REPRESENTATIONAL TRANSFORMATIONS IN PROBLEM-SOLVING

Session F2G TEXT TO DIAGRAM TO SYMBOL: REPRESENTATIONAL TRANSFORMATIONS IN PROBLEM-SOLVING W. Michael McCracken1 , Wendy C. Newstetter 2

Abstract Central to engineering problem solving is what we call representational transformation. Such transformations are built upon community-sanctioned practices often referred to as “back of the envelope” calculations. First a problem statement (text) is translated into a sketch (diagram) which visually articulates the essential problem parts. Mechanical models and free-body diagrams are instances of this first transformation. The qualitative model is then transformed into a set of mathematical formulae (symbols), which drive the problem solution. Thus, the problem is solved using three types of representational systems: textual, diagrammatic and symbolic. At each step the engineer translates information from one representational system to another, enacting an abstract cultural algorithm. The knowledge necessary to undertake these transformations is described in this paper in the context of multi-literacies. We propose that a large part of learning engineering problem solving is in fact learning the relationships between the multiple languages of problem solving. Index Terms education, learning, literacy, problemsolving,

three types of representational systems: textual, diagrammatic, symbolic (TDS). As shown in Figure 1, at each step in the TDS sequence, the engineer translates information from one representational system to another.

Problem Prompt T

Transformation 1 D

Transformation 2 S

Textual Account The problem statement made up of words and sentences delivered in written or oral form

Diagrammatic Account A visual diagram or schematic of the essential problem elements as they relate to each other. “Body of interest”

Symbolic Account Mathematical equations

REPRESENTATIONAL TRANSFORMATIONS IN PROBLEM SOLVING A common techniques employed in engineering problem solving is a practice referred to as “back of the envelope” calculations. Engineers of all types use this technique to gain insight into the dominating or critical issues within a complex problem. Once identified, the engineer can perform more detailed analysis to drive the problem solution. As cognitive scientists, we are concerned with the complex cognitive strategies required to undertake such calculations. When we look at this problem solving practice from the outside, what we see is data being translated from one symbol system to another in the enactment of an abstract cultural algorithmic process. First a problem statement (Text -T) is translated into a sketch (Diagram-D) which visually articulates the essential problem parts. Block and free-body diagrams are instances of this first transformation. This diagrammatic or qualitative model is then transformed into a set of mathematical formulae (Symbol-S), which drive the problem solution. Thus, the problem is solved using 1 2

FIGURE 1 The TDS Sequence To Back-Of-Envelope Problem Solving

The knowledge necessary to undertake these representational transformations is central to engineering practice. This paper explores the cognitive issues in these transformations and their implications for engineering classrooms. We are wondering where and how students learn to “see” into a problem using these three semiotic systems. Are these systems explicitly taught or are they just picked up over time? We are concerned with where students learn to transform texts to diagrams, to identity problem parts inscribed in the diagram and to appropriately annotate the visual model using mathematical inscriptions. Such transformations rely on manipulations of both internal and external representations, which are strongly bound in experts. In novices, these relationships are tenuous and often

W. Michael McCracken, Georgia Institute of Technology, College of Computing, Atlanta, GA 30332-0280, [email protected] Wendy C. Newstetter, Georgia Institute of Technology, School of Biomedical Engineering, Atlanta, GA 30332, [email protected]

0-7803-6669-7/01/$10.00 © 2001 IEEE October 10 - 13, 2001 Reno, NV 31 st ASEE/IEEE Frontiers in Education Conference F2G-13

Session F2G non-existent. It seems that engineering educators attempt to build these relationships in problem presentations by drawing sketches on the board to help the students visualize the elements of the problem and to relate the analysis steps to the sketch. Textbooks follow the same strategy and provide students with the visual models, which they use to get to the next step. But is this the right strategy? Should this first transformative move from text to qualitative diagram be taken for granted while putting all emphasis on the third transformation, from diagram to formulae? In this paper, we attempt to provide a framework for thinking about teaching and learning back-of-the envelope calculations. We begin by proposing literacy as critical to engineering learning. We show how multiple literacies are required to do engineering problem solving. We interrogate each move in the TDS sequence and identify the kinds of cognitive skills and knowledge required to undertake it. We then examine whether practices in a mechanical engineering classroom support the development of such forms of transformative expertise.

LITERACY DEMANDS IN ENGINEERING PROBLEM SOLVING We propose that engineering problem solving demands multiple forms of literacy. By taking this stand, we embrace an expanded notion of literacy that has been proposed by recent literacy researchers. What counts as literacy in any group is visible in the actions members take, what they are oriented towards, for what they hold each other accountable, what they accept or reject as preferred responses of others, visual language and how they engage with, interpret and construct text.”[1] The representations of engineering problem solving--textual, diagrammatic and symbolic-- each represent a form of literate action dependent on the ability to both comprehend and generate a variety of semiotic 3 forms. And each semiotic form---the textual, the visual and the symbolic--has its own grammatical conventions and semantic base. For example, in schematic diagrams of electrical engineering there are standard visual components like inductors that make up the “grammatical” base of the visual language. There is a syntax such that certain elements can collocate and others cannot. Each of these elements has a denoted meaning which community members all understand. In that sense, the visual account of a problem in back-of the envelope calculations represents a semiotic system in its own right. When engineering students are learning to solve problem in thermodynamics or digital signal processing, they are simultaneously learning the literate practices that community members act with and are oriented towards. Literacy researchers have begun to look at the literacy 3

demands of science and math curricula and have found they rely on the use of multiple literacies. Lemke characterizes these demands as "the ability to use the complex representational apparatus of scientific reasoning, calculation and practice. Professional science, today as for the past few centuries, makes extravagant use of not only a technical verbal language, but also of mathematical, graphical, diagrammatic, pictorial, and a host of other modalities of representation" [2] Moreover O’Halloran in analyzing the literacy demands in mathematics asserts that "at times these codes [symbols, graphs and charts, and natural language] alternate as the primary resource for meaning, and also interact with each other to construct meaning” [3]. For a novice, the construction of meaning from multiple languages is the confound. Unwittingly science and math education often ask students to learn more than one language at a time, the equivalent of trying to simultaneously master Russian, Swahili and French. We assert that such literacy demands are similar in engineering problem solving. This is particularly clear when we examine the transformational representations in back-of-the-envelope calculations.

COGNITIVE STRATEGIES IN ENGINEERING PROBLEM SOLVING: PROBLEM RECOGNITION, FRAMING AND SYNTHESIS The TDS sequence addresses the external representations that are used to drive problem solving forward. At the same time, we need to understand the cognitive strategies that undergird the TDS structure. We put forward a model of problem solving strategies an engineer uses to solve a problem that consists of four categories undertaken in three phases as shown in Figure 2. The three phases of problem solving are problem recognition, problem framing and problem synthesis. The problem recognition phase entails gathering facts and data found in a textbook or real world problem statement and developing initial understanding. The problem recognition phase attempts to categorize the problem as a certain type. The semiotic system of this phase is primarily textual. This categorization allows movement to the next phase-- problem framing. In this phase the engineer generates assumptions and hypotheses to simplify the problem and sketches a potential solution, The semiotic system of the problemframing phase is primarily diagrammatic. Problem synthesis attempts to validate the problem-framing move by mathematically modeling the solution. The results of problem synthesis drive iteration through the three phases if required. The semiotic system of the synthetic phase is symbolic. Having shown how the TDS model of semiotic systems maps onto engineering problem solving, we can enter the space of novices learning engineering problem solving.

Semiotics as the science of signs aims to account for all meaningful sign systems in a society. The “sign” is the ba sic unit in which meaning and form coalesce.


Session F2G Facts/Data Stable elements that inform the model creation derived from problem statement and inquiry

Problem Recognition

Assumptions/Hypotheses Axiomatic laws, definitions, approximations, idealizations and assumptions

Cartoon A real world diagram or schematic of the essential problem elements as they relate to each other. “body of interest”

Problem Framing

Quantitative Solution A model used to obtain a solution

Problem Synthesis

FIGURE 2 A Simplified Engineering Problem Solving Process

DO WE SUPPORT STUDENTS IN BECOMING PROBLEM SOLVERS ? As mentioned in the introduction, we are concerned with how and whether the different semiotic systems used in problem solving are taught and practiced. We are also interested to learn whether students are asked to undertake problem recognition and problem framing in classroom activities. When we have visited classrooms and looked at engineering textbooks, we find that diagrammatic representations, assumptions and simplifying details, and possibly queues or hints for solution are generally provided. This suggest that students have few if any opportunities to generate back of the envelope calculations thereby learning all of the semiotic systems of problem solving. Our initial observations were of graduate biomedical engineering students not using the multiple languages of engineering problem solving. The students seemed to rely on a verbal problem description to frame the problem and attempted to solve the problem qualitatively. They appeared to be unable to transform the problem into diagrammatic and then symbolic forms to quantitatively solve the problem. That observation led us to look at engineering classrooms more carefully to understand how and if students are gaining mastery of the multiple languages of problem solving. To clarify our concerns we first describe our observations of the biomedical engineering students. Novice biomedical engineering problem solving Biomedical engineering students had to analyze a shaken baby syndrome legal case from the view of a biomedical engineer. Their job was to determine from the medical evidence and from a biomedical engineering viewpoint

whether the baby had been injured due to shaking. The students in general did not solve the problem as the faculty anticipated they would because they. did not effectively transform the verbal description of the problem into a model of a mechanical system and then analyze the properties of the problem to determine if the forces produced could have caused the damage to the baby. After observing the students' failure, we offered the problem to two mechanical engineering faculty and found, quite surprisingly, a consistent and successful solution method applied by the faculty. The faculty problem solving method was to initially sketch the baby as a mechanical system. Once the problem had been framed, the sketch prompted the recall of a set of symbolic formula that were applied to solve it. Our observation of the students was that they were unable to transform the verbal problem into a model of an analyzable system. Was the problem worded to somehow dissuade the students into thinking it was a mechanical system? Was the inability to sketch potential views of the problem not part of the student's repertoire? Without sketching, did the students not see the interaction between the simple equations of mechanics and the baby? These observations are what prompted us to look more closely at what goes on in undergraduate classrooms. We chose an introductory thermodynamics class as our initial entry. Novice thermodynamics problem solving We undertook a preliminary study of activities in a thermodynamics class to tease out if there is support for students constructing multi-literacy proficiency. We observed classroom activities of both the professor and the students and interviewed the professor of the course. The class we observed was a junior level introductory thermodynamics class in mechanical engineering. We observed the class during the earlier portion of the semester. Introducing the thermodynamic states of a system and its many properties: The student view The professor begins the lecture by noting the complexity of a three-dimensional surface that represents pressure as a function of temperature and volume for a particular substance. The professor shows a transparency of the example surface and discusses the relationships between pressure, volume and temperature. Since the professor mentioned to the students that the figure on the transparency is from the text for the class, some of the students refer to it in their books and seem to be trying to make sense of the figure in relation to the lecture. The professor proceeds to describe a means of simplifying the three-dimensional figure by rotating it in your mind and looking at two-dimensional cuts of the figure. The cuts represent holding one of the variables constant. At the same time, the professor is introducing relevant terms, such as saturation, liquid-vapor mixtures, vapor dome, saturation dome, quality and saturation tables. The professor then begins describing quality, which is a measure of the ratio of vapor versus


Session F2G liquid under particular constraints. The professor draws a graphical representation of a container with liquid and vapor to show the students a model of quality, and then immediately returns to the symbolic form of deriving quality. Throughout the lecture the professor reminded the students that "the secret to solving these types of problems is recognizing what type of problem it is". "Know where you are, and how to get to the solution. Setting the problem up is the key." This lecture prompted many questions on our part, some of which remain unanswered. • In the eyes of the students, is the three-dimensional diagram a graphical representation of a formula or a model of some thermodynamic system? Is it an abstraction without meaning, or is it a model with meaning? How do the students develop a deep understanding of the relation between a verbal account of how substances behave under variable conditions and a graph? • Are the students able to rotate and cut a three-D image in their minds as suggested by the professor? Is it just another mathematical abstraction or are the students able to construct meaning that it is a graph that is describing the behavior of a substance? • Are the introduced terms being related by the students to anything being explained? Are the terms being used as queues to previously solved problems or categories of problems to help frame new problems? • Is the picture of vapor and liquid used in the quality portion of the lecture explaining anything? Is the picture a sketch that helps frame a problem (an aid in transformation) or is it merely an analog of the verbal problem statement? • Is the professor's ability to move from a conceptual verbal description of a problem directly to the mathematics of its solution impeding the students' ability to understand the steps needed to get from a verbal description to the framing of the problem to its eventual solution?

problems? Do the students see the symbolic forms of solving problems as just mathematics or do they see them as symbolic models of the behavior of substances? Introducing the thermodynamic states of a system and its many properties: The professor's view After our classroom observations, we interviewed the professor to gain an understanding of his perspective. We started the discussion with asking him to describe the process he was trying to teach the students. Essentially he repeated the comment he had made in the class several times. You must understand where you are and what you have, to be able to solve these problems. We inquired further, and he proceeded to draw for us his view of the problem space of introductory thermodynamics. The problems boil down to energy or mass balance problems. The hard part is seeing what the problem is and framing it into a form that can be solved. That is the big step. That is what is hard for the students to learn. Once it is framed, it is relatively easy to plug the values into the equations and solve it. We asked how does he relate the abstract forms of problems offered in class to real problems on real systems? Or does he? I can see a jet engine or turbine in my head. Though those systems are much more complicated than anything we look at in class, a simplified view of one of them can help me understand what is going on in the problem. We found that the professor essentially draws a sketch as an internal representation (in his head), and uses that to reason about the characteristics of the problem and to frame it. We asked if he could move from problem statements directly to the mathematical symbology of the solution space. Once I see what the problem is (referring to what type of problem it is), I can move to the equations. For water and some other substances I am familiar with, I know whether or not I am in the saturation region just by looking at the numbers. For substances I am not familiar with, I often need to go to the tables and look things up.

DISCUSSION

The literacy demands on the students are relatively Through this preliminary investigation we have answered a evident, though to a thermodynamics expert possibly unseen few questions, but have generated many more. Though there probably because these skills have become tacit for him. is much research into science and math learning, the view of The demands are both on making meaning of the individual it being at least partially a problem of multi-literacy is languages (textual, diagrammatic, and symbolic) and making relatively unique. If engineering problem solving is moving meaning through the interaction of the three language types. from recognition to framing and then synthesis, and meaning The students are dealing with increasing their vocabulary. making is confounded by multiple languages that at times The meanings of the terms are often explained in terms of have primary responsibility for meaning and at other times symbolic or graphical representations. Can a person have interactive responsibility for meaning, how can we understand what a vapor dome is without drawing a support students' learning of these languages? If the process graphical representation of the p/v behavior of a substance? of solving problems has to do with transforming Are the students developing an understanding of the model representations, which is similar to translating between of behavior of substances under different conditions and can spoken languages, how do we teach students how to make the model be related to textual problem descriptions? Are these transformations? An additional complexity is that the the students developing a set of queues or indices into representations of engineering problem solving are not previously solved problems that help them frame new 0-7803-6669-7/01/$10.00 © 2001 IEEE October 10 - 13, 2001 Reno, NV 31 st ASEE/IEEE Frontiers in Education Conference F2G-16

Session F2G directly translatable, nor are they meant to be. Lemke best describes it as, "…. we do not have so much an exact translatability among verbal statements, mathematical formulas, and visual-graphical or material-operational representations as a complex set of co-ordinating practices for functionally integrating our uses of them. These coordinating practices must be learned in each case as a difficult and specialized form of multi-literacy"[4]. An engineering expert "sees" the meaning of a symbolic equation as a model of a physical device. An engineering expert uses sketches to visually queue internal and external views of a problem and its potential solutions. An engineering expert makes meaning of verbal problem descriptions by transforming them into symbolic and diagrammatic forms. An engineering expert sees these three languages as one language, as an expert translator can automatically move from one language to another. The combined TDS and problem solving models help us understand what is happening in the thermodynamics class and how those types of activities may have impacted the Biomedical Engineering students. If engineering problem solving is moving from recognition to framing and then synthesis and relies on the TDS representations to construct meaning, how is that taught, or is it taught? What we see in engineering classrooms is a reliance on eliminating the transformative moves from text to diagram, and thus the recognition and framing phases of problem solving. A reliance on synthesis with its associated symbolic representations dominates. There are many valid reasons for this. First, novices can easily become overwhelmed with too many representations. Also novices need to learn the manipulation of the mathematical representations first. Further novices need to see how multiple problems are solved prior to being exposed to ill-constrained problems. If we agree that these "fundamentals" must be taught first, when are the students taught to recognize and frame problems? How do they learn the languages of engineering problem solving and their relations to the phases of problem solving? When are they taught back of the envelope calculations? Our observations of the Biomedical Engineering graduate students lead us to believe students are not exposed to problem recognition and framing activities.

[1]

Castanheira, M.L., et al., Interactional ethnography: An approach to studying the social construction of literate practices. Linguistics and Education, 2001. 11(4): p. 353-400.

[2]

Lemke, J., Multimedia literacy demands of the scientific curriculum. Linguistics and Education, 1998. 10(3): p. 247-271.

[3]

O'Halloran, K.L., Classroom Discourse in Mathematics: A Multisemiotic Analysis. Linguistics and Education, 1998. 10(3): p. 359-388.

[4]

Lemke, J., ed. Multiplying meaning:Visual and verbal semiotics in scientific text. Reading Science: Critical and functional perspectives on discourses of science, ed. J.R. Martin and R. Veel. 1998, Routledge: London.

SUMMARY Though we have generated questions and few answers, we believe that understanding the relationships between representations and their transformations in the context of language learning may offer some new methods of teaching engineering. Students appear to separate the languages of problem solving and as a result have difficulty making meaning of the relations and interactions between the semiotic systems that support engineering problem solving.
