4 work provides a theoretical and empirical basis for understanding how the use ... important role played by lists during the early development of writing (Goody, 1977). .... becomes a major part of the physics curriculum, then students will be ...... I will have somewhat more to say on this topic in Chapter 7, but I will make one.
The Symbolic Basis of Physical Intuition A Study of Two Symbol Systems in Physics Instruction by Bruce Lawrence Sherin B.A. (Princeton University) 1985 M.A. (University of California, Berkeley) 1988
A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Science and Mathematics Education in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY
Committee in charge: Professor Andrea A. diSessa, Chair Professor Rogers P. Hall Professor Daniel S. Rokhsar Professor Barbara Y. White 1996
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The Symbolic Basis of Physical Intuition A Study of Two Symbol Systems in Physics Instruction
© 1996 by Bruce Lawrence Sherin
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Abstract The Symbolic Basis of Physical Intuition: A Study of Two Symbol Systems in Physics Instruction by Bruce L. Sherin Doctor of Philosophy in Science and Mathematics Education University of California, Berkeley Professor Andrea A. diSessa, Chair This dissertation is a comparative study of the use of two symbol systems in physics instruction. The first, algebraic notation, plays an important role as the language in which physicists make precise and compact statements of physical laws and relations. Second, this research explores the use of programming languages in physics instruction. Central to this endeavor is the notion that programming languages can be elevated to the status of bona fide representational systems for physics. I undertook the cognitive project of characterizing the knowledge that would result from each of these two instructional practices. To this end, I constructed a model of one aspect of the knowledge associated with symbol use in physics. This model included two major types of knowledge elements: (1) symbolic forms, which constitute a conceptual vocabulary in terms of which physics expressions are understood, and (2) representational devices, which function as a set of interpretive strategies. The model constitutes a partial theory of “meaningful symbol use” and how it affects conceptual development. The empirical basis of this work is a data corpus consisting of two parts, one which contains videotapes of pairs of college students solving textbook physics problems using algebraic notation, and one in which college students program computer simulations of various motions. The videotapes in each half of the corpus were transcribed and analyzed in terms of the above model, and the resulting analyses compared. This involved listing the specific symbolic forms and representational devices employed by the students, as well as a measurement of the frequency of use of the various knowledge elements. A conclusion of this work is that algebra-physics can be characterized as a physics of balance and equilibrium, and programming-physics a physics of processes and causation. More generally, this
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work provides a theoretical and empirical basis for understanding how the use of particular symbol systems affects students’ conceptualization.
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Acknowledgements This work, to a great extent, grew out of my interactions with my advisor, Andrea A. diSessa. He shaped my views on thinking and learning generally, and also contributed directly to many of the ideas in this thesis. My views have also been strongly influenced by Rogers Hall and Barbara White. Rogers gave me many new ways to think about representational artifacts—really, more than I could handle—and Barbara kept me anchored to the world of physics instruction. Thanks also to Daniel Rokhsar, for his physics-eye view. Many other members of the EMST/SESAME community helped me along the way. Thanks especially to the members of the Boxer Research Group, for listening to me talk about this work for several years, and giving helpful feedback. Thanks also to the members of the Representational Practices Group and the ThinkerTools Group for the added perspective provided by their own research. I could never have finished—or taken so long—without my friends to distract me. Thanks to the Geebers for Geebing, the thing that they do best. My parents have never been anything but supportive in my academic pursuits, a fact that I really do appreciate. Thanks also to my grandparents, my sister Heidi, my parents-in-law, all my brothers-in-law, sisters-in-law, nieces, nephews, and everybody else for generally being a mellow bunch. My wife Miriam, in a thousand ways, made this work possible. She did the last-minute tasks that I couldn’t do, and kept me sane and happy. If she’s proud of me, then I’m satisfied.
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Chapter 1. Introduction and Major Themes 4 26 319 64 9 6 33
fattened kids fattened goat-heifers pastured sheep pastured he-goats pastured ewes pastured goats weaned lambs…
Circa 2500 BCE, an inhabitant of Mesopotamia—probably a man—stands working on a clay tablet, writing in cuneiform. He is recording information concerning livestock to be presented in trade. The marks he makes are not a careful transcription of speech, but they are also not a haphazard collection of information. The information is represented in a specific form, as a list. This ancient man is making and using an external representation. To a present day individual, the making of such a list is a mundane event. We make lists before going shopping and to keep track of the tasks we need “to do.” But the timeless and ubiquitous nature of written lists does not suggest that they are simple and unimportant constructions; on the contrary, it implies that they have a power that transcends time and task. Why are lists so useful that they are not only written on sheets of paper, they have even been etched on clay tablets? Some of the answers to this question are relatively straightforward. The list is useful to our ancient Mesopotamian because he can use it later as an aid to memory. In general, the use of written lists allows individuals to remember long lists of information with relative ease, as compared with committing the same information to memory. In addition, lists, such as shopping lists, can serve as resources during future action. While at the supermarket, we can procure the items on our list one by one, checking them off as we go. Furthermore, there is the striking fact that the list above is a real one: This particular list of information was actually written by some inhabitant of Mesopotamia during the third millennium, BCE (It is taken from Goody, 1977). Thus, the information has accomplished the trick of traveling through time several millennia so that we can read it, a striking feat! Some aspects of the power of lists and other external representations are somewhat less easy to recognize and explain. In The Domestication of the Savage Mind, Jack Goody discusses the important role played by lists during the early development of writing (Goody, 1977). According 6
to Goody, writing—and lists, in particular—did not simply constitute a means of performing the same tasks, such as remembering existing information, more easily and efficiently. The changes introduced by written lists were not only changes in degree or extent. Rather, he argues, the development of lists changed people’s practices as well as the very nature of knowledge and understanding. My concern here is to show that these written forms were not simply by-products of the interaction between writing and say, the economy, filling some hitherto hidden ‘need,’ but that they represented a significant change not only in the nature of transactions, but also in the ‘modes of thought’ that accompanied them, at least if we interpret ‘modes of thought’ in terms of the formal, cognitive and linguistic operations which this new technology of the intellect opened up. (p. 81)
The point is this: As our list-writer works, he must bend his thought and action to the framework provided by the list. For this reason, the thoughts of our list-writer, during the moments of composing the list, are in part shaped by the structure of the list. Thus, the product of this activity is also in part determined by the list structure. Goody even argues that the use of lists led to the development of scientific taxonomies and eventually to the development of science as we know it. Step forward to the 20th century. Lists and other external representations are still quite popular. External representations are an important part of everyday activity, as well as of work in almost all disciplines. Here we turn to the scientific disciplines; in particular, we turn to the domain of physics. Within the discipline of physics, equations and related symbolic expressions are an extremely important external representational form. Physicists write equations and manipulate symbols in order to perform extended and complex computations. Furthermore, physicists use equations as a means of making precise and compact expressions of physical laws, which then appear in academic papers and in textbooks. As with Goody’s lists, the implications of equation-use in physics extend to the subtle and profound. Equation-use is so pervasive in physics that, for the novice physics student, it may be hard to escape the intuition that equations ARE physics knowledge. And it may seem like “doing physics” means nothing more than using and manipulating symbols. This may be more than a foolhardy misconception, however. At least one physicist believes that there is some truth behind this appearance. In The Character of Physical Law, physicist Richard Feynman, expresses a similar intuition (Feynman, 1965): The burden of the lecture is just to emphasize the fact that it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding in mathematics. I am sorry, but this seems to be the case. (p. 39)
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Why not tell me in words instead of in symbols? Mathematics is just a language, and I want to be able to translate the language. … But I do not think it is possible, because mathematics is not just another language. Mathematics is a language plus reasoning; it is like a language plus logic. (p. 40) The apparent enormous complexities of nature, with all its funny laws and rules, each of which has been carefully explained to you, are really very closely interwoven. However, if you do not appreciate the mathematics, you cannot see, among the great variety of facts, that logic permits you to go from one to the other. (p. 41)
In these quotes, Feynman does not quite say that physics knowledge is nothing but equations, but he is making a related claim. Here, he expresses the intuition that, in some manner, it is not possible to fully understand physics without being an initiate into the physicist’s use of mathematics and the associated symbols. Feynman tells his students that he cannot explain “in words instead of symbols.” To understand physics, you must have the experience of living in the tight net of physical relations expressed as equations, going from one to the other. You have to write equations and solve them, derive one equation from another, and from this experience develop a sense for the properties of the equations and the relations among them. From Goody’s lists to Feynman’s equations, the power of external representations is manifest. They allow us to easily remember long lists of facts for a duration that is essentially unlimited. And they allow us to perform extended, context-free manipulations to compute results. Furthermore, these authors point to the possibility of more subtle implications of external representations for the nature of human knowledge. The first, an historical view, posits a role for external representations in an historically developing body of knowledge. The second view boils down to a physicist’s intuition that an appreciation of the beauties of physics is somehow tied up with their expression in the language of mathematics. Similar notions have appeared in many forms, and in a variety of literature. Jerome Bruner spoke of external representations as a type of “cultural amplifier” (Bruner, 1966). Whorf hypothesized that the vocabulary and syntax of a language shape the speaker’s view of reality (Whorf, 1956). Lev Vygotsky was the father of a branch of psychology for which the internalization of external signs plays a fundamental role in the development and character of thought (Vygotsky, 1934/1986). And there is an extensive body of literature devoted to determining the effect of literacy on thought (e.g., Goody & Watt, 1968; Scribner & Cole, 1981). For each of the above researchers, knowledge and external representations are inextricably related. But the diversity of views here will require some sorting, and the intuitions call for expression in theoretically precise terms. Here, I will not adopt the claims or perspective of any of 8
these researchers, or add directly to any of their research programs. Unlike these anthropologists and historians, I will not look across cultures or time for differences in thought or the nature of knowledge associated with the use of various symbol systems. Instead, I will use the methods of cognitive science to build a model of symbol use within a particular domain. Then, given this model, I can begin to tentatively address some of the long-standing issues concerning external representations and human knowledge.
A Tale of Two (Representational) Systems The inquiry to be described here will be built around a comparative study of two symbol systems within the domain of physics. The first of these is the traditional system of algebraic notation. By “algebraic notation” I mean to include the full range of equations that physicists write. Within the practice of physics, algebraic notation plays an important role as the language in which physicists make precise and compact statements of physical laws and relations, and in which they solve problems and derive new results. Here we will be particularly concerned with the use of algebraic notation as it occurs in introductory physics instruction. The use of the second representational system is novel: This research explores the use of programming languages in physics instruction. Central to this endeavor is the notion that programming languages can be elevated to the status of bona fide representational systems for physics. This means treating programming as potentially having a similar status and performing a similar function to algebraic notation in physics learning. It means that instead of representing laws as equations, students will represent them as programs. This proposal—to replace algebraic notation with a programming language—would certainly constitute a significant change in the nature of physics and physics instruction. If programming becomes a major part of the physics curriculum, then students will be engaged in very different activities: They will be programming instead of solving traditional physics textbook problems. And the knowledge that students take away from this new curriculum will be substantially different, at least in some superficial respects. Students will learn programming and programming algorithms, rather than some derivational and problem solving strategies that are particular to textbook physics problems. In addition, we must also keep in mind the more profound possibilities raised by the likes of Goody and Feynman. Goody might tell us that we must expect to alter the nature of physics knowledge in a more fundamental manner. And Feynman might worry that an approach to 9
physics that passes through programming may deprive students of an appreciation of the “beauty of nature.” These observations are more than interesting asides; they have serious implications for how the comparative study at the heart of this inquiry must be understood. If external representations are inextricably tied to the nature of physics knowledge, then it will not be reasonable to think of programming as a new representation for doing the same old jobs. In Goody’s terms, representations do not just fill some need, they also shape and determine the nature of the knowledge. For this reason, it will not be precisely correct to ask whether programming or algebra does certain jobs better. I cannot ask which of the systems better “represents” physics knowledge. I cannot even really ask which of these representational systems will lead to better student learning of physics. In my view, there is no physics that exists separate from these representational systems and the associated tasks, that can be represented more accurately or taught better. As an alternative, I thus propose to ask and answer the following questions: How does “programming-physics” differ from “algebra-physics?” And: Is “programming-physics” a respectable form of physics? These questions will provide the focus and anchor for this inquiry. Two practices for introductory physics instruction There is still another complication to be considered. I cannot assume that these symbol systems, somehow defined prior to their application to physics, uniquely determine the associated practices. In other words, for example, given any programming language, there are many possible practices of programming-physics that one could build around that programming language. For this study, I must therefore pick out a particular algebra-physics and a particular programmingphysics for study. Specifying a practice of algebra-physics for study does not pose a significant difficulty. Since I am interested in programming as an alternative for physics instruction, I will examine algebraphysics as it is practiced by students in traditional introductory physics courses. Because of the great uniformity that exists across such traditional courses (which still make up most physics instruction), this constitutes a relatively well-defined and uniform practice. Perhaps the most prominent aspect of this practice is an emphasis on solving a certain class of problems. A problem and solution, taken from the textbook by Tipler (1976), is shown in Figure Chapter 1. -1. This problem is wholly typical of those that students solve during the first weeks of an introductory
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course. Note that it employs a few equations (notably v = vo + at and ∆x = vot + 12 at 2 ) that students are expected to have learned during this period. A ball is thrown upward with an initial velocity of 30 m/s. If its acceleration is 10 m/s2 downward, how long does it take to reach its highest point, and what is the distance to the highest point?
v = vo + at 0 = 30 m s + −10 m s 2 t
(
t=
)
30 m s
= 3.0s 10 m s 2
(
)
∆x = vot + 12 at 2 = (30 m s)(3.0s ) + 12 −10 m s 2 (3.0s )2 = +90m − 45m = 45m Figure Chapter 1. -1. A typical problem and solution (from Tipler, 1976).
For the case of programming, there is less to build on, and there is certainly no uniformly accepted method of employing programming languages in physics instruction. In this work, I will draw on a practice of programming-physics that was developed by the Boxer Research Group at U.C. Berkeley, and was implemented in 6th grade and high school classrooms (diSessa, 1989; Sherin, diSessa, and Hammer, 1993). For the purposes of this study, this practice was transported into a laboratory setting, and it was employed with university students who had already completed two semesters of introductory physics. The resulting complications will be discussed later. Unlike the problem-based approach typical of traditional introductory physics courses, the courses developed by the Boxer Research Group were all simulation-based; rather than solving standard physics textbook problems, the students spent their time programming simulations of motion. Figure Chapter 1. -2 shows an example program, which simulates the motion of a dropped ball. This simulation is written in the Boxer programming environment (diSessa, Abelson, and Ploger, 1991). At first glance, the external representations that appear in this figure probably seem very different than the equations in Figure Chapter 1. -1—and they are! Nonetheless, this research will uncover a surprising degree and kind of similarity. The Boxer programming language and the practice of programming-physics employed in this study will be described in some detail in Chapter 8.
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Instructions Make a realistic simulation of the motion of a ball that is dropped from someone's hand. Data
drop setup change change change repeat
pos 0 vel 0 acc 1 23 tick
pos
vel
acc
253
23
1
Data
Data
Data
tick
Doit
change pos
pos + vel Doit
change vel
vel + acc Doit
fd vel dot Doit Data
Data
Figure Chapter 1. -2. Simulation of a dropped ball.
Finally, I want to note that, just looking at Figure Chapter 1. -1 and Figure Chapter 1. -2, it is evident that each of these practices involves more than a single external representation—they involve more than just algebraic notation and programming statements. For example, both of the above figures have textual elements, and the programming figure shows a representation of the ball’s motion involving dots. Furthermore, even more external representations are involved in these practices than appear in these figures. For instance, algebra-physics uses graphs and various types of diagrams. Nonetheless, I have chosen to name these two practices after algebra and programming because my attention will be focused tightly around these particular symbol systems. In addition, I believe that these symbol systems play a critical role in constraining the nature of their respective practices. The advantageous properties of this study Why is this particular comparative study the right place to explore the relation between external representations and understanding? First, it is apparent that symbol use plays a substantial and important role in the discipline of physics, both in instruction and in the work of practicing experts. The process of learning physics traditionally involves a great deal of writing equations and
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manipulating them to solve problems, and equations are the target form for physicists’ laws. Because symbol use plays such an important role in physics, this study is pretty much guaranteed to encounter some of the more interesting interactions between knowledge and external representations. Second, although algebraic notation plays a central role in physics, it is not as integrated in everyday activity as natural language; thus, its role should be somewhat easier to fathom. The point is that describing the relationship between literacy and thought (Goody & Watt, 1968), or thought and language (Vygotsky, 1934/1986) is probably a much harder problem than describing the relationship between equations and physics understanding. We can thus think of this project as a first and easier step toward these other, more ambitious undertakings. Third, the use of a comparative study has some benefits. I will be able to do more than speculate about how physics built around an alternative symbol system might differ; I can actually uncover some specific differences. It is also important that the two practices we are comparing are not altogether different. If the two practices were too divergent, what type of comparisons might we make? How, for example, would one compare physics and chemistry? Finally, this study has interest beyond the more theoretical issues I have described. One of the aims of this project is to explore the feasibility of a novel instructional approach for physics. Preliminary research has found that the use of a programming language in physics instruction is a promising alternative and that programming languages may even have certain advantages (as well as certain disadvantages) over algebraic notation. The study described here will contribute to the investigation of this novel instructional approach.
Overview of the Approach Taken down to a single driving notion, this study can be seen as an exploration of the possibility of replacing algebraic notation with a programming language in physics instruction. But, when the question is viewed with a little care, such a study blows up to a much larger scale, and it is evident that a number of subsidiary questions must be answered. First, we cannot possibly understand the implications of replacing algebra with programming if we do not have some notion of the role played by algebraic notation in traditional physics learning. So, already this study has doubled in size; before even starting to look at the use of programming in physics, we have to go back and understand certain aspects of traditional physics learning using algebraic notation. 13
And we have seen that it is probably not going to be good enough to think simply in terms of replacing algebraic notation with a programming language. Such a switch may have far-reaching effects on the nature of physics knowledge and understanding. So, we really cannot do anything like a simple comparison, there is some tricky business about the relationship between knowledge and external representations. This is a problem to overcome, but it is also a positive feature of this inquiry: One of the outcomes of this study will be some insight into this complex relationship. In order to compare algebra-physics and programming-physics I proceeded as follows. I began by collecting a data corpus consisting of two parts, one which contains observations of students engaged in the practice of algebra-physics and another in which students were engaged in programming-physics. The question then becomes: How can we go about comparing these two sets of observations? The key to producing a useful comparison of algebra-physics and programming-physics based on these observations was the construction of a model of some aspects of symbol use in physics. This model, which I will preview below in a moment, is a model in the spirit of cognitive science; I attempted to account for a certain class of human behavior (symbol use in physics) by hypothesizing that people possess and employ certain elements of knowledge. This model is the central result of this research and the key to getting at the goals of this inquiry. Viewed at a certain level, this model will be general enough to describe both varieties of symbol use under consideration in this study. Thus, it can provide the categories for understanding and comparing the observations of algebra-physics and programming-physics in the data corpus. In addition, once we have the model, we can look at it with more general issues in mind. In the remainder of this section I will briefly describe the data corpus and the model of symbol use on which my analysis is based. The data corpus, in brief All of the subjects in this study were UC Berkeley students enrolled in “Physics 7C.” Physics 7C is a third semester introductory course intended primarily for engineering majors of all persuasions. The fact that these students were in Physics 7C implies that they had already completed two semesters of instruction, Physics 7A and 7B. Many also had some instruction during high school.
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The subjects were divided into two distinct pools, an “algebra” pool and a “programming” pool, with each pool consisting of 5 pairs of students. All experimental sessions were conducted in a laboratory setting and these sessions were videotaped. Students in the algebra pool worked with their partner at a blackboard to solve a pre-specified set of problems. Most of these problems were fairly traditional textbook problems, but a few more unusual tasks were also included. The pairs typically solved the problems in 5 1/2 hours spread over 5 sessions. The result was a total of 27 hours of videotape of these students using algebraic notation to solve physics problems. A subset of this data, 10 hours, was selected for more focused analysis. Programming pool subjects worked in pairs at computers. The students were first given about four hours of training, usually requiring 2 two-hour sessions. Following the training, the students were asked to program a set of simulations of physical motions. They typically required 4 additional two-hour sessions to complete these tasks. In sum, the result was a total of 53 hours of videotape of students programming simulations. A subset of this data, 16 hours, was selected for more focused analysis All of the videotaped sessions selected for focused analysis were carefully transcribed and the resulting transcripts were analyzed. The examples given in upcoming chapters are all taken from these transcripts, and the model described is drawn from this analysis. However, I will not describe the data corpus or my analysis techniques in more detail until Chapter 6. That chapter will also include an account of the rationale for the above design. Overview of the model: The Theory of Forms and Devices, in brief As I mentioned above, the model will be a theory in the spirit of cognitive science, which means it will posit knowledge elements to explain certain behaviors. The scope of the model will be restricted in a number of senses. Most importantly, I will attempt to describe only a small subset of the knowledge necessary to provide a complete account of symbol use in physics. This subset is particularly important, I will argue, in the construction of novel symbolic expressions and the interpretation of expressions, both in algebra-physics and programming-physics. Furthermore, I will argue that this small subset of knowledge is precisely where our attention must be focused in order to get at the relation between physics knowledge and external representations. The model of symbol use will be built around two major theoretical constructs, what I call “symbolic forms” and “representational devices.” I will now briefly discuss each of these constructs. (A more complete discussion can be found in Chapters 3 and 4.)
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On Symbolic Forms An important contention of this work is that initiates in algebra-physics are capable of recognizing a particular type of structure in physics expressions. More specifically, they are able to see expressions in terms of a set of templates that I refer to as “symbol patterns,” each of which is associated with a simple conceptual schema. The knowledge element involving this combination of symbol pattern and schema I refer to as a “symbolic form” or just as a “form,” for short. In a sense, symbolic forms constitute a conceptual vocabulary in terms of which physics equations are written and understood. For example, one of the most common forms is what I call “BALANCING .” In the conceptual schema associated with BALANCING , a situation is schematized as involving two influences, such as two forces, in balance. Furthermore, the symbol pattern associated with BALANCING involves two expressions separated by an equal sign: =
BALANCING
For illustration, imagine an example in which a student is solving a problem involving a block that has two opposing forces acting on it. Because the block is not moving, the student writes the expression F1 = F2. Now, the question is, how precisely did the student know to write this equation? The answer I will give to this question is that the BALANCING form has been engaged and, when it is engaged, it pins down what the student needs to write at a certain level. In this case, it specifies that the student needs to write two expressions separated by an equal sign, with each side of the equation associated with one of the two influences in balance. Furthermore, this process can also work in reverse; a student can look at an equation, like F1 = F2, and see it in terms of the BALANCING
form.
Symbolic forms, in part, develop out of symbolic experience and then become a way of experiencing the world of symbols. Because of the developed ability to be sensitive to the patterns associated with forms, the symbolic world of experience is more structured and meaningful for the physics initiate than one might first expect. On Representational Devices In general, a number of different symbolic forms may be recognized in any particular physics expressions. The question then arises: What determines, at any particular time, the forms that a student recognizes in an expression? The answer, I will attempt to show, is that initiates in physics possess a repertoire of interpretive strategies that I call “representational devices.” Roughly, 16
representational devices set a stance within which symbolic forms are engaged. The term “device” here is intended to call to mind “literary devices,” for reasons that I hope to make apparent in Chapter 4. I will argue that there are three major classes of representational devices. In one of these classes, which I call Narrative devices, an equation is embedded in an imaginary process in which some change occurs. For example, suppose that a student has derived the expression a=F/m, which gives the acceleration of an object in terms of the mass and the force applied. The student can then imagine a process, for example, in which the mass increases, and infer from the equation that the acceleration must decrease. This would be an interpretation involving a Narrative device. The other two classes of representational devices are what I call Static and Special Case devices. In Static devices, an equation is seen as describing a moment in a motion. For example, an equation may be seen as true only at the apex of a projectile’s trajectory. In Special Case devices, a stance is set by considering some restricted set of cases to which the equation applies. For example, a student may consider the behavior of an equation in an extreme or limiting case. In sum, the Theory of Forms and Devices describes a collection of knowledge that allows initiates in physics to see a certain type of meaningful structure in physics expressions. Two specific types of knowledge are involved in this collection: symbolic forms, which correspond to a set of meaningful structures that students can recognize in expressions, and representational devices, which constitute a repertoire of interpretive stances. Primarily, it is by looking at the forms and devices associated with algebra-physics and programming-physics that I will compare these two practices. In the remainder of this chapter, I will now work toward situating this work in relation to broader issues and in relation to existing research. I will begin, in the next section, by attempting to provide a definition of “external representation.” Then I will present a more in depth discussion of a central issue that I have so far only touched upon: the relation between knowledge and external representations. Finally, I will discuss how this work relates to existing research concerning physics learning and instruction.
A Definition of ÒExternal RepresentationÓ Much of what I say in this document will be specific to the external representational practices described in the previous section. But I will also want to argue for some generality of my 17
observations, especially when these observations are framed in broader terms. However, in order to make these claims of generality there is some additional work that must be done. If I want to say that my observations apply more broadly, then I need to say something about the scope of application. In particular, it would be helpful to have a definition of “external representation.” So, what is an “external representation.” Let’s take a moment, first, to think about the type of definition that is required here. My project is to build a model of some knowledge associated with symbol use in physics. In essence, I am going to be providing a kind of description of a class of human behaviors. Thus, the definition we need is a definition that characterizes this class of behavior, that specifies it in contrast to other behaviors. Finding such a definition may very well be difficult, since human activity is complicated and messy. In fact, there is no reason to believe that there is a simple definition that lines up with our intuitions concerning what constitutes a representation and still specifies a relatively circumscribable class of human activity. An alternative approach, taken by a number of other researchers, is to define external representation without explicit reference to behavior or cognition. They begin, instead, by defining a “representational system.” For example, in a typical version of such a definition, a representational system is defined in terms of correspondences between a “representing” and a “represented world.” This type of definition of representation hooks up with our intuitive notion of representation quite clearly—a representation is something that “stands for” something else. Such correspondence definitions have their uses, and I will follow this line for a while in Chapter 7. However, I believe that this is not the right type of definition for the present project. Since I am building a model of symbol use, I need a definition that slices off a class of human activity. Let’s take a stab at this type of definition. I will define an “external representational practice” as a class of human activity that involves: 1.
A set of externally realizable marks. These marks will be organized, in some manner, either in space or through time. They may be arrangements of objects, marks on paper, sounds, or even smells.
2.
Conventions concerning the making of marks. There will be conventions—socially dictated regularities in human behavior that are shared by many individuals—pertaining to how the marks are organized in space and time, and how the making of marks is integrated with other aspects of activity.
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3.
Conventions concerning action associated with existing arrangements of marks. People who have learned to participate in the practice will be sensitive to distinctions in the arrangement of marks. This means that their action in the presence of the external representation—what they say and do because of and with a particular collection of marks—will depend sensitively on conventionally defined distinctions in the arrangement of the marks.
To see that this definition can actually do some work, let’s look at what it rules out: •
A random pile of stones out in the middle of nowhere. This is not an instance of human activity so it’s not even covered by the definition.
•
Some children make a haphazardly arranged pile of stones. This violates requirements (2) and (3) of the definition. To the extent that there are conventions, they are not widely shared by many individuals.
•
A table is set prior to a meal. In this case, requirement (2) is met since there are conventions concerning the arrangement of the objects. However, requirement (3) is not met; fine distinctions in the arrangement of objects are not important for future action.
•
The entrails of a goat are used to predict the future. Here, there is an arrangement of physical marks that is interpreted, so requirement (3) is met. But the act of arranging the marks is not, presumably, a normal component of the practice, so requirement (2) is not met and this is not a representational practice. Note that this is not really a technical definition and I am relying on the reader’s intuitions, to
a great extent, to apply it. Furthermore, the agreement of specific instances with the requirements of the definition may be a matter of degree rather than clearly establishable fact. For example, a sophisticated diner may know that the outermost utensils in a place setting are always supposed to be used first. Thus, the arrangement of objects is significant for human action and dining becomes a representational practice. My first response to this is that maybe this is not such a ridiculous conclusion. To the extent that the arrangement of silverware communicates information from the table setter to the diner, perhaps it is reasonable to think of this as a very simple representational practice. My second response is to again appeal to the type of definition that I am trying to create. Since I am attempting to slice off a class of human behavior, there are certain to be some fuzzy
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edges. The best I can really shoot for is a rough characterization of this class, something that gives the reader a sense for the scope of the conclusions to be drawn here. The hope is that the intuitively central cases (equations, graphs, writing) are clearly included by the definition, and that intuitively excludable cases (the haphazard arrangement of stones) are excluded. Thus, this definition does not specify properties of external representations that make them absolutely unique among artifacts; most notably, I have not tried to argue that external representations have the special property that they refer to or stand for ideas or other elements of the environment. Instead, I have characterized a representational practice as involving the assigning of significance to fine distinctions in the arrangements of elements of the environment. Furthermore, the conventions for assigning this significance are shared among individuals. On these dimensions, the difference between a representational practice and other activity is a matter of degree; the most typical instances of representational practices involve a sensitivity to many and numerous fine distinctions, with the associated conventions shared by many individuals.
Relations Between Knowledge and External Representations I am working from the premise that replacing algebraic notation with programming must change the nature of physics knowledge and understanding. But, to this point, I have only proffered some vague notions and rough suggestions as to the manner in which knowledge and external representations are related. I have Goody’s notion that lists shape knowledge and Feynman’s intuition that an appreciation of physics is tied up with its mathematical expression. But, more precisely, what can it mean to say that knowledge is inextricably representational? In what ways might physics knowledge and external representations be related? Here I will take some preliminary steps toward answering these questions. To do this, I will lay out a very basic framework on which we can hang some of our intuitions concerning relations between knowledge and external representations. However, I will not push too hard on these intuitions and I certainly do not intend this discussion to be exhaustive in its stating of relations between knowledge and external representation. My purpose is only to make a basic accounting of some of these relations within a simple framework. Direct versus indirect effects of external representations In order to introduce the framework on which the rest of the discussion in this section depends, I will begin with an important distinction between “direct” and “indirect” influences of 20
external representations on the knowledge of individuals. This distinction is illustrated nicely by some competing viewpoints concerning the effects of literacy on thought. To begin, I turn to Goody and Watt’s seminal article The Consequences of Literacy (Goody & Watt, 1968). In this article, Goody and Watt’s agenda was to react against what they perceived to be the over-extreme cultural relativism of the time. To respond to this relativism, they proposed that a number of significant differences existed between “oral” and “literate” societies. Most importantly for our concerns here, they presented a number of strong hypotheses concerning the effects of literacy on human thought. The arguments of Goody and Watt rest largely on the analysis of historical cases. For example, they claim that the development of the specific type of alphabet used by the Ancient Greeks was responsible for—or, at least, allowed—many of the important intellectual achievements of the time. Included is the claim that the development of this type of literacy allowed the development of a new sort of logic and even “scientific” thought. Two researchers, Sylvia Scribner and Michael Cole, proposed to directly test these claims. In The Psychology of Literacy, Scribner and Cole describe an anthropological research project involving the Vai people of Northwestern Liberia (Scribner & Cole, 1981). For the purposes of studying the consequences of literacy for individuals, the Vai provided an opportunity for a potentially illuminating case study. The interesting fact about the Vai culture is that there are three widely practiced forms of literacy, as well as widespread non-literacy. First, the Vai employ an independently invented phonetic writing system—Vai script—that is not used elsewhere. No formal institutional settings exist for the propagation of this script; rather, it is transmitted informally among the Vai. In addition, there are Qur’anic schools where the Arabic script is learned, and English writing is taught in the context of formal western-style schooling. Scribner and Cole’s framing idea was that the Vai culture provided a natural laboratory in which the effects of literacy could be studied separately from the effects of schooling. Their method involved the administering of various psychological tests to individuals with differing literacies and with differing schooling backgrounds. Through such a study, Scribner and Cole reasoned, they could isolate the consequences of literacy for the mental abilities of individuals. The results of this research, at least with regard to encompassing cognitive consequences, were generally negative. There is nothing in our findings that would lead us to speak of cognitive consequences of literacy with the notion in mind that such consequences affect intellectual performance in all tasks to which the human mind is put. Nothing in our data would support the statement quoted earlier that reading and
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writing entail fundamental “cognitive restructurings” that control intellectual performance in all domains (Scribner & Cole, 1988, p. 70).
Literacy, it seemed, did not lead Vai individuals to perform better on measures of general cognitive ability, such as tests of logic or categorization tasks. However, Scribner and Cole did find differences in more specific “skills” associated with each of the individual forms of literacy. For example, it turns out that letter writing is one of the most common uses of the Vai script. For this reason, Scribner and Cole hypothesized that Vai literacy would require and foster the development of some specialized skills not required by oral communication. In writing, meaning is carried entirely by the text. An effective written communication requires sensitivity to the informational needs of the reader and skill in use of elaborative linguistic techniques. (Scribner & Cole, 1988, p. 66)
In fact, Scribner and Cole did find a correlation between Vai script literacy and communication skills. Similarly, because the use of the Arabic script was associated with memorization tasks in Qur’anic learning, they predicted that this Arabic literacy would be associated with increased memory skills. This hypothesis was born out, though only in the types of memory tasks that were similar to those found in the Qur’anic schools. So Scribner and Cole’s results seem to indicate individual differences due to literacy at the level of specific skills, but not at the level of more general cognitive capacities. Goody’s response to these results was that, given the nature of the study performed, these results are not at all surprising: That is to say, we did not expect the ‘mastery of writing’ (of whatever form) to produce in itself an immediate change in the intellectual operations of individuals … But if we are referring to an operation like syllogistic reasoning, the expectation that ‘mastery of writing’ in itself would lead directly to its adoption is patently absurd. The syllogism, as we know it, was a particular invention of a particular place and time; there were forerunners of a sort in Mesopotamia just as there were forerunners of Pythagoras’ Theorem; nor on a more general level is the process of deductive inference unknown in oral societies. But we are talking about a particular kind of puzzle, ‘logic’, theorem, that involves a graphic lay-out. In this sense the syllogism is consequent upon or implied in writing. However its use as distinct from its invention does not demand a mastery of writing. Once invented, it can be fed back into the activities of individual illiterates or even non-literates just as the same individuals can be taught to operate the arithmetic table or, as Scribner and Cole point out, to decode Rebus writing. (Goody, 1987, p. 221)
Goody (1987) believes that psychological tests that compare individuals in a society should not be expected to uncover the more general effects predicted by Goody and Watt (1968). According to Goody, we should expect the stronger consequences of literacy to be manifested only at the level
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of societies, not at the level of individuals. Thus, we should not expect to see any strong differences between Vai individuals due to differences in literacy. To clarify this point, Goody makes a distinction between what he calls “mediated” and “unmediated” effects of literacy (Goody, 1987). The idea here is that the consequences of literacy for the thought processes of individuals need not arise through the direct use of writing by individuals. Instead, the consequences of literacy may play themselves out at a society-wide level and then be “fed back” into the thought of individuals. An individual need not even be literate to be affected. Given this viewpoint, Goody and Watt’s original hypotheses need not be manifested as differences between individuals in the Vai society. There are many other extant criticisms of the hypotheses of Goody and Watt, such as the claim that their arguments tend toward an inappropriate “technological determinism” (Street, 1984). But my purpose here is not to engage in a thorough analysis of particular hypotheses for the consequences of literacy. Instead, my goal has been to use this debate as a context for illustrating that hypotheses concerning the effects of external representations on individual knowledge may be of two sorts: We can posit direct effects, which arise due to an individual’s own participation in a representational practice, and indirect effects, which do not necessarily arise from actual use of the representation. The main purpose of the simple framework shown in Figure Chapter 1. -4 is to highlight this distinction. Representational Practice Direct Effects Literate Individual External Representation
"Products"
Indirect Effects
Cultural Mediation
Literate or Non-Literate Individual
Figure Chapter 1. -4. An individual’s knowledge can change through interaction with an external representation. In addition, the products of this interactions may be “fed back” into the knowledge of literate as well as non-literate individuals.
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I will take a moment to walk the reader through Figure Chapter 1. -4. To begin, we have a “literate” individual, defined as such by their ability to participate in a representational practice. Following the structure provided by this practice, this individual interacts with an external representation. Participation in this practice may have a direct (unmediated) affect on the knowledge or thought processes of the literate individual. In addition, there may be some outcome of this interaction, a “product” whose nature is shaped by the external representation and the associated practice. This product may then be processed by society and ultimately fed back into the knowledge of individual literates or non-literates. Also note that the products of representational activity may feed back into a reshaping of the representational practice itself. This simple framework will guide us in the discussion that follows. In the remainder of this section I will go into slightly more detail concerning some particular notions regarding individual knowledge and external representations. Where appropriate, I will relate these notions to the framework presented in Figure Chapter 1. -4. A Òmore powerful systemÓ One stance that we can take is that the literate individual and external representation, interacting as in Figure Chapter 1. -4, constitute a system whose properties we can study. The point here, which has been made by many researchers, is that this system may in some ways be “more powerful” than the individual acting alone. Before saying how this stance reflects on the relationship between external representations and knowledge, I want to take a moment to explore this view. The “more powerful system” notion has many roots, one of which is in the work of Jerome Bruner (Bruner, 1966). Bruner argues that cultures provide individuals with a number of “cultural amplifiers,” artifacts and techniques that amplify the capacities of individuals, both physical and mental. More recently Norman (1991), following Cole and Griffin (1980), argued that the amplifier metaphor is somewhat misleading. Norman does allow that a person-artifact system can indeed have enhanced capabilities when compared with an individual acting alone—in fact, this observation is central for him—but he does not believe that this enhancement is accurately described as a process of “amplification.” Instead, he argues that artifacts that enhance human cognitive capabilities do so by changing the nature of the task that an individual must perform, not by “amplifying” individual action.
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One common way in which the system view is applied is to describe a person-artifact system as having an increased memory capacity or increased knowledge base to draw on. Consider the system consisting of a literate person together with a pencil and a sheet of paper. In a sense, this system may be said to possess a “better memory” than the person acting alone. In fact, at least one author, Merlin Donald, takes the system view, together with an emphasis on memory-enhancing properties of external representations, as a central way to understand the functioning of the modern human mind. In The Origins of the Modern Mind, Donald attempts to lay out, in broad sweep, the evolution of the human mind (Donald, 1991). His story posits three stages in this evolution. The first stage, associated with Homo Erectus, is the development of “mimetic culture.” According to Donald, Homo Erectus was distinguished from apes in its ability to “mime, or re-enact, events.” During the second stage, associated with the appearance of Homo Sapiens, speech developed. In Donald’s scheme, both of these stages involved true biological evolution in the sense that there were concomitant changes in human biology. The third stage in Donald’s evolution of the mind is distinguished from these first two in that, rather than involving any biological changes, it involved the development of what Donald calls “external symbolic storage.” Donald’s external symbolic storage is precisely what we have been calling external representations. For Donald, the “storage” feature gets the lion’s share of his emphasis: External symbolic storage must be regarded as a hardware change in human cognitive structure, albeit a nonbiological hardware change. Its consequence for the cognitive architecture of humans was similar to the consequence of providing the CPU of a computer with an external storage device, or more accurately, with a link to a network (p. 17).
To be fair, Donald allows a somewhat more complex role for external representations than a simple memory-aid story. Because his external symbolic storage supplements short-term as well as long-term memory, its presence can alter the nature of short-term thinking processes. To Donald, it is as if the very “architecture” of the mind has been augmented; he treats external representations as comparable to an additional supply of memory that just happens to be connected through the visual channel. So what does the “more powerful system” view tell us about the relation between external representations and the knowledge of individuals? The answer is that it does not tell us much. For the simple reason that the focus of a system analysis is on the system and not on an individual, it does not require many presumptions about how the thought processes of individuals must change. However, the fact that the system view allows external representations to profoundly enhance 25
cognition, without requiring changes in the nature of thought processes, does have important implications for this discussion. Because it can predict strong effects without positing any major, or even permanent, changes in the thought processes of individuals, the system view frees us to take the stance that there are no such important changes. Incidentally, symbol use in physics is often understood through the lens of the “more powerful system” view. The idea is that when equations are written on a sheet of paper, they can be manipulated following a relatively simple set of rules. Thus computations and derivations, which might be difficult or impossible otherwise, are made easy or, at least, possible. Again, the point is that this does not necessitate any strong influence on the abilities of individuals in the absence of external symbols. The person-representation system is capable of performing remarkable computations, but these remarkable computations still only require the person to perform a relatively simple task, the rule-based manipulation of symbols. The Òresidual effectsÓ of symbol use Restricting ourselves to the system view really begs the question of whether there are any residual effects on individuals due to the use of external representations. So now let us consider this question explicitly. Suppose a person is in interaction with an external representation as in Figure Chapter 1. -4. What types of residual effects might there be due to the direct interaction of person and representation? Here I consider a few possibilities. Knowledge is adapted by and for symbol use Recall that, as I explained above, Norman argued that a person-artifact system gets its enhanced abilities by allowing the person to perform an alternative, easier task. The important observation here is this: Although the artifact allows a person to perform a task that is somehow easier, the individual may still need to learn or otherwise develop new capabilities in order to perform this alternative task. Thus we see an opening for the possibility of residual effects. The knowledge and capabilities of an individual may need to be changed and reorganized for the specific tasks required by representational practices. In fact, the positive results found by Scribner and Cole can be understood in this way. The representational practices associated with Vai script, such as letter-writing, could require individuals to develop certain skills. It is interesting to note that these required changes in what an individual must learn are not always seen in a positive light. To illustrate, I begin with an analogy. Imagine that a person spends a significant amount of time using some sort of physical support, such as a crutch. It is possible 26
that certain portions of this person’s physique will gradually be modified. Some muscles may atrophy. Or, for example, if the person is using a wheelchair, the muscles in their arms might grow stronger while the muscles in their legs grow weaker. In an analogous manner, Plato worried, in the Phaedrus, that the use of writing would dampen a scholar’s ability to memorize information (Plato, trans. 1871). Similar observations can be made for the case of symbol use in physics. If students can learn to remember and use certain equations, they may never need to learn the conceptual information that is somehow embodied in these equations. They can know the equations and not their meanings. In this way, the use of external representations could have a direct effect on the nature of an individual’s understanding of physics, literally what an individual “knows.” A student may, in a sense, just know enough to fill in the gaps between the equations and to use equations effectively. The point I am trying to make here is, I believe, relatively uncontroversial. If we spend time engaged in a particular variety of symbol use, then our knowledge and capabilities will be adjusted for the requirements of that variety of symbol use. We will learn what we need, and not learn what we do not need. In this way, the knowledge of an individual may be adapted by and for symbol use. Of course, in any particular case, we can argue about the type and degree of adaptations, but the general point still holds. A symbolic world of experience Now I want to propose a second way to think about possible residual effects on individual symbol users due to direct interaction. This viewpoint is not logically distinct from that presented just above, but I believe that announcing it will help us to recognize and understand some additional possible relations between knowledge and external representations. Here, I connect to Feynman’s intuition that one must have a certain type of mathematical-symbolic experience in order to appreciate the beauty of nature. I summarize this viewpoint as follows. Start with the (admittedly vague) presumption that individual knowledge derives from experience. For example, much of our physics-relevant knowledge comes from our experiences living in the physical world. Before anybody takes their first physics course, they know what a physical object is, just because they are a person living in the world. They have held objects, kicked them, painted them, etc. Now here is the key move associated with this viewpoint: We can simply take the experience of manipulating symbols to be a
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new class of experience. Working with equations—reading them, writing them, manipulating them, talking to other people about them—is a kind of experience that need not be any less meaningful than our experience in the physical world. And this experience forms part of the basis for our understanding of physics. Instrumental Psychology and internalization In speaking of a “symbolic world of experience,” I make it clear that the character of human experience is shaped by humans. In part, our experience consists of a world of our own devising. One theory of psychology, the “Instrumental Psychology” of Lev Vygotsky, puts this idea at its center. For Vygotsky, it is essential that individuals are subject to stimuli not only from the “natural environment,” they also actively modify their stimuli. Furthermore, the tools for modifying and mastering the environment are inventions that have been perfected over the course of human history, and which are somewhat specific to the culture in which the individual is embedded. Strikingly, for Vygotsky, not only is it the case that the knowledge of individuals is somehow affected by stimuli of human devising, these stimuli happen to be the origins of the higher, uniquely human forms of mental activity. External, social activity is internalized and gives human thought its particular character. In addition, spoken language—a specific external representation—is assigned a special role in this process. For Vygotsky, it is language that gives order to the initially undifferentiated stream of infant thought. In mastering external speech, the child starts from one word, then connects two or three words; a little later, he advances from simple sentences to more complicated ones, and finally to coherent speech made up of a series of such sentences; in other words, he proceeds from a part to the whole. In regard to meaning, on the other hand, the first word of the child is a whole sentence. Semantically, the child starts from the whole, from a meaningful complex, and only later begins to master the separate semantic units, the meanings of words, and to divide his formerly undifferentiated thought into those units. The external and the semantic aspects of speech develop in opposite directions—one from the particular to the whole, from word to sentences, and the other from the whole to the particular, from sentence to word. (Vygotsky, 1934/86, p. 219) [My emphasis.]
Vygotsky takes us close to what is perhaps the strongest form of “direct effect” one can imagine, the view that thought is, in some sense, an internalized version of external symbolic activity. For example, someone who adopts this most extreme of views might assert that human thought is internalized speech. While Vygotsky does not quite propose this extreme view, he does believe that the use of external signs is primarily responsible for the shaping of the highest levels of individual thought.
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Indirect effects of symbol use In the preceding sub-section I discussed residual effects on the knowledge of individuals due to direct interactions with external representations. Although they will not be of primary concern in the rest of this work, I want to briefly consider some hypotheses concerning indirect effects. The general form of these hypotheses is that the use of certain external representations shapes a society’s knowledge products in a certain way, and these knowledge products are then internalized by individuals. Again, these are indirect in the sense that the knowledge of an individual may be affected even if they are not literate with the external representation in question. One example of such an indirect effect I call “epistemic structuring.” To illustrate I turn once again to Jack Goody. In The Domestication of the Savage Mind, Goody extends his discussion of lists from purely administrative lists, such as the one with which I began this chapter, to lists designed for a more academic purpose. In particular, he describes the “Onomasticon of Amenope,” which was composed by an Egyptian around the end of the Second Millennium BCE. Apparently, this document was the result of an attempt to make a truly exhaustive list of “all things that exist.” It bears the title: Beginning of the teaching for clearing the mind, for instruction of the ignorant and for learning all things that exist: what Ptah created, what Thoth copied down, heaven with its affairs, earth and what is in it, what the mountains belch forth, what is watered by the flood, all things upon which Re has shone all that is grown on the back of the earth, excogitated by the scribe of the sacred books in the House of Life, Amenope, son of Amenope. (p. 100)
For Goody, the fact that this document takes the form of a list is essential; the representational form has the effect of shaping the ideas that are represented: We can see here the dialectical effect of writing upon classification. On the one hand it sharpens the outlines of the categories; one has to make a decision as to whether rain or dew is of the heavens or of the earth. At the same time, it leads to questions about the nature of the classes through the very fact of placing them together. How is the binary split into male and female related to the tripartite division between man, stripling, and old man? (p. 102)
Collins and Ferguson (1993) would say that the list is functioning as an “epistemic form,” a “generative framework” for knowledge. Goody’s point is that the mere fact of listing leads to reflection and progress in elaborating the categories used for describing the world. The idea here is that the “product” of the person interacting with the list is shaped by the nature of the list structure. Then, once this “product” is produced, it can be fed back into the knowledge of individuals, both literate and non-literate. Furthermore, after it is processed by
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society, the product need not even be in the form of a list; distinctions can be taught to nonliterate as well as literate individuals, without restriction to the list structure. It is important to note that these proposed indirect effects can vary greatly in how profoundly they purport to alter the nature of individual cognition. For example, we could imagine taking one of Amenope’s specific distinctions and teaching it to someone. This would not lead to a very profound change in the nature of that individual’s thought. However, some of the indirect effects proposed by researchers suggest more general changes in the nature of thinking. Above I mentioned Goody and Watt’s hypothesis that the invention of the Greek alphabet allowed the development of techniques for reasoning logically—techniques that could be learned by any individual. Similarly, other authors have argued that the appearance of literacy led to generally important intellectual achievements. David Olson, for example, makes some of his own very strong hypotheses concerning consequences associated with the development of writing (Olson, 1994). According to Olson, an important feature of writing, in contrast to spoken language, is that it must be interpreted without many of the contextual cues that are available during verbal discourse. For this reason, it is much harder for writers to convey their intended “illocutionary force.” Therefore, Olson argues, writing must contain additional clues that convey the author’s stance to what is said. For example an author must tell the reader whether they mean their statements to be taken literally or as metaphor; and an author must tell a reader whether they intend to be presenting an hypothesis or a fact. Olson’s big leap is that, for this reason, the development of writing had to be accompanied by developments in epistemology; the categories available for describing knowledge had to be made explicit and refined. It is in this manner, he argues, that categories like “fact” and “hypothesis” were born. With these new categories available, the cognitive processes of individuals—even non-literate individuals—would be generally enhanced. Summary: Where I will stand In this section I have briefly discussed some hypotheses concerning deep relations between external representations and the knowledge of individuals. Centrally, I distinguished the direct effects that arise from an individual’s use of an external representations with indirect effects, which can even influence non-literates. In this project, it will be helpful to think of my focus as being on
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direct, non-mediated effects of external representations. I will be interested in the knowledge and abilities that must develop specifically for symbol use in physics. Furthermore, I will lean heavily on the notion that physicists develop an intuition—they get a feel for physics—through participation in what I have called “the symbolic world of experience.” I will argue that, to know physics as physicists know it, one must be an initiate into the full world of physics practice, including learning to write and manipulate symbols, and learning to say things with and about equations in the manner of a physicist. Note that this view implicitly presumes that experience with symbols is not any less meaningful or important than other types of experience, such as daily experience interacting with objects in the physical world. In fact, a major part of the work of this document will be to illustrate that, for physicists and even for mildly advanced physics students, an equation is an element in a highly structured and meaningful world. However, my own orientation stops somewhat short of strong versions of the internalization view. I do not believe that symbol use is uniquely responsible for the development of human thought. Nor do I believe that direct interaction with external representations tends to engender broad changes in the character of the thought of individuals, such as a move toward more “logical” or “scientific” thought. External representations are a meaningful and important element of our experience, but activities with external representations are not unique in giving human thought its character. Finally, it should be noted that, throughout the discussion in this section, I have essentially presumed that it is useful to speak of knowledge as something that is localized in the minds of individuals. Though, to some, this may seem like an obvious and unproblematic move, some researchers have questioned whether it is appropriate and scientifically productive. Researchers such as Greeno and Moore (1993) and Hall (1996) take the stance that knowledge is most profitably understood as a theoretical entity that spans person and elements of the environment. Note that, since the environment may include external representations, this view provides a quite difference sense in which knowledge and external representations may be related. I will not adopt the stance that knowledge must be interactionally defined in this manner; throughout this document, I will talk as if knowledge is something that can exist localized in individuals. I will have somewhat more to say on this topic in Chapter 7, but I will make one additional point here. The notion that knowledge must be interactionally defined is an element of
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some versions of a more general viewpoint that is known under such names as “situated action.”1 Situated action, as compared to traditional cognitive science, tends to emphasize the influence of social and historical factors on individual behavior, as well as the importance of the immediate environment. In allowing that knowledge is, in some sense, dependent on specific practices, including the use of cultural artifacts such as equations, I hope to be taking into account many of the concerns that are central to situated action. Again, I will have more to say on this subject later.
Concerning Physics Learning and Instruction A central goal of this inquiry is to contribute to research in physics learning and instruction. We want to know what it means to understand physics, and why students do or do not succeed in learning this topic. Of course, the emphasis here remains on symbol use, how the using of equations contributes to and is part of physics understanding. In this section, I will describe how the current project ties into work that has already been done.
Figure Chapter 1. -6. A typical misconception.
Intuitive knowledge and the learning of physics In the early 1980’s, a number of studies appeared that seemed to indicate that physics instruction was, stated simply, a disaster. Although students could solve many of the problems that are typical of introductory instruction, such as the one shown in Figure Chapter 1. -1, they apparently could not answer some very basic qualitative questions. For example, a typical question asked students what happens when a ball is shot out of a spiral tube (Figure Chapter 1. -6). In response, students frequently stated that the ball continues to move in a curved path after leaving the tube (e.g., McCloskey, 1984). This answer is not correct; once the ball leaves the tube, it
1
See, for example, the special issue of the journal Cognitive Science that is devoted to this topic (ÒSituated Action,Ó 1993). 32
moves in a straight line, continuing in the direction that it had at the moment it exited the tube. These alternative answers were attributed to “misconceptions” or “alternative conceptions” by researchers.2 The presence of misconceptions such as this one is shocking because it suggests that students lack an understanding of some of the most fundamental aspects of Newtonian physics. For example, Newton’s first law states that: Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon. (Newton, 1687/1995)
What this means is that, if you go out into outer space and throw a baseball, the baseball continues forever, moving with the same speed and in the same direction that you threw it. It never slows down and it never turns, unless it happens to come near to a planet or asteroid. It does not matter how the ball is launched; the ball doesn’t care and it doesn’t remember. Even if it is launched from a spiral tube, once the ball leaves the tube, it just continues in a straight line. Thus the spiral tube misconception is in contradiction with one of the first things that students learn in their introductory physics classes, Newton’s first law. And such errors appear to be commonplace. Halloun and Hestenes (1985b), in a study that employed a multiple choice test given to hundreds of university physics students, found that 44% of these students exhibited the belief that a force is required to maintain motion. (Though individual students tended to answer questions inconsistently.) Nearly half, it seems, did not learn Newton’s first law! These results led to the obvious question: What is the problem with physics instruction? Researchers hypothesized that part of the answer was that students spent the vast majority of their time engaged in solving problems. And it was possible to get good at solving these problems without really understanding the content of the equations that you were using. I will return to this hypothesis below. But the over-emphasis on problem solving could not be the whole story. The fact that objects continue moving in the absence of forces seems like a straightforward notion, even if students were not given much time to mull it over. This fact even seems like something that can be memorized, whether or not you believe it. What is so intrinsically difficult about the notion that objects just 2
For a variety of accounts of student difficulties see, for example: R. Cohen, Eylon, and Ganiel, 1983; Halloun and Hestenes, 1985a & 1985b; McDermott, 1984; Trowbridge and McDermott, 1980 & 1981; Viennot, 1979. Also, see Smith, diSessa, and Roschelle (1993) for a critique of the misconceptions
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keep moving in a straight line? In some manner, it seemed, students were actually resistant to learning physics. What researchers came to realize was that students’ existing knowledge of the physical world has a strong impact on the learning of formal physics. In fact, students start physics instruction with quite a lot of knowledge about the physical world. Furthermore, this knowledge is based in quite a number of years of experience, so it might well be resistant to change. So, one of the major outcomes of this “disaster” research was the realization that students have prior knowledge of the physical world that is relevant to physics learning. The study of this prior knowledge—often called “intuitive physics” or “naive physics” knowledge—has become an endeavor of its own. When it takes its simplest form, naive physics research simply catalogs student’s alternative conceptions (e.g., Halloun & Hestenes, 1985b). Further research attempted to go beyond mere lists of misconceptions to develop generalizations about the nature of student beliefs. These generalizations were directed at answering two distinct questions: 1. Can intuitive physics be formalized? 2. How is intuitive physics actually realized as knowledge in people? The first question takes student’s answers to misconception-style questions and searches for a specification of some underlying physics that these answers reflect. To understand the nature of this approach, imagine that it was applied to the statements of expert physicists. In that case, the assumption is that this methodology would produce Newtonian physics (Newton’s three laws, literally) as the formalization of a physics underlying expert answers. However, in searching for a similar formalization of intuitive physics, researchers must make many questionable assumptions. In particular, they must interpret students’ answers to questions as reflecting more general beliefs. In so doing, they beg the question of whether a formalization exists. Furthermore—and this is the point of the second question—even once you have found a formalization, you still do not know how this physics is realized as knowledge in people. For example, even if we know that expert physics formalizes as Newton’s laws, we still know very little about what knowledge experts actually have. Most of the first formalizers, such as Halloun and Hestenes (1985b) and Viennot (1979), gloss this issue with statements like: “[students show] the
perspective. 34
belief that, under no net force, an object slows down” (Halloun & Hestenes, 1985b). What is left often by such assertions is how such a belief is realized as knowledge. Do students actually possess propositional knowledge of this form? Or is there some more fundamental type of knowledge that causes students to behave, in certain circumstances, as if they have this particular belief? Throughout this manuscript, I intend particularly to build on the work of one researcher who is self-consciously directed at answering the second question. In his Toward an epistemology of physics, Andrea diSessa sets out to describe a portion of intuitive physics knowledge that he calls the “sense-of-mechanism” (diSessa, 1993). The idea is that elements of the sense-of-mechanism form the base level of our intuitive explanations of physical phenomena. As an example, diSessa asks us to think about what happens when a hand is placed over the nozzle of a vacuum cleaner. When this is done, the pitch of the vacuum cleaner increases. According to diSessa, the way people explain this phenomena is they say that, because your hand is getting in the way of its work, the vacuum cleaner has to start working much harder. The point is that this explanation relies on a certain primitive notion: Things have to work harder in the presence of increased resistance if they want to produce the same result. diSessa’s program involves the identification of these primitive pieces of knowledge which he calls “phenomenological primitives” or just “p-prims,” for short. (The p-prim that is employed to explain the vacuum cleaner phenomenon is known as “OHM’S P -PRIM .”) The word “phenomenological” appears in the name of this knowledge element in part because they are presumed to be abstracted from our experience. OHM’S P -PRIM , for example, is abstracted from the many experiences we have in the physical world in which things have to work harder in the presence of increased resistance. diSessa lists a variety of p-prims, including p-prims pertaining to force and motion (e.g., FORCE AS MOVER , DYING AWAY ) and p-prims pertaining to constraint phenomena (e.g., GUIDING , ABSTRACT BALANCE). Furthermore, during physics instruction, the senseof-mechanism is refined and adapted, and comes to play an important role in expert physics understanding. I will have more to say about diSessa’s theory in later chapters, particularly in Chapter 5. Problem solving in physics Contemporaneous with this research into conceptual understanding and naive physics, a second and essentially disjoint set of studies were examining how students—and, to a lesser extent, expert physicists—solve textbook physics problems (Bhaskar & Simon, 1977; Chi,
35
Feltovich, & Glaser, 1981; Larkin, McDermott, Simon, & Simon, 1980). In its earliest incarnations, these studies of physics problem solving were strongly influenced by prior research into human problem solving, conceived more generally. For example, some early research in artificial intelligence and cognitive science examined how people approached the solution of the famous “Tower of Hanoi” puzzle. (Simon, 1975, is a typical example.) The Tower of Hanoi puzzle involves a set of rings that can be moved among three pegs (Figure Chapter 1. -7). The object of the puzzle is to move all of the rings from the left hand peg to the far right hand peg. The catch is that at no time can a larger ring be on top of a smaller ring. Furthermore, only one ring may be moved at a time.
Figure Chapter 1. -7. The Tower of Hanoi puzzle
An essential feature of this puzzle is that, at any time, there are no more than a few possible moves. You can only move the topmost ring on any peg to one of the other pegs. And many such moves will be prohibited because of the rule that larger rings cannot be on top of smaller rings. Furthermore, the current state and desired end-state of the puzzle are clearly and simply defined. In this early cognitive science literature, the solution of the Tower of Hanoi puzzle was described as a search through the space of the possible states of the puzzle. From this viewpoint, the most interesting question to be answered was exactly how a person goes about searching this space; in other words, we would want to know how, at any time during the solution of the puzzle, a person chooses among the few possible available moves. What Simon (1975) found was that problem-solvers could be well described by a relatively simple strategy, what he called “meansends analysis.” In means-ends analysis, a problem solver chooses the move that appears to most decrease the perceived difference between the current state and the goal state. Initially, this research was extended to physics problem solving in a quite straightforward manner (Bhaskar & Simon, 1977; Larkin et al., 1980). The notion was that moves in the space of equations written on a sheet of paper could be treated much like moves in the solution of a puzzle. More specifically, physics problem solving was seen to work like this: The student or expert reads the problem and takes note of the quantities that are given in the problem and the quantities that are needed. Then the problem solver writes, from memory, equations that contain these various quantities as variables. Once the equations are written, the search can begin; the equations are
36
manipulated until a route from the given quantities to what is given is obtained. The problem shown in Figure Chapter 1. -1 provides a simple example. In that problem, the student is given the initial velocity, vo , the acceleration, a, and asked for the time. Once the equation v = vo + at is written, it can simply be solved for the time, t. Of course, in more complicated problems, some type of strategy is needed to find your way through the space of equations from the givens to the solution, just as means-ends analysis was needed in the Tower of Hanoi problem. Larkin and colleagues (1980) found the interesting result that novices work by “backward chaining:” They start with the quantity that they want to know, then write down an equation in which that quantity appears. Each of the unknowns in this new equation then becomes a new quantity to be determined, and the process iterates. In contrast, experts were found to “forward chain;” that is, work forward from the quantities given by writing equations in which these quantities appear, and then using these equations to find new quantities. ÒUnderstandingÓ and problem solving I am now approaching the point where I will say how the current inquiry fits in with all of this prior research. First, notice that there are some obvious limits in the above problem solving research. As I have presented it, the physics problem solving research does not posit much of a role for anything like “understanding” during problem solving and the use of equations. The story is simply that you see the quantities that appear in the problem, you dredge equations out of memory and write them down, then you manipulate the equations according to simple rules to find an answer. So where is “understanding?” Larkin and colleagues (1980) themselves comment: The major limitation we see in the current models is their use of an exceedingly primitive problem representation. In fact, after the initial problem representation, these models work only with algebraic quantities and principles described by algebraic equations. Thus, in a sense, they have no knowledge of physics, but only the algebraic representation. This may actually not be too bad for capturing the performance of novice solvers, who have little knowledge of physics, but much more of algebra. However, it is certainly inadequate for capturing the work of more competent solvers.
But “understanding” may be tricky to define and to recognize. Perhaps, knowing equations and knowing how to execute the above problem solving process is exactly what constitutes understanding. Even if we adopt this improbable—and, I think, unpalatable—stance, it is still possible to find some quite obvious and definite lacks in the above view of equation use. Most obvious of these lacks is the fact that, if equations are only written from memory and are not “understood” in any deeper sense, then how is it possible that people ever compose completely new equations, corresponding to newly discovered principles and never-before-modeled systems? 37
There are some obvious ways out of this bind. One could maintain, quite reasonably, that the above model of problem solving is only designed for a certain regime of problem solving involving well-known principles and systems. But, if we accept this view, then we are simply left with the new problem of describing the unaccounted for regimes of equation use and problem solving. Furthermore, one might maintain that the composition of new equations is something that is only done rarely and by scientific geniuses, like Einstein and Feynman. The rest of us can then make relatively mechanical use of the equations that they develop for us. I will attempt to show, in succeeding chapters, that this is simply not how things are. The “naive physics” researchers may seem to have an advantage here, since they discuss behavior that, at least intuitively, appears closer to “understanding.” But these researchers have yet to tell us precisely how intuitive physics and conceptual knowledge is employed during the use of equations. In what way do physicists make use of the sort of qualitative understanding that these researchers are concerned with during problem solving? Is there any relation or are there simply two sets of things to learn to become a physicist? This is where I come in. In a sense, this inquiry will combine these two areas of research that, to this point, have remained relatively disjoint. I will be examining phenomena akin to misconceptions, naive theories, and p-prims, but in the context of problem solving and equation use. Strikingly, we will see that there is actually a very strong connection between intuitive physics knowledge and equation use; equations are understood and generated from knowledge that is very closely related to intuitive physics knowledge. Describing this tight link between equations and understanding is the central focus of this work. Before proceeding, I want to note that the above account of the problem solving literature is a little unfair. The same researchers have subsequently elaborated their work to account for some of the difficulties that I have mentioned. A notable example is Larkin (1983), which I will discuss extensively in Chapter 3. Also, the substantial success of the problem solving literature deserves some emphasis. These researchers were very successful in building models that accurately described subjects’ behavior in the regimes of problem solving that they studied. Thus, it is likely that future research can profitably build on this work. On the teaching of Òconceptual physicsÓ I want to close this section with a note concerning how the results of the above research on physics learning, particular the disastrous results reported by the misconceptions research, have
38
been interpreted as a prescription for improving physics instruction. The fact that intuitive physics exists has led to the notion that it should be directly addressed in some way. Take the example of the spiral tube misconception and students’ failure to understand the implications of Newton’s First Law. Why not take the time to emphasize the (qualitative) principle that objects continue to move in a straight line with a constant speed in the absence of intervening forces? This prescription—emphasize the qualitative principles at the expense of problem solving—has been taken up in a variety of courses that I will refer to as “Conceptual Physics” courses.3 These courses operate under the principle that they should emphasize the “conceptual” content of physics. In practice, this means that there is little problem solving, and few equations. Though, of course, this varies greatly from course to course. Given the viewpoint I am beginning to lay out here, it is clear that we should at least take the time to reconsider this prescription for instruction. I have proposed replacing algebraic notation with a programming language. Conceptual Physics courses make a similarly strong move, they replace equations with plain language statements. If knowledge is inextricably representational, then what are the consequences if the equations are removed altogether? What would Feynman say? I believe that the consequences are not necessarily bad; they may very well not be worse than the consequences of my proposal to use programming languages. But I do want to make the point that we need to understand what we’re doing if we choose to teach physics in the manner of Conceptual Physics. I challenge the assumption that—in physics or any domain—we can separate the “conceptual” from the symbolic elements of a practice for the purposes of instruction. In removing equations from the mix, we change the constitution of understanding. This does not imply that we cannot teach physics without equations. However, it does imply that equation-free courses will result in an understanding of physics that is fundamentally different than physics-asunderstood-by-physicists. Again, I must make clear that I do not mean to critique Conceptual Physics courses, rather I hope to better understand the consequences of omitting equations. In fact, in the vein of these more “conceptual” techniques, there is already a history of striking successes and exciting proposals. For example, White and Horwitz designed and implemented a middle school curriculum based around computer models. The students in this curriculum performed better on
3
See, for example, the textbook ÒConceptual PhysicsÓ (Hewitt, 1971). This textbook has been reprinted in a number of more recent additions. 39
many tasks than students in a traditional high school course (White, 1993b). Clement, through his technique of “bridging analogies,” has had success in training expert intuition (Clement, 1987). And I suspect that even many working physicists could learn a great deal from Haertel’s impressive reworking of the entire topic of electricity and magnetism in qualitative terms (Haertel, 1987). All of this published research is in addition to the many instructors at universities and high schools that are enjoying success in designing and teaching Conceptual Physics courses. My work here will help us to better understand the character of conceptual physics, so that we can understand and better judge the results of such an approach, in clear contrast to alternatives. Incidentally, the instructional move embodied in Conceptual Physics courses appears to be common in much of the recent innovation in mathematics and science instruction. “Teaching for understanding” seems to frequently involve a scaling back in the use of formalism. I do not intend necessarily to question whether this is a productive instructional maneuver. However, if the concerns expressed here are valid, then this sort of “teaching for understanding” may lead to new sorts of understanding of mathematics and science.
Chapter Overview Part 1. Algebra-physics and the Theory of Forms and Devices The remainder of this dissertation is divided into two major parts. In Part 1 I will describe the Theory of Forms and Devices, drawing on examples from the algebra-physics portion of the data corpus. Thus, the first part of the dissertation will perform the dual roles of describing the theory and applying it to describe algebra-physics. Part 1 is divided into six chapters, Chapters 2 through 7. In Chapter 2 I take some steps preparatory to the introduction of the major theoretical constructs in the following chapters. This chapter will include some accounts of students using equations, accounts which are designed simply to show that students can use equations in a meaningful manner. At the end of this chapter I will also introduce the notion of “registrations.” In Chapter 3 I will embark on a detailed discussion of symbolic forms. I will explain the nature of symbolic forms in more detail, and I will individually discuss each of the specific symbolic forms needed to account for the observations in the algebra data corpus. This will include numerous examples from the data. In addition, I will argue against some competing viewpoints, including what I call the “principle-based schemata” view. 40
Chapter 4 introduces and describes representational devices. As with symbolic forms, I will discuss each of the individual devices needed to account for the observations in the data corpus and I will draw examples from the data to illustrate my points. Along the way, I will have reason to discuss what it means to “interpret” a physics equation. Importantly, we will see that the “meaning” of a physics expression, as it is embodied in students’ interpretive utterances, is highly dependent on the concerns of the moment. There is no single, universal manner in which meaning is attached to equations. In Chapter 5, I will speculate on the development of the knowledge system described in the preceding chapters. My basic tactic here will be to draw on existing accounts of students’ knowledge as it is found prior to instruction, namely the research on intuitive physics knowledge that I described above. Most prominently, I will draw on diSessa’s theory of the sense-ofmechanism. By comparing diSessa’s account to my own model, I will be able to comment on the relationship between new and old knowledge, and I will construct hypothetical developmental paths to describe how symbol use may contribute to the development of physics understanding. One important result of this discussion will be the observation that expert “physical intuition” is sensitively dependent on the details of the symbolic practices of physics. In Chapter 6, I will present the results of applying the model systematically to describe the whole of the algebra data corpus. In that chapter, I will describe the algebra corpus in some detail, as well as the iterative process through which the data corpus was analyzed. In addition, the chapter will contain an attempt to characterize aspects of the algebra corpus in a quantitative manner. One reason for this quantitative characterization is to provide the reader with a means of drawing conclusions concerning the prevalence of the phenomena that I describe, without having to read the entirety of the data corpus. For example, through this quantitative characterization, we will arrive at a measure of how often students construct novel equations and interpret equations. Both of these phenomena turn out to be quite common. I will also present results concerning the frequency of individual forms and representational devices. The reader should be warned that, although the analysis process I will describe in Chapter 6 is meticulous and detailed, it does not provide an exact and rigorous method of extracting my model from the data. Instead, it is a highly heuristic process, and the recognition of specific knowledge elements in the data should be understood as rooted in my intuition and theoretical predispositions, as well as in the data itself. For this reason, much of the rhetorical burden of arguing for my view must be carried by Chapters 2 through 5.
41
In Chapter 7—the last chapter of Part 1—I will reflect on the theory, draw out some implications of the theory, and I will attempt to generalize the observations of the preceding chapters outside the immediate concerns of symbol use in physics. Included in this chapter will be a discussion of the implications of the preceding results for the relation between physics knowledge and external representations. In addition, I will contrast my own theoretical position with a variety of other theoretical accounts. Part 2. Programming-physics, its analysis, and a comparison to algebra-physics In Part 2, the discussion turns to programming-physics. This is the first time that programming will appear following this introductory chapter. The primary concerns of this part of the dissertation are to apply the Theory of Forms and Devices to describe programming-physics, and to use the theory to compare programming-physics and algebra-physics. Chapter 8 introduces the reader to the particular practice of programming-physics that was the subject of this study. I will describe the programming language used and give examples to illustrate the types of programs written by students. In Chapter 9, I apply the Theory of Forms and Devices to programming-physics. This will include a discussion of individual forms and devices with examples drawn from the programming data corpus. Particular attention will be paid to the forms and devices that do not also appear in algebra-physics. Chapter 10 presents the systematic analysis of the programming data corpus. The content of this chapter will be parallel to that of Chapter 6, in which I described the systematic analysis of the algebra corpus. In Chapter 11, I will use the analyses of algebra-physics and programming-physics in terms of forms and devices to compare these two practices. To briefly preview the results, I found that the distribution of forms in programming-physics was substantially similar to that in algebra-physics; however, an important new class of forms appeared that was completely absent in the algebra corpus. These new forms relate to schematizations of processes. With regard to representational devices, there were some interesting differences in the frequency with which the various classes of devices were employed. Finally, in Chapter 12, I will summarize the thesis and trace the instructional implications of this work. This will include a discussion of the implications of this work for traditional physics
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instruction. I will argue that, given an improved understanding of how students use and interpret equations, we may be able to recognize and remediate some types of student difficulties. Finally, not least among the educational implications of this work is the fact that it considers the possibility of a substantially altered practice of instruction, built around student programming. In the last chapter, I will finally calibrate the strengths and weaknesses of this alternative approach, to the extent that this study allows.
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Chapter 2. Prelude to Theory
Rij − 12 gij R − Λgij = 8πGT ij In the years around 1916, Albert Einstein presented the world with a new theory of gravitation. And a truly new theory it was. In the existing Newtonian theory, gravity was “explained” by the simple statement that there is an attractive force between every two masses. Pick any two masses, m1 and m2, separated by a distance r. Then each of those masses feels a force, directed toward the other mass, of size
F=
Gm 1 m 2 r2
In Einstein’s theory, the nature of gravity was wholly reconceptualized. Rather than explaining gravity by an appeal to some unexplained “action at a distance,” Einstein reduced gravity to the tendency of matter to distort space. An object, like the sun, reshapes the space around it. Then when other objects—like the Earth—travel through this distorted space, their motion is directed by the shape of space, just like a marble moving around the inside of a curved metal bowl. At the heart of this new theory is Einstein’s “Field Equation,” written at the head of this chapter, which relates the curvature of space to the distribution of matter. What we need to note about this equation is that, somehow, a portion of Einstein’s reconceptualization of gravity is embodied in this expression. Einstein began with the notion that matter distorts geometry, mixed in the old Newtonian theory and some essential principles, and produced a new equation, one that did not exist before.4 As we saw in Chapter 1, the existing literature concerning physics problem solving is not capable of accounting for Einsteinian behavior of this sort. In that literature, problem solving is dependent on a database of known equations, so it cannot explain the development of wholly new
4
Physicists might complain that textbooks do, in fact, contain ÒderivationsÓ of EinsteinÕs field equation. However, a close examination of these ÒderivationsÓ reveals that they do not begin with accepted physical laws and then perform logical manipulations to derive consequences of these laws. Rather, the Field Equation is derived in part from some assumptions concerning what a geometric theory of gravity must look like, such as what quantities the curvature of space may reasonably depend on. 44
equations, or modifications based on conceptual considerations. However, it may not be fair or appropriate to think of this as a deficiency in that literature or even to think of it as a task for education research. The focus of the problem solving literature (and of this study) is on physics learning and instruction. Our job here is to describe what students do and what we can reasonably expect them to learn in an introductory course. And it’s possible that the construction of novel equations is solely the province of physical geniuses, like Einstein and Feynman, or at least expert physicists. Even if such invention does not require special mental prowess, it may nonetheless depend on possessing some esoteric skills. Thus, the rest of us, including introductory physics students, should perhaps only expect to use the equations that the experts write. If this is the case, then accounting for this behavior may be less important for educationally directed research, both because it is not really relevant to learning, and simply because it is relatively rare. In the next sections of this chapter, I will take the first step toward showing that this view is untenable. Equations have a perceived meaningful structure for all initiates in physics, even introductory physics students. We will see that students can and do construct novel equations, and they use equations in a meaningful manner, flexibly constructing their own interpretations. In this chapter, I will begin by presenting some examples designed only to show that students can, at least sometimes, construct and interpret equations. These examples will also provide a first taste of what these behaviors look like. Then, in the latter part of this chapter, I will take some of the first steps toward a theoretical account of this behavior. In subsequent chapters I will complete the theoretical account and I will show through systematic quantitative analysis that these phenomena are actually quite common.
Young Einsteins Of all the tasks that I asked students to do, students found a task that I call “The Shoved Block” problem the most exasperating. They found it exasperating, not because it is difficult, but because it is a seemingly easy problem that produces counter-intuitive results. In the Shoved Block problem, a block that rests on a table is given a hard shove. The block slides across the table and eventually comes to rest because friction between the block and the table slows the block down. The problem then asks this question: Suppose that the same experiment is done with a heavier block and a lighter block. Assuming that both blocks are started with the same initial velocity, which block travels farther?
45
v = vo v = vo
Figure Chapter 2. -1. A heavier and a lighter block slide to a halt on a table. If they are shoved so as to have the same initial speed, they travel exactly the same distance.
The counter-intuitive answer to this question is that the heavier and lighter block travel precisely the same distance (refer to Figure Chapter 2. -1). In a later chapter I will describe how students used equations to obtain this result. However, for the present purposes, I want to describe how the dissatisfaction of a pair of students with this counter-intuitive result led them to invent their “own brand of physics.” Students actually express two competing intuitions concerning the Shoved Block. The most common intuition is that the heavier block should travel less far. The reason given is that this is true because the heavier block presses down harder on the table, and thus experiences a stronger frictional force slowing it down. The alternative intuition is that the heavier block should travel farther, since it has more momentum and is thus “harder to stop.” In fact, it is because these two effects precisely cancel that the heavier and lighter block travel the same distance. The pair of students I am going to discuss here adopted the latter intuition, that heavier blocks should travel farther. And they clung to this intuition, even after they had used equations to derive the “correct” result. This was especially true of one student in the pair, Karl5, who argued that it was simply “common sense” that the heavier block should travel farther. Karl
Yeah, that's true. But I still say that the heavier object will take the longer distance to stop than a lighter object, just as a matter of common sense.
His partner, Mike, was somewhat less worried by the counter-intuitive result, but he was willing to stop and discuss what might be changed in their solution. Eventually, Karl came to the conclusion that “in real life” the coefficient of friction probably varies with weight in such a way that heavier blocks travel farther. Karl
I think that the only thing that it could be is that the coefficient of friction is not constant. And the coefficient of friction actually varies with the weight.
The coefficient of friction is a parameter that determines the size of the frictional force. Students are taught in introductory physics classes that this parameter can be treated as a constant that
46
depends only on the nature of the materials involved, such as whether the objects are made out of wood or steel. Given the counter-intuitive result, Mike and Karl were willing to reconsider the assumption that the coefficient of friction is a constant. After some very minimal cajoling, (I said only “Do you want to try putting in maybe some hypothetical weight dependence to that”), they set out to compose their own expression for the coefficient of friction. Karl began by expressing their core assumption: Karl
I guess what we're saying is that the larger the weight, the less the coefficient of friction would be.
To obtain their desired result—that heavier masses travel farther—they needed to have the coefficient of friction decrease with increasing weight. In other words, friction must work less on a heavier mass. Over approximately the next nine minutes, they laid out some additional properties that they wanted their new expression to have. Here’s an excerpt (refer to Appendix A for a key to the notations used in transcripts): Karl
Well yeah maybe you could consider the frictional force as having two components. One that goes to zero and the other one that's constant. So that one component would be dependent on the weight. And the other component would be independent of the weight.
Mike
So, do you mean the sliding friction would be dependent on the weight?
Karl
Well I'm talking about the sliding friction would have two components. One component would be fixed based on whatever it's made out of. The other component would be a function of the normal force. The larger the normal force, the smaller that component. So that it would approach a - it would approach a finite limit. It would approach a limit that would never be zero, but the heavier the object, the less the coefficient of friction at the same time. (…)
Mike
I don't remember reading that at all. [laughs]
Karl
See, I'm just inventing my own brand of physics here. But, if I had to come up with a way if I had to come up with a way that would get this equation to match with what I think is experience, then I would have to - that's what I would have to say that the=
Mike
Actually, it wouldn't be hard to=
Karl
=the coefficient of friction has two components. One that's a constant and one that varies inversely as the weight.
In this passage, Karl outlines what he wants from the new expression. He decided that the coefficient of friction should have two components. One of these components is independent of weight, the other decreases with increasing weight. After some fits and starts, Mark and Karl, settled on this expression for the coefficient of friction: µ
µ = µ 1 + C m2
5
All student names are fictitious. 47
Here µ1, µ2, and C are constants and ‘m’ is the mass. This expression does have some problems, most notable of which is the fact that the coefficient of friction tends to infinity as the mass goes to zero. But this expression does capture much of what Mike and Karl wanted to include, it decreases with increasing mass, and it approaches a constant as the mass becomes infinitely large. The most important thing to note about this expression is that you will not find it in any textbook. Mike and Karl did not learn it anywhere, and they didn’t simply derive it by manipulating equations that they already knew; instead, they built this expression from an understanding of what they wanted it to do. As Karl says, he was “just inventing his own brand of physics.” Their expression is not as fancy and subtle as Einstein’s Field equation, but it is clearly a non-trivial, novel construction. It seems that the construction of novel equations is not only the province of experts like Einstein. The question for us to answer in the upcoming chapters is: How did they do this? How did they know to write a ‘+’ instead of a ‘×’ between the two terms? How did they know to put the m in the denominator? These are precisely the behaviors we need to be able to explain; we want to know what knowledge gives students the ability to connect conceptual content and equations. After a few more preliminaries, that is what we will do.
Sensible Folk Episodes like the last one were a little uncommon in my data corpus, though not really rare. In fact, like Mike and Karl, all of the pairs in my study demonstrated an ability to compose new expressions at some point during their work, and without obvious difficulty. Nonetheless, these episodes were uncommon because most of the physics problems that students are asked to solve do not require them to invent novel equations. In this next example, I am going to describe an episode that should appear almost painfully mundane to those who are familiar with introductory physics. The important fact about this episode is that it includes an instance of a student interpreting an equation. Such episodes are not uncommon because, although students to not need to invent totally new expressions very often, they do frequently need to interpret equations that they derive. In this episode, Jack and Jim were asked to solve the following problem, in which a mass hangs at the end of a spring: A mass hangs from a spring attached to the ceiling. How does the equilibrium position of the mass depend upon the spring constant, k, and the mass, m? 48
This problem was extremely easy for them and they spent only about 2 1/2 minutes working on it. Their board work is reproduced in Figure Chapter 2. -2.
k
x
F=-kx m F=ma
kx=mg x=
mg k
Figure Chapter 2. -2. Jack and Jim's solution to the mass on a spring problem.6
Jack and Jim began by explaining that there is a force from gravity acting downward, a force from the spring acting upward, and that these forces must be equal for the mass to hang motionless: Jim
Mmm. Well, there's the gravitational force acting down. [w. Arrow down; F=ma next to arrow.] And then there is
Jack
a force due to the spring // holding it up. // a force due to the spring which I believe is,, [w. F=-kx] Is equal
Jim to that.
Jack and Jim know that the force of gravity on an object is equal to mg where m is the mass of the object and g is the gravitational acceleration, a constant value. They also know that springs apply a force proportional to their displacement. If a spring is stretched an amount x from its rest length—the length it likes to be—then it applies a force kx, where k is a constant known as the “spring constant.” Jack and Jim went on to equate these two forces, writing kx=mg. Then they solved for the displacement x to obtain their final expression: x=
mg k
Then Jim immediately explained that this is a completely sensible result: Jim
Okay, and this makes sort of sense because you figure that, as you have a more massive block hanging from the spring, [g. block] then you're position x is gonna increase, [g.~ x in diagram] which is what this is showing. [g.~ m then k in x=mg/k] And that if you have a
6
All diagrams are my own reproductions of student work. In creating these reproductions I attempted to retain the flavor of student drawing while rendering important elementsÑespecially symbolic expressionsÑclearly. 49
stiffer spring, then you're position x is gonna decrease. [g. Uses fingers to indicate the gap between the mass and ceiling.] That why it's in the denominator. So, the answer makes sense.
Jim says that this expression makes sense for two reasons. First, he tells us that the equation says that, if you increase the mass (with k held constant) then x increases. Second he considers the case where the spring is made stiffer; i.e., k is increased. In this case, he says, the equation implies that x will decrease. Presumably, these observations make sense to Jim because of knowledge that he has concerning the behavior of springs. I consider this statement by Jim to be an “interpretation.” Jim is not merely providing us with a literal reading of the symbols on the page; he does not simply say “X equals M G over K.” Rather, he tells us what the equation “says.” For example, one thing that the equation “says” is that if you increase k then x decreases. Even this very simply interpretation has a number of interesting characteristics that are worth mentioning. In saying that as k increases x decreases, Jim is imagining a very particular process: k is increasing rather than decreasing, and m and g are parameters that are implicitly held fixed. Describing and accounting for these particulars is part of the job of this study. I would like to know: What do interpretations look like, in general? Do they always involve holding some parameters fixed and varying others? Furthermore, how does Jim know what the equation says? How does he know, by looking at the equation, that if k increases then x will decrease? Interpretations of this sort are a simple and important phenomenon that are not included in existing accounts of equation use in physics. Interpretation has no place in a model in which equations are just written from memory and then manipulated to produce a numerical result. This might be acceptable if this behavior was rare or unimportant, but that is not the case. To those familiar with introductory physics, Jim’s interpretation will appear so obvious and common that it is almost unnoticeable. Later on, the systematic analysis will show just how frequently these interpretations appear in my data corpus. In fact, almost every problem solving episode involves some interpretations. And students interpret, in some manner, a relatively high percentage of the equations that they write. So, in addition to any prima facie importance of these phenomena, there is the simple fact that interpretations are extremely common in students’ problem solving, and therefore should be taken into account in any model.
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A Feel for the Numbers The above examples begin to illustrate what it means for students to use equations meaningfully. Mike and Karl composed an expression from an idea of what they wanted from the expression. And Jim told us what an equations “says.” In both of these examples, the equations were, in a sense, directly meaningful for the students; the meaning was somehow seen in the structure of the equations itself. The ability to check the sensibility of results is very important. “Is my work correct?” and “Does this equation make sense?” are extremely useful questions to be able to ask and answer when working with equations. The point I want to make with this next example is that equation users have many methods available for determining if what they have written is correct; they do not always need to see meaning directly in equations. These alternative methods of checking the sensibility of expressions will not play as important a role in this inquiry. I will not attempt to model the knowledge involved in these behaviors. However, I want to do one brief example here in order to help clarify what I intend to exclude. In this episode, Jon and Ella were working on a very standard problem: Suppose a pitcher throws a baseball straight up at 100 mph. Ignoring air resistance, how high does it go?
Jon and Ella’s path to the solution of this problem was very short. They began by writing, from memory, an equation: v 2f = vo2 + 2ax This equation applies to cases in which an object moves with constant acceleration, either speeding up or slowing down. It happens that gravity has (approximately) the effect of causing objects to decelerate at a constant rate as they travel up into the air, so this equation is appropriate for the task. Here, vf and vo are the final and initial velocity of the object, a is the acceleration, and x is the distance traveled. After the ball is thrown, it steadily decelerates at the fixed rate determined by gravity until, at the peak, it stops momentarily. Thus, the ball’s maximum height can be found by substituting a value of zero for vf and then solving for x. This is precisely what Jon and Ella did, and they wrote
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vo2 =x 2a When they substituted numerical values for vo and a, Jon and Ella obtained a result of 336ft for x. Then they began to wonder if this is a reasonable result: Ella
Yeah, okay. I guess that’s pretty reasonable.
Jon
Well that’s like a - a football field. [g. =336ft.]
Ella
Yeah. How far,,
Jon
A hundred miles an hour is,, (1.0) If you were in a car going a hundred miles an hour,
Ella
[laughs] straight up into the air.
Jon
straight up into the air, would you go up a football field? (0.5) I have no idea. Well it doesn’t sound - it’s on the right order of magnitude maybe.
Ella
Yeah. Okay.
(5.0) Jon
I mean, it’s certainly not gonna go (0.5) thirty six feet. That doesn’t seem far enough. Cause a hundred miles an hour is pretty fast.
Ella
Hm-mm.
Jon
It’s not gonna go three thousand feet.
So, Jon and Ella concluded that 336 feet was a reasonable answer, or at least the right “order of magnitude.” This method of checking the sensibility of a result is quite different than Jim’s interpretation of an equation. Here, Jon and Ella did not look at the nature of the final symbolic expression, rather they examined the numerical result that this expression produces. Once they obtained the numerical result, they were able to draw on a wide variety of knowledge to get a feel for whether this result was correct. The ball is thrown at roughly the speed that a car travels and it goes about as high as a football field, does this make sense? Well, they argued, it is a little hard to tell but “it’s on the right order of magnitude.” Students have a selection of alternative routes available for checking the sensibility of their work. As in the first two episodes I presented, they can see meaning “directly” in equations. However, they also have some less direct methods available for checking whether their work makes sense. For example, like Jon and Ella, they can consider the reasonableness of numerical results. Or they can look at their expressions and see if they have the “right units.” As I mentioned, these alternative routes will be less important in the work that follows and will not be a central focus of my modeling efforts.
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Steps Toward a Model The above episodes, especially the first two, are intended to provide a first introduction to the type of phenomena that I want to understand, as well as to show that they exist. In the first example, Mike and Karl composed their own novel expression, starting from an understanding of what they wanted from that expression. The observation that students have this capability has implications beyond contexts where students have to invent their own equations. If students can build equations as meaningful structures, why can’t they see existing equations, composed by others, in terms of the same meaningful structures? The answer is that they can, and this is precisely what was illustrated in the second episode. Jim told us what an equation said. Furthermore, the second example provided a glimpse into what it means to “interpret” an equation. The task now is to build a model of symbol use in physics, in the spirit of cognitive science, that incorporates these important phenomena. Before proceeding, it is worth a moment to comment on what is involved in “the spirit of cognitive science.” One way that cognitive science proceeds is by making models that are designed to make predictions and provide accounts of human activity—what people say, do, and think. Furthermore, these models are of a particular sort, they generally posit the existence of internal mental structures, including entities called “knowledge.” In spite of the fact that I will be working within this tradition, I will not develop a model that is detailed enough that it can be “run” to generate predictions. I will posit the existence of knowledge elements that are to be understood as elements of such a model. But to actually “run” the model, in any sense, we would have to fill in many missing details with common sense and intuition. This is required because the creation of anything approaching a complete model would be very difficult given the present early stage of this research. Note that this boils down to self-consciously applying a different tactic than that taken by the existing physics problem solving literature. In that literature, certain phenomena are disregarded and simplified situations are studied so that a runnable model is possible. Such an approach has great merits, but simply will not do for my needs. We want to press on what “meaningful” symbol use is, which precludes our ignoring phenomena like interpretation. This does not mean that I am not throwing things out for the sake of simplicity. But preserving the phenomena that I am interested in, even in a simplified form, is enough to make the creation of a runnable model very difficult. As we will see, the interpretation of equations is a complex and idiosyncratic 53
business, sensitively dependent on details of context, and hence is not amenable to description by any simple models. Let me pin down a little further exactly what phenomena I propose to model and where my attention will be focused. First, my model of symbol use in physics should primarily be understood as a model that applies in the very moments of symbol use, when an individual is actually working with an expression. The model will address questions such as: How does a student know to write “+” instead of “×”? And: What kinds of things does a student say when they have an equation in front of them? Interpersonal and developmental (learning-related) phenomena will be treated as secondary, although I will do some tentative extending of the model in each of these directions. Second, my attention will be focused on only one particular slice of the knowledge associated with symbol use in physics; prototypically, this is knowledge that is particularly evident when novel expressions are constructed and expressions are interpreted. I will not worry about, for example, the knowledge needed to correctly perform syntactic manipulations of expressions, nor will I worry about how students know which equations to pull out of memory for a given task.7 This focus is justified because of the primary concern of this work, which is to understand the relation between understanding and symbol use. The presumption is that, to understand that relation, our effort is best focused at the places where symbols and understanding, in a sense, come together. First step: Registrations Now we are finally ready to take some first steps in the direction of a model of symbol use in physics. Let us begin by imagining a specific situation, a situation that is prototypical of our concerns. A student is looking at a sheet of paper that has, written on it, a single equation. The student has some particular interest in this equation, perhaps wondering if it’s written correctly, perhaps wondering what its implications are. To take a first step toward constructing our model, we can begin with a fundamental question: What does the student “see” when she looks at this equation? On what features of the page of expressions does her behavior depend?
7
Chi et al. (1981) present evidence that students and experts can directly recognize problems as being of a specific type and then equations are associated with particular problem types. This is one model of how students draw on known equations from memory. 54
This may seem like a simple or silly question, but it is not altogether trivial. To illustrate, think for a moment of the experience you have when looking at a word processing document on a computer display. The text that you see is actually, at its lowest level, made up of a number of small dots, pixels. Does this mean that you are seeing pixels when you look at the document on a computer display? In a sense, yes. But, in an important sense, you also “see” letters. Notice that there is no effort involved in going from the pixels to the letters, you just see the letters directly. In fact, if you are reading a printed version of this document, the letters are also probably formed from dots, but you are certainly not conscious of those dots as you read. And there are other things that you “see” when you look at a text document. You can see words, groups of words, paragraphs, and perhaps arbitrary groupings of text. You can even see the entire block of text on a page as a unit. Similar points have been made in a variety of literature with diverse concerns. It is a wellknown result that an account of perception which maintains that we simply perceive a neutral array of colors is inadequate. Rather we see a world of objects and edges, a partly processed world of things. I am not going to attempt at all a survey of the literature that has made related points. However, I will borrow a term from Roschelle (1991). Roschelle uses the term “registration” to describe the way people “carve up their sensory experience into parts, give labels to parts, and assign those labeled parts significance.” This provides us with a way to rephrase the question we are currently concerned with: How does our hypothetical student “carve up” her experience of the equation written on the sheet of paper? Here is a preliminary list of some of the more important registrations with which I will be concerned: 1. Individual symbols. If our hypothetical student is looking at a sheet of paper where the equation v = vo + at is written, she may see the letter ‘v’ as an individual symbol. This is in contrast to seeing some dots, or two straight lines that meet at a point. The student could also see the letter ‘t’ or a ‘+’. 2. Numbers. Students can also see numbers. Jon and Ella saw the number “336” that they had written on the blackboard. Note that this number can also be seen as comprised of three individual symbols. 3. Whole equations. Our hypothetical student can see the equation v = vo + at as a single unit.
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4. Groups of symbols. Arbitrary groups of symbols can also be seen as units. For instance, if I write an expression for the total momentum of two objects as, P = m1v1 + m2v2, then m1v1 can be seen as the momentum of one of the objects. Terms in expressions and symbols grouped in parentheses are frequently seen as units, but arbitrary and complicated groupings are not uncommon. 5. Symbol Patterns. Finally, part or all of an expression may be seen in terms of a number of “symbol patterns”—patterns in the arrangement of symbols. These patterns will be discussed extensively in the next chapter, along with many examples, so I will just give a brief explanation here. Roughly, symbol patterns can be understood as being templates in terms of which expressions are understood and written. In general, these are relatively simple templates. For example, two symbol patterns that students see in expressions are: =
+
+
…
The symbol pattern on the left is a template for an equation in which two expressions are set equal. These two expressions can be of any sort—the
can be filled in with any expression.
Similarly, the symbol pattern on the right is a template for an expression in which a string of terms are separated by plus signs. Note that symbol patterns differ from what I have called groups of symbols because, in the case of groups, the arrangement of symbols within the group is unimportant; the group is simply treated as a whole. For patterns, the arrangement of symbols constitutes a recognizable configuration. Symbol patterns will play a key role in my explanation of how students interpret expressions and compose novel expressions. So, with this discussion of registrations, we have taken a first step into the physicist’s world of symbols. What does a physicist or physics student see when looking at a sheet of equations? They do not see an organic array of curving lines and marks; they see symbols, equations, terms, and groups of various sorts. Furthermore, we will see that, for a physicist, a sheet of physics equations is even more structured and meaningful than this first account implies. In the next chapter, we dive hard into this world of structure and meaning.
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Chapter 3. A Vocabulary of Forms
Rabbit or duck?
Art and Illusion, the well-known book by E. H. Gombrich concerning art and pictorial representation, begins with a discussion of what Gombrich calls “the riddle of style” (Gombrich, 1960). For Gombrich, the riddle is this: “Why is it that different ages and different nations have represented the visible world in such different ways? Will the paintings we accept as true to life look as unconvincing to future generations as Egyptian paintings look to us?” Gombrich rejects the view that the history of art can simply be understood as a development toward more accurate representations of reality. However, Gombrich balances this rejection with an acceptance of a certain kind of historical development. He believes that, over time, artists have developed an increasingly varied and flexible “vocabulary of forms.” “But we have seen that in all styles the artist has to rely on a vocabulary of forms and that it is the knowledge of this vocabulary rather than a knowledge of things that distinguishes the skilled from the unskilled artist.” Every artist and every person that looks at art learns the current vocabulary by looking at existing paintings. Then, new paintings are created from this vocabulary and other paintings are seen in terms of the vocabulary. The point I want to extract here is that the forms that we see when looking at a painting are not somehow inherent in the painting, or even in the painting and the “raw” perceptual capabilities of humans. People learn—and we, as individuals, learn—to see the forms, and then we bring this capability with us when we look at new drawings and paintings. This point is especially evident in some more marginal cases. Consider the simple drawing at the top of this page, which I have taken from Gombrich. Is it a smiling duck? Is it a cute, but wary rabbit? Whatever it is, this drawing consists of a very simple arrangement of lines. But when we look at it we see more than those lines; to us, the image is much more than a simple sum of its parts. If it was a simple sum of its parts, how could it possibly be both a rabbit and a duck? Because we have learned to see certain forms, we see a meaningful image. In a sense, we bring the rabbit and duck with us.
57
I can now add a basic point to the discussion of registrations I presented at the end of the last chapter. The registrations available in any page of expressions is not only a function of the marks on the paper, it also depends on the (learned) capabilities of the person looking at the display. To see letters, for example, you have to know the alphabet. This will be my focus in this chapter: I will describe the knowledge that people acquire that is associated with certain registrations in symbolic expressions.
What is a Symbolic Form? I am now ready to define the first major theoretical construct in my model of symbol use in physics, what I will refer to as “symbolic forms” or just as “forms,” for short. A symbolic form is a particular type of knowledge element that plays a central role in the composition and interpretation of physics expressions. Each form has two components: 1. Symbol Pattern. Each form is associated with a specific symbol pattern. Recall that symbol patterns are one of the types of registrations I described in the previous chapter. A symbol pattern is akin to a template for an expression and symbolic expressions are composed out of and seen through these templates. To be more precise, forms are knowledge elements that include the ability to see a particular symbolic pattern. 2. Schema. Each symbolic form also includes a conceptual schema. The particular schemata associated with forms turn out to be relatively simple structures, involving only a few entities and a small number of simple relations among these entities. These schemata are similar to diSessa’s (1993) “p-prims” and Johnson’s (1987) “image schemata,” which are also presumed to have relatively simple structures. For example, as I discussed earlier, diSessa describes a particular p-prim known as “Ohm’s p-prim” in which an agent works against some resistance to produce a result. This p-prim can be understood, in part, as a simple schematization involving just three entities, in terms of which physical situations are understood. Although I will discuss both p-prims and image schemata in later chapters, the view that I will describe shares many more of the commitments of diSessa’s theory. One other point about forms is worthy of note at this very early stage. The schema component of forms can allow specific inferences. I will understand this as working as follows: Generic
58
capacities (that I will not describe) act on the schemata to produce inferences. I will say more about this in a moment. To illustrate forms, I want to return to the very first example that I presented from my data corpus. In Chapter 2, I described an episode in which Mike and Karl constructed their own expression for the coefficient of friction: µ
µ = µ 1 + C m2 When presenting this episode, I commented that this was an equation that you would not find in any textbook; Mike and Karl had composed this equation on their own. Now we are ready to fill in the first pieces of the puzzle in accounting for how they accomplished this feat. Recall that the core of Mike and Karl’s specification for this equation was the notion that the coefficient of friction should consist of two parts, one that is constant and one that varies inversely with the weight: Karl
=the coefficient of friction has two components. One that's a constant and one that varies inversely as the weight.
I want to argue that two forms are evident in this specification, the first of which I call PARTS -OF -AWHOLE.
Like all forms, PARTS -OF -A-WHOLE consists of a symbol pattern and a schema. The symbol
pattern for PARTS -OF -A-WHOLE is two or more terms separated by plus (+) signs. And the schema has entities of two types: There is a whole which is composed of two or more parts. This schema can be seen behind Karl’s above statement; he says that the coefficient of friction consists of “two components.” In other words, the coefficient of friction is a whole that has two parts. Furthermore, the equation has the structure specified in the symbol pattern; it has two terms separated by a plus sign. To describe the various symbol patterns that we will encounter, I will employ a shorthand notation. For PARTS -OF -A-WHOLE I write the symbol pattern as: PARTS -OF -A-WHOLE
[
+
+…]
Here the ‘ ’ character refers to a term or group of symbols, typically a single symbol or a product of two or more factors. The brackets around the whole pattern indicate that the entity corresponding to the whole pattern is an element of the schema. The shorthand notation for symbol patterns is briefly summarized in Appendix D.
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The second form in Karl’s statement is what I call PROPORTIONALITY MINUS or just PROP-. The schema for PROP- involves two entities: One quantity is seen as inversely proportional to a second quantity. In Karl’s specification for this expression, it is the second component of the coefficient of friction that is inversely proportional to the weight. As he says, there’s one component that’s constant and one that “varies inversely as the weight.” The symbol pattern for PROP- is relatively simple. When a student sees PROP- in an expression, all they are attending to is the fact that some specific symbol appears in the denominator of an expression. I write this as: … …x…
PROP-
Here, x will usually correspond to a single symbol. In Mike and Karl’s expression, the m is the symbol of interest and, in line with the PROP- form, it appears in the denominator of the second term. I want to digress for a moment here to make a couple of points about the forms we have seen thus far. First, it may seem as though the list I am building will simply end up with a form corresponding to each syntactic feature of algebraic notation. But this is not the case. We will see that there are limits to what features people are sensitive to, and that identical patterns can correspond to more than one form. Note, for example, that
PROP-
constitutes a very specific
hypothesis concerning what features are consequential for students during equation construction and interpretation. All that is relevant for PROP- is that the symbol in question appears in the denominator of the expression; it does not matter whether the quantity is raised to a power or whether there are other factors surrounding it. With regard to this form, such details are not consequential; both 1/m and 1/m2 can be seen as PROP-. Second, I want to say a little about the inferences permitted by these two forms, just to give the flavor of what I have in mind. Here is a sampling of inferences permitted by
PARTS -OF -A-
WHOLE:
•
If one part changes and the other parts are constant, the whole will change.
•
If one part increases and the other parts are constant, the whole will increase.
•
If one part decreases and the other parts are constant, the whole will decrease.
•
If the whole is necessarily constant and one part increases, then another part must decrease.
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And here is a sampling of inferences permitted by
PROP-:
•
If the denominator varies, the whole varies.
•
If the denominator increases, the whole decreases.
•
If the denominator decreases, the whole increases.
Some of these inferences may be playing an implicit role in Mike and Karl’s construction of their equation for the coefficient of friction. Recall that Mike and Karl began with the requirement that µ must decrease with increasing weight: Karl
I guess what we're saying is that the larger the weight, the less the coefficient of friction would be.
Inferences of the above sort could confirm that this requirement is met. If the weight is increased, PROP-
tells us that the second term in Mike and Karl’s expression for µ must decrease. Then PARTS -
OF -A-WHOLE
permits the inference that if the second part is decreased the whole also decreases,
since the first term is constant. Thus, the whole coefficient of friction decreases if the weight is increased. Two additional forms also played a role in Mike and Karl’s building of their novel equation and I will just comment on them briefly here. The first of these forms has to do with the ‘C’ in front of the second term. This factor was inserted by Mike and Karl almost as an afterthought, at the very end of the episode. I take this to be a case in which the “COEFFICIENT” form is in play: COEFFICIENT
[x
]
In COEFFICIENT, a single symbol or number, usually written on the left, multiplies a group of factors to produce a result. A number of features are typical of the coefficient part of the schema, and I will explain these features later in this chapter. For now, I just want to comment that the coefficient is frequently seen as controlling the size of an effect. This, roughly, was Mike and Karl’s reason for inserting the coefficient. As Karl wrote, he commented: Karl
You find out that uh you know the effect is small, or you find out the effect is large or whatever so you might have to screw around with it.
So, for Mike and Karl, the factor of C was a way for them to tune their expression. Finally, there is one important part of Mike and Karl’s expression that I have yet to comment on; I have not said anything about the equal sign and the µ that appears on the left side of the
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equation. This constitutes an application of one of the simplest and most ubiquitous forms, the IDENTITY
form. x = […]
IDENTITY
In the IDENTITY form, a single symbol—usually written on the left—is separated from an arbitrary expression by an equal sign. I will have more to say about IDENTITY later in this chapter; at this point, however, I just want to note that identity transparently allows the inference that whatever is true of the entity corresponding to the whole right hand side is also true of the entity on the left. In the case of Mike and Karl’s equation for the coefficient of friction, since the whole expression on the right decreases with increasing mass, µ can also be presumed to decrease with increasing mass. In the previous chapter I asked: How did these students know to write a ‘+’ instead of a ‘×’ between the two terms? And: How did they know to put the m in the denominator? Now I have given the answers to these questions; or, at least, I have given a particular variety of explanation. I have said that, to compose a new equation, these students had to essentially develop a specification for that equation in terms of symbolic forms. Once this was done, the symbol patterns that are part of these forms specified, at a certain level of detail, how the expression was to be written. In addition to being knowledge for the construction of the expression, the forms are also the way that the equation is seen and understood by Mike and Karl. Just as we can see a rabbit or a duck in the drawing at the head of this chapter, Mike and Karl see IDENTITY
PARTS -OF -A-WHOLE
and
when they look at their expression for µ. For Mike and Karl, their equation is not just a
collection of marks, or even a collection of unrelated symbols, it has a meaningful structure.
Forms and a More Traditional Problem As I have mentioned, the episode discussed in the previous section is somewhat atypical. When solving problems in an introductory physics course, it is rare that students must build elaborate expressions from scratch. In this section, I want to turn to a more typical episode, involving more familiar expressions. In addition, this episode is interesting because I believe it may provoke skeptical readers to argue that knowledge other than forms is responsible for the students’ ability to write the expressions that appear. Thus, I will need to do some work to argue that forms are, nonetheless, a central part of the knowledge involved here.
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In this example, I am again going to look at the work of Mike and Karl. The reason for sticking with these students is that I want there to be no doubt that the students we see doing the more typical work described here have the abilities demonstrated in the above section. In this episode, Mike and Karl were working together at a blackboard to answer the following question: For this problem, imagine that two objects are dropped from a great height. These two objects are identical in size and shape, but one object has twice the mass of the other object. Because of air resistance, both objects eventually reach terminal velocity. (a) Compare the terminal velocities of the two objects. Are their terminal velocities the same? Is the terminal velocity of one object twice as large as the terminal velocity of the other? (Hint: Keep in mind that a steel ball falls more quickly than an identically shaped paper ball in the presence of air resistance.)
This problem asks about a situation in which a ball has been dropped under the influences of gravity and air resistance. Gravity is pulling the ball down and, as the ball rushes through the air, the resistance of the air tends to oppose the downward fall. When the ball is first dropped, it initially accelerates for a while—it goes faster and faster. But eventually, because of the opposition of air resistance, the speed levels off at a constant value. This maximum speed that the ball reaches is called the terminal velocity. The question that this problems asks is: How does the terminal velocity differ for two objects, one of which is twice as large as the other?
(b)
(a)
(c) Fair
Fair
Fgravity
(d)
Fgravity
Fgravity
Fair
Fgravity
Figure Chapter 3. -1. (a) An object is dropped. (b) & (c) As it speeds up, the opposing force of air resistance increases. (d) At terminal velocity, the forces of air resistance and gravity are equal.
One way to understand what’s going on here is that there are two forces acting on the ball, a force down from gravity and a force from air resistance acting upward. Initially the ball is not moving and the force from air resistance is zero. But gradually the ball speeds up and the force from air resistance grows until, at terminal velocity, it’s exactly equal to the force from gravity (refer to Figure Chapter 3. -1). Now let’s look at what Mike and Karl had to say about this problem. They began by commenting that the hint given in the problem “basically answers the question,” since it gives away the fact that the terminal velocity of a heavier object is higher. The discussion below
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followed, in which, Mike and Karl agreed that what they called the “upward” and “downward accelerations” of a dropped object must be equal when it reaches terminal velocity: Karl
So, we have to figure out (0.5) what, how do you figure out what the // terminal velocity is.
Mike
// terminal velocity is. Well, okay, so, you have a terminal velocity, You have terminal velocity, that means it falls at a steady rate. // Right, which means the force opposing it // Right, which means
Karl Karl
It means it has an upward acceleration that's equal to the downward acceleration.
Mike
Ri:::ght.
Following this discussion, Mike drew the diagram shown in Figure Chapter 3. -2, talking as he drew.
f(v)=
Fa
v m Fg Figure Chapter 3. -2. Mike and Karl’s air resistance diagram. Mike
Air resistance. You have a ball going down with a certain velocity. A for:::ce,, You reach terminal velocity when air resistance, the force from the air resistance is the same as the force from gravity. [g. points to each of these arrows as he writes, with spread thumb and forefinger indicating size] But, we also know that force of air resistance is a function of velocity. [w. f(v)=] Right?
In this last comment, Mike reiterates the notion that the influences of air and gravity must be equal at terminal velocity, but here he talks in terms of opposing forces, rather than opposing accelerations. Notice that Mike’s drawing is in agreement with the account I presented in Figure Chapter 3. -1. Finally, in the next passage, Mike and Karl compose an expression for air resistance: a(t ) = −g +
f (v) m
Mike
So, at least we can agree on this and we can start our problem from this scenario. [g. Indicates the diagram with a sweeping gesture] Right? Okay? So, at any time,, At any time, the acceleration due to gravity is G, and the acceleration due to the resistance force is F of V over M. [w. g + f(v)/m] This is mass M. [w. m next to the mass on the diagram]
Karl
Ah:::.
Mike
Right.
Karl
Yeah.
Mike
Okay, // now they're opposing so it's a minus.
Karl
// (So) as your mass goes up,, [g. m in equation]
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Mike
So, this is negative G. [g. down, makes a negative sign in front of the g; w. arrow up labeled "+" near the top of the diagram] Positive direction. You have negative G plus F of V over M [gestures down and then up]. That's you're acceleration at any time. Right?
Karl
Well, wait. You want them to have,,
Mike
This is the acceleration at any time T. [w. a(t) next to the expression]
In this last passage, Mike and Karl’s conceptualization of the situation in terms of two competing influences takes shape as an equation. The construction involves a few forms: OPPOSITION,
COMPETING TERMS,
and IDENTITY. COMPETING TERMS
±
OPPOSITION
– x = […]
IDENTITY
We encountered the
IDENTITY
±…
form in the previous episode. COMPETING TERMS and OPPOSITION
are new, so I will explain them briefly: COMPETING TERM s. Each of the competing influences corresponds to a term in the equation. As I will discuss later, this is one of the most common ways that physicists see equations, as terms competing to have their way. OPPOSITIO n. The two competing influences are competing in a specific manner, they are opposing. Opposition is frequently associated with the use of a negative sign. “Now they’re opposing so it’s minus.”
A Fundamental Contrast: Forms versus Principle-Based Schemata The basic presumption underlying my account of Mike and Karl’s above work is that they conceptualized the situation as involving competing influences and these competing influences get associated with terms in equations. But notice that there are aspects of the equation that this does not explain. In particular, I have not said anything about how they knew to write f(v)/m for the term corresponding to air resistance. Likely this has to do with the equation F=ma, an equation that Mike and Karl certainly remember. They stated that the force of air resistance depends on velocity, which led them to write this force as f(v). Then, since they were talking about opposing accelerations, they used F=ma to write the acceleration as a=F/m. So, at the least, forms cannot be the whole story in this construction episode—and they will not be the whole story in most construction episodes. More formal considerations and remembered equations must also play a role. In my rendition of the above episode, a remembered
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equation was used to fill in one of the slots in a symbol pattern. However, the above episode is open to some interpretations that are significantly different than the one I provided. Most notably, one might argue that Mike and Karl have memorized the equation:
∑F=ma Here, the ‘∑’ symbol means ‘summation.’ This equation says that the sum of all the forces on an object is equal to the mass of the object multiplied by its acceleration. Thus, it may be possible to see Mike and Karl’s work as solely a straightforward application of this equation; they write the sum of the forces and then divide by the mass to get the acceleration. This alternative viewpoint is an important enough (and strong enough) alternative that I want to devote significant effort to discussing it and contrasting it with the view I am developing in this thesis. In addition, working through this alternative will help to clarify what is particular about my own viewpoint. But, I will not try to argue that this alternative view is incorrect, or that it is even strictly competitive with the one I am developing in this thesis. In fact, I believe that a more complete description of the knowledge involved in expert symbol use would include knowledge of multiple sorts, including knowledge more closely tied to formally recognized principles, as well as symbolic forms. However, I will try to argue that some student behaviors are better explained by symbolic forms and that symbolic forms—and the related behavior—persist into expertise, complementing more formal aspects of knowledge. A version of this alternative view was worked out by Larkin (1983). Larkin begins by distinguishing what she calls the “naive” and “physical” representations. (Here, when Larkin says “representation” she is talking about internal mental representations, not external representations.) In Larkin’s scheme, everyone—novice AND expert—constructs a naive representation as a first step toward analyzing a physical situation. A naive representation involves only “familiar” entities and objects; it essentially uses our everyday vocabulary for describing the world. In contrast, the physical representation, which is only constructed by physicists, “contains fictitious entities such as forces and energies.” Once the physical representation is constructed, the writing of an equation is apparently relatively easy. Larkin tells us that: “Qualitative relations between these entities can be ‘seen’ directly in this physical representation, and these qualitative relations then guide the application of quantitative principles.” To this point, Larkin’s view may be said to share many broad similarities with my own. There is a qualitative mental representation that guides the construction of an equation. But Larkin soon 66
takes a step that strongly differentiates her view. She says: “The physical representation of a problem is closely tied to the instantiation of the quantitative physics principles describing the problem.” In contrast, in my own view, forms are not closely tied to specific physical principles, such as Newton’s Laws. Let’s take a moment to see how Larkin’s model works. According to Larkin, experts possess a number of schemata that are directly associated with physical principles. These schemata contain what she calls “construction rules” and “extension rules.” The construction rules act on the naive representation to produce the physical representation. And the extension rules act on an existing physical representation to add to that representation. Larkin provides two examples of these schemata, the “Forces Schema” and the “Work-Energy Schema.” These schemata are quite closely related to physical principles, as these principles would be presented in a physics textbook. For example, Larkin says that the Forces Schema “corresponds to the physical principle that the total force on a system (along a particular direction) is equal to the system’s mass times its acceleration (along that direction).” This is essentially Newton’s Second Law, F=ma. The construction rules in this schema correspond to “force laws,” which are the laws that allow a physicist to compute the forces on an object given the arrangement of objects in a physical system. So the idea is this: A physicist looks at a physical system and sees the arrangement of objects involved. The Forces Schema then provides force laws that allows the physicist to determine the forces (a particular kind of fictitious entity) acting on the objects. Then the physicist plugs these forces into the equation ∑F=ma and solves for whatever is desired. Note that this equation, ∑F=ma, is a template of a sort, and it allows the construction of equations that are not literally identical to those seen before. But, it is has particular limits on its range of applicability: the entities in the sum must be forces and they must be set equal to ma. Of course, as I have already suggested, it may be possible to apply Larkin’s account to describe Mike and Karl’s work on the air resistance task. This would go as follows: Using the construction rules provided by the activated Forces Schema, Mike and Karl write the force on each object; -mg is the force of gravity, which is negative because it acts in the downward direction, and the force of air resistance is simply taken to be f(v). These forces are then summed and set equal to ma: ma = −mg + f (v ) (Presumable, Mike and Karl do this step mentally, since it doesn’t appear as one of the equations they write.) Finally, this equation is divided by the mass, m, to produce an equation for a. 67
I must emphasize that one crucial difference between Larkin’s view and my own is the fact that her schemata are closely tied to physical principles. (This is in addition to some more global differences in orientation.) For her, it is crucial that the influences in competition are forces, since Newton’s laws only talk about forces. And the above equation is only written because the schema tells us to—it’s a simple matter of “plugging-in.” In contrast, in my view, the situation is generically schematized as competing influences, a notion that is supposed to cut across physical laws. Many types of entities, other than forces, may compete. For this reason, I will refer to Larkin’s account as the “principle-based schemata” view. The modifier “principle-based” is important here since, as we have seen, a type of schema also appears in my own model. However, in this section, I will at times simply refer to Larkin’s view as “the schema view.” Now that I have presented the principle-based schemata view and contrasted it with my own view (the “forms” view), I want to take steps toward arguing for my own account of some episodes. In the remainder of this section I will discuss a few episodes that are specifically selected to contrast these two views. In addition, in later sections and later chapters I will argue for the usefulness of my viewpoint on other grounds. Step 1: Students do, at least sometimes, see symbolic forms The first step I want to take is to point out that I have already presented enough evidence to show that the schema view cannot be the whole story. In particular, there is no way for Larkin’s schema view to explain Mike and Karl’s ability to construct their own equation for the coefficient of friction. µ
µ = µ 1 + C m2 This is because there are simply no principles or existing equations to guide this construction. Furthermore, once we have accepted that Mike and Karl have abilities that transcend what is allowed by the principle-based schemata model, we have to live with this observation in all accounts that we give of all types of problem solving. Both of the above two episodes involved the same pair of students, Mike and Karl. And, just because they were working on a more familiar task, there is no reason to believe that their form-related capabilities will simply go away or not be employed. Suppose, for a moment, that something like Larkin’s Forces Schema is responsible for
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Mike and Karl’s work on the air resistance task. Even if this is the case, because Mike and Karl have the ability to see meaningful patterns—forms—they may, at any time, be seeing these patterns in equations that they write. I want to illustrate this point with an analogy. Suppose that we decided to teach some mathematicians a little physics, so we teach them to write equations and solve problems. Mathematicians already know a lot about equations. They know a linear equation when they see one, and they can differentiate a linear equation from a quadratic equation. Thus, even if they learn to write the expression v = vo + at from memory, they may still always see this equation as a linear relation between the velocity and the time. The equation has a structure that a mathematician can recognize. I am trying to make an analogous point about forms. The coefficient of friction episode tells us that Mike and Karl can sometimes compose expressions from symbolic forms. They may thus see these forms in any equation that they write. If this is true, then the principle-based schemata view cannot be “the whole story.” Step 2: Deviations from the ÒschemaÓ model in the air resistance example Mike and Karl’s work on the air resistance task warrants a closer look. Recall that they bounced between describing the situation in terms of competing accelerations and in terms of competing forces. This must, to some degree, be because Mike and Karl are aware that force and acceleration are formally interchangeable via F=ma. However, I also take this as evidence that a view of the situation in terms of opposing influences is the core driving element of these students’ work; thus, the specific nature of these influences is not of utmost importance to them at all stages of the solution attempt. This observation really cuts to the heart of the difference between the forms model and the principle-based schemata model. I am arguing that Mike and Karl’s work is driven—at least, in part—by an understanding tied to a notion of opposing influences that cuts across principles, not by a schema tied solely to forces and Newton’s laws. For those who are less familiar with traditional physics practice, it is worth noting that Karl’s use of the terms “upward” and “downward acceleration” would probably be considered inappropriate by many physicists. To a physicist, the term “acceleration” is usually reserved for describing actual changes in the speed of objects, not for virtual changes that may be canceled out by other virtual changes. Thus, Mike and Karl’s work departs somewhat from a strict and careful
69
application of traditional methods that follow a universal schema associated with physical principles. Finally, notice that, as they work, Mike and Karl have interpretations readily available. For example, Mike says that one acceleration is “opposing” the other. The fact that such interpretive comments are ready-to-hand suggests that these notions are playing a role in these students’ work. Step 3: Students do not rigorously deduce from principles As the final step in my argument against the schema view, I want to present a battery of evidence designed to further refute the notion that students (and experts) work by slavishly following standard techniques and rigorously deducing from principles, even when these techniques are available and appropriate. The purpose of this data is not only to argue against certain aspects of the principle-based schemata view, but also to provide counter-evidence for a broad range of opposing views and to give a feel for the character of student work. I will begin with one more example concerning Mike and Karl; here, I look at their work on the “spring” task, in which a mass hangs motionless at the end of a spring. In Chapter 2, I briefly examined Jack and Jim’s work on this task. The incident I want to recount actually took place during Mike and Karl’s final session, which was in part used to wrap-up some issues and debrief the subjects. During this session, Mike and Karl had returned to the spring task, and they had written the equation: Fs = Fg This equation says that, when the mass hangs motionless, the force upward due to the spring must equal the force of gravity pulling down. Since this was a wrap-up session, I probed Mike and Karl to see how, and in what way, they related this equation to known principles. I began by asking if they could somehow include the equation F=ma in their solution. What I had in mind was that they would say that, because they know that the sum of the forces is equal to ma, and the mass isn’t accelerating (a = 0), the sum of the forces must be zero: Fs + Fg = 0 . Furthermore, if we are willing to adjust some sign conventions, then this expression can be rewritten as Fs = Fg . Immediately following my initial probe, Mike and Karl complained that it simply did not make sense to employ F=ma in the solution of this problem: Mike
It's not very natural.
Karl
We're looking at a static system.
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So, I tried another approach. I pressed them as to how they knew that the forces of the spring and gravity must be equal. Mike
How do I know F S equals F G? Because it's in equilibrium.
Karl
It's defined that way. [They both laugh]
Mike
Because it's in equilibrium, right, so the forces they have to, um, add up to zero.
Bruce
Why do the forces have to add up to zero?
Mike
To be in equilibrium. [laughs] We're going in circles aren't we. If it's in equilibrium there's no acceleration, is that what you're looking for?
In the above passage, Mike and Karl first seem to assert that it is essentially obvious that these forces must be equal. But then, at the end of the passage, it seems like Mike has latched onto what I had in mind, he says: “If it's in equilibrium there's no acceleration, is that what you're looking for?” Strikingly, Mike’s further work made it clear that he was still not on my wavelength. He went on to assert that the acceleration due to each force must be equal and wrote as = ag Apparently, he had in mind using F=ma simply to find an acceleration associated with each of the forces, not to establish the dynamical truth of their equality. Mike
There’s acceleration but they cancel, right? I mean there is - when there’s force on it and there’s no - the result - there’s no resulting acceleration. There’s no net acceleration, maybe that’s a better way to put it. (3.0) Cause one is accelerating one way [g. up] and the other force is accelerating downward.
So, it seems that I had still not succeeded in getting Mike and Karl to state the argument that I had in mind. Nonetheless, to this point in the episode, it is still possible that Mike and Karl were well aware of how their equation followed form F=ma, but that they simply did not know what I was looking for. Perhaps, for example, the explanation from F=ma is simply too simple to be worth mentioning. To explore this possibility, I finally just told them what I had in mind, giving the argument I presented above. Mike and Karl’s reaction to this made it clear that this was an entirely novel idea to them. Karl was especially astounded and pleased by my little explanation of how Fs = Fg followed from F=ma and a = 0. In fact, as the session continued and we moved on to other issues, he occasionally returned to this issue to comment on how much he liked my explanation: Karl
I like - I like - this made sense to me because this took away the “well, this is true just because.”
Bruce
Right.
Karl
And we get a lot of “just becauses.”
Mike
Uh-huh.
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Karl
This explained why.
What I am hoping that this episode makes very clear is that fundamental principles like F=ma are not always lurking secretly behind student’s work, simply unsaid. It is true that this episode cannot quite confirm the kind of generality I have asserted for forms; for instance, it is possible that Karl has something like a Forces Schema that is specialized for static situations. But this episode can at least eliminate the most economical version of the principle-based schemata model, in which all work flows from schemata associated with the most basic fundamental principles. Of course, even if we accept that Mike and Karl’s work is not driven by principle-based schemata, one could point out that these students are simply not experts and maintain that, in expertise, behavior tends toward the description given by the principle-based-schemata view. It is very likely that experts and more advanced students would be able to quickly produce the argument that the equality of the two forces in the spring task follows from F=ma. Nonetheless, we must keep in mind that these students are at least at an intermediate level of expertise. They have completed two full semesters of physics and they can very reliably solve problems like the spring task. The fact that this sort of behavior continues into a stage in which there is practiced and reliable performance suggests that the associated knowledge may linger into expertise. Adding to the plausibility of this statement is the observation that symbolic forms constitute a useful type of knowledge. This knowledge is necessary, for example, in the construction of completely novel expressions, such as Mike and Karl’s expression for µ. Because there is an important niche for this knowledge in expertise, it is thus more plausible that it will continue to play a role as students progress toward more expert behavior. For the sake of completeness, I should mention that I would explain Mike and Karl’s construction of the relation Fs = Fg by appeal to what I call the BALANCING form. BALANCING
=
In BALANCING , two competing influences are seen as precisely equal and opposite. Note, once again, the central point here: The BALANCING form is more general than any assertions about balanced forces. Any types of influences can be balanced and, once they are seen as balanced, a relation in line with the BALANCING symbol pattern can be written.
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Some additional examples Mike and Karl were quite typical of my subjects; as a whole, the students almost never worked by a strict following of traditional techniques that follow from principles. Furthermore, when students did appear to work from principles—or, at least, maintained that they were working from principles—these applications of principles often seemed to play the role of an after-the-fact rationalization for work that was already done. And even these rationalizations often involved some hand-waving. Let’s look briefly at a few examples. First, I return to Jack and Jim’s work on the spring task. Recall that they began by writing the force due to the spring as F=-kx and the force due to gravity as F=ma. Then, in a passage that I did not cite earlier, they proceeded to equate these two forces, writing kx=mg: Jack
So, okay, so then these two have to be in equilibrium those are the only forces acting on it. So then somehow I guess, um, (3.0) That negative sign is somewhat arbitrary [g. F=-kx] depending on how you assign it. Of course, in this case, acceleration is gravity [g. a in F=ma] Which would be negative so we really don't have to worry about that [g.~ F=-kx] So I guess we end up with K X is M G. [w. kx=mg]
The delicate issue that Jack is trying to deal with is, if these forces are going to be equated, then what happens to the negative sign in F=-kx? Jack knows that he should somehow end up with kx=mg, but he makes a pretense of being careful; he says that the acceleration due to gravity is also negative so the negative signs cancel. This is not quite up to the standards of a rigorous argument; such an argument would require that Jack carefully associate the signs of terms and directions on the diagram. Jack does not need to do this however, because he knows the answer before he starts. This sort of hand-waving concerning signs is extremely common in my data corpus. If anything, it is the norm. Finally, I want to talk about some student work on a task I have yet to mention, what I call the “Stranded Skater” problem. Peggy Fleming (a one-time famous figure skater) is stuck on a patch of frictionless ice. Cleverly, she takes off one of her ice skates and throws it as hard as she can. (a) Roughly, how far does she travel? (b) Roughly, how fast does she travel?
What happens in this situation is that, when Peggy throws the skate, she moves off in the direction opposite to the direction in which the skate was thrown. I am going to talk about the work of two pairs on this problem, Jack and Jim and Alan and Bob. Jack and Jim began by announcing that “it's basically a conservation of momentum problem.” Then they immediately went on to write the expression
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m pf v1 = ms v2 where m pf is the mass of Peggy Fleming and ms is the mass of the skate. Here, Jack and Jim are implicitly using the relation P=mv, where P stands for the momentum of an object. The question is, how did Jack and Jim know to simply equate the momenta of the skater and the skate? I believe that they see this problem in terms of BALANCING . After the skate is thrown, the movement of the skater and skate must somehow balance out.
Figure Chapter 3. -3. Alan and Bob’s diagram for the Stranded Skater problem.
After they had written this relation, Jack moved to rationalize it in terms of the principle of conservation of momentum. He wrote the following expression and said: m pf v1 + ms v2 = 0 Jack
Actually, it should probably be … [w. MpfV+MsV=0] This case we'll assume that this velocity would be a negative velocity [g. above v of skate] so that we can go to that. [g. first equation] Cause one's gonna go one way, one's gonna go the other.
So, in this episode, Jack began by writing a relation that he believed to be true and then moved to, after the fact, rationalize that relation in terms of known principles. The principle applied here is the conservation of momentum, which says that the initial and final momenta of a system must be equal in the absence of any externally applied forces. In this case the initial momentum is zero, since everything in the system is at rest before the throwing of the skate. Thus, an expression for the total final momentum can be correctly equated to zero. This is what Jack does in the above equation. I want to note that it was common for students to see the Stranded Skater problem in terms of BALANCING as well as in terms of a related form, what I call CANCELING (0 =
-
Ê
). Alan and
Bob began their work on this problem by writing the expression: 0 = ms vs − m pv p Alan
And we expect her to move to the left to compensate (her) mass of Peggy times the velocity of Peggy. [w.mod 0=MsVs–MpVp]
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Here, Alan argues that Peggy must move in the direction opposite that of the skate in order to “compensate” for the motion of the skate. The important point here is that the use of a negative sign between these terms is not derived from a principle or from mechanically following techniques. Rather, it is introduced based on an understanding of what is going to happen and what the expression should look like. In my view, Alan sees “competing influences” that cancel and he makes sure that his expression is in line with this symbolic form. Final comments on principle-based schemata In this section I have contrasted my model with a model that is truly different, what I have called “the principle-based schemata” view. This contrast is extremely important because it gets to the heart of some of the essential features of my model. Allow me to elaborate briefly. There is a conventionalized version of physics knowledge that is presented in physics textbooks. But it is an error simply to presume that, when people learn physics, the knowledge they acquire somehow has the same structure as the conventionalized knowledge presented in a textbook.8 Physics knowledge, as realized in individuals, may mirror textbook physics, but it may not; the question must be settled empirically. I have attempted to argue that at least one slice of physics knowledge does not have the same structure as textbook physics. In particular, the knowledge constituted by forms is not organized like a textbook; it is not tightly related to basic physical principles. Instead, notions like BALANCING are central. And these notions cut across physical principles—you can have balanced forces OR balanced momenta. However, principle-based knowledge must have been playing some role in the episodes I presented above. For example, even if we accept that some students understand the Stranded Skater problem in terms of BALANCING , we still have to explain how they knew to balance momenta, in particular. I believe that this depends, in part, on students ability to recognize problem types and to know what principles apply to those problem types.9 This observation implies that the knowledge employed in symbol use, in its full and finished expert form, must certainly have components that are tied to physics principles. In fact, I believe that there is likely to be knowledge that is somewhat like principle-based schemata. This returns me again to the point that a full model of symbol use in physics will be quite complicated, more
8 9
Andrea A. diSessa has made this point in a number of articles. See, for example, diSessa, 1993. See, for example, Chi et al., 1981. 75
complicated than I can deal with here. So, I do not mean to discard the principle-bases schemata view, but I do hope that I have shown that there are cracks in this model—some pretty big cracks, in fact—that make room for other important classes of knowledge, and that greatly broaden the range of student activity that we can expect and explain. To sum up this section, we have seen that, first, there are episodes that the principle-based schemata model simply cannot handle. These episodes constitute evidence for the kind of knowledge that I am calling “symbolic forms.” Furthermore, I have argued that, because this knowledge exists, it is plausible to presume that it gets employed during more traditional types of symbol use. In addition, I presented a number of examples to illustrate that students do not always work by rigorous deduction from principles; instead, they wave their hands and rationalize to get the answer that they already know they want. This constitutes evidence that forms play a role in what, at first glance we might take to be the strict following of principle-based tradition.
Forms: A Primer In this section I am going to discuss all the forms that are necessary to account for the behavior of students in my data corpus. Because this knowledge system is not organized according to the principles of physics—and because it apparently has no other simple structure—there are no shortcuts for describing the breadth of forms, I have to describe each type of form and give examples. However, for the reader that forges ahead, this section is not without its rewards. There are many interesting details associated with particular forms, as well as important and subtle points to be made along the way. In addition, the broad scope provided by this overview is important. The hope is that it paints a picture of a physicist’s phenomenal world of symbols—how physicists and physics students experience a page of expressions. Keep in mind that this list is intended only to account for the particular data in my algebra corpus. If I had given students a different set of tasks, looked at a different population, and certainly if I was concerned with a different domain, the symbolic forms in this primer would be different. Nonetheless, I believe that this list captures much of the breadth as well as the most common and central forms of introductory algebra-physics. Adding tasks would likely discover more, but my belief is that these additions would primarily allow us to fill in some less common forms around the fringes of the form system. I hope to support this viewpoint when I present the systematic analysis in Chapter 6.
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In saying that this list of forms “accounts” for my data corpus, I do not mean to imply that the relation between this list and my data is straightforward. In Chapter 6, I will describe the procedure whereby I make contact with my data. However, I must warn the reader that, even there, I will not be presenting an exact procedure for extracting this list of forms from my transcripts. Instead, the procedure I will describe is highly heuristic, and the resulting categories depend heavily on theoretical arguments, as well as the data corpus itself. Although there is no very simple structure to the forms knowledge system, it is possible to organize forms according to what I call “clusters.” 10 Within a given cluster, the various schemata tend to have entities of the same or similar ontological type. For example, forms in the Competing Terms Cluster are primarily concerned with influences. In addition, the forms in a cluster tend to parse equations at the same level of detail. For example, some forms tend to deal with equations at the level of terms, while others tend to involve individual symbols. A complete list of the forms to be discussed here, organized by cluster, is provided in Figure Chapter 3. -4. In addition, a list of these forms with brief descriptions is given in Appendix D.
10
In using this term I am following diSessa, 1993. I will make close contact with diSessa and his use of this term in Chapter 5. 77
Competing Terms Cluster
Terms are Amounts Cluster PARTS -OF -A-WHOLE
[
+
–
BASE ± CHANGE
[
± ∆]
BALANCING
=
WHOLE - PART
[
-
CANCELING
0=
COMPETING TERMS
±
OPPOSITION
± …
SAME AMOUNT
–
Dependence Cluster DEPENDENCE […x…] NO DEPENDENCE SOLE DEPENDENCE
…]
]
=
Coefficient Cluster COEFFICIENT
[x
]
SCALING
[n
]
[…] […x…]
Other
Multiplication Cluster INTENSIVE •EXTENSIVE x × y EXTENSIVE •EXTENSIVE
+
IDENTITY DYING AWAY
x=…
[e− x…]
x×y Proportionality Cluster
PROP+
…x… …
PROP-
… …x…
x y CANCELING(B) …x… …x… RATIO
Figure Chapter 3. -4. Symbolic forms by cluster.
The Competing Terms Cluster To a physicist, a sheet of equations is, perhaps first and foremost, an arrangement of terms that conflict and support, that oppose and balance. The Competing Terms Cluster contains the forms related to seeing equations in this manner, as terms associated with influences in competition. Frequently—though not always—these influences are forces, in the technical sense. The forms in the Competing Terms Cluster are listed in Table Chapter 3. -1. COMPETING TERMS
±
OPPOSITION
–
BALANCING
=
CANCELING
0=
± …
–
Table Chapter 3. -1. Forms in the Competing Terms Cluster.
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We have already encountered a number of forms from this cluster. During Mike and Karl’s work on the air resistance task, they composed the equation:
a(t ) = −g +
f (v) m
I argued that their construction of this equation involved the COMPETING TERMS and OPPOSITION forms. Mike and Karl schematized the situation as involving two opposing influences, sometimes understood as forces and sometimes as accelerations. Recall that it was the
OPPOSITION
form that
was associated with Mike’s insertion of a minus sign: “now they're opposing so it's a minus.” Many of the interesting details here, in fact, have to do with how students select signs (+ or –) for the terms in their Competing Terms expressions. In cases in which these forms are in play, students frequently draw diagrams and associate the signs of influences with directions in the diagram. For example, a student may decide that influences acting to the right are positive, and influences acting to the left are negative. We also encountered some instances of BALANCING in previous episodes. I maintained that BALANCING
was responsible for the equating of the forces in Mike and Karl’s work on the spring
task; the force up due to the spring and the force down due to gravity were seen as “in balance.” In addition, I argued that Jack and Jim were BALANCING the momenta of the skater and the skate in the Stranded Skater problem when they wrote: m pf v1 = ms v2 The last form in this cluster, CANCELING, was encountered briefly in Alan and Bob’s work on the skater problem: 0 = ms vs − m pv p Alan
And we expect her to move to the left to compensate (her) mass of Peggy times the velocity of Peggy. [w.mod 0=MsVs–MpVp]
The CANCELING and BALANCING forms are quite closely related. Both of these forms involve equal and opposite influences. However, they do differ in some important respects. First, with respect to their symbol patterns, CANCELING is usually written with terms separated by a minus sign and set equal to zero, rather than as two terms separated by an equal sign. In addition, the associated schemata differ in a subtle manner. The schema associated with the BALANCING form involves only two entities, the two influences that are in balance. In contrast, CANCELING can involve three entities; two entities are 79
combined to produce a net result that happens to be zero. Furthermore, the relations between the entities in CANCELING frequently include a weak type of ordering of the two competing influences. Sometimes, for example, one of the influences is seen as a response to the other influence. This type of relationship is evident in Alan’s statement just above. It is as if Peggy’s motion is a response to the throwing of the skate; she throws the skate and then moves “to compensate.” BALANCING is an extremely common and important form, and thus merits a little further discussion here. BALANCING is important because it provides the central dynamical constraint of many problem solving episodes. For example, in Mike and Karl’s work on the spring problem described above, it was BALANCING that was really answering the “why” question behind all their work. Why do we write Fs = Fg ? Because the forces must balance. The next couple of examples involve student work on a task I call the “Buoyant Cube” problem. In this problem, as shown in Figure Chapter 3. -5, an ice cube floats at the surface of a glass of water, and the students need to find how much of the ice cube is below the surface. An ice cube, with edges of length L, is placed in a large container of water. (a) How far below the surface does the cube sink? (ρice = .92 g/cm3; ρwater = 1 g/cm3)
L
L
Figure Chapter 3. -5. The Buoyant Cube Problem.
In brief, this problem is typically solved as follows (refer to Figure Chapter 3. -6). Two forces act on the floating cube, the force downward from gravity, Fg , and a buoyant force upward from the water, Fb . In order for the cube to be motionless, these two forces must be equal. Furthermore, the force of gravity is equal to the weight of the cube, which is the product of the density of ice, the volume of the cube, and the acceleration due to gravity: ρi gVcube . And the buoyant force is equal to the weight of the water that is displaced by the cube: ρw gV water . If the amount that the cube sinks below the surface is given the label s, then the volume of water displaced is equal to L2s. With these relations in place, it is possible to solve for the value of s as shown in Figure Chapter 3. -6.
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Fb = ρ wgV water Fb = Fg ρ wgV water = ρ i gVcube
( )
( )
ρ wg L2s = ρ i g L3 ρ s= i L ρw s =.92L
S
Fg = ρ igV cube Figure Chapter 3. -6. A solution to the Buoyant Cube problem.
For almost all students, this problem strongly suggests BALANCING . Frequently they even oblige us by saying the word “balance:” Jack
Um, so we know the mass going - the force down is M G and that has to be balanced by the force of the wa=
Jim
=the displaced water.
What is truly striking, is that students seem to really possess the sense for
BALANCING
at quite a
fundamental level. In Mark and Roger’s work on this task, Roger carried most of the load, with Mark expressing confusion along the way. After Roger produced a solution, Mark tried to rationalize Roger’s work for himself. In particular, Mark was a little mystified by Roger’s first expression, in which he had equated the weight of the ice cube and the weight of the displaced water. W w = Wi Mark ultimately satisfied himself that this equation makes sense with this comment: Mark
We know one is gonna float. And then just sort of, I guess, I don't know from like all our physics things with proportionalities and everything, that we'll have to have something equal, like the buoyancy force should be equal to the weight of the block of ice. [g. pulling down block]
Here, Mark does not appeal to forces or anything like the notion that the net force must be zero. Rather he simply states that, if the block is going to float, then something has got to balance: “we’ll have to have something equal.” Instead of being an appeal to principle, Mark’s argument is an appeal to experience. Furthermore, the experiences he appeals to are of a special sort: “…from like all our physics things with proportionalities and everything, that we'll have to have something equal… .” Mark, like all other physics students, has a wealth of experience with equations, writing expressions, setting 81
things equal, and manipulating to find results. From this symbolic body of experience, Mark has developed the sense that setting things equal is a sensible, intrinsically meaningful action, and that it is typical physics. I will return to this point in later chapters. The final point I want to make about BALANCING is that its prevalence may actually extend to overuse; students sometimes employ BALANCING in a manner that could be called inappropriate. To illustrate, I will return to the Shoved Block problem, which I discussed briefly in Chapter 2. Recall that this problem involves two objects that are shoved in such a way that they both start moving with the same initial speed. The question is, given that one block is twice as heavy as the other, which block travels farther? As I mentioned earlier, the correct answer is that both blocks travel the same distance. The derivation of this fact can be quite short. (Refer to Figure Chapter 3. -7). After the shoved block is released the only force acting on it is the force of friction, which is F f = µmg . Since this is the only force, it alone is responsible for any acceleration that the block undergoes as it slides along the table. For this reason, this force can simply be substituted for F in the equation F=ma to give the equation µmg = ma . Then the m that appears on each side of the equation can be canceled to produce the final expression µg=a. Direction of motion
F f = µN = µmg Friction
∑ F = ma F f = ma µmg = ma µg = a
Figure Chapter 3. -7. A solution to the Shoved Block problem.
The important thing to note about the final result is that this expression for acceleration does not involve the mass. This means that it simply does not matter what the mass is, the shoved block always travels the same distance. We will have reason to return to this problem a number of times, but, for now, I just want to consider a single interesting episode. Alan and Bob had produced a solution very similar to the one presented in Figure Chapter 3. -7. In brief, their work looked just right. However, after they were done, Alan pointed to the equation µmg=ma and added this little addendum to their explanation: µmg=ma Alan
What Bob didn't mention is that this is when the two forces are equal and the object is no longer moving because the force frictional is equal to the force it received.
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I believe that this interpretation is clearly inappropriate. When Alan looked at the equation µmg=ma, he saw two influences in balance, the “force frictional” and the “force it received.” More specifically, he sees this equation as specifying a condition for an end state, in which the block comes to rest because these influences balance. However, as I have mentioned, once the block is released, only one force acts on it, the force due to friction. In other words, only one side of this equation corresponds to an influence! Thus, Alan is seeing BALANCING in an inappropriate context. The Proportionality Cluster When a physicist or physics student looks at an equation, the line that divides the top from the bottom of a ratio—the numerator from the denominator—is a major landmark. Forms in the Proportionality Cluster involve the seeing of individual symbols as either above or below this important landmark. Earlier in this chapter I argued that one of these forms,
PROP-,
was involved
in Mike and Karl’s construction of their novel expression for the coefficient of friction: µ µ = µ1 + C 2 m Karl
The coefficient of friction has two components. One that's a constant and one that varies inversely as the weight.
The idea was that Karl’s statement specified that the second component of their expression should be inversely proportional to the mass. Thus, the PROP- form was invoked and the ‘m’ symbol was written in the denominator of the second term. Table Chapter 3. -2 contains a list of all the forms in the Proportionality Cluster. PROP+ PROP-
…x… … … …x…
x y CANCELING(B) …x… …x… RATIO
Table Chapter 3. -2. Forms in the Proportionality Cluster
We also encountered some other clear examples of
PROP+
and PROP- in Chapter 2, though
they were not noted at the time. I have in mind the interpretation that Jim gave of the final result of the spring problem:
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x= Jim
mg k
Okay, and this makes sort of sense because you figure that, as you have a more massive block hanging from the spring, [g. block] then your position x is gonna increase, [g.~ x in diagram] which is what this is showing. [g.~ m then k] And that if you have a stiffer spring, then your position x is gonna decrease. [g. uses fingers to indicate the gap between the mass and ceiling] That why it's in the denominator. So, the answer makes sense.
In the first part of this interpretation, Jim sees the expression in terms of the PROP+ form, x is seen as directly proportional to the mass: “as you have a more massive block hanging from the spring, then you're position x is gonna increase.” And, in the second part of the interpretation, the expression is seen through the lens of PROP-: “if you have a stiffer spring, then you're position x is gonna decrease.” There were many instances in which students interpreted final expressions of this sort by announcing proportionality relations. For example, when solving the air resistance problem, students typically derived an expression very similar to v=
mg k
for the terminal velocity. In this relation, k is just a constant. To cite an instance, after Alan and Bob derived this equation, Alan commented: Alan
So now, since the mass is in the numerator, we can just say velocity terminal is proportional to mass [w. Vt∝m] and that would explain why the steel ball reaches a faster terminal velocity than the paper ball.
And, concerning the same relation, Mark said simply: Mark
So the terminal velocity of the heavier object would be greater.
Now I will move on to a form we have not before encountered, the RATIO form. In the RATIO form, the symbol pattern involves an individual symbol in both the numerator and denominator of an expression. The key to the RATIO form is that the ratio is seen as a comparison—it’s a comparison of the quantities that appear on the top and the bottom of the ratio. In most cases, the quantities involved are of the same type; for example, they may be both masses or both velocities. To see an example, I return to Alan and Bob’s work on the Stranded Skater problem. Recall that they began their work on this problem by writing this expression. 0 = ms vs − m pv p
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With this expression in hand, they proceeded to solve for v p , the velocity of Peggy: vp =
ms vs mp
Then Alan commented: Alan
Well the mass of the skate [g. Ms] is approximately small - it’s smaller. It’s probably much smaller than mass of Peggy. [w. Ms
