ZDM Mathematics Education (2011) 43:617–620 DOI 10.1007/s11858-011-0369-7
EDITORIAL
Learning, teaching, and using measurement: introduction to the issue John P. Smith III • Marja van den Heuvel-Panhuizen Anne R. Teppo
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Accepted: 13 September 2011 / Published online: 24 September 2011 Ó FIZ Karlsruhe 2011
This issue presents a collection of empirical research reports that have examined different aspects of the learning, teaching, and use of measurement. The work reported addresses measurement as an important domain of school mathematics, including vocational education, and measurement in use in various occupations and workplaces. The collection is diverse in many ways, as characterized below. Though the focus of many articles is the measurement of space (length, area, or volume), attention is also given in some to non-spatial quantities such as time, weight, and money. The appearance of this issue in ZDM reflects the concern felt in many countries that measurement is an important elementary mathematical and scientific competence, but one that—as evidence considered below suggests—appears to be poorly learned. Weak learning of measurement—particularly of the conceptual principles that underlie measurement procedures—undermines students’ ability to learn and understand more advanced mathematical and scientific content and hence their access to important kinds of skilled work—both professional and not. The research reported in this issue will not solve that problem. Instead, the issue targets a more modest goal: That more researchers across the globe J. P. Smith III (&) Michigan State University, 509C Erickson Hall, East Lansing, MI 48824, USA e-mail:
[email protected] M. van den Heuvel-Panhuizen Utrecht University, Freudenthal Institute, PO Box 85.170, 3508 AD Utrecht, The Netherlands e-mail:
[email protected] A. R. Teppo PO Box 570, Livingston, MT 59047, USA e-mail:
[email protected]
will reconsider the importance of measurement (in school and out), its place in elementary mathematics, and the need to pursue research that will produce partial answers the basic question, ‘‘why are we doing so poorly teaching and learning measurement?’’ We hope these partial answers, as they assemble, will help curriculum developers design more potent materials, teachers teach the measurement content more effectively, and assessment professionals develop more revealing assessments of learning. In this introduction, we seek to orient the reader to the collected articles in two ways. First, we briefly review some of the issues that make measurement ‘‘basic and fundamental’’ content in mathematics and science, in order to orient and frame the inquiries reported in the articles. We also identify some of the principal themes pursued and central results reported in the articles. While this overview is approximate, leaving out important messages particular to individual articles, it is offered to the reader as a partial ‘‘roadmap’’ to the issue—and as motivation to explore further.
1 Why is measurement important content in elementary mathematics? Measurement, the coordination of continuous quantity and number, has a long-standing and important place in mathematics. Spatial measurement, the coordination of space and number, dates from human kind’s initial efforts to understand and master the physical world (Lehrer, 2003). Though many countries expect elementary mathematics curriculum and teaching to support students’ measurement learning, typically beginning with length, measurement is even more fundamental to the content and practice of science (Michaels, Shouse, & Schweingruber,
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2008). In both mathematics and science, the measurement of tangible and directly experienced quantities (e.g., mass/ weight, length, volume/capacity) leads quickly to the measurement and study of composite quantities in both mathematics (e.g., speed as a rate) and science (e.g., density, force). For these reasons, measurement is a foundational competence for all fields and careers that build upon mathematical and scientific knowledge. Also, measurement may be unique in its inherent meaningfulness among mathematical topics. Where the hallmark of mathematics is the abstraction of structure from patterns observed in the physical and social world (Steen, 1990), measurement— particularly spatial measurement—remains strongly connected to the measurer’s physical world. For that reason, measurement is among the most sensible, contextually situated, and practical domains of mathematics for students. But as a growing body of research has indicated (Baturo & Nason, 1996; Chappell & Thompson, 1999; Clements & Bright, 2003; Irwin, Vistro-Yu, & Ell, 2004; Zacharos, 2006), measurement is poorly learned in school classrooms in many countries. There is also evidence that elementary teachers who are responsible for guiding students’ measurement learning struggle with shallow understanding themselves (Menon, 1998; Simon & Blume, 1994). The interpretations offered by researchers examining these results share a common theme: Students perform relatively well on well-practiced tasks and very poorly on tasks that are conceptually simple but frame measurement in nonstandard ways. Students around the world, it seems, have been successful learning the standard procedures of measurement (the typical focus of classroom instruction) without learning the conceptual principles that stand behind and justify those procedures (Irwin, Vistro-Yu, & Ell, 2004; Stephan & Clements, 2003). Factors related to curricular content and teachers’ knowledge and instructional practices have been suggested as root causes (Menon, 1998). In at least some countries, this pattern of poor learning is also the result of less classroom attention to the measurement of continuous quantities than to developing students’ understanding of base-10 number and arithmetic operations. Weaker attention to measurement is more than a difference in instructional time and textbook pages. Number is built on the foundation of counting discrete quantities—collections of objects that are physically separated. By contrast, measurement addresses questions of ‘‘how much?’’ and ‘‘how much more?’’ for initially continuous quantities—typically length, time, and volume/capacity in the early grades. In measurement, a continuous quantity becomes discrete when we choose a suitable unit and iterate that unit to determine the number of copies required to exhaust the quantity. Its measure is the count of those units.
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The predominant attention to discrete quantities, counting, and number, place value, and the arithmetic of whole numbers over the measurement of continuous quantities appears to contribute to students’ challenges in learning measurement. Not only do they suffer from limited time on measurement tasks, but they also apply a discrete quantity orientation to measurement tools and representations of space, e.g., counting marks on rulers to determine the lengths of objects and counting dots on grid paper to determine the area of shapes (Blume, Galindo & Walcott, 2007; Kamii & Kysh, 2006; Nunes & Bryant, 1996). Moreover, they are also ill-prepared to draw on continuous quantities, especially spatial quantities, when more advanced mathematical topics, such as multiplication and division of rational numbers, call for them. Similarly, the coordination of discrete and continuous quantity necessary to achieve a deep understanding of calculus is difficult to navigate when students’ knowledge of continuous quantities and measurement is weak. When students move into skilled work or technical careers, measurement remains ubiquitous, but often removed from view and direct physical activity. Technological interfaces and systems produce measures of quantities that are central to work enterprises, but in ways that introduce many layers of intermediate process and computation between the workers and the measured quantity. The embedding of measurement processes in complex technological systems raises the importance of measurement knowledge in workers who use these systems and at the same time creates substantial challenges for workers’ interpretation of the measurements these systems produce.
2 What aspects of teaching, learning, and using measurement are explored in the issue? The ten articles that comprise this issue collectively represent a diverse set of theoretical and methodological approaches, phenomena under study (e.g., curriculum, assessment, student thinking, teaching, workplace practices), national contexts, and central themes and results. There are large-scale empirical analyses of student achievement, intensive qualitative analyses of the thinking of small numbers of students, and experiments that compare student outcomes from different instructional experiences. There are analyses of the written curriculum and classroom-based studies that focus on the ‘‘enacted’’ curriculum—how the teaching of measurement actually unfolds. And there are two studies that replace classrooms with workplaces to examine how measurement practices are carried out in support of productive work. The articles are ordered roughly by the age of the learners reported in
Learning, teaching, and using measurement
the articles, from the beginning to the end of schooling and then into the workforce. But among these different analyses, some commonalities are evident. These are not the only meaningful connections that can be drawn between two or more articles, but they are worthy of note. Assessing achievement and characterizing measurement knowledge Given the pattern of poor achievement and learning around the globe, it is not surprising that researchers have looked more carefully at what is learned (and not) and when it is learned. The Marja van den Heuvel-Panhuizen and Iliada Elia article, for example, reports on the particular competencies of length measurement mastered by kindergartners and on the efficacy of reading them picture books highlighting length measurement to develop these competences. They also investigated the components of these children’s length measurement performance. Research to assess measurement achievement requires the analysis of different categorizations of measurement knowledge and items to assess that knowledge, either prior to or after assessment. Such analysis is also shown in the articles authored by Jasmin Hanninghofer and colleagues and by KoSze Lee and John Smith. Hanninghofer and colleagues’ analysis of German primary school students’ responses to measurement items led them to replace a standards-based items classification with a more effective distinction between instrumental knowledge and measurement sense. Lee and Smith applied the tripartite distinction between conceptual, procedural, and conventional knowledge to the textual treatment of length measurement in two countries, revealing a strong emphasis on measurement procedures in both. Measurement in relation to other mathematical domains In most elementary mathematics textbooks, measurement is a specific topic area, taught beside but in less depth than base-10 number and arithmetic. This view follows and may strengthen the separation between discrete and continuous quantities outlined above. An alternative approach sees measurement of continuous quantities as a suitable, even desirable entry point into other mathematical topics areas, including number and operations. Richard Lehrer and colleagues show how length measurement can serve as an entry point for meaningful work in statistics that focuses on the key constructs of variation and measures of variation. The work reported by Jeffrey Barrett and colleagues illustrates how the comparison of spatial quantities can support the development of notions of ratio and eventually algebraic reasoning. Arthur Bakker and colleagues’ analysis of measurement competencies in a variety of skilled occupations also reveals an intimate relation between measurement and knowledge and reasoning in other mathematical domains.
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Learning trajectories for measurement As the collected international body of research on students’ achievements and struggles in learning measurement continues to grow, in number and depth, some researchers have begun to develop multi-year learning pathways (or trajectories) that characterize the nature of students’ progress from initial to more sophisticated understandings (Sarama & Clements, 2009; Van den Heuvel-Panhuizen & Buys, 2008). Julie Sarama and colleagues’ article and the Barrett et al. article report such learning trajectories, provide empirical support for them, and show how these trajectories have been revised to accommodate empirical testing. Sarama et al. discuss length measurement in the early years; Barrett et al. investigate length and area measurement across the elementary years. While many studies (including some in this issue) target measurement of a single quantity, the Barrett et al. article provides evidence for an educational approach that attends to common conceptual issues that appear across quantities. Estimation in measurement Curriculum and instruction in measurement typically focus on the use of physical and computational tools to produce relatively exact measures. But many everyday situations where measurement is needed and/or useful either require or naturally call for estimation processes that produce rough and inexact measures—suitable to the measurer’s needs. Two articles explore the nature of estimation processes in length measurement. Kuo-Liang Chang and colleagues’ article shows how the presentation of length estimation tasks in US textbooks structure some parts of the estimation process for students, while leaving others open to teachers’ and students’ interpretation and decision. The partially open character of estimation tasks creates both opportunities and potential pitfalls for learning. Zahra Gooya and colleagues’ study of Iranian secondary students’ methods for estimating lengths in familiar physical contexts shows the individual and inventive character of their estimates that combine the visual application of personal units with features of the physical situation. Measurement in the workplace Measurement, especially the measurement of space, is a common component of work in many skilled, non-professional occupations (Millroy, 1992; Masingila, 1994; Smith, 2002). Educators cite use in the world as a principal motivation for learning to measure and understanding measurement. But as both articles in this issue that address measurement at work show clearly, the measurement skills that are taught and valued in school are quite different from those used and valued in many workplaces. One common thread reported in the articles from Philip Kent and colleagues and Arthur Bakker and colleagues is that measurement at work is mediated by digital technologies that separate the measurer
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and the measured material much more so than in schools. The Bakker et al. article shows the demand for measurement can be uneven across occupational categories and how complex it can be when it appears. The case studies reported by Kent and colleagues clearly show that measurement is valued as one, but only one component in workplace decisions that aim to optimize production processes. That is, measurement is an activity that appears in much more complex reasoning and problem solving contexts than is usually the case in school.
References Baturo, A., & Nason, R. (1996). Student teachers’ subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31, 235–268. Blume, G. W., Galindo, E., & Walcott, C. (2007). Performance in measurement and geometry from the perspective of the principles and standards for school mathematics. In P. Kloosterman & F. K. Lester (Eds.), Results and interpretations of the 2003 Mathematics Assessment of the National Assessment of Educational Progress (pp. 95–138). Reston, VA: National Council of Teachers of Mathematics. Chappell, M. F., & Thompson, D. R. (1999). Perimeter or area? Which measure is it? Mathematics Teaching in the Middle School, 5, 20–23. Clements, D. H., & Bright, G. (2003). Learning and teaching measurement: 2003 Yearbook. Reston, VA: National Council of Teachers of Mathematics. Irwin, K. C., Vistro-Yu, C. P., & Ell, F. R. (2004). Understanding linear measurement: A comparison of Filipino and New Zealand children. Mathematics Education Research Journal, 16, 3–24. Kamii, C., & Kysh, J. (2006). The difficulty of ‘‘Length 9 width’’: Is a square the unit of measurement? The Journal of Mathematical Behavior, 25, 105–115. Lehrer, R. (2003). Developing understanding of measurement. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research
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J. P. Smith III et al. companion to principles and standards for school mathematics (pp. 179–192). Reston, VA: National Council of Teachers of Mathematics. Masingila, J. O. (1994). Mathematics practice in carpet laying. Anthropology and Education Quarterly, 25, 430–462. Menon, R. (1998). Preservice teachers’ understanding of perimeter and area. School Science and Mathematics, 98, 361–367. Michaels, S., Shouse, A. W., & Schweingruber, H. A. (2008). Ready, set, SCIENCE!: Putting research to work in K-8 science classrooms. Washington, DC: National Academy Press. Millroy, W. L. (1992). An ethnographic study of the mathematical ideas of a group of carpenters. Journal for Research in Mathematics Education, monograph #5. Reston, VA: National Council of Teachers of Mathematics. Nunes, T., & Bryant, P. (1996). Children doing mathematics. Malden, MA: Blackwell Publishers. Sarama, J., & Clements, D. H. (2009). Early childhood mathematics education research: Learning trajectories for young children. New York: Routledge. Simon, M. A., & Blume, G. W. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25, 472–494. Smith, J. P., III (2002). Everyday mathematical activity in automobile production work. In M. E. Brenner & J. N. Moschkovich (Eds.), Everyday and academic mathematics in the classroom (Journal for Research in Mathematics Education Monograph 11) (pp. 111–130). Reston, VA: National Council of Teachers of Mathematics. Steen, L. A. (Ed.). (1990). On the shoulders of giants: New approaches to numeracy. Washington, DC: National Academy Press. Stephan, M., & Clements, D. (2003). Linear and area measurement in prekindergarten to grade 2. In D. Clements & G. Bright (Eds.), Learning and teaching measurement: 2003 Yearbook (pp. 3–16). Reston, VA: National Council of Teachers of Mathematics. Van den Heuvel-Panhuizen, M., & Buys, K. (Eds.). (2008). Young children learn measurement and geometry. Rotterdam: Sense Publishers. Zacharos, K. (2006). Prevailing educational practices for area measurement and students’ failure in measuring areas. Journal of Mathematical Behavior, 25, 224–239.
