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But the use of math in physics is not so simple or sequential as this diagram may tend to indicate. The physics and the math get intimately tangled, as we have seen in the examples above.

For many professional physicists, the evaluation part—the part showing where the mathematical mapping does not work—is the most important one. Correcting partial models and inventing new ones are the place where physics is at its most creative. But the figure serves to emphasize that our traditional way of thinking about using math in physics classes may not give enough emphasis to the critical elements of modeling, interpreting, and evaluating.

Physics instruction tends to provide our students with ready-made often over-simplified toy models, and we may be exasperated—or even irritated—if students focus on details that we know to be irrelevant to the toy model we are considering.

Formulas: Physics Formulas and Math Formulas

We tend to let them do the mathematical manipulations in the process step, but we rarely ask them to interpret their results and even less frequently ask them to evaluate whether the initial model is adequate. When they do not succeed on their own with complex problem solving, we tend to pander by only giving simple problems, on which success is not evidence of problem-solving expertise. We often do not recognize what is complex in a problem for a student and that makes it hard to design appropriate and effective problems. Our examples show that physicists as well as other scientists and engineers often use ancillary physical knowledge—often implicit, tacit, or unstated—when applying mathematics to physical systems.

Interestingly enough, a similar idea is valuable to linguists trying to understand how we put meaning to words—semantics. Developments in the linguistics and semantics literature help us to begin to build a terminology to be able to better describe the difference between the expectations of the cultures of physicists and mathematicians.

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To understand how we make meaning using the language of mathematics in the context of physics, let us consider what is known about how people make meaning using language in the context of daily life. Although researchers have not entirely come to an overarching synthesis, they have many ideas that are valuable in helping us make sense of how we make meaning. We offer an exceedingly brief summary of a rich and complex subject, selecting those elements that are particularly relevant.

Embodied cognition: Meaning is grounded in physical experience. Encyclopedic knowledge : Ancillary knowledge creates meaning. Contextualization : Meaning is constructed dynamically.

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Embodied cognition refers to the interaction of complex cognitive functions with basic physics experience—our sensory perceptions, motor functions, and how they are tied to cultural contexts. Infants learn what a shape is by coordinating their vision and touch. Toddlers learn the names of these shapes by associating words they hear to objects. The thesis of embodied cognition states that ultimately our conceptual system is grounded in our interaction with the physical world: How we construe even highly abstract meaning is constrained by and is often derived from our very concrete experiences in the physical world.

Note that embodied cognition is not a reference to the cognitive activity that is obviously involved in performing sensorimotor activities. The grounding of conceptualization in physical experience and actions also extends to higher cognitive processes such as mathematical reasoning. Many of the sophisticated ideas and formulations in mathematics are intricately entwined with the physicality of our being. A symbolic form blends a grammatical signifier—a mathematical symbol template —with an abstraction of an understanding of relationships obtained from embodied experience—a conceptual schema.

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The boxes indicate that any symbol may be in the box—we are only talking about the grammatical structure of the mathematical representation. The conceptual meaning put to it is that something can be considered to be made up of parts. This is something with which we have both direct physical experience and an abstract schema of parts and whole.

The principle of encyclopedic knowledge implies that we understand the meaning of words not in terms of terse definitions provided in a dictionary but in reference to a contextual web of concepts perhaps represented by other words that are themselves understood on the basis of still other concepts.

Instead, these linguists argue that words are always understood with respect to frames or domains of experience…. Evans and Green , p. For most mathematicians and even high school students , Eq.

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With a knowledge of labeling conventions, x and y are interpreted as variables capable of taking on many different values, while m and b are interpreted as constants. With this addition, the equation takes on the meaning of a relation between the independent variable x and the dependent variable y.

Additionally, the assumed constancy of m implies that the equation refers to a straight line.

The constants now take on additional mathematical meaning: m as the slope of the line and b as the intercept on the y- axis, bringing in ideas from graphs. Meaning construction draws upon encyclopaedic knowledge…and involves inferencing strategies that relate to different aspects of conceptual structure, organization and packaging… [Emphasis in original.

We can now understand that the mathematicians who ignore the variable names are attaching meaning to the equation within the domain of simple functions and variables, while the physicists who ignore the implications of functional dependence are interpreting the equation within the domain of coordinate systems adding physical meaning to the variables.

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Similarly, the idea of symbolic forms does not suggest a 1-to-1 correspondence between a particular equation and a symbol template or between a symbol template and a conceptual schema. Making of meaning with equations shares at least three key commonalities with meaning-making in language: an embodied basis, the use of encyclopedic knowledge, and contextual selection of that encyclopedic knowledge for meaning-making. The ideas in Sect.

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Embodied cognition: Phenomenological primitives tie basic physics reasoning to embodied experience. Encyclopedic knowledge : Manifold productive resources are used dynamically. Contextualization : Activation depends on conceptual, epistemological, and affective factors. These are knowledge elements learned, often at a very young age, about how the world works. Two of their core aspects are obviousness and irreducibility —p-prims are activated easily and directly, and, as far as the user is aware, they have no structure. In a similar way, Sherin found that upper-division physics undergraduates commonly construct novel equations to model physical situations through their intuitive understanding rather than the application of formal physics rules or principles.

This way of leveraging the physical interpretation of the situation to affect how the mathematical equations are generated, used, and interpreted reflects physics disciplinary expertise and stands in contrast to just formally processing the mathematical syntax. P-prims form a subset of the knowledge that individuals can bring to bear in understanding physical situations. Just as some subset of our ancillary encyclopedic knowledge is applied to make meaning of language, individuals bring some subset of their resources to make meaning in physics.

Because of the manifold possible meanings, learning physics involves refining patterns of activation of and connection to our encyclopedic knowledge base to build a coherent and stable knowledge structure that aligns with the canonical knowledge and reasoning of the discipline of physics. The apparatus was shown at the beginning of a recitation section, and the mechanism explained by a graduate teaching assistant.

Students were working in groups, and each group of four students were given four tapes of different lengths containing six dots as shown in Fig. Multiple groups of students were seen to transition quickly from one interpretation to another. There are two important features to note from this common reasoning trajectory. First, in different moments, a group of students can connect different cognitive resources to reach different interpretations about what the different lengths mean.

Second, the groups draw on different pieces of their encyclopedic knowledge depending on how they have contexualized the task in response to the different cues in different parts of the lesson to make two different and mutually exclusive meanings of the same objects. A crucial part of the resources framework is the observation that resources are general—neither right nor wrong until the context and use is specified. We argue that although both meanings are correct ways of interpreting an equation in physics, opportunistically and productively blending physical meaning with the mathematical syntax evidences more expert-like reasoning in the discipline of physics.

Understanding an equation in physics is not limited to connecting the symbols to physical variables and being able to perform the operations with that equation. An important component is being able to connect the mathematical operations in the equation to their physical meaning and integrating the equation with its implications in the physical world. In this section, we illustrate the differences in ways that meaning could be attached to an equation by analyzing excerpts of two clinical interviews with students in an introductory physics class for engineers Kuo et al.


Both students can use the equation satisfactorily for solving problems, but the encyclopedic meaning ascribed to the equation by Alex is different than that ascribed by Pat. I: Right. So suppose you had to explain this equation to a friend from class. How would you go about doing that? You can find the velocity. I: Ok. And um, if you were to explain this equation to a friend from class, how would you go about explaining this? Furthermore, we see that this interpretation of the equation affects its use in problem solving.

Later in the interview, Alex and Pat use the equation to solve a problem about differences in the speeds of two balls thrown from a building at the same time with different initial velocities. Alex uses the equation as a tool to compute final velocities given the initial velocity, time, and acceleration. Pat on the other hand uses the equation much more as an expert physicist might, reaching the answer without needing to plug in numbers and carry out arithmetic calculations, and she exhibits an expert-like understanding of why the result should be what it is, based on the meaning that she assigns to the structure of the equation.