A ROLE FOR SIMULATIVE MENTAL MODELS IN SCIENCE CONCEPT REPRESENTATION: THE CASE OF INSTANTANEOUS SPEED
A DISSERTATION
SUBMITTED TO THE SCHOOL OF
EDUCATION AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
Jonathan Todd Shemwell June 2011
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This dissertation is online at: http://purl.stanford.edu/mq237pv6788
© 2011 by Jonathan T. Shemwell. All Rights Reserved. Re-distributed by Stanford University under license with the author.
This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Richard Shavelson, Primary Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Jonathan Osborne
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.
Daniel Schwartz
Approved for the Stanford University Committee on Graduate Studies.
Patricia J. Gumport, Vice Provost Graduate Education
This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in
University Archives.
iv Abstract
This study addresses the question of what kinds of cognitive processes are needed in order to represent science concepts. Its empirical focus is on how students represent an important concept in introductory physics, instantaneous speed. However, its theoretical application is to science concept representation more generally. It
advances the tentative claim that there are many science concepts for which cognitive representation—and hence, understanding—depends upon depictive mental
simulation.
The function of depictive representations as instruments of thought is most easily explained within mental model theory (Johnson-Laird, 1983; Nersessian, 2008). Mental models are organized units of representations that are structural, behavioral, or functional analogs of the physical world (Craik, 1943). The representations within mental models can be either language-like or depictive. Some depictive mental models are also simulative mental models. In simulative mental models, the thinker reasons by transforming or animating mental depictions in the ways that reflect possible
transformations in what is represented (Nersessian, 2008). Schwartz and Black‘s (1996a) gear rotation studies show how people can reason with simulative mental models. In these studies, people imagined gears rotating in a system to infer which direction a particular gear would turn. As an example of the function of simulative mental models in scientific thinking, Clement and Steinberg (2002) show how a student constructs mental images of movement to conceptualize electric current.
Knowledge Domain. My empirical focus was on the potential for a simulative mental model to be a productive conceptual resource for understanding a difficult concept within the mathematics of motion: instantaneous speed. Instantaneous speed is a foundational concept in Newtonian mechanics, yet many introductory students find it very difficult to grasp (Dykstra & Sweet, 2009; Trowbridge & McDermott, 1980b). For instance, students find it difficult to relate the speed at some point on the trajectory of an accelerating object (i.e., the instantaneous speed) to the average speed over a segment of trajectory that contains that point (Trowbridge & McDermott,
v
1980b). Figure 1 is an assessment item I designed to measure how well students could establish this relationship qualitatively.
The marble is speeding up as it rolls down the ramp. It starts to the left of point A and goes past points A and B, which are 5 centimeters apart. It takes 1 second to go from A to B.
As the marble passes by point A, its speed is: More than 5 centimeters per second Equal to 5 centimeters per second Less than 5 centimeters per second Explain why you chose this answer.
Answer: Less than 5 cm/second; because if it’s average speed is 5, it must start out slower than 5 and end up faster.
Figure 1. Assessment item for comparing instantaneous to average speed. Students often find it difficult to relate instantaneous speed (e.g., the speed at point A) to average speed (the speed for the segment AB).
When answering the question posed in Figure 1, many students respond incorrectly that the speed at point A is 7 cm/sec. They often give this answer even when acknowledging that the speed is changing, a fact that suggests that they understand the physical situation at a course level, but they lack the conceptual resources to reason precisely about speeds at points on the trajectory.
Hypothesis. According to my hypothesis, the difficulty that students encounter on tasks like Figure 1 and many others in the literature is that they are ―stuck‖ with a mental model of speed as something that is constant over a finite length of trajectory. This non-simulative mental model is a powerful resource for conceptualizing average speed, but it gets in the way when students have to think about speeds at points. The structure of this length model is much informed by Piaget‘s detailed account of what he took to be the developmental progression of children‘s understanding of speed (Piaget, 1946/1970). Piaget described the central conceptual entity for formal thinking about speed as being a spatial representation of a length of trajectory from starting
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point to endpoint. He showed how the relationships between speed, distance and time are bundled into the length representation. Essentially, faster is longer distance, or faster is shorter time. As a result of this bundling, thinkers reason very efficiently about speed relationships once the model is instantiated. Piaget‘s spatial representation fits the description of a mental model (length model) very well because the structural relationships between variables in the phenomenon are preserved in the structure of the representation.
The length model has two obvious shortcomings for thinking about
instantaneous speed: (1) it is a finite length and not a point; (2) speed is thought of as being constant over the length. Given these shortcomings, I intuited that an alternative mental model would be needed to help thinkers set aside the length model and
conceptualize speed as something happening at a point in space. In this hypothesized model, speed is bound to a depictive mental simulation of motion. That is, speed is a magnitude (fast or slow) associated with a mental simulation of fast or slow object motion over a short distance centered on a point in the visual field. I refer to this alternative model as the motion model. The reader can try out the motion model by looking at Figure 1 and projecting an image of an object moving slowly just as it passes A, and faster as it passes an imaginary point midway between A and B.
Design. My hypothesis predicted that instruction in which learners imagined an object‘s movement across a point in space would facilitate their construction of the motion model and thus enable them to conceptualize instantaneous speed more
effectively than learners who did not have this experience. I tested this prediction in an experiment with 36 ninth grade students engaged in hands-on learning in which they made many speed measurements of a marble rolling on a flat track. One of the two learning conditions supported students in visualizing the marble‘s motion as it crossed a point in space (i.e., supported the essential cognitive process for constructing the motion model); the other learning condition supported the length model of speed. Students in both length and motion conditions completed the same set of speed measurements, but they used different tools for conceptualizing them. These ―different‖ tools were actually two different configurations of a photo-gate timer, a
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typical school laboratory apparatus (Figure 2). Photo-gate timers often feature two gates with light sensitive detectors to measure how long it takes an object to go from the first gate to the second (Figure 2 - center). This dual gate configuration was intended to support thinking consistent with the length model. An alternate
configuration of photo-gate timers can measure speed by tracking how long a moving object of known size blocks the light beam of a single photo-gate (Figure 2 - right). In this case, an object will travel its own length during the time the light beam is
interrupted. This single gate configuration was intended to discourage thinking with the length model because there was no obvious starting or ending point. At the same time, it was meant to facilitate motion model construction by encouraging students to mentally simulate the displacement of the marble as a motion across the beam.
Procedure. Students participated in the study one at a time, using either single or dual photo-gates, depending on experimental condition. I guided them through the learning activities using a detailed instructional protocol. On the face of things,
students were investigating what happened to a marble rolling on a ramp. For instance, they found out whether a marble released higher on the ramp was faster at the bottom than a marble released lower down. However, the real force of the treatment came from making speed measurements with either single or dual photo-gates. All
measurements took place on a flat section of track at the bottom of the ramp on which the marble rolled at a constant speed, so the idea of different speeds at different points was not presented or made salient within the measurement process. In order to
encourage students to construct the relevant mental models, I periodically prompted Figure 2. Two ways to use photo-gates. With dual gates (center), timer starts when the beam at A is interrupted and stops when B is interrupted. With a single gate (right), the timer starts when the beam is first interrupted and stops when the interruption ends. Drawing at left by Grace Shemwell.
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them to verbally describe or draw pictures of ―how you use the photo-gate(s) to obtain the distance and time of the marble.‖ These verbalizations and drawings also served as descriptive measures of the cognition involved in the two learning conditions.
Results. On a posttest consisting of seven items mostly similar to the question posed in Figure 1, there were statistically significant condition differences in students‘ understanding of instantaneous speed. These differences were traceable to thinking observed during learning that was consistent with either length or motion models. Dual gate students often answered items like Figure 1 incorrectly, reasoning about overall distance and time. However, they frequently acknowledged that speed was changing. Thus, they knew their thinking was not adequate, but they lacked the representational faculty (i.e., the mental model) to describe differences in speed precisely. During learning, dual gate students‘ drawings and verbal descriptions of gate use were dominated by representations of the distance between the photo-gates. This shows that they thought about their speed measurements as times
pertaining to lengths of track.
In the single gate condition, half the students did significantly better than dual gate students at posttest, and half did worse. Those doing better were consistently able to tell which points on the trajectory should be faster or slower than the average speed. Those doing worse reasoned only about average speed and did not even acknowledge that speed was changing. During learning, single gate students‘ verbal descriptions and drawings of photo-gate use featured two kinds of variation that were strong predictors of posttest scores. First, students who described the displacement of the marble as a movement through the light beam scored higher at posttest than those who did not talk about this movement but merely recalled that the diameter of the marble was the displacement. Thus, in contrast to dual gate students, successful single gate students thought about their speed measurements as times pertaining to movements, instead of times pertaining to lengths. Second, students whose drawings were well structured, showing the change in position across the photo-gate beam precisely, did better at posttest than those whose drawings did not have this precision. Taking students‘ drawings to be reflections of their internal depictive representations, I inferred that
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students were only successful in learning if they generated a high quality depiction of the marble‘s movement in the mind‘s eye. In general, only those students who both verbalized displacement as motion and drew finely structured drawings were consistently able to distinguish instantaneous speed at posttest.
Conclusions. Considering the above evidence together, I concluded that instruction engaging single gate students in depictive mental simulation of the marble‘s movement across the light beam helped them to generate simulative mental models that were useful for representing instantaneous speed on the subsequent posttest.
Discussion. The present study shows that a simulative mental model can play an important role in helping students represent a difficult science concept. It makes plausible the more expansive hypothesis that simulative mental models are often useful for representing science concepts. This hypothesis has the potential to extend current theoretical accounts of how science concepts are represented. For instance, it extends diSessa‘s (1983; 1993) account of conceptual thinking as stemming from irreducible elements of conceptual thought called phenomenological primitives, or p-prims. Considering the example of Minstrell‘s (1982) finding that invoking a spring‘s force can help students conceptualize a table‘s force, DiSessa (1983; 1993) describes a p-prim, springiness (force arising from squeezing or deformation) as being the
knowledge element by which the normal force is conceptualized. In the simulative mental model hypothesis, invoking the primitive ―springiness‖ is further elaborated as constructing a mental model in which the notion of force is dependent upon depictive mental simulation. This more elaborate description of springiness is useful because it specifies a modal cognitive process, depictive mental simulation, that is either
necessary for (or at least contributes to) the representation of force. As a result of this specification, the simulative mental model hypothesis leads to novel predictions for student learning. For instance, it predicts that increasing students‘ general facility with depictive mental simulation will improve their ability to conceptualize the normal force. Of course, further studies are needed in order to sustain or refute the claim that simulative mental models should be recognized as a distinct form of concept
x
representation. However, the payoff from this line of inquiry could be considerable in the form of a more informative account of the cognitive processes that science
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Acknowledgements
I am grateful to many people for helping me get to this point. I owe a special thanks to Rich Shavelson. You supported me, Rich, through every step and misstep. I‘ll do my best to live up to your example in future.
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Table of Contents
Abstract ... iv
List of Tables ... xv
List of Illustrations ... xvii
Chapter 1 – Introduction and Rationale ... 1
Chapter 2 – Research Literature and Hypothesis ... 8
The Knowledge Domain ... 8
Student Conceptions of Speed with Accelerated Motion ... 10
Two Theoretical Accounts of Speed Thinking ... 15
Hypothesis ... 21
Chapter 3 – Methods ... 28
Context and Participants ... 28
School Context ... 28
Participant Characteristics ... 29
Recruiting Participants ... 30
Assigning Participants to Condition ... 30
Procedures ... 31
Instructional Materials ... 31
Overview of Instructional Protocol: Rationale ... 35
Overview of Instructional Protocol: Procedures ... 39
In-briefings and Out-briefings ... 46
Instrumentation and Data Analysis ... 47
Sources of Data ... 47
Instrumentation ... 50
Procedures for Data Analysis ... 57
Two Bodies of Evidence: What Students Learned and How ... 63
Chapter 4 – Evidence from Pre- and Posttests ... 64
Part 1: Students‘ Overall Reasoning about Instantaneous Speed ... 64
Student Reasoning on Average-Instantaneous Items ... 65
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Comparing Pretest Scores of Students in the Entire 9th Grade Physics Students to
Scores of the Thirty-six Students in the Experimental Study ... 68
Students‘ Reasoning about Instantaneous Speed ... 68
Part 2: Effects of Experimental Conditions ... 73
Student Learning As Evidenced by Posttest Scores and Pre-Post Difference Scores ... 73
Levels of Understanding for Average-Instantaneous Comparison ... 75
Discussion of Experimental Effects ... 76
Further Evidence Needed in the Form of Cognitive Process Data ... 81
Chapter 5 – Evidence of Cognitive Processes during Learning ... 83
Summary of Evidence ... 83
Evidence Bearing on the Motion Model ... 83
Evidence Bearing on the Length Model ... 86
The Evidence in Detail ... 88
Evidence Strand 1: Qualitative Reasoning about Speed, Distance, and Time ... 88
Evidence Strand 2: Students‘ Explications of Photo-gate Use ... 89
Strand 2 Part 1 – Student Verbalizations of Displacement ... 89
Strand 2 Part 2 – Student Drawings of Displacement ... 107
Evidence Strand 3: The Performance Assessment ... 110
Conclusions ... 116
Conclusions about the Motion Model ... 116
Conclusions about the Length Model ... 118
Chapter 6 – The Broader Hypothesis for Science Concept Representation ... 119
The Hypothesis Statement: Simulative Mental Models Can Account for Understanding of Many Different Science Concepts ... 120
The Hypothesis Applied to Three Science Concepts ... 121
The Marginal Utility of a Simulative Mental Model Hypothesis ... 133
The Contribution of the Present Hypothesis Relative to Other Simulative Mental Model Hypotheses: Expanding on Concept Representation ... 134
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In Closing: Why Research on Concept Representation with Simulative Mental
Models? ... 140
Appendix A – Instructional Protocol ... 142
Appendix B – Video Images and Script ... 152
Appendix C – Pretest ... 154
Appendix D – Posttest ... 157
xv List of Tables
Chapter 2
Table 1. The intuitive and formal stages in the development of Piaget‘s length conception, together with sub-stages and transition states………
17 Table 2. Predictions resulting from the motion model portion of the hypothesis. 25
Table 3. Predictions from the length model portion of the hypothesis…………. 27
Chapter 3 Table 1. Major instructional activities, Day 1………... 40
Table 2. Sequence of graded hints for helping students see how the photo-gate(s) could be used to measure displacement and elapsed time………. 44 Table 3. Major instructional activities, Day 2………... 46
Table 4. Elements of in- and out-briefings designed to ensure independent participation………... 47
Table 5. Sources of data for student‘s initial knowledge state, learning process, and final knowledge state……….. 49 Table 6. Coding scheme for average-instantaneous comparison items ………… 53
Table 7. Coding scheme for rollercoaster prediction……… 54
Table 8. Example verbal protocol for explication………. 60
Table 9. Coding definitions for types of displacement statements……… 62
Table 10. Coding definition for attributes of photo-gate drawings………... 63
Chapter 4 Table 1. Numbers of students at three levels of performance on average-instantaneous items, with mean score cut points………... 65 Table 2. Numbers of students at three levels of ramp or rollercoaster prediction………... 66 Table 3. Correlations between ramp and rollercoaster prediction scores and
mean scores for average-instantaneous items at pretest and posttest……… 67 Table 4. Number of students at three levels of ramp prediction on after-posttest interviews……….
67 Table 5. The statistical significance of the condition by pretest level interaction for mean scores for the three types of posttest items………
75 78
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Table 6. Predictions resulting from the motion model portion of the hypothesis (reprinted from Chapter 2, Table 2)……….. Table 7. Predictions from the length model portion of the hypothesis (reprinted from Chapter 2, Table 3)………...
79 Chapter 5
Table 1. Predictions from the length model component of the hypothesis (reprinted from Chapter 2, Table 3) together with evidence bearing on those predictions……….
87
Table 2. Numbers of follow-up prompts and displacement statements by
condition, summed over both explanations………... 91 Table 3. Experimenter‘s decision to issue one or more follow-up prompts after each student‘s initial response when verbally explicating photo-gate use……… 93 Table 4. Pearson correlations between posttest scores and number of
displacement statements (length or motion) in each explanation, broken out by statement type and prompting………... 96 Table 5. Three levels of DM score within the single gate condition, with
example explanations and calculations………. 105 Table 6. Frequency of displacement representations in student drawings
showing how to use the photo-gate(s) to obtain distance and time………... 109 Table 7. Correlation matrices, posttest scores with performance assessment
prediction and observation scores………. 111 Table 8. Photo-gate use errors during performance assessment observation
phase……….. 115 Table 9. Predictions resulting from the motion model portion of the hypothesis (reprinted from Chapter 2, Table 2)……….. 117 Chapter 6
Table 1. The essential elements of the simulative mental models for four
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List of Illustrations
Abstract
Figure 1. Assessment item for comparing instantaneous to average speed... v
Figure 2. Two ways to use photo-gates ...………. vii
Chapter 2 Figure 1. Average speed vs. instantaneous speed………..………... 10
Figure 2. Assessment item for comparing instantaneous to average speed... 12
Figure 3. Trowbridge and McDermott‘s instantaneous speed task………... 15
Chapter 3 Figure 1. Photo-gate timer………. 32
Figure 2. Ramp used for most learning activities……….. 33
Figure 3. Photo-gate simulator……….………. 34
Figure 4. Marble and pad……….………. 35
Figure 5. Main item types for pre- and posttests……….………. 51
Figure 6. Rollercoaster used for the performance assessment………... 55
Chapter 4 Figure 1. Average speed vs. instantaneous speed………..………... 70
Figure 2. Exercise for learning mathematical patterns in kinematics…………... 72 Figure 3. Mean posttest scores by experimental condition and pretest
level………... 74 Figure 4. Posttest scores for average-instantaneous items by experimental
condition and pretest level………. 76 Chapter 5
Figure 1. Single gate post test scores related to talk, drawings, and
performance assessment representational errors………..……... 85 Figure 2. Number of displacement statements by experimental condition
and type of prompting………..……... 94 Figure 3. Dual gate posttest scores vs. number of prompted length
statements………. 98 Figure 4. Performance assessment prediction by experimental condition
and pretest level………... 112
xviii Chapter 6
Figure 1. The force exerted by a table………... 123
Figure 2. Bridging cases showing table deformation……….…………... 124
Figure 3. Refraction of light passing from air to water………. 127
Figure 4. Assessing understanding of the refraction………. 128
Figure 5. Connected wheels analog………... 129
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Chapter 1 – Introduction and Rationale
How do people represent science concepts like osmosis, ion, or centripetal force? When learners explain what it means for an insect to have an abdomen or what happens when a wave moves along a taught string, what kinds of images, verbally encoded meanings, or sensory-motor impulses flash through their minds? Philosophers have wrestled with these kinds of questions for ages, and psychologists and cognitive scientists have toiled over them for decades. So have science teachers and education researchers. Why? Because the question of how students can represent science concepts lies behind the question of how they can learn them. Without some idea, however tacit, of the nature of the internal representations that students must construct, teaching for understanding is a shot in the dark.
The present study addresses the question of what kinds of cognitive processes are needed in order for learners to represent science concepts. Its empirical focus is on the cognitive processes inherent within two related concepts in introductory physics. However, its theoretical application is to science concept representation more
generally. It advances the tentative claim that there are many science concepts for which understanding depends upon a format of representation which requires the cognitive process of depictive mental simulation.
To consider a concept like fluid pressure or hormone and simply ask, ―What does the cognitive representation for this concept consist of?‖ is to pose a question that is philosophically and empirically untenable. However, questions about some aspect of a cognitive representation, or hypotheses describing to some degree a
representation‘s format, content, or function can be both empirically tractable and highly fruitful. An exemplar for both tractability and fruitfulness comes from the study of mathematics learning. I refer to the classic set of studies described by Griffin, Case and Siegler (1994) of young children‘s learning of number concepts. These authors hypothesized that number values are most effectively represented as a mental number line (see especially Case & Okamoto, 1996). In Griffin et al.‘s studies, kindergarteners played many different board games that incorporated spatial layouts and counting procedures that emphasized the hypothesized linear spatial representation. After
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playing the games for 20 minutes a day for a period of four months, children
developed much better understanding of number concepts than students in a matched control group. The differences in learning spanned multiple measures, including measures of transfer, and the effects of the intervention continued to be observable at the end of the next year of schooling. Later work by Siegler (2003) further confirmed the core hypothesis that the key internal representation of number values is a mental line (Siegler & Opfer, 2003; Siegler & Ramani, 2009). This hypothesis now explicitly informs the design of early mathematics curricula (Clements & Sarama, 2007;
Starkey, Klein, & Wakeley, 2004).
In science education, there is no coherent body of work that builds over many studies a detailed theoretical description of students‘ internal representations of a specific set of concepts, as Griffin, Case and Siegler have done in mathematics. However, many different researchers have taken important steps in the direction of such theoretical descriptions. Some, for instance, have shown what kinds of external representations or analogs are necessary for supporting conceptual understanding (Clement, 1993; Gentner & Gentner, 1983; Smith, Snir, & Grosslight, 1992). Others have described aspects of internal representations by observing the cognitive output of students or scientists engaged in conceptual thinking (Clement & Steinberg, 2002; Clement, 2009; diSessa, 1983; diSessa, 1993).
Studies of science thinking and learning like those mentioned above have brought us far in understanding science concept representation, but we still have far to go. One aspect of representation that needs to be better understood is format: whether (or under what conditions) representations should be thought of as being depictive and therefore analog, as opposed to consisting of language-like propositions. In Schwartz and Black‘s (1996a) definition, a representation is depictive if it retains perceptual aspects of its referent. A mental image is one example. A depictive representation is analog in that the representation incorporates structure which mimics the structure of what it represents.
A classic set of studies addressing the representation of force exemplifies the format question. Clement (1993), following Brown and Clement (1987; 1989), and
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Minstrell (1982), showed that in order for students to understand that a rigid table can exert an upward force on a book that lies on top of it, they had to think of the table as being flimsy, yielding to the downward pressure of the book. Furthermore, Clement showed that it was not sufficient for students to simply know, as a fact, that the table yielded. Rather, they had to work through a set of ―bridging cases‖ that illustrated the yielding. This result suggests that students must construct internal representations of the yielding or deformation of a perceptually rigid surface like a table in order to conceptualize its exertion of force. Moreover, the crucial role played by the illustrations of yielding suggests that the format of the required internal
representations is depictive as opposed to language-like. However, Clement, whose focus was more on exploring the effectiveness of the bridging cases, and less on the nature of the representations involved, did not advance the hypothesis that the force representations suggested by the bridging cases were depictive. However, he did note that the bridging cases ―appear to work with knowledge representations that are qualitative, physical intuition schemas, not at a level that uses formal notations‖ (Clement, 1993, p. 1252).
diSessa (1983;1993), in part interpreting Brown, Clement, and Minstrell‘s work, extended Clement‘s account of force representation by explicitly hypothesizing that representation of contact forces (e.g., the table‘s upward force on the book) depends on representation of elastic material deformation as a result of contact. However, diSessa, who was not attempting to construct a theory of concept
representation per se, did not address the question of representational format. In his account, force arising from elastic deformation was simply a thought element which either occurred or did not occur. As a result, despite the tantalizing suggestion of mental depiction inherent in Brown and Clement‘s bridging cases, the question of whether depictive representation is sufficient or necessary for conceptualizing force remained unanswered.
There are good reasons for researchers to avoid questions about the format of concept representations, or at least approach such questions obliquely. One is
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incorporate a depictive format, it can be very hard to collect convincing evidence that the cognitive process of mental depiction is actually occurring (but see Clement, 2003; Nersessian, 2002; Roth, 2001). Another reason for avoiding the study of
representational format is that there are vast numbers of science concepts to be
considered. A program of intensive study of each separate one would be like trying to empty an ocean with a pail. This problem of vastness is exacerbated by the fact that science concepts are rarely, if ever, unitary conceptual entities from the thinker‘s standpoint (or even theoretically, as with wave-particle duality). Rather, any one science concept demands many different forms of representation, depending on the theoretical framing involved and the situational context (diSessa, 1993; Gupta,
Hammer, & Redish, 2010). For this reason, an intensive study of representation in one context might address only a small piece of the cognitive apparatus needed for robust understanding of a given concept.
Despite the objections just stated, there are at least four reasons that studies of representational format are worthwhile to undertake. First, although there endless science concepts (and aspects of concepts) to be learned, only some of them pose the severe challenges to learning which merit intensive study. Certain of those concepts, such as the one featured in the present study, are unsolved problems in science pedagogy, which, because they are foundational to further learning, play havoc with students‘ progress through the larger curriculum. The existence of such extremely troublesome concepts doubtless contributes to debates within the teaching community about whether all students should be expected to learn science at the same level (Dyson, 2001; Ellse & Osborne, 2004). If visions of science for all (Hehn & Neuschatz, 2006) are to be substantially fulfilled, then the representation of certain ―most difficult‖ concepts must be understood much better than it currently is.
The second reason to study concept representation, and within that representational format, is that it is sensible to expect that theoretical accounts of representation for one concept and context can inform accounts of representation for other concepts and contexts. Thus, a series of studies of individual concepts is not necessarily a process of infinite regress. Rather, a small web of theoretical descriptions
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could be developed that would inform researchers‘ and practitioners‘ understanding of how myriad concepts are represented. P-prim theory (diSessa, 1983; 1993) is a case in point.
Thirdly, it is at least plausible that representations for certain classes of concepts have similar formats and content structures. For instance, it is possible that depictive representation of movement is needed for representing both springiness and electric current. Such commonalities suggest an overlapping set of cognitive processes for generating representations of seemingly unrelated concepts. If these commonalities were found to exist, then instruction to support understanding could be reorganized to be much more coherent than it currently is. The core cognitive processes that support representations could become the instructional objectives, and diverse concepts and their representations could become the contexts in which these core processes are instantiated. This situation would be analogous to Griffin et al.‘s board games, which incorporated many different surface features but shared the common objective of encouraging children‘s linear-spatial representation of number. With such a transformation in objectives, the field of science education could take a big step forward in the effort to put aside the teaching of science as ideas in isolation, and focus instead on the core thinking that is needed to engage with those ideas.
Fourthly and finally, new ways of both describing and observing the format of representations continue to open up. On the theoretical front, the theory of mental models which was once in some confusion (Rips, 1986) has been broadened and clarified (Johnson-Laird, 2004; Nersessian, 2008; Thagard, 2010). Armed with the latest theoretical and empirical apparatus, it is becoming possible to ask the more direct questions about the contents of representations that researchers only a decade before may have reluctantly set aside (Clement, 2009).
The specific purpose of the present study is to show that there is an important role for the cognitive process of depictive mental simulation in the representation of science concepts. I define depictive mental simulation as manipulating or transforming perceptual representations such as visual images, as in Shepard and Metzler‘s classic studies (Shepard & Metzler, 1971). Depictive mental simulation enables concept
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representation because of the role that it plays in simulative mental models. A simulative mental model is a depictive mental simulation that incorporates the same functional (or transformational) constraints as the entity or process it represents (Nersessian, 2008).
To illustrate the role of depictive mental simulation in concept representation, I focus on students‘ learning of two related concepts from introductory physics. The concepts, average speed and instantaneous speed, come from the domain of
kinematics, or the mathematics of motion. The study‘s hypothesis describes the two concepts as relying on different types of mental models. The mental model for average speed incorporates a static (non-simulative) depictive representation, the essential aspects of which were first described in detail by Piaget (Piaget, 1946/1970). The representation for instantaneous speed is a simulative mental model. That is, it is a mental manipulation of an organized set of depictive representations, and the concept of instantaneous speed is embedded in this manipulation. The empirical method of the study is to test predictions about learning that devolve from the hypothesis and thereby make claims about the cognitive processes— especially processes of depictive mental simulation—that inhere in the representation of instantaneous speed as distinguished from average speed. The broader purpose behind this method is to advance the idea that concept representation in science may often require depictive mental simulation.
In the next chapter of this work, I develop the research hypothesis describing the cognitive processes needed for representation of average and instantaneous speed. The development begins with a review of what is known about student conceptions of speed, from Piaget‘s in-depth developmental studies in the 1930‘s and 40‘s, to
misconception studies in the late 1970‘s, to research on classroom learning in the late 2000‘s. Next, I examine two theoretical perspectives on ―speed thinking‖ in light of this body of evidence. I draw from both of these perspectives, together with the theory of mental models, to present my own hypothesis describing the mental models needed to represent instantaneous and average speed as distinct concepts. The hypothesis leads to several research questions and predictions about student learning with which the chapter concludes.
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In Chapter 3, I describe the methods by which the study was conducted and the data analyzed. At the heart of these methods is a true experiment with two learning conditions, each supporting one of the two hypothesized mental models.
In Chapters 4 and 5, I present the results of the experiment and discuss what they reveal about student representations. Chapter 4 provides the experimental results: what students learned in each condition. Chapter 5 provides evidence of student
thinking during learning that sheds light on how the experimental results came about. Both chapters incorporate evidence-based arguments about the cognitive processes needed for representing average and instantaneous speed. The body of evidence supports the study‘s hypothesis that the cognitive process of depictive mental simulation contributes to the understanding of instantaneous speed, and that the representation (or at least this one representation) is a simulative mental model.
In Chapter 6, I sketch the more general hypothesis for science concept representation, arguing that simulative mental models with characteristics similar to the one featured in the present study can be instrumental in representing many different science concepts. I build this hypothesis by constructing a generalized account of the simulative mental model for instantaneous speed and then providing arguments for three different instantiations of this generalized account as three distinct simulative mental models, each representing a different science concept. I conclude the dissertation with a call for several categories of further research to test and refine this general hypothesis.
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Chapter 2 – Research Literature and Hypothesis
In this chapter, I develop the research hypothesis by reviewing the literature bearing on the representation of the focal concepts, average speed and instantaneous speed. I begin by introducing the focal concepts and establishing their importance to the larger curriculum. I then review extant research on conceptions and
misconceptions that learners tend to exhibit in their thinking about these concepts. Next, I outline two theoretical accounts of speed thinking and critique them according to how well they explain the existing evidence of student conceptions as well as their utility for framing further hypotheses. In light of this critique, I then present my own hypothesis, which utilizes mental model theory, especially the theory of simulative mental models, to introduce a novel conceptual entity for thinking about instantaneous speed. Finally, I lay out the predictions devolving from the hypothesis that the study is designed to address.
The Knowledge Domain
This study was conducted within the general domain of kinematics, or the mathematics of motion. Kinematics is an important topic within introductory physics because it furnishes part of the conceptual foundation for Newtonian mechanics. Incorporating as it does many concepts that were far from obvious to great thinkers of antiquity and the middle ages (Sherry, 1986), it is known to be one of the most
challenging topics in the introductory physics curriculum (Hewitt, 1999). Within kinematics, this study addressed how students think about and learn one of the more difficult concepts: instantaneous speed1 for accelerated motion. This concept was considered in the context of the simplest possible case of speeding up in a straight line, and in combination with the related concept of average speed.
The notion that that an accelerating object has a distinct speed at each instant of time is nearly axiomatic in kinematics instruction. The same is true of the idea that,
1
The term ―velocity‖ is often used by physicists to encapsulate the idea of speed combined with direction. I was only interested in velocity for an object accelerating in a straight line without reversal of direction. Hence, the term instantaneous speed is a sufficient and more precise description of the topic under study.
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for each finite interval of motion, there is an average speed determined by the overall distance traveled and the overall elapsed time. Figure 1 is an illustration from an introductory high school curriculum that presents both kinds of speeds and shows their relationship. The figure shows that the rock dropped from the cliff starts with an instantaneous speed of zero and steadily speeds up as it falls. At the end of the first second, it has an instantaneous speed of about 10 meters per second (m/s). However, since the instantaneous speed builds up from zero to 10 m/s, the average speed during this second must be less than 10 m/s. In the special case2 presented here, the average speed, 5 m/s, is half of the final instantaneous speed. However, the more important point for my purposes is that the average speed is less than the final speed and more than the initial speed.
2 In the case of constant acceleration, the average speed is the numerical average of the starting and
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Figure 1. Average speed vs. instantaneous speed. During the first second of fall, the rock travels 5 meters, so its average speed is 5 m/s. At the end of the first second the instantaneous speed, by contrast, is 10 m/s, as shown on the second speedometer. Reprinted by permission from Hewitt, P.G., Conceptual physics: The high school physics program. Concept-development practice book, page 15, 1992. Copyright 1987 Addison-Wesley Publishing Company.
Student Conceptions of Speed with Accelerated Motion
In situations like Figure 1, learners must think of speeds during intervals of time as distinct from speeds at instants of time for accelerated motion. What kinds of thinking do they tend to exhibit when they try to conceptualize these quantities? The following review shows that they tend to be very good at thinking about speeds over intervals and very poor at thinking about speeds at instants.
Conceptions of speeds during intervals. When asked to consider a speed for some finite interval in a situation with accelerated motion, people generally perform a simple mental transformation to aid their thinking. They gloss over the fact that speed
Speed reading starts at zero
After falling 1 second, the speed reading is 10 m/s, but the distance fallen is 5 m
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is changing and conceptualize the speed as a constant during the interval they are considering. Piaget (1946/1970) showed that most children 13 or 14 years old could do this very well. (Much more will be said of Piaget‘s work in the next section.) You, the reader, may wish to try out the transformation on Figure 1 by considering ―the different speeds‖ for the three depicted intervals. You‘ll think of each one as being faster than the previous, but you‘ll also think of each interval as having a constant speed.3 For many purposes in kinematics this conception of accelerated motion as constant speed over some length is a very helpful one. For instance, considering that ―the speeds‖ for the first three intervals are 5, 15, and 25 m/s, you can begin to understand what it means for acceleration to be constant: for each interval, ―the speed‖ increases by the same amount, 10 m/s. Your conception of the speeds as being constant for these different lengths, together with metrification of these speeds, helps you concentrate on the changes in speed from one interval to the next.
Getting stuck in constant speed thinking. Sometimes, learners need to switch
out of constant speed thinking and into thinking about speeds at instants of time or points on a trajectory. As an illustration, consider again the first second of the rock‘s fall in Figure 1. Using your constant speed conception, you can easily think of this interval as having ―a speed‖ (i.e., an average speed) of 5 meters per second. However, it can be difficult to let go of this conception of ―a speed‖ and reason that early within the interval, the rock moves more slowly, and near the end of the interval, faster. The problem is that students often fail to make this switch in thinking. Instead, they continue to rely on their constant speed conceptions. Figure 2 is an assessment item that I developed to test for this tendency to fixate on constant speed when there is a need to switch to thinking about different speeds at different places. The situation is isomorphic to that of Figure 1 in that the marble speeds up from some initial speed to a final speed with an average speed in between the two. The student has to reason that, although the average speed for the interval is 5 centimeters per second, the marble rolls slower than this early in the interval and faster later on.
3 If you did this and doubt that you were thinking in terms of constant speed, try again but look at any
one of the intervals and visualize the rock speeding up during the interval. You‘ll feel a quite different kind of thinking swing into gear.
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The marble is speeding up as it rolls down the ramp. It starts to the left of point A and goes past points A and B, which are 5 centimeters apart. It takes 1 second to go from A to B.
As the marble passes by point A, its speed is: More than 5 centimeters per second Equal to 5 centimeters per second Less than 5 centimeters per second Explain why you chose this answer.
Answer: Less than 5 cm/second; because if it’s average speed is 5, it must start out slower than 5 and end up faster.
Figure 2. Assessment item for comparing instantaneous to average speed. To answer this question correctly, the student must switch out of constant speed thinking and conceptualize the marble as moving more slowly at A and faster at B. Many students have difficulty making the switch.
To indicate the strength of students‘ predilection to get stuck in constant speed thinking when responding to the question posed in Figure 2, I will provide here a preview of a small portion of data from the present study. The data consists of 85 suburban ninth grade students‘ responses to this question. About 50% of the students answered the question incorrectly by saying that the speed at the beginning of the interval (at point A) was the same as the average speed, 5 centimeters per second. The students who gave these incorrect answers did not acknowledge the changing speed of the marble in any way, despite evidence from other assessment items that they
understood at some level that the marble was speeding up. Trowbridge and
McDermott (1980a) point out a similar tendency among college students to focus on average speed and ignore instantaneous speed when both conceptions are salient.
Intrusion of constant speed thinking (speed-as-length intrusion). The
assessment item in Figure 2 incorporates strong cues for constant speed thinking that make it hard to switch over to thinking about different speeds at different places. Research by Dykstra and Sweet (2009) indicates that inappropriate constant speed thinking also extends to situations where the cues for it are not so strong. These
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authors describe a surprising occurrence when middle school students in their study were asked to walk with accelerated motion aided by a sonic ranging motion detector. This device gives a real-time plot of an object‘s (or a person‘s) position or velocity with respect to time. The students were asked to walk in front of the motion detector while speeding up gradually. Their written instructions were to ―start from rest and walk away from the motion detector while speeding up. Repeat your motion several times until the velocity graph shows a smooth steady increase‖ (pp. 471, 472). The authors found that many students had difficulty with this part of the task in that they did not actually speed up gradually. Instead, they walked slowly for a time, then shifted to a higher speed:
The students would walk with a steady velocity or would walk slowly and then suddenly walk with a greater speed. When coaching each other how to move, they would say things like, ―Walk slow and then go fast.‖ (Dykstra & Sweet, 2009, p. 472)
It appears that, when asked to think about changing speed, Dykstra and Sweet‘s students thought in terms of different constant speeds occurring over
perceptible lengths. The authors described these students‘ thinking as the ―snapshot‖ conception of speed. I interpret it as an inappropriate intrusion of the conception of speed as something that is constant over a finite length into what should have been thinking about continuously changing speed. I call this the speed as length intrusion. In a separate task in which children described the motions of accelerated objects such as the vertical drop of a bean bag, or the motion of a can of food rolling down an incline, a large proportion of students exhibited snapshot conceptions (possibly, speed-as-length intrusions). There was some evidence that college students in the same study (although performing a different task) were also highly prone to these intrusions. Conceptions of speeds at points. The tendency to exhibit speed-as-length intrusions described above, together with the finding that students tend to get stuck in their constant speed thinking when it is cued, suggests that the conception of constant speed over a finite interval is a robust and potentially dominant conception within students‘ thinking about accelerated motion. However, a general predilection for
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constant speed thinking is only one side of the problem when it comes to thinking about different speeds at different points. The other, complementary side seems to be that many students do not have (or cannot access) a good way of thinking about an object moving slightly faster at one point than at another point. This aspect of the problem can be illustrated by considering again the preview of data from the present study showing that 50% of suburban ninth grade students answered the question posed in Figure 2 without acknowledging the fact that the speed was changing during the depicted interval. Another 40% of these students also answered this question incorrectly, but they gave very specific indications that they knew this answer to be inconsistent with the fact that the marble was speeding up, or else they answered correctly but gave a poor explanation for why the answer was correct. In my
interpretation, these students knew that their constant speed thinking was incorrect, but they did not have the requisite conceptual resources to reason effectively about the difference between the speed at A and the speed during the interval from A to B.
A classic study by Trowbridge and McDermott (1980b) provides further evidence that learners often have very weak conceptions of speeds at points on an accelerated trajectory. They found that when college students were asked to reason qualitatively (in terms of relative magnitudes) about instantaneous speeds, they
responded as if they had only a vague idea of what this concept meant, despite the fact that they could recite and use its mathematical formula. In one of their tasks, students were asked to compare the speed of a ball rolling at constant speed on a flat track to that of a ball rolling on a parallel track uphill on a slight incline and therefore slowing down (Figure 3). The uphill ball, B, was initially faster than the flat-track ball, A, and passed it. However, as the uphill ball slowed down, the flat-track ball became the faster of the two, so A passed B in turn. Students were shown the demonstration several times and asked ―do these balls ever have the same speed?‖ The answer to this question is that the balls have the same speed when B has slowed down just enough to match A‘s speed at a single instant. Of special note here is the fact that there could be only one time in which the balls had the same speed, since B started out faster than A and ended up slower.
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Figure 3. Trowbridge and McDermott‘s instantaneous speed task. When asked if balls A and B ever had the same speed, many students answered that the speeds were the same twice: at the points at which one of the balls would pass the other. Students who clearly understood that B started out faster and ended up slower than A did not deduce that there could be only one place where the speeds were the same. Reprinted by permission from Trowbridge, D. E., & McDermott, L. C., American Journal of Physics, vol. 48, page 1023, 1980. Copyright 1980 American Association of Physics Teachers.
Many of the students responded to the question by saying there were two different places for which the speeds were the same. These were the positions where one of the balls passed the other. This is a surprising answer, considering the fact that children as young as five will readily say that an object that is overtaking another object must be going faster. It seems that these college students were reasoning poorly at least in part because they could not conceptualize faster and slower speeds at different places on the uphill ball‘s path.
Two Theoretical Accounts of Speed Thinking
In this section, I present two largely opposing theoretical accounts of the thinking that underlies the conceptions of speed just described. One account is older than the other, and much more elaborated with regard to the specific conceptual issues addressed in this study. After presenting both accounts, I review some limitations of each for the purpose of better understanding how people think about and learn instantaneous and average speed.
Account I – Piaget and the length conception. By far the most
comprehensive theoretical treatment of speed thinking to date was published in book form by Jean Piaget in 1946 and translated to English in 1970 (Piaget, 1946/1970). In this work, Piaget explained how the mature thinker‘s conception of speed developed from initial intuition to a formal conception and finally to operations on that
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conception. He argued, using qualitative evidence from a progression of interview tasks, for the evolution of a single conceptual entity in which the relations between speed, time and distance were coordinated. This conceptual entity was a representation of a linear element of space (a length) associated with an elapsed time. It was meant to progress through a series of stages starting out as essentially direct perception and ending up as a symbol that could be utilized in higher order (e.g., mathematical) systems of thought. Piaget assumed that in its final form, the length conception would symbolize speed as an intensive quantity (i.e., a quantity that retains its essence when divided into smaller and smaller elements).
Development of the length conception. Piaget gave most of his attention to the
first two stages of what he presented as a three-stage progression of speed
development: (1) intuitive; (2) formal; (3) formal operational. The first two stages are shown in Table 1, together with sub-stages and transitional states. Children in the intuitive stage would only say that one object was faster than a slower one if it was seen to overtake it on a parallel track. Once they reached the formal stage, children could reason that a longer path in the same time meant a higher speed, and they could determine this without observing the overtaking event. This conception is referred to as longer is faster in Table 1. There are two longer is faster sub-stages, denoting two levels of abstraction. For students at sub-stage longer is faster I, two objects had to be on straight, parallel paths in order for them to say that the one with the longer path would have to be faster. Students at longer is faster II could apply this conception correctly to nonparallel or curved paths. Evidently, children at this sub-stage had developed a more abstract notion of longer paths.
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The intuitive and formal stages in the development of Piaget’s length conception, together with sub-stages and transition states
Description. An object is faster than another when…
Transition state
Sub-stage Stage Age It can be seen to overtake the other on a
parallel path
Perceived overtaking
Intuitive 5 It travels a greater length in the same
time on a parallel path causing the thinker to imagine that overtaking must occur
Visualized overtaking
On straight, parallel paths only It travels a greater length in the same time
It travels the same length in a shorter time
Longer is faster I
Formal 8
On curved or nonparallel paths
It travels a greater distance causing the thinker to mentally translate nonlinear distance into length
Translation of distance to length On curved or nonparallel paths
It travels a greater distance
Longer is faster II
Formal 11
Piaget argued that the progression through the sub-stages reflected the
evolution of a single entity as opposed to a succession of different entities. He backed this argument by demonstrating that each sub-stage had roots in the previous one. Part of this evidence is provided in Table 1 in the form of two different transition states. The first transition state, visualized overtaking, shows that the sub-stage longer is faster - I has roots in the earlier sub-stage, perceived overtaking through the act of visualizing the unseen overtaking that must occur on parallel paths of different lengths for two objects that start and end at the same time. The existence of this transition state suggests that the formal conception of longer is faster is rooted in the earlier
conception of overtaking. The second transition state, from longer is faster I to longer is faster II, is evidenced by students mentally straightening curved and nonparallel tracks in order to relate speed and time to distance. This transition state suggests that older children‘s robust conception that faster provides for greater distance (or, with the
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same distance, less time), rests upon an internal representation of length, not an abstract representation of distance:
…the distance traveled (lengths or intervals in space between starting and stopping points) and the time taken (duration or interval in time between the simultaneous moments of departure and the simultaneous moments of arrival) are directly correlated in the form of speed. (Piaget, 1946/1970, p. 132)
Thus, Piaget saw the mental representation of length, explicit in longer is faster I, and implicit in longer is faster II as a incorporating a coherent relational structure between speed, length, and time. The spatial format of the length representation, and the notion that relational structure functions as part of the representation are points I will return to later in this chapter when I present my own hypothesis.
As Piaget saw it, the spatial representation of length was not the conception utilized by mature thinkers, but a stepping stone toward that conception. This comes out in his discussion of the results of the final interview task in the book, which is the only one of the tasks that did not deal with constant speed motion. This task featured a drawing of a skier about to accelerate down an inclined plane. The skier was shown to pass through four different intervals marked with flags equally spaced along the incline. The interviews showed that children 13 and 14 years old could reliably utilize their length conceptions of speed to describe the skier‘s motion as accelerated by treating the different sections of the incline as successive lengths with different constant speeds. Piaget interpreted these operations on the length representation (setting them end-to-end, each with a different speed) as early steps on the way to ―a progressively more operational spatio-temporal construction‖ (p. 275). That is where he left the matter with regard to empirical data, but his further remarks indicated that he expected the length conception of speed to continue to gain in abstraction and therefore take on an increasingly symbolic character and function. Inherent within this view seems to have been the assumption that the length conception would, at some point, be abstract enough to represent speed at a point in space or an instant in time.
The length conception in extant curricula. Whether or not Piaget assumed
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length conception, precisely that view was acted on by successors Trowbridge and McDermott, and later by Rosenquist and McDermott, who developed what is arguably the most elaborate curriculum currently available for teaching about instantaneous and average velocities. Trowbridge and McDermott (1980b) described the essence of students‘ difficulty with thinking about instantaneous speed to be ―the problem of extending the concept of speed over a finite interval of time to the case of an infinitesimal time interval‖ (p. 1021). The pedagogical challenge, as according to these authors, was to develop the length conception into the dimensionless abstraction that Piaget assumed would develop. Many years after Trowbridge and McDermott‘s studies, Rosenquist and McDermott (1987) published a paper illustrating several methods for helping students develop the abstraction. In general, these methods encouraged students to focus their reasoning with the length conception on very small segments of trajectory, and compare different segments to each other to see
differences in speeds. They then appealed to students to think of the lengths as infinitesimally small yet different from one another for changing speed. Some years after Rosenquist and McDermott described these methods, McDermott published a selection of them within a formal curriculum that continues to be in wide circulation (McDermott, Shaffer, & University of Washington Physics Education Group, 2002).
Account II – knowledge in pieces. Andrea diSessa‘s theory of
phenomenological primitives (p-prims) addresses speed thinking briefly, and to a sufficient degree to frame an alternative to Piaget‘s theoretical account (diSessa, 1993). diSessa breaks Piaget‘s central conceptual entity, speed as length, into a
number of different primitives. For instance, the p-prim less distance takes less time is potentially a different thought element than higher speed takes less time.4 diSessa‘s general view that conceptual thinking stems from diverse and fragmented knowledge pieces runs counter to Piaget‘s preference for a primary conceptual structure. The difference in the two views is of critical importance because it predicts different routes for conceptual development which, in turn lead to very different approaches to
instruction. As I have shown, in Piaget‘s view, the conception of constant speed over
4 DiSessa does not attempt a comprehensive list. The p-prim higher speed takes less time is my own
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finite length (length conception), once developed, undergoes further transformation, eventually taking on the character of a language-like symbol that can handle all kinds of speed thinking. In diSessa‘s view, the knowledge fragments for constant speed do not undergo transformation but retain their original form and function. Development occurs in shifting of priority among fragments and by the activation of additional fragments that incorporate aspects of speed thinking that are missing in the constant speed elements. However, the particular fragments needed for instantaneous speed are unspecified.
Comparing the utility of Piaget’s and diSessa’s theoretical accounts. Although, in general, diSessa‘s manifold view of conceptual knowledge is more consonant with existing evidence of conceptual thinking than Piaget‘s account of a single entity, Piaget‘s length conception does explain some of the existing data very well. His argument that a hard-learned spatial representation embeds the relational structure between the speed variables specifically addresses the fact that people are especially good at reasoning about constant speed motion. Moreover, the spatial nature of this entity explains the problems with people‘s reasoning about accelerated motion, previously discussed, of getting ―stuck‖ on constant speed thinking when length is cued, and intrusion of speed-as-length thinking when length is not cued. By contrast, the theory that speed thinking depends on a loose collection of knowledge fragments does not explain the preference for thinking of constant speed in terms of perceptible lengths. The obvious problem with Piaget‘s account shows up in the fact that very often the length conception does not develop into an entity that can handle speeds at points in space. The most compelling evidence for this lack of development can be seen in Trowbridge and McDermott‘s studies in which students reasoned poorly about instantaneous speed (Trowbridge & McDermott, 1980a; 1980b). The same students shown to have great difficulty with instantaneous speed in these studies were entirely proficient at Piaget‘s tasks requiring formal operational thinking about accelerated motion with the length conception. Evidently, these students‘ length conceptions had not become increasingly abstract as Piaget would have expected. A trivial explanation for this occurrence is that the length conception stalls in further development for some
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people and not others. However, there is no good reason why this stalling should occur when, in general, most people do reach the requisite formal operational stage of
reasoning with the length conception.
diSessa‘s manifold view of conceptual thinking provides a very simple and highly plausible explanation for why learners can conceptualize average speed but not instantaneous speed: the two concepts depend on different conceptual entities. This view suggests that the remedy to the problem of getting stuck in constant speed thinking is to activate a rival set of cognitive processes. Piaget‘s theoretical account, by contrast, has no remedy other than to learn the length conception better. However, the theory of phenomenological primitives as currently articulated has important limitations for developing a rich account of how people think about average and instantaneous speed. Unlike Piaget‘s detailed description, the p-prim theory is sparsely articulated for speed thinking, and says almost nothing about instantaneous speed. So even though the theory implies that there should be p-prims for instantaneous speed, at present, nobody knows what they are or where to look for them. Another limitation is that the architecture of the p-prim theory is set up to address the format of knowledge representation. For this reason, the theory can do little to inform, or be informed by, hypotheses specifying formats of cognitive processes by which ideas are represented and the ways in which different representational formats might play out in thinking and learning.
Hypothesis
My own account of speed thinking drew on both Piaget‘s and diSessa‘s theoretical perspectives. Following the general direction of p-prim theory, I assumed that thinking about average speed and instantaneous speed rely on different conceptual entities. Also, I conjectured that these entities are very simple knowledge elements with grain sizes similar to those of p-prims. I utilized Piaget‘s account of speed thinking in turn as the basis for the default conceptual entity for constant speed
thinking. However—and this is the essential point to be made in the present paper—in order to conjecture an alternate entity, I relied on a theoretical account of thinking that
