifact to particular states at particular moments in the discourse. On a large scale, this means that teachers create a pedagogical progression from states that are more prototypical, easily interpretable, or prerequisite to interpreting other states, to states that are less prototypical, more difficult or complex to interpret, or in some way dependent upon
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previously-learned states. On a small scale, such as in the short episodes of instructional discourse we have been examining here, controlling states of the artifact means that teachers can set the artifact to a state that readily affords anchoring of a particular conceptual blend (such as aligning the hands vertically for the Clock Quarters blend), or they can set it to a source state that, when a relevant blend is actually run, will lead to the desired goal state (as when the teacher moves the long hand to the 12 to start a counting operation). In addition to setting the artifact to particular states at key moments in the discourse, teachers can also change the artifact’s state in a controlled fashion to support the unfolding of an anchored conceptual process, such as moving the long hand from number to number while counting. Controlling states of the artifact ensures that the appropriate material anchors are available at relevant moments in the instructional interaction, both to support the performance of particular tasks and to support the overall progression of learning. The second thing that teachers do to guide learners through the process of building and running conceptual blends is that teachers add additional material structure to the artifact to support particular conceptualizations. They do this in several ways. One is by substituting other artifacts that have additional structure built into them, such as a teaching clock with a number scale for the minutes. Another is by making opportunistic use of physical structure already present, such as laying a pointing stick across the clock face. Yet another is by creating physical structure where none previously existed, such as drawing a line across a clock face diagram. But the sources of additional structure do not stop there. Teachers also add material structure by enacting structure with their bodies. Sometimes they use gestures to draw virtual lines, circles, or arrows (directional lines) on the artifact. At other times, they substitute anchors by using a body part or held object as a proxy for an artifact structure such as a clock hand, so that the body part anchors the conceptual element normally anchored by the clock part. These body-part anchors show up especially when teachers gesture in the air, away from the artifact, either because the artifact is not present or near at hand or because, for whatever reason, the teacher shifts gesture space away from the artifact (the reasons for such shifts remain to be explored). An example of this from our video data is when a 3rd grade teacher, rather than retrieving the teaching clock that was just out of reach and resetting it, used her own arms in place of clock hands, in effect showing the analog clock time with her body.27 Once a virtual object is set up in the discourse, it can be referenced indexically (pointed to) or acted upon by additional gestures, annotating it with virtual lines, circles, and so on, as for a real object, so long as it remains relevant and active in working memory (presumably so long as it is not obscured by subsequent gestures in the same space or by major shifts in the discourse, although this also 27
Gestures in the air have been examined in gesture research where the referenced objects are not physically present, for example when participants explain abstract subjects like mathematics (Núñez, in prep.; Smith 2004) or narrate previously viewed cartoons (McNeill 1992; see Liddell 1998, for an analysis of such gestures using conceptual blending theory).
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remains to be explored). Gestural actions do differ from actual manipulations of the clock face or the addition of other physical structures in that they leave no physical trace. Although they are ephemeral, they are nevertheless powerful because they occur at precisely the moment when they are relevant to the ongoing conceptualization. Finally, teachers talk as they gesture, using another highly patterned material structure (sound waves traveling through the air) to prompt conceptual operations that relate to the material states and actions being played out in front of the learner.
5.4.5 Diagramming the construction of meaning This focus on how teachers guide conceptualization raises an important point about the diagrams in section 5.3. As is typical for conceptual blending analyses, the construction of meaning diagrammed here is idealized. It represents what a typical participant with the appropriate background who is attending to the discourse might be expected to construct. Of course, there is no guarantee that all of the students are attending to the teacher’s talk and actions, that all share and can activate the conceptual models the teacher refers to (and that they do not activate other models instead and thus form their own idiosyncratic blends), that all understand the goal of the activity, and so on. Although all students can be expected to operate with the same conceptual capacities (e.g. for conceptual mapping and blending) and all are members of the same classroom community, the outcomes of this process of guided conceptualization can still differ, leading to meanings which the teacher might see as misinterpretations of her talk and actions or as errors in reading the clock time. Those students who already have some understanding of the topic and who anticipate where the teacher is going with her instructional discourse would be more likely to construct relevant meanings like those diagrammed here, while students with less understanding and who do not anticipate the conceptual destination would be more likely to get lost along the way or to arrive at different interpretations, missing the relevant meaning. When a teacher misspeaks, mis-points, or performs some action that activates a more familiar but inappropriate conceptual model, a majority of students may be led down the same garden path, producing a shared understanding that is not particularly relevant or that may even be incorrect for the activity of telling the time. Although idealized, the diagrams of meaning construction are constrained in several important ways: (1) by the content, coordination, and timing of observed speech, gestures, and manipulations of artifacts; (2) by other visible details of the interaction, including how participants orient toward one another and where they direct their gaze; (3) by structure that is visibly present in the setting; (4) by conceptual resources known to be present in the community of practice (based on the ethnographic study); and (5) by the way that analyses of different episodes of activity inform one another, revealing common patterns. This last constraint enriches the detail of individual analyses while it boosts the analyst’s confidence in the veracity of what is being uncovered.
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5.4.6 Guided conceptualization and communication Finally, it seems unlikely that the process of guided conceptualization described here is limited to formal teaching situations. It may even characterize much of human communication. Certain characteristics of instructional activity do make it more likely to produce clear examples of this process: the intentional pedagogical structuring of the setting and situation, the teacher’s control of the unfolding discourse (including control of the sequence of actions involved in performing the cognitive activity), and the systematic and controlled way in which teachers annotate artifacts with additional objects or markings, gestures, and accompanying speech as they guide students step-by-step through physical and conceptual operations. In most everyday discourse, control is more evenly distributed, resulting in more turntaking, restarts, breakdowns, and repairs. This breaks the phenomenon into fragments, making it harder to see the ways in which our actions, gestures, and speech structure one another’s conceptual processes. Nevertheless, guided conceptualization is still at work, shaping the meanings we construct. This is a distinctive feature of human communication.
5.5 Conclusion The construction of meaning from the clock face is a process that unfolds through time. It begins with the setting of a goal, such as reading the current time or reading the interval between the current time and some other reference time (the anticipated start or end of an event). It then continues through a process of situated seeing: selective looking at structures on the clock face while instantiating conceptual blends that support the generation of relevant inferences from the positions of the long and short hands. The order of this looking varies, as does the nature of the conceptual models and resulting blends, depending on whether an absolute or relative time reading is being formed. The process of situated seeing and running the blend ultimately leads to the construction of an output: a time reading in absolute or relative form or, in some cases, a time interval. If the output is not in the desired form for reporting, it has to be transformed—another conceptual process. Finally, the output has to be related to the system of time measurement (hours, minutes, and seconds) and to our experiential time sense (how long a minute lasts, what portion of the day the time falls in, what kinds of activities typically occur then, etc.) in order for it to be meaningful. The lessons examined here have focused on the central portion of this time-reading process: selective looking at the clock face, instantiating relevant conceptual blends and generating appropriate inferences, and thereby producing a time output in absolute or relative format. In this chapter we have focused on the teacher’s actions, gestures, and speech, and how they guide conceptualization in the learner. In the next chapter, we pick up the topic of the learner’s performance, errors, and conceptual change.
6
STUDENT TIME-TELLING, ERRORS, AND C O N C E P T UA L C H A N G E
In their lessons, teachers teach students how to instantiate local functional systems for telling time. These systems, composed of clock and clock-reader, take a state of the clock face as input and generate a time-reading (in absolute or relative form) as output. There is more than one path from input to output, so while students practice the components of time-telling—selective situated seeing, conceptual mapping and blending, and so on—they are also practicing assembling these components into functional time-telling systems to perform the task at hand. Teachers help students in at least two different ways: (1) guiding conceptualization through artifact manipulations, annotations, gestures, and speech; and (2) providing sequential control of the actions that instantiate the time-telling functional system. In turning over responsibility to the students for performing the task, teachers may retain sequential control, leaving students to do the conceptual work for each step, or they may guide students through the conceptual work of a given step and then ask students what to do next, leaving them with responsibility for sequential control. As students develop proficiency, they gradually take over both of these functions, guiding their own conceptualization (perhaps aided by their own self-talk and gestures) as they proceed through a self-directed process of reading the time. Often, they perform successfully. At other times, they produce incorrect time readings. The previous two chapters have looked at the elements of correct performance and how teaching puts them in place. In this chapter, we will look at student performance, sources of error, and the nature of underlying conceptual change. In particular, we will explore the effects of using inappropriate conceptual models or mis-mappings and examine the nature of conceptual changes that can occur after periods of successful performance.
6.1 Components of correct performance To understand what can go wrong, let’s begin by reviewing what has to happen for things to go right. In assembling a functional system that produces a correct output, students need to match the form of the time expression, the order of looking, and the appropriate conceptual models, as shown in Table 6.1. When these do not match, students can still produce a correct output by applying an additional transformation. An example of this is the analog clock-reader in the Bock et al. (2003) study who appeared to use the order of looking and conceptual models for
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absolute time and then to convert the output to relative time. As we will see in this chapter, students may even produce a correct output when applying an inappropriate conceptual model, depending on the state of the clock. For other clock states, the same model will produce an incorrect output. It also happens that students produce either incorrect time readings or no output at all (the functional system breaks down) when the time expressions, order of looking, and conceptual models are mismatched, one or the other is missing, or an inappropriate expression or model is brought into the system in place of an appropriate one. As students assemble the proper elements and engage them in a workable sequence, they also need to impose image-schematic structure appropriately and to make the proper mappings between conceptual models and structures on the clock face. Failure to do so can produce output of the correct form but with incorrect content, such as a time reading that is off by one unit in the hour or minutes. We will explore these kinds of errors, but before we do so, let’s take a moment to consider the frequency of various clock-reading errors reported in research on children’s time-telling.
6.2 Sources of error According to the studies discussed in chapter 5, what are the common errors children make when reading the clock? The most comprehensive study of children’s time-telling, Friedman and Laycock (1989), did not include analysis of the types of errors children made. Siegler and McGilly (1989) found in their study of 8- to 9year-old children reading analog clocks that fully one third of the children’s errors were misreading of the hour and that these errors occurred almost exclusively when the minute hand was on the left side of the clock. The implication is that children misread the current hour as the number the short hand is closest to. Another onethird of children’s errors consisted of being off by one minute when reading 1minute times. This error had two apparent causes: rounding the time to an adjacent 5-minute mark when it was actually one minute before or after (actually a useful strategy for everyday time-telling), and (slightly less often) miscounting the time. Miscounting errors peaked when the minute hand pointed three minutes past a 5minute mark. Assuming that the likelihood of errors increases with the length of the count, this result is consistent with children counting up to read the time unless the minute hand is just one tick mark from the next 5-minute reference point, in which case they count down; the other possibility is that children count from the nearest major tick mark and make more errors when counting down by two than when counting up by two. Some combination of both is likely. Finally, about one-eighth of children’s errors involved confusing the hour and minute hands. This could be a perceptual error, e.g. mistakenly identifying the longer hand as shorter, especially when the hands are at different orientations and differ little in their length. It could also be an association error, correctly identifying the hand as long or short but then associating it with the wrong time component and dial, and thus reading the time incorrectly. (In a pilot study, an adult clock reader called the short hand “the big
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hand” several times; in the debriefing she said that she called it the big hand “because it was fatter” and because she “wanted to associate it with the bigger numbers.”). The mentally retarded teenagers (with a mean mental age of 6.5 years) in Case et al.’s (1986) study made similar errors, but some also read the minutes and hours from the same scale, namely the displayed hour numbers (e.g. reading 9:15 as “nine three”), or failed to read the minutes at all. Springer (1952) found that preschool children were also likely to miscoordinate their counting with the numbers on the clock face, and, in some cases, to confuse one number with another. Some children’s errors appear to be simple guessing: this seems to be an attempt to rely upon the primary strategy of retrieval (direct naming)—the first strategy that children learn for reading hour times—accompanied by a failure to resort to a counting strategy when retrieval fails. Adult clock-readers also make errors, and Case et al. found that one-third of these errors involved miscalculating 1-minute times by one unit. A higher proportion of these 1-minute errors occurred on the left side of the clock face. In Case et al.’s study, adults preferred to read times in relative form (in sharp contrast to the Bock et al. study done seventeen years later), so the higher proportion of 1minute errors on the left side could be explained by the reversal of direction for incrementing and decrementing from the nearest 5-minute mark. In some cases, incrementing and decrementing were clearly confused, as when a clock-reader read 6:33 as “six thirty-seven.” In rare cases, the hour was decremented along with the minute. In this study, another one-fourth of errors involved misreading 5-minute times by one unit (probably a recognition or naming error, since adults rarely count to identify 5-minute marks), one-fourth involved confusing the long and short hands (a perceptual or association error), and nearly one-sixth involved misreading the hour by one unit (Case et al.’s example suggests a proximity effect, as seen in children’s clock-reading). These studies give us some idea of the frequency and kinds of errors that occur; what remains is to account for the most frequent errors in terms of the conceptual models and mappings that have been at the center of our analyses.
6.2.1 Using an inappropriate conceptual model Field observations, teacher reports, and the studies discussed above all indicate that the most common error in children’s time-telling is misreading the hour when the long hand is on the left side of the clock face. As the long hand approaches the 12, the short hand moves ever closer to the number for the upcoming hour, even touching it before the hour officially changes (as I write this the time happens to be 10:58 on my watch and the short hand appears to point directly to the 11). The apparent cause of these errors is the tendency to read the hour as whichever number the short hand points to—something children are explicitly told to do when they first learn to read hour times (e.g. “When the big hand points to the 12 and the little hand points to the 9, it’s nine o’clock.”). This gets children into trouble when they start to read non-hour times. If the hour hand actually pointed to the number of the current hour, the short hand would point directly to a number for a while and then jump to
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the next number (like the second hand on my watch, which jumps from tick mark to tick mark) rather than sweeping a continuous path around the dial. This confusion surfaces frequently in children’s clock-drawing: when given a non-hour time and asked to draw the hands on a clock face (a common school exercise), children frequently draw the short hand pointing directly to the number named in the time reading (e.g. to the 10 for “ten forty-five” or to the 11 for “quarter till 11”) rather than to a location between two numbers.28,29 The problem has to do with the conceptual model for pointing, which, in everyday experience, is bound up with the image schema for proximity. Recall from chapter 4 that interpreting a point involves imposing center-periphery, extension, and source-path-goal image-schematic structure. We scan away from the center (origo/source) along the extended structure (body part or proxy) and continue the trajectory (path) of the scan until we encounter some likely referent (goal). This referent may be found in direct contact with the end of the pointing structure or it may be located some distance away, in which case we are forced to search the space around the extension of the point. This search is guided by the proximity schema, and so if there are two possible referents, we are likely to choose the one closest to the trajectory of the point. On the clock face, if we are able to perceive the hour dial (using the part-whole schema to link the short hand and the ring of numbers), then we understand that the referent pointed to by the hour hand must be one of these numbers around the periphery of the dial. When the hour hand points somewhere between two numbers, the search for a referent triggers the all-too-familiar proximity schema, inclining us to select as referent the number closest to the extension of the point when in fact we should impose an image schema appropriate to the particular form of the time reading, namely the container schema for absolute time or the source-path-goal schema for relative time (Figure 6.1). What is pointed to by the hour hand is not, in fact, a number, but a location on the periphery of the dial, somewhere along the number line that forms the hour scale; how this location is to be related to the adjoining numbers (through the imposition of image-schematic structure) depends upon the form of the 28
Adults also make this error. I recently attended a presentation in which the researcher showed us samples of the clock faces children with focal brain lesions were asked to read. The clock faces consisted of preprinted circles and numbers, with hand-drawn arrows for clock hands. The clock face for 9:30 had the short arrow pointing halfway between the 9 and 10, as expected, but the clock face for 9:15 had the short arrow pointing directly to the 9, forming a straight line with the long arrow pointing to the 3. Quite by accident, the children in the study were being asked to read an impossible clock state.
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In my observations in schools, I have noticed that children are more likely to draw the short hand correctly for :30 times than for other non-hour times. When teaching children to read half-hour times, teachers I observed all emphasized the position of the short hand as pointing halfway between two numbers or as having moved halfway from one number to the next, a motivation for the “half past” reading. This did not occur for other non-hour times: at ten minutes past the hour, the short hand has moved one-sixth of the way from one number to the next, but teachers are unlikely to call attention to this because “a sixth past” is not a conventional way of reporting the time.
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time reading. Although proximity is, in this sense, never the appropriate image schema for reading the hour, whether for absolute or relative time, it does happen to produce the correct output under certain conditions. Consider four possible combinations of clock state and time reporting format: (1) long hand on the right side of the clock (first half of the hour) for absolute time; (2) long hand on the right side of the clock for relative time (past); (3) long hand on the left side of the clock (second half of the hour) for absolute time; and (4) long hand on the left side of the clock for relative time (till). For the first two sets of conditions (Figure 6.2(a)), when the long hand is on the right side, using the proximity schema to read the hour happens to produce the correct output because the hour hand is closer to the prior number, which coincides with both the current hour (identity of the container) and the previous reference hour (source of the path of motion). For the third set of conditions (Figure 6.2(b)), when the long hand is on the left side while reading absolute time, using proximity to read the hour produces an incorrect reading in which the hour is one unit too high (4:45 in place of 3:45). This tendency becomes exacerbated as the long hand moves up the left side of the clock, bringing the short hand ever closer to the upcoming number (increasing the sense of proximity), then in contact with it (touching, the most proximal form of pointing), and then seeming to point directly at it (close to its center) just before the hour changes. In recent work by Bock et al. (in press), this is where contemporary American undergraduates switch to relative time reporting, as if activation of the proximal hour (corresponding to the goal) overpowers the current hour used in absolute time reporting. Under the fourth set of conditions (also Figure 6.2(b)), with the long hand on the left side while reading relative time, the proximity schema again happens to produce the correct output. Why? Because it is conventional when the long hand is on the left to reference the relative time to the upcoming hour (goal), and proximity produces the same number in the reading. In fact, this is the one way in which proximity is appropriate to the hour reading: in choosing to read the time relative to the nearest hour, which seems to have been the fashionable way of reporting the time before the widespread use of digital watches. The impact of using the proximity schema to read the hour under the four conditions shown in Figure 6.2 is to produce seemingly correct readings 75% of the time. This is a fairly high success rate and one likely to lull teachers into the belief that students are mastering clock-reading when in fact they are making a fundamental conceptual error. Only when a particular set of conditions arises, namely when the students are reading absolute times while the long hand is on the left side of the clock, do errors suddenly appear in the output. Reconsidering the psychological studies described above, it’s not simply that students make more errors during the second half of the hour. They could be making the same error with equal frequency throughout their time-telling; it’s just that the errors become noticeable only under these conditions. Of course, it may still be the case that students do make additional errors reading times in the second half of the hour, due to the larger numbers being manipulated or confusion engendered by the misalignment of absolute and relative time-reading on the left side of the clock (as discussed in
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chapter 4). A clue that the errors may be related to an inappropriate conceptual model (or, perhaps, to an appropriate model with a consistent mis-mapping, as discussed in the next section) is the systematicity of the errors. If students consistently misread the hour under the same set of conditions (when reading absolute time with the long hand on the left side of the clock) and if these readings deviate in a consistent way (are always one unit too high), then this provides a clue that students may be making a consistent conceptual error that runs through all of their timetelling while remaining largely undetected. I believe that hidden errors such as these permeate much apparently functional cognitive activity, waiting to be unmasked when conditions are just right (or just wrong, as the case may be).30 What of the students who master reading the current hour for absolute time using the container schema but then fail to switch to the source-path-goal schema for reading relative time? This situation is depicted in Figure 6.3. Using the container schema to read the hour produces the correct number in the reading for quarter past because the current hour happens to coincide with the prior reference hour (source), which the students should be reading. For quarter till, reading the current hour instead of the upcoming reference hour (goal) produces a detectable error: an hour reading that is one unit too low. Recall that in the lessons analyzed in chapter 5, the teacher did not provide any instruction for how to read the hour when introducing ‘quarter past’, nor did she call attention to the hour reading for naming the time as ‘X fifteen’. Because reading absolute time receives primary emphasis in early time-telling instruction and because learning to read the current hour correctly demands using the container schema in place of proximity, students who are successfully using the container schema to read the current hour are likely to continue to do so when reading the hour with ‘quarter past’. This overgeneralization error remains hidden until the students are asked to read ‘quarter till’. Meanwhile, less advanced students who persist in using the proximity schema for all clockreading actually produce the correct hour number for both ‘quarter past’ and ‘quarter till’ and thus appear more adept at relative time reading when, in fact, they have less conceptual understanding than their classmates. In both cases, the students are using inappropriate conceptual models (i.e. imposing the wrong image-schematic structure), but only the errors made by the students over-using the container schema stand out. Recall from the lessons analyzed in chapter 5 that the teacher did call attention to the hour reading when teaching “a quarter till seven” by pointing to the hour number (goal) that coincided with the future state when the long hand finished its sweep to the top of the clock—but the teacher’s only explicit instruction for how to read the hour occurred when teaching students to read the same clock state as “six forty-five.” In this case, the teacher was concerned that students read the current hour correctly for absolute time, using the container schema, and that they not rely upon proximity, which produces the most visible student errors. 30
Hutchins (personal communication) claims to be able to support this hypothesis with evidence of latent errors in airline pilots’ conceptual understanding of aircraft automation systems. Think about that the next time you fly.
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The impact of using the inappropriate image schema to read the hour is summarized in Table 6.2. Here, correct uses are marked “OK,” while conceptual errors are marked “hidden” or “ERROR” depending upon whether the error distorts the output. In this table, we see clearly that in the case of the systematic proximity error, three-quarters of the errors remain hidden, while in the case of overgeneralization of the container schema, half of the errors remain hidden. Under both conditions, when errors do surface in the output, they do so only when the long hand is on the left side of the clock, partially explaining the high preponderance of errors observed and reported for times in the second half of the hour. Finally, it is true that students could tell time successfully by using the container schema to read the current hour for absolute time and then still using the proximity schema to read the reference hour for relative time, so long as they always referenced the relative time to the nearest hour (i.e. read times as ‘past’ when the long hand was on the right side of the clock and as ‘till’ when the long hand was on the left side of the clock). Is this a hidden error? It could be argued that it is. First, it misses the conceptual linkage that motivates the meaning of “past” or “till.” More importantly, if this past/till convention were ever violated, such as reading the time as three quarters past the hour or forty minutes till the hour (which might be the case if one of the reference hours were made particularly salient), then proximity would produce the wrong reference hour reading. This sort of unusual situation could actually force attention to the fact that “past” and “till” reference the source and goal of motion, resulting in a reconceptualization of ‘past’ and ‘till’ times more generally. We see here an example of a deeper (and in this case, more correct) understanding of conceptual relationships occurring after a period of apparent successful performance. The reconceptualization results in no visible changes in performance when reading relative times, but it does afford other kinds of reasoning about relationships between the current time and reference times more generally, such as the times of appointments, departures, arrivals, etc., that do not coincide with the hour. Because conceptual errors can remain largely hidden during long periods of apparent success, lurking as latent errors beneath the cover of everyday circumstance, it may take an unusual situation to make such errors overt. Such a situation can lead to a breakdown in performance, necessitate reconceptualization for recovery, and provide the opportunity for that reconceptualization to change the understanding inherent in future performances of an activity thought to have been mastered long ago.
6.2.2 Mis-mapping Even when students use the appropriate conceptual model, they may still make errors in mapping from the conceptual model to the material artifact. An example of this type of error that occurred multiple times in different classrooms was a consistent mis-mapping of the counting model onto the clock face, which caused students to read the time ‘X:15’ as “X twenty.” Recall that counting involves coordinating a sequence of verbal labels with actions on a set of objects. Counting by five involves saying “five, ten, fifteen,” and so on, while touching (or pointing to,
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or fixating one’s gaze on) a series of objects in turn, each object being a set with five elements. To count successfully using touching or pointing, the counter needs to follow a path of motion that encompasses every to-be-counted object but touches each only once, and this is accomplished by imposing a source-path-goal image schema with one object as the source or starting point for the count, the remaining objects encompassed by the path of the count, and a final to-be-counted object as the goal or endpoint of the count. The verbal label that coincides with touching or pointing to the goal object becomes the final count or tally and corresponds to the quantity of items in the counted group. In the case of counting the minutes on the clock face to read an absolute time, the choices of starting point and path are highly conventionalized: start at the top of the clock and count the minutes clockwise, stopping the count when you reach the location pointed to by the long hand. This fixes the starting point (source) and path, but leaves the stopping point (goal) variable. How does this go wrong in clock-reading? The common mis-mapping in this case is actually a mis-coordination of the verbal labels with the actions on the clock face. Recall the clock-reading lessons involving counting that were examined in chapter 5 (sections 5.3.2 and 5.3.4). In these examples, the starting point or source of the counting motion was the top of the clock, where the long hand pointed at the 12, but the first label, “five,” was not uttered until the pointing finger (or longhand proxy) completed its jump to the 1, the first stopping point in the series of objects to be counted, which were the series of numbers on the clock face. Students have been observed to mis-coordinate their counts by saying “five” when they touch the starting point at the top of the clock, then continuing with “ten” when they touch the 1, and so on, arriving at a count of “twenty” when they touch the 3 (where the long hand is pointing), and thus reading the time ‘X:15’ as “X twenty.” In this type of error, students have the correct conceptual model and even have the correct counterparts for the mappings—the series of numbers on the clock face as the objects to be counted, sequential touching as the action of counting, and the location pointed to by the long hand as the goal or endpoint of the count—but they miscoordinate the verbal labels with the starting point of the count (the source of the counting path) so that when they arrive at the goal they produce a counting label that is one place too high in the verbal sequence, resulting in a time reading five minutes higher than the displayed time. The correctness of the conceptual model and its relation to counterparts on the clock face support a performance that seems correct to the performer, but a simple mis-mapping produces an error in the output.
6.2.3 Using an inappropriate procedure So far, the errors we have considered are conceptual errors: applying inappropriate conceptual models or mapping them incorrectly. Another, more subtle form of conceptual error is misunderstanding the conditions under which a particular procedure can be used and thus applying it inappropriately. This type of error appeared in 3rd-graders’ attempts to solve time problems that involved relating one time to another, such as finding the interval from the time displayed on the
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clock to the reference time for some future event (the time from now to time Y). 3rd-grade students had learned that they could use subtraction to find the minutes till the upcoming hour. Their procedure was to read the current time (say 10:42) and then to write a subtraction problem in which they subtracted the current number of minutes from 60, the total number of minutes in an hour. For our example, they would write a vertically aligned subtraction problem for 60 minus 42 and then solve the problem using borrowing (crossing out the 6 in the tens place and writing ‘5’ and changing the 0 in the ones place to ‘10’, then carrying out the subtraction of the ones and tens). This procedure produces a result of 18, the number of minutes remaining in the current hour, which is also the number of minutes till the next hour. In subsequent interviews, when the time of the upcoming event Y was moved past the hour, say to 11:15, many students persisted in applying the subtraction algorithm. They wrote the subtraction problem 15 minus 42, which they should not be able to solve since they have no experience with negative results. Some students stopped at this point, unsure how to proceed, while others borrowed from a non-existent hundreds place (rewriting 1 as ‘11’ even though there were no hundreds to borrow from) or just subtracted the smaller number from the larger (either just in the tens place or by rewriting the whole problem as 42 minus 15). This produced various results—73, 33, and 27 (the correct answer is 33—surprise! another hidden error). Students who misapplied the subtraction algorithm remained undisturbed by answers that made little sense (such as 15 minus 42 equals 73, a difference larger than 42 (larger even than the sum of 15 and 42) and larger than the number of minutes in an hour. The difficulty, of course, is that the students are applying place-10 algorithms to a base-60 minute counting system. A similar problem occurs for hour differences because of the modulo-12 hour counting system: the difference between 11 a.m. and 3 p.m. is 4 hours, but 3 minus 11 is not 4. Once again, under restricted conditions, these procedures do consistently produce correct results. The restrictive condition is that the interval between the current time and the reference time not cross an hour boundary. If the current time is 12:20 and the time of the future appointment is 12:45, then the subtraction problem 45 minus 20 produces the result 25, which is indeed the number of minutes till the appointed time. Because the place-10 algorithms do work within hour boundaries, a problem involving a time interval that crosses an hour boundary, such as the 10:42 to 11:15 problem mentioned earlier, could be solved by breaking it into two intervals, one from the current time to the upcoming hour (60 minus 42 is 18 minutes) and one from the hour to the reference time (quickly recognizable as another 15 minutes). These intermediate results can then be added to produce the full interval of 33 minutes. Constructing this type of solution depends upon recognizing the boundary problem and understanding the underlying relationships well enough to construct a procedure for working around it, something that appears to exceed the abilities of most of the 3rd-grade time-tellers interviewed. A more direct strategy for solving this type of problem is to use the clock face as a support and to count the interval from the current time to the appointed time (or vice versa), in effect visualizing the clock state for the future time and imposing a source-path-goal schema to count the number of tick marks in
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between the current and future positions of the long hand. Students who were weaker in their time-telling overall tended to rely singly upon the subtraction procedure, even across hour boundaries, while those with greater mastery of timetelling were more likely to vary their procedure depending upon the state of the clock, e.g. recognizing the interval from 2:50 to 3:20 (visualized on the clock) as a half hour—rather than breaking the problem into sub-problems, subtracting, and adding the results. Here we see the beginning of expert knowledge: being able to apply different strategies depending upon the situation and task, the particular times being compared, and the relevant affordances of the clock display. This is where procedural knowledge crosses into conceptual knowledge. Clearly, an important aspect of solving such problems is determining the applicability or non-applicability of particular algorithms. Determining applicability is more than a matter of simply recognizing a set of conditions (the “if” side of an “if/then” rule). It involves relating situational affordances, such as how the times being considered appear on the clock display, to underlying conceptual relationships, such as those relating times on different sides of an hour boundary. This interrelation of situational affordances with conceptual knowledge enables the construction of an ad hoc procedure like separating a time problem into two sub-problems, one on either side of the hour boundary, thereby creating a set of conditions under which a familiar algorithm can be applied, and then recognizing the conceptual relations that determine how the results should be combined to produce a solution.
6.2.4 Making a procedural or memory error Finally, there is another source of incorrect time readings that is not a conceptual error, namely procedural or memory errors such as miscounting (omitting a verbal label, missing an object, counting the same object twice) or making a basic arithmetic error (45 minus 23 is 12). Unless done systematically, these are simple mistakes rather than the kinds of errors that reflect fundamental conceptual misunderstandings. The first three forms of errors—using inappropriate conceptual models, mis-mapping, or applying procedures inappropriately—are the most critical when attempting to instantiate a local functional system to solve a time problem.
6.3 Discussion 6.3.1 The prevalence of hidden errors We attribute error when someone reports what we know to be an inaccurate result. This is error manifested in the output of a cognitive process. When such errors appear, we search for underlying causes, and sometimes, when the learner performs an action we know to be improperly timed, improperly executed, or simply inappropriate for the activity at hand, we believe we have found the cause of error. When we have no overt behavior to draw from in diagnosing error, we may ask the
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learner to talk aloud while performing the activity or to give a post-hoc explanation of the reasoning behind the result. The examples discussed above argue for a rethinking of what counts as error. Important conceptual errors occur not only when the output is inaccurate but also quite often—indeed much more often in our examples—when the output is entirely correct. These are what I call “hidden errors.” It is an error to apply an inappropriate conceptual model even when it happens to produce a correct result. Hidden errors may permeate much of the learning process and may indeed even continue after reasonable mastery appears to have been achieved. A likely source of such errors is overgeneralization of conceptual models and mappings that have already proven successful under other conditions. An example of this is when the pointing model used to read the hour for hour times (such as “four o’clock”) gets extended via the familiar association of pointing with proximity to the reading of the current hour for other times, producing apparently correct output for ‘three fifteen’ but incorrect output for ‘three forty-five’, mistakenly read as “four forty-five.” Both readings involve the same error, use of a proximity schema instead of a container schema, but in the first case, the error is hidden, while in the second, it is exposed. Examples such as these argue for the importance of testing performance under a variety of conditions (such as reading times during both the first and second half of the hour), of creating novel situations that test understanding of conceptual relationships (such as asking students to draw the clock hand positions for “a quarter till ten thirty”), and of using methods that open the reasoning process to scrutiny (such as think-aloud protocols and posthoc explanations—while being careful to treat such rationalized accounts, typically constructed after-the-fact to satisfy the perceived demands of the rhetorical situation, as additional data rather than as definitive). The examples also point out the importance of treating incorrect output as evidence of error rather than as the error itself, and of considering the possibility that the actual error may extend beyond the conditions that produced the overt manifestation. For example, to assume that reading times during the second half of the hour is harder than during the first based only on the higher rate of incorrect readings overlooks the possibility that the learner might be committing a systematic error that the states of the clock during the second half of the hour simply make detectable. Finding and eliminating any systematic but otherwise hidden errors would be an important step toward isolating sources of error that operate solely when the long hand is on the left side of the clock face. For this reason, slower reaction times rather than error in the output might be a better index of processing difficulty, while error in the output might provide better evidence of a potentially inappropriate conceptual model or mis-mapping that could extend to other conditions where its effects are undetectable.
6.3.2 The relationship between performance and understanding The use of an inappropriate conceptual model is a fundamental error because it results from ignorance of the conceptual relationships that motivate application of
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that model to the situation at hand. Using proximity to read the current hour for absolute time reflects a basic lack of understanding of what the current hour is, when it changes on the clock face, and, most importantly, why. This “why” gets at the relationship between the act of reading the time from a clock and the act of interpreting such a reading. We have learned that reading the time involves imposing image-schematic structure (part-whole) to perceive each relevant dial, applying the conceptual model for pointing (using center-periphery, source-path-goal, and extension schemas to structure a visual scan), and then either imposing imageschematic structure (proximity, container, or source-path-goal) or mapping from another conceptual model (e.g. a divided circle) to read the state of the dial. These choices relate to the form of the time reading, relative or absolute. So why, for example, does one use a source-path-goal schema to read the hour for relative time rather than a container or proximity schema? The answer has to do with the conceptual relationships involved in interpreting a clock reading. To say that it is “a quarter to six” means not only that the long hand has moved three quarters of the way from the 12 clockwise around to the 12 again and has one-quarter of the path left to traverse, and that once the long hand reaches this state the short hand will point to the 6. “A quarter to six” also means that a period of time corresponding to three quarters of an hour (or forty-five minutes) has passed since it was five o’clock and that another quarter of an hour (or fifteen minutes) will pass before it will be six o’clock. This understanding comes from relating the clock reading to the system of time measurement—days divided into 24 hours (two sets of 12 hours on the clock) with each hour divided into 60 minutes—and then relating this to the embodied sense for how long an hour, quarter-hour, or minute lasts. A novice clock reader may know to impose the source-path-goal schema to read “a quarter to six,” but to know why the source-path-goal schema should be used, and why one reads the goal and not the source, depends upon understanding the relationship between states of the clock and the system of time measurement and how these change as time passes. Interpreting a time reading like “a quarter to six” also involves additional conceptual relationships, those linking five and six o’clock to a portion of the day (morning or evening) and to the kinds of activities that commonly take place at such times. These relationships are a focus of instruction before the particulars of clock-reading are even explored. Clearly, conceptual understanding informs performance, especially with respect to the application of conceptual models and procedures under various conditions. Using rich conceptual understanding to motivate the application of particular strategies to the given circumstances (or, when possible, to change the circumstances to afford the use of a particular strategy) would be an apt characterization of expert performance. But in the case of novices, successful performance can precede understanding of the conceptual relationships that make the performance work. In fact, consistently successful performance, punctuated by occasional error, may provide the optimal conditions for such conceptual understanding to develop. Consider the simple act of reading ‘quarter past’, as depicted in Figure 6.4. When students are introduced to ‘quarter past’ in the 1st-grade time-telling lesson
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discussed in chapter 5 (section 5.3.1), they immediately practice reading several ‘quarter past’ times in a row. Interestingly, because all of these times begin with ‘quarter past’, the only actual clock-reading involved is the reading of the reference hour, which the students seem able to do with little effort (it is likely that they are using either the proximity or container schema, both of which are inappropriate but happen to produce the correct output, as they have yet to practice using the sourcepath-goal schema to read relative times). The teacher has created a set of conditions under which the students can perform successfully without actually having to read ‘quarter past’. Of course, nothing prevents them from doing so, and the teacher has just spent the last few minutes teaching them to see the upper right clock-quarter, which would encourage such a reading. By performing successfully under these conditions, what the students do minimally is to build up an association between the label “quarter past” and the position of the long hand pointing due right at the 3, as shown in Figure 6.4(a). Those who attended closely to the lesson, who learned to see the upper right clock-quarter via the conceptual blend the teacher constructed, and who are able to maintain this situated seeing during the practice that follows, will achieve something more: by performing successfully, they will reinforce an association between the label “quarter past” and the upper right clock-quarter, as shown in Figure 6.4(b). Maintaining this situated seeing during the structured practice increases the likelihood that this conceptual blend will be instantiated in the future when the same position of the long hand is encountered. The hope is that when students see the long hand in this position, they will do more than simply recall a label: they will recognize the long hand as bounding a region of space on the clock face (a container), relate this region to its counterparts via the part-whole schema, and thereby instantiate the full Clock Quarters blend and “see” the upper right clock quarter. Why bother to develop this conceptual blend when a simple association would do? Because the Clock Quarters blend affords inferences unavailable to students who rely on the simple association of “quarter past” with orientation or pointing at the 3. Examples of such inferences are that there is one quarter till the half hour, three quarters till the next hour, and that a quarter contains 15 minutes. The next step in the sequence of practice in the lesson mixed ‘quarter past’ times with other times familiar to the students, namely hour and half-hour times. In this case, students did indeed have to read the ‘quarter past’ portion of the reading in order to distinguish it from the other times, but they could continue to do so by relying upon the simple association with orientation or pointing at the 3. In fact, students who rely on this association could do so indefinitely and continue to produce the correct output when reading ‘quarter past’ times. What might cause these students to eventually conceptualize ‘quarter past’ in terms of the Clock Quarters blend? First, repeated instruction (guided conceptualization) by the teacher and by peers might get these students to see the Clock Quarters blend and then to make use of it. Second, instruction in ‘quarter till’ (the very next lesson) would afford the possibility of comparing the ‘quarter past’ and ‘quarter till’ readings and perhaps discovering a conceptual relationship, although this too might not occur until students had relied for some time on the independent associations of “quarter
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past” with the long hand pointing at 3 and “quarter till” with the long hand pointing at 9. Third, novel situations, such as the teacher claiming that 10:45 is “a quarter past ten thirty,” might trigger a reconceptualization of ‘quarter past’ that would then generate a new level of understanding for the more familiar reading. The blended space constructed by the teacher in the ‘quarter past’ lesson can be enriched further through the addition of motion, as shown in Figure 6.4(c). Indeed, this was exactly the case for the ‘quarter till’ lesson taught the very next day (and examined in section 5.3.3) when the addition of motion was necessary to generate the future space for reading the upcoming reference hour. This level of understanding incorporates the source-path-goal image schema. Here, “quarter past” refers to the position along a path of motion, expressed relative to the source or starting point of that motion as a reference point. At “quarter past,” the long hand has traveled one quarter of the path from the top of the previous hour (pointing at the 12) to the top of the next hour (also pointing at the 12). It is in fact the understanding of ‘quarter past’ in terms of motion along a path that motivates the correct image schema for the hour portion of the reading. Once ‘quarter past’ is conceptualized in terms of motion away from the 12 (the source), then it makes sense to read the hour in terms of motion away from its source: the previous reference hour (the previous “o’clock” time). Reading the long hand in terms of the motional source-path-goal schema while persisting in reading the short hand in terms of the static container (or even proximity) schema ignores the conceptual linkage between the two, which is manifested physically in the mechanical linkage of the hands by a geared assembly. The static part-whole conceptualization of ‘quarter past’ shown in Figure 6.4(b), which was used in the ‘quarter past’ lesson analyzed in chapter 5, served the teacher’s initial purpose of providing a conceptual basis or motivation for the expression “quarter past” while teaching students to perceive the quarter-hour dial in the clock face. It also turns out to be useful because it relates portions of the clock-circle to portions of the hour, providing a conceptual underpinning for perceiving time intervals on the clock. The teacher did not appear to be concerned about the effect of this static conceptualization on the reading of the reference hour, probably because she did not anticipate any errors in the output, any such errors being likely to remain hidden. Only when she introduced ‘quarter till’ in the next lesson did the teacher have to add motion to the conceptualization, forming the ‘quarter till’ counterpart to the conceptualization shown in Figure 6.4(c), because this motion provided the motivation or conceptual basis for reading the hour correctly: as the goal of motion. So which conceptualization is best? It would be a mistake to assume that the rich motion-based conceptualization shown in Figure 6.4(c) should be applied under all circumstances. A simple association like that in Figure 6.4(a) requires no mapping and is therefore economical and efficient in practice. The static conceptualization of Figure 6.4(b) is immediate, stable, and likewise somewhat economical: the clock face anchors the Clock Quarters blend in the present moment, supporting spatially-based inferences about temporal relationships without the need to construct a full integration network of past and future spaces. The motion-based conceptualization
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of Figure 6.4(c) is the most dynamic, bringing in alternate time spaces and affording mental simulation of clock motion to reason about the passage of time. None of these conceptualizations is wrong in any way, but they differ from one another in terms of the complexity of the conceptual integration networks involved and the kinds of inferences they afford. In any given moment of activity, the exact network constructed can be expected to depend upon task demands, available conceptual and material resources, social expectations, and simple ingrained habit. An expert would be someone who could make use of any or all of these conceptualizations to construct the optimal network for the situation at hand, instantiating a time-telling functional system of minimal complexity that produces an output which the expert believes with reasonable confidence to be correct. This confidence derives from the sense of appropriateness, coherence, and reliability of the conceptual and material resources employed. This analysis illustrates how conceptual understanding can change over time, even after a sustained period of correct performance. It also opens the way for a hypothesis: all things being equal, proficient performers will tend to rely on the simplest conceptualization that works for the task at hand, switching to more complex conceptualizations only when the situation demands it. I call this principle “pragmatic parsimony.” It may be countered by a second tendency, related to functional fixedness(Duncker 1945), in which performers tend to rely upon the conceptualization they have used most frequently in similar situations in the past, even when a simpler conceptualization might be available. The two principles are not mutually exclusive and are likely to interact, making it difficult to test these predictions experimentally.
6.4 Conclusion In a local functional system for time-telling, the artifact (a clock) contributes a set of constraints. These are realized in the material state of the clock face, whose changing state is governed by mechanisms that (when working properly) keep the set of constraints aligned with the system of time measurement anchored to the celestial event of the sun’s apex at noon over the Greenwich observatory. In instantiating a local functional system that operates on these clock constraints to construct a reading of the time, the time-teller bears responsibility for generating an appropriate sequence of actions that will result in a correct time reading in one of the conventional formats. This sequence of actions includes both physical actions (e.g. looking at a hand) and mental actions (e.g. imposing image-schematic structure and/or mapping from a conceptual model). The time-teller also bears the responsibility for making sure that these actions are well-founded, i.e. that they respect the relations that bind together the parts of a time reading and that connect such a reading to the broader system of time measurement and the world of human activity.
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No one is born knowing how to instantiate such a system and how to ensure its integrity. Developing the ability to do so depends upon a social process in which more-experienced others take responsibility for controlling the learner’s sequence of actions while provoking in the learner the experience of the conceptual operations involved, a process I have called “guided conceptualization.” By prompting, guiding, and anchoring the learner’s construction of meaning, a teacher makes it possible for the learner’s brain to adapt to the experience through synaptic change, making it more likely that similar conceptualizations could be generated in the future without the teacher’s prompting and, ultimately, without the same degree of material support. Ultimately, teachers cede to learners the responsibility for both controlling the sequence of actions and for ensuring the integrity of the conceptual operations involved, including the appropriateness of imposed image-schematic structures, the applicability of particular conceptual models, the correctness of the mappings, and the soundness of the inferences generated. This idealized process is not foolproof. Typically, errors arise. Learners confuse the order of actions, perform some that are inappropriate, or omit others that are necessary. Learners impose the wrong image-schematic structure and apply inappropriate conceptual models (or simply fail to apply the appropriate ones). They coordinate input spaces incorrectly and mis-map from one to another. They fail to make adequate use of material anchors to stabilize conceptual relations. They construct less useful or relevant blends, or construct useful ones but elaborate them inadequately, failing to arrive at the desired inferences. Even when the functional system appears to function and to produce the correct output, learners may be making fundamental errors whose effects appear only under certain conditions. In our most telling example, using the wrong image schema to read the hour produced noticeable distortions in the output in only 1 of 4 possible combinations of clock state and time-reporting format. Such errors may permeate performance for an extended period, especially when their observable effects are rare or minimal in impact. Viewed this way, few human performances would be expected to be entirely error-free. Changing conditions can unmask a hidden error that has long remained undetected, providing an opportunity—sometimes a necessity—for a change in conceptual understanding. Even when the functional system is entirely functional and when appropriate operations produce correct outputs, there remains the possibility for further conceptual change. Such change modifies the range of inferences available in performing the present activity and opens the way to instantiating functional systems for new, related activities. A change in conceptual understanding in a wellfunctioning system is likely to be triggered either by discovering relationships between new conceptualizations and familiar ones or by unusual circumstances in which the usual functional system fails somehow to support completion of the activity. Our example showed how the conceptual understanding of ‘quarter past’ can change through the realization of the conceptual relationship between ‘quarter past’ and ‘quarter till’ and through unusual situations where the expression ‘quarter past’ is used in an unconventional way. Changes in the conceptualization of ‘quarter
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past’ afford new inferences, such as those relating arcs on the clock-circle to intervals of time. These conceptual changes can occur after long periods of successful reading of ‘quarter past’ times, opening the way to new functional systems for solving other time problems. Experts should have multiple ways of conceptualizing each clock state and the ability to develop a conceptualization appropriate to the situation and task at hand and likely to generate the desired inferences. Finally, this view of the role of conceptualization in the instantiation of functional systems for performing cognitive activities provides the foundation for a hypothesis: humans tend to use the simplest conceptualization (i.e. the simplest conceptual integration network) that works for the task at hand, constructing more complex conceptualizations (more complex integration networks) when pressed to do so by the circumstances. This pragmatic parsimony hypothesis may conflict with functional fixedness. This remains to be tested.
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CONCLUSION: FUNCTIONAL SYSTEMS, G U I D E D C O N C E P T UA L I Z A T I O N , A N D T H E FUTURE OF COGNITIVE SCIENCE
7.1 Cognition as a dynamic interactive process Our journey through time-telling and time-telling instruction has touched on many themes in contemporary cognitive science: cognitive ecologies, cognitive artifacts, functional systems, conceptual mappings and conceptual integration networks, material anchors for conceptual blends, environmentally coupled gestures, situated activity, errors (both detectable and hidden), conceptual change, and the uniquely human activity of direct instruction combining the sequential control of action with guided conceptualization. By integrating distributed cognition with cognitive semantics, a picture of cognition as a dynamic interactive process begins to emerge. Cognition has at least a twofold dynamic.31 The first dynamic relates to the ‘cognition as computation’ approach that originated cognitive science as a field of study. A computational view is defensible only if the unit of analysis extends beyond the brain to include the material and social environment as well as how present cognitive activity incorporates products of previous activity (which may trace back over many generations) (Hutchins 1995, ch. 9). The study of such processes is the province of distributed cognition. The second dynamic relates to the process of meaning-making that is at the center of the human experience but that has been neglected in the cognition-as-computation approach. Meaning-making returns to center stage in the study of the processes of conceptualization, the province of cognitive semantics. Computation and conceptualization: one relates to how cognitive activities are accomplished by humans thinking and acting in the world, while the other relates to the meanings that humans construct as they engage in those activities. Explaining both is essential to a full understanding of human cognition. Such an understanding can only be achieved by examining the dynamics of how meaning is constructed in real instances of situated cognitive activity and by 31
A third dynamic, that of affect, is not discussed in this dissertation. Affect is likely to impact the two dynamics discussed here (perhaps the second more than the first) as well as playing its own unique role in modulating how the human organism responds to others in its environment. Affect has been left out of computational accounts of cognition, and its role in conceptual mapping remains greatly under-explored.
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constructing theoretical accounts that link observable actions to internal conceptual operations. From a computational point of view, we have seen that telling time involves instantiating a local functional system built around a specially constructed artifact, a clock. This computational system takes a given state of the clock as input and generates a time reading in a standard format as output, answering the question “What time is it?” by applying a set of constraints and executing a series of operations in a step-by-step fashion. The computation is actually accomplished by interacting with the material artifact in a particular way: sequentially directing attention to various artifact structures, instantiating relevant conceptual blends, and using these to construct a complete time reading. This process includes imposing image-schematic structure and mapping from familiar conceptual models while using the material structure of the clock face to anchor the blended spaces used to perform cognitive work. As the human participant in the functional system—the one who binds the system together, provides the impetus, and makes it function—the clock-reader is engaged in an active process of meaning construction: activating conceptual knowledge, building mental spaces and cross-space mappings, selectively projecting and integrating to form blended spaces at human scale, generating inferences, transferring inferences to target spaces that are the focus of discourse, and so on. These meaning-making operations—including mental simulation and affective responses that accompany it (anticipation, frustration, pleasure)—form the subjective experience of the person engaged in the computational activity of reading the time. Conceptual operations are the essence of meaning-making; they are how we make sense of the world. What is amazing is that humans are able to orchestrate these conceptual operations so systematically, in interaction with the world and one another, that they can instantiate local functional systems for generating sophisticated and dependable computational outcomes. Telling time is but one manifestation of this profound ability.
7.2 The lifespan of cognitive functional systems Where do cognitive functional systems come from? Historically, they develop within evolving cognitive ecologies as part of the cultural process. Human activities and the values placed on them, the tools and artifacts constructed to support those activities, the conceptual models developed by participating in them, the changing social organizations and relations among participants, and so on, all apply selective pressures that shape the interrelated web of artifacts, practices, and the meanings that accompany them. In contrast with a strictly biological ecology, a cognitive ecology changes both by local adaptation and by design (Hutchins 1995, ch. 8). Specialized artifacts are intentionally constructed to solve specific problems but then become opportunistically exploited in unforeseen ways, often by taking advantage of the conceptual relations they make apparent and stable through material anchoring of
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conceptual blends. As problems broaden in scope, local functional systems come into contact with one another, bringing their artifacts and practices into systems of coordination that lead to regularization and standardization across a more widely dispersed geography (quite literally in the move from local sundial time to global clock time zones). How do such functional systems continue across multiple generations of human participants? Material artifacts wear out but can be repaired or replaced by new ones. More problematic is the perpetuation of useful conceptual models and sequences of actions. Because humans die and new humans are born, these need to be renewed somehow in each new generation. A central process for this renewal is the peculiarly human activity of direct instruction. This kind of instruction is more than mere demonstration and imitation (although these are powerful sources of learning): it is a process of structuring the sequence of activity while guiding the learner’s conceptualization. Guided conceptualization creates experiences for learners to which their brains can adapt through neural change, making the recreation of such experiences more likely and eventually, with sufficient time and practice, enabling learners to guide their own actions and conceptual operations as they instantiate such functional systems in the future. Engaging in such instruction means applying valuable resources of the cultural group—expertise, tools, labor, and so on—that could be applied elsewhere for more immediate benefit; instruction is thus likely to focus on perpetuating functional systems that are highly valued by the cultural group.
7.3 The use of speech, gesture, and the material environment in guided conceptualization Guided conceptualization means using the available semiotic resources— speech, the body, and the world—to provoke in the learner the experience of constructing relevant meanings (conceptualizations) during the course of activity. Language (spoken or signed) clearly plays a crucial role here: activating conceptual models; building past, present, future, and counterfactual spaces; providing construals; profiling elements and relations; prompting mapping schemes; and so on. But language is only part of the picture. Gesture plays a key role, too, especially in tying conceptualization to the material world by enacting conceptual entities, highlighting material structures, and superimposing conceptual elements over their material anchors. This linking role of gesture can be understood only by examining simultaneously how gesture couples with the material environment and how it relates to co-occurring speech. What counts as information may emerge only in the interrelation of gesture, environment, and speech rather than in what is represented in each of these media alone. While the episodes analyzed in chapter 5 illustrate the importance of gesture in guiding the construction of meaning, they fail to exhaust the range of gesture’s possibilities, including interacting with implied objects not actually present and providing body-part surrogates for conceptual objects when
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other material anchors are not employed. The approach to meaning construction taken in this dissertation, rooted in conceptual integration theory and the role of material anchors, can accommodate these uses of gesture, but this remains to be elaborated in further research. Finally, the material environment itself, including artifacts specially constructed for the activity at hand and other structures that just happen to be present in the setting, is exploited to anchor conceptual elements in meaning-making, reasoning, and problem-solving. Such structures may be incorporated fleetingly in the dynamic flow of discourse (as gestures are), or they may be employed in a more sustained fashion when engaging in a multi-step computational operation. From this perspective, speech (sound waves propagated through the air), gesture (particular movements of the body, especially the face and hands), and the physical environment are all material structures that impact conceptualization, prompting, guiding, and anchoring conceptual processes.
7.4 The dynamics of learning How do learners co-participate in functional systems during instructional interactions, and how does their participation change over time? By attending to the teacher’s actions, gestures, and speech, following the teacher’s lead, and responding to the teacher’s prompts, learners join in the sequence of actions that leads to a functional outcome while simultaneously (it is hoped) engaging in the construction of meanings associated with those actions. In doing so, learners rely on teachers to provide the sequential organization of activity and to prompt and guide the relevant conceptual mappings and integrations until the learners, through experience, develop the conceptual resources to guide their own actions and meaning-making. These activities, too, unfold within a particular cognitive ecology. An individual learner may, for example, lack a conceptual model presupposed by the teacher and thus be unable to respond to the teacher’s prompting and guiding in the appropriate way. For that matter, a learner may simply attend differently, notice different things, activate different conceptual models, and construct entirely different conceptual blends that may not coincide with the goals of instruction. Even when appearing to perform the activity entirely on their own, learners may rely on conceptualizations that are less rich than those the teacher has attempted to construct, and they may come to rely on these simpler conceptualizations for some time before problem situations demand inferences that these conceptualizations fail to support. Learners may even rely upon conceptualizations that are conceptually inappropriate, failing to appreciate the underlying relationships that motivate their use or non-use, but which seem correct because they produce correct output under most conditions. These errors may remain hidden, becoming evident only under particular combinations of circumstances. Even after an extended period of apparently successful performance, the discovery of conceptual relationships that link conceptual models used in different aspects or phases of an activity can lead learners to deeper levels of conceptual understanding and thus
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greater flexibility in the instantiation of functional systems to solve new problems. As such expertise emerges, learners may still tend to rely on the simplest conceptualization that works for the task at hand, constructing more complex conceptualizations only when the situation warrants. Alternatively, they may tend to rely on the conceptualization used most frequently in similar situations in the past, even when a simpler conceptualization might be sufficient. Further research is needed to see whether ‘pragmatic parsimony’ or ‘functional fixity’ dominates or whether the two interact and under what conditions.
7.5 Material anchors in cognition and instruction Further research is also needed on the way that teachers (and others) create material structure and then interact with it in instruction. The perennial presence of chalkboards and whiteboards in classrooms—even when these rooms are outfitted with the latest technology for projecting prepared visual aids—provides evidence of the importance of such on-the-fly creations in instructional discourse. Teachers pick up chalk or a marker and create representations, gesturing over them and annotating them in close coordination with accompanying speech. This appears to be a process of guided conceptualization like that exhibited in the lesson episodes analyzed in chapter 5, so it should be amenable to similar methods of study and be able to be accounted for within the same conceptual framework. Such immediatelyconstructed representations also anchor conceptual elements and relations but have properties that distinguish them from durable artifacts like the teaching clock: they tend to be more schematic in nature, representing only pertinent elements and relations while omitting other details; they are created element-by-element in a process co-extensive with the unfolding discourse, rather than appearing on the scene as whole objects; and they lend themselves to annotation in ways that physical artifacts often do not (a teacher is unlikely, for example, to draw on the face of a teaching clock because this would mar the artifact for future use). Using the framework developed here to analyze teaching episodes involving the creation of material structure should help us to deepen our understanding of the relations between material supports and conceptual processes in cognition and to gain a greater appreciation for the role these interactions play in instruction. What about additions or deletions of supporting structure in the artifacts themselves? Sometimes additional supports for cognitive activity are built right into the structure of a common artifact, as in Figure 7.1. To the left is the simple clock face we have been considering. In the center is a diagram of the face of the first marine chronometer, a highly accurate timepiece famously used to find the longitude at sea. This timepiece has additional material structures not found on the simple clock face: a ring of small arabic numerals, distinguished from the hour numbers by both size and type, which provides support for reading the minutes; and arrowheads on the indicators which highlight their directionality and support application of the conceptual model of pointing. To the right in Figure 7.1 is a diagram of a learning
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artifact used in one of the 2nd-grade classrooms in the study: a modified clock face, about 4 inches in diameter, with manipulable hands that are not geared together. Each student had such an artifact and could manipulate the hands independently to set the clock face to times specified by the teacher. This artifact also has numbers for a minute scale, distinguished from the hour numbers by both size and color. Color is used to link each number scale to the appropriate clock hand (red for the hour hand and numbers, blue for the minute hand and numbers), supporting perception of the two most important dials embedded in the clock face (the quarterhour dial is not highlighted). Each clock hand is also labeled with the word ‘hour’ or ‘minute’, supporting connection of the embedded dials to the system of time measurement. These added structures—numbers, colors, and labels—all help the clock-reader make appropriate conceptual groupings and mappings. When such supports are not built into the structure of the artifact, teachers may attempt to add them, either as annotations directly on the artifact or, quite commonly, as annotations on a separately constructed diagram of the artifact (mentioned above as meriting further research). The opposite of adding material anchors is dropping them, as illustrated in Figure 7.2. Many watches, because of their small size or style (influenced by their role as a fashion accessory), dispense with the hour numbers or tick marks or both. Some even eliminate the circular shape, as shown in the depiction of a wristwatch at the bottom right of Figure 7.2. Ultimately, everything can be dropped from the clock face except the barest structures needed to anchor a time reading: two indicators of distinguishable length moving in coordination with the system of time measurement, and some type of landmark to orient the clock face. In the wristwatch shown in Figure 7.2, the orientation is provided by the diamond at the top where the ‘12’ would normally appear, this being the most salient reference point in timetelling. The orientation is also maintained by the way the watch is worn on the wrist (with the set knob to the right) and by how the wrist is held in front of the eyes during time-telling, making even the diamond landmark somewhat redundant. How is it possible to read the time correctly with so little support? The answer to this question has multiple parts, all relating to the extensive experience of the time-teller. Part of the answer is that familiar structures can be filled in from imagination when their physical counterparts are not present, so long as they can be anchored by the structures that are there, just as the Micronesian navigator discussed in chapter 2 can anchor a sidereal compass in imagination by fixing one corner of it to the position of the rising sun. Part of the answer relates to the multiplicity of conceptualizations available to the experienced time-teller. These conceptualizations reinforce one another and provide flexibility in time-reading, making it possible to adapt to unusual displays. For example, the little hand pointing to the right, pointing to an imagined 3, pointing to the third tick mark clockwise from the top, pointing 90 degrees, and so on, are all associated with a reading of “three o’clock.” And part of the answer (especially relevant to reading a minimalist display like the wristwatch in Figure 7.2) is a shift from controlled processes like counting to more automatic processes like pattern recognition, in this case recognizing positions of the individual clock hands
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that normally align with major tick marks or, for salient times like the hour or halfhour, recognizing the entire configuration of the clock hands as a distinct pattern. The recognition of such patterns is aided by the tremendous reduction in possible patterns resulting from the mechanical gearing of the clock hands (e.g., producing just 12 patterns for the salient “o’clock” times) and the frequent recurrence of these patterns—twice a day, every day, at regular intervals—in the many clock faces that surround us. From these considerations, it follows that one way of testing developing expertise is to remove material supports and to see whether the performer can still accomplish the activity. This seems to happen quite naturally when moving from a specialized teaching artifact like the color-coded clock face, depicted at the right of Figure 7.1, to the artifact normally encountered in the environment, a classroom clock with a simple face like the one depicted at the left of Figure 7.1. But it can also result from intentional manipulation by the teacher, e.g. creating clock face diagrams with selectively deleted elements and seeing how these deletions impact time-telling performance. Dropping anchors disrupts the stabilization provided by the material supports. Somewhat experienced learners can compensate for this by adding their own material supports (e.g. writing in the numbers), filling in structure from imagination (such as a number normally located at a particular position), and/or shifting strategies (e.g. shifting to a backup counting strategy to identify the tick mark being pointed to when no number is present). Novices have less experience with the artifact, making it harder to fill in structure from imagination. They also have less well-developed and ingrained conceptual models, making it harder to maintain important conceptual relations without the stabilization provided by material anchors. Finally, novices are likely to rely on a single familiar strategy and be less able to switch to another strategy when the circumstances no longer support the familiar one.
7.6 Material artifacts, conceptual models, and mental imagery In our discussion of how material artifacts and highly entrenched conceptual models stabilize cognitive processes, we should take a moment to highlight an important difference between the two. A material structure is a physical entity with a fixed form (although it may have moving parts). A conceptual model has no fixed form, although it is a structuring resource. A conceptual model has topology, relating distinct elements, but its form becomes fixed only when it is used to structure a particular conceptualization. This difference is key to understanding the interplay of conceptual processes with the material world: the conceptual model provides the relations among elements which we use to make sense of the world, while the material world (through perception) fixes the form of the conceptualization and maintains it through time so we can selectively alter our attention, as we engage in a process of reasoning or computation, without losing the relations we are reasoning or computing over. Engaging in such a process may involve physically
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interacting with the artifact (pointing from one location to another, tracing a path, or manipulating the artifact), shifting gaze (looking at structures or locations in sequence or scanning along a path), covertly shifting attention, or engaging in mental simulation. Once conceptual elements have been mapped to material anchors, conceptual relationships can be manipulated by physically manipulating the material anchors associated with conceptual elements. When the pointing stick is laid across the clock face, for example, it superimposes a conceptual dividing line over the clock circle. Layering material anchors layers conceptual constraints, which is a way of physically enacting a computational process. Consider the problem of positionfixing in ship navigation (Hutchins 1995). A navigation chart has many layers of material anchors for conceptual elements already built into its physical structure. These are the residue of past cognitive activities. In addition, lines of position serve as material anchors for the conceptual representation of the ship’s direction relative to some known landmark. Drawing lines of position on the navigation chart layers these material anchors and their associated conceptual elements, which are constraints on the possible position of the ship. This layering of constraints produces a computational output: a fix of the ship’s current position. When the material world cannot be drawn upon to anchor a conceptualization, it may be replaced (with greater or lesser degrees of success) by a vivid mental image of a familiar environmental structure. For this reason we can use the method of loci to remember a sequence of ideas associated with landmarks along a familiar path by merely walking the path in imagination rather than actually walking the path in the physical world. Whether the anchoring is provided by the physical world or by visualization, it still occurs in a perceptual space that is an input to a conceptual integration network. In the first case, the perceptual space is driven primarily by sensory data from the physical world, while in the second, it is driven primarily by activity within the brain reactivating sensorimotor areas. Brain imaging studies show this kind of activation when subjects engage in imagined experience, supporting such an account (see, e.g., Barsalou 1999for a discussion of this evidence and how it relates to theories of concepts). Short of actual hallucination, however, mental imagery is less vivid than sensory-driven perception, as evidenced by lower levels of activation in primary sensory areas of the brain. Perhaps more importantly, because of its lack of connection to the immediate physical environment, mental imagery is also much less persistent (that is to say, it is not continuously renewed by further perception). This makes it difficult to manipulate imagistic anchors or to layer them on top of one another. Nevertheless, mental imagery of highly familiar objects and environments may have sufficient detail and immediacy to anchor a step at hand in many cognitive activities. This discussion of the possible role of mental imagery in anchoring conceptual operations, although emphasizing internal mental processes, is still fully in keeping with a distributed view of cognition. Cognition is distributed not only across the individual and aspects of the material and social world; it is also distributed within the individual in the interaction of different areas of the brain. Taking a systemslevel view of cognition makes it possible to examine processes of anchored
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conceptualization within the same theoretical framework whether the stabilization results from sensory perception of the external world or from reactivation of sensorimotor areas within the brain (a process made possible only by previous experience interacting with the physical world). It is, in a sense, an ecumenical approach to the study of cognition.
7.7 The internal/external boundary and the future of cognitive science This ecumenical view has other implications for the study of cognition as a dynamic interactive process. If cognitive activities are accomplished through the instantiation of functional systems, then it ought to be possible for a system to continue to be functional—working from available inputs to produce desired outputs—while changes occur within the system that modify structures both internal and external to the individual and thus change the relations between them. Within the brain, processes of adaptation form new connections that increase the likelihood of reproducing a sequence of words, meanings, or actions (see, e.g., Hutchins 1995, ch. 7), instantiating relevant conceptual models, or producing mental imagery of familiar objects and events. This creates a sense of internalization, even though nothing actually moves from the world into the head. The counterpart process, every bit as important, is creating or modifying physical structures in the world to support cognitive activities, including constructing material anchors for conceptual elements and relations used in processes of reasoning and problem-solving. This might seem like externalization, but it is less a matter of moving ideas from the head into the world than it is a matter of modifying the world to support the smooth unfolding of cognitive processes, bringing the internal and external into alignment. The idea is to keep the functional system functioning while improving its performance, robustness, and efficiency. One way to do this, from a computational point of view, is to build computational relations right into the structure of the artifact, locking out many possible errors and changing the actor’s role within the functional system from mental computation to physical manipulation and perception of the output (Hutchins 1995). This is a powerful strategy, but a more pervasive strategy might be to support the performance of conceptual operations by providing material anchors for conceptual elements and using material structures to fix the form of conceptual relations. The two are clearly related and may often go hand-inhand. Meanwhile, the experience of interacting with the world builds neural assemblies whose activity provides new structuring resources (conceptual models), stabilizing resources (mental imagery of highly familiar objects or environments), and processing abilities (automatization through pattern recognition and association, including associations that guide sequences of actions). These strengthen functional systems while reducing the dependence on material supports for successful performance (though not, in the case of time-telling, ever eliminating them entirely).
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From a computational point of view, the internal/external boundary is irrelevant, so long as the functional system takes the relevant inputs and produces appropriate outputs. From a cognitive science point of view, the boundary is interesting—but not as any kind of limit on what counts as a legitimate object of study. Rather it is interesting because we want to understand what changes occur on each side of the boundary and how these relate to one another as cognitive functional systems evolve with repeated instantiation or adapt to changing circumstances. The ecological study of cognition as a dynamic and evolving process spanning parts of an organism, other organisms, and the physical environment, should be the mission of our research. A major goal of this research should be the delineation of phenomena that interconnect internal and external structures in cognitive processes. This dissertation has undertaken such a project. Telling time is a single domain of everyday cognitive activity, albeit an important one. Will the approach taken here work for other domains, including those with sophisticated technology and complex social distributions of activity? Are the phenomena of guided conceptualization—particularly the use of gestures and cogesture speech to guide conceptual mappings, the use of material structures to anchor conceptual elements, and the construction and elaboration of blended spaces to generate inferences—common to instruction in quite different domains and settings? How do these phenomena compare in instructional vs. non-instructional interactions? As is often the case, the present dissertation spawns more questions than it answers, opening the way to future research.
APPENDIX: FIGURES, TABLES, AND TRANSCRIPTS
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