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Even if students can make the blend, interpret physics correctly in mathematical symbology and graphs, they still need to be able to apply that knowledge in productive and coherent ways. As instructors, we can show our solutions to complex problems in class. We can give complex problems to students as homework. But our students are likely to still have trouble because they are missing a key element of making sense of how we think about physics: How to tell the story of what's happening., Comment: 7 pages, 10 figures

Learning to use math in physics involves combining (blending) our everyday experiences and the conceptual ideas of physics with symbolic mathematical representations. Graphs are one of the best ways to learn to build the blend. They are a mathematical representation that builds on visual recognition to create a bridge between words and equations. But students in introductory physics classes often see a graph as an endpoint, a task the teacher asks them to complete, rather than as a tool to help them make sense of a physical system. And most of the graph problems in traditional introductory physics texts simply ask students to extract a number from a graph. But if graphs are used appropriately, they can be a powerful tool in helping students learn to build the blend and develop their physical intuition and ability to think with math., Comment: 6 pages, 7 figures, supplementary materials 24 pages

At many physics departments a significant fraction of teaching is in support of engineers and scientists in other majors. These service courses are thus an automatic crucible of interdisciplinary interaction, and at times, strife. For example, the traditional algebra-based physics course is often considered by both biology faculty and students as having little relevance to their discipline. To address this issue, our multi-disciplinary multi-university team has been negotiating the role of a physics in the curriculum of life-science students; In NEXUS/Physics we have designed a class that stresses traditional physics skills but in contexts chosen to better meet the needs of life science students. Non-standard topics include chemical energy, diffusion and random motion, and thermodynamics with careful discussions of entropy, enthalpy, and Gibbs free energy. Explorations into how physics intertwines with an engineer's curriculum suggests places where analogous negotiations could lead to substantial modifications of physics courses for engineers that substantially enhance their value for engineering students., Comment: Invited talk presented at the Frontiers in Mathematics and Science Education Research Conference, 1-3 May 2014, Famagusta,North Cyprus

When students are learning to use math in physics, one of the most important ideas they need to learn is that equations are not just calculational tools; they represent relationships between physical variables that change together (covary). How much a change in one variable or parameter is associated with a change in another depends on how they appear in the equation: their functional dependence. Understanding this sort of relationship is rarely taught in introductory mathematics classes, and students who have not yet learned to blend conceptual ideas with mathematical symbols may not see the relevance and power of this idea. We need to explicitly teach functional dependence as part of our effort to help students to learn to use math productively in science., Comment: 5 pages, 4 figures

An important step in learning to use math in science is learning to see physics equations as not just calculational tools, but as ways of expressing fundamental relationships among physical quantities, of coding conceptual information, and of organizing physics knowledge structures. In this paper I discuss the role of basic anchor equations in introductory physics and show some examples of how to help students learn to use them., Comment: 6 pages, 4 figures

Learning to create, use, and evaluate models is a central element of becoming a scientist. In physics, we often begin an analysis of a complex system with highly simplified or toy models. In introductory physics classes, we tend to use them without comment or motivation. Some students infer that physics is irrelevant to their understanding of the real world and are discouraged from making the cognitive blend of physics concepts with math symbology essential for making sense of physics. In this paper, I discuss the often hidden barriers that make it difficult for our students to accept and understand the value of toy models, and suggest instructional approaches that can help., Comment: 6 pages, 5 figures

Making meaning with math in physics requires blending physical conceptual knowledge with mathematical symbology. Students in introductory physics classes often struggle with this, but it is an essential component of learning how to think with math. Teaching dimensional analysis (DA). figuring out what measurements were combined to create a symbolic quantity, is a valuable first step in helping them learn to appreciate this difference. In this paper I discuss some of the issues associated with learning dimensional analysis and show some ways we can modify our instruction to help. This paper is one of a series on how to help students develop the scientific thinking skills required for learning to use math in science., Comment: 5 pages, 3 figures

Learning to use math in science is a non-trivial task. It involves many different skills (not usually taught in a math class) that help blend physical knowledge with mathematical symbology. One of these is the idea of quantification: that physical quantities can be assigned specific numbers. A second is to develop an intuition for scale. One way to help students develop these skills is to teach estimation: the ability to consider a physical situation and put reasonable approximate numbers to it, Comment: 5 pages, 2 figures

The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even to the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general scientific thinking skills that are often taken for granted (and rarely taught) in physics classes. In this paper, I give an overview of my analysis of these additional skills. I propose specific tools for helping students develop these skills in subsequent papers., Comment: 6 pages, 2 figures

Equations are about more than computing physical quantities or constructing formal models; they are also about understanding. The conceptual systems physicists use to think about nature are made from many different resources, formal and not, working together and inextricably linked. By blending mathematical forms and physical intuition, physicists breathe meaning into the equations they use. In contrast, in physics class, novice students often treat mathematics as only a calculational tool, isolating it from their rich knowledge of the physical world. We are interested in cases where students break that pattern by reading, manipulating, and building equations meaningfully rather than purely formally. To find examples of this and explore the diversity of ways students combine formal and intuitive resources, we conducted problem-solving interviews with students in an introductory physics for life sciences class. During the interviews, we scaffolded student use of strategies which call for both formal and intuitive reasoning, such as "examine the extreme cases". We use the analytic framework of epistemic games to model how students used the strategies and how they accessed problem-solving resources, and we present evidence that novice students using these strategies accessed more expert-like conceptual systems than those typically described in problem-solving literature. They blended physical intuition with mathematical symbolic templates, reconceptualized the nature of equations, and distinguished similar functional forms. Once introduced to a strategy, students applied it to new scenarios and found new types of uses for it, acknowledging it as a useful, general purpose problem-solving technique. Our data suggests that these strategies can potentially help novice students learn to develop and apply their physical intuition more effectively., Comment: 27 pages, 2 figures, submitted to a forthcoming book

A primary goal of physics is to create mathematical models that allow both predictions and explanations of physical phenomena. We weave maths extensively into our physics instruction beginning in high school, and the level and complexity of the maths we draw on grows as our students progress through a physics curriculum. Despite much research on the learning of both physics and math, the problem of how to successfully teach most of our students to use maths in physics effectively remains unsolved. A fundamental issue is that in physics, we don't just use maths, we think about the physical world with it. As a result, we make meaning with math-ematical symbology in a different way than mathematicians do. In this talk we analyze how developing the competency of mathematical modeling is more than just "learning to do math" but requires learning to blend physical meaning into mathematical representations and use that physical meaning in solving problems. Examples are drawn from across the curriculum., Comment: 14 pages, 8 figures, GIREP 2015 Conference paper

Energy is an abstract science concept, so the ways that we think and talk about energy rely heavily on ontological metaphors: metaphors for what kind of thing energy is. Two commonly used ontological metaphors for energy are energy as a substance and energy as a vertical location. Our previous work has demonstrated that students and experts can productively use both the substance and location ontologies for energy. In this paper, we use Fauconnier and Turner's conceptual blending framework to demonstrate that experts and novices can successfully blend the substance and location ontologies into a coherent mental model in order to reason about energy. Our data come from classroom recordings of a physics professor teaching a physics course for the life sciences, and from an interview with an undergraduate student in that course. We analyze these data using predicate analysis and gesture analysis, looking at verbal utterances, gestures, and the interaction between them. This analysis yields evidence that the speakers are blending the substance and location ontologies into a single blended mental space., Comment: 20 pages, 14 figures. Accepted to International Journal of Science Education. The first two authors contributed equally to the data analysis and production of this paper

Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don't just use math, we make meaning with it in a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction., Comment: 27 pages, 9 figures

As interdisciplinary courses are developed, instructors and researchers have to grapple with questions of how students should make connections across disciplines. We explore the issue of interdisciplinary reconciliation (IDR): how students reconcile seemingly contradictory ideas from different disciplines. While IDR has elements in common with other frameworks for the reconciliation of ideas across contexts, it differs in that each disciplinary idea is considered canonically correct within its own discipline. The setting for the research is an introductory physics course for biology majors that seeks to build greater interdisciplinary coherence and therefore includes biologically relevant topics such as ATP and chemical bond energy. In our case-study data, students grapple with the apparent contradiction between the energy released when the phosphate bond in ATP is broken and the idea that an energy input is required to break a bond. We see students justifying context-dependent modeling choices, showing nuance in articulating how system choices may be related to disciplinary problems of interest. This represents a desired endpoint of IDR, in which students can build coherent connections between concepts from different disciplines while understanding each concept in its own disciplinary context. Our case study also illustrates elements of the instructional environment that play roles in the process of IDR., Comment: 16 pages

Teaching about energy in interdisciplinary settings that emphasize coherence among physics, chemistry, and biology leads to a more central role for chemical bond energy. We argue that an interdisciplinary approach to chemical energy leads to modeling chemical bonds in terms of negative energy. While recent work on ontological metaphors for energy has emphasized the affordances of the substance ontology, this ontology is problematic in the context of negative energy. Instead, we apply a dynamic ontologies perspective to argue that blending the substance and location ontologies for energy can be effective in reasoning about negative energy in the context of reasoning about chemical bonds. We present data from an introductory physics for the life sciences (IPLS) course in which both experts and students successfully use this blended ontology. Blending these ontologies is most successful when the substance and location ontologies are combined such that each is strategically utilized in reasoning about particular aspects of energetic processes., Comment: 11 pages, 4 figures

In an Introductory Physics for Life Science (IPLS) course that leverages authentic biological examples, student ideas about entropy as "disorder" or "chaos" come into contact with their ideas about the spontaneous formation of organized biological structure. It is possible to reconcile the "natural tendency to disorder" with the organized clustering of macromolecules, but doing so in a way that will be meaningful to students requires that we take seriously the ideas about entropy and spontaneity that students bring to IPLS courses from their prior experiences in biology and chemistry. We draw on case study interviews to argue that an approach that emphasizes the interplay of energy and entropy in determining spontaneity (one that involves a central role for free energy) is one that draws on students' resources from biology and chemistry in particularly effective ways. We see the positioning of entropic arguments alongside energetic arguments in the determination of spontaneity as an important step toward making our life science students' biology, chemistry, and physics experiences more coherent., Comment: 9 pages

Physics Education Research (PER) applies a scientific approach to the question, "How do our students think about and learn physics?" PER allows us to explore such intellectually engaging questions as, "What does it mean to understand something in physics?" and, "What skills and competencies do we want our students to learn from our physics classes?" To address questions like these, we need to do more than observe student difficulties and build curricula. We need a theoretical framework -- a structure for talking about, making sense of, and modeling how one thinks about, learns, and understands physics. In this paper, I outline some aspects of the Resources Framework, a structure that some of us are using to create a phenomenology of physics learning that ties closely to modern developments in neuroscience, psychology, and linguistics. As an example of how this framework gives new insights, I discuss epistemological framing -- the role of students' perceptions of the nature of the knowledge they are learning and what knowledge is appropriate to bring to bear on a given task. I discuss how this foothold idea fits into our theoretical framework, show some classroom data on how it plays out in the classroom, and give some examples of how my awareness of the resources framework influences my approach to teaching., Comment: 18 pages, 11 figures, AAPT 2013 Oersted Award Lecture

Energy is a complex idea that cuts across scientific disciplines. For life science students, an approach to energy that incorporates chemical bonds and chemical reactions is better equipped to meet the needs of life sciences students than a traditional introductory physics approach that focuses primarily on mechanical energy. We present a curricular sequence, or thread, designed to build up students' understanding of chemical energy in an introductory physics course for the life sciences. This thread is designed to connect ideas about energy from physics, biology, and chemistry. We describe the kinds of connections among energetic concepts that we intended to develop to build interdisciplinary coherence, and present some examples of curriculum materials and student data that illustrate our approach., Comment: 11 pages

Much recent work in physics education research has focused on ontological metaphors for energy, particularly the substance ontology and its pedagogical affordances. The concept of negative energy problematizes the substance ontology for energy, but in many instructional settings, the specific difficulties around negative energy are outweighed by the general advantages of the substance ontology. However, we claim that our interdisciplinary setting (a physics class that builds deep connections to biology and chemistry) leads to a different set of considerations and conclusions. In a course designed to draw interdisciplinary connections, the centrality of chemical bond energy in biology necessitates foregrounding negative energy from the beginning. We argue that the emphasis on negative energy requires a combination of substance and location ontologies. The location ontology enables energies both "above" and "below" zero. We present preliminary student data that illustrate difficulties in reasoning about negative energy, and the affordances of the location metaphor., Comment: 4 pages, submitted to PERC 2013 Proceedings

Teaching physics to biologists requires far more than making the course for engineers mathematically less rigorous and adding in a few superficial biological problems. What is needed is for physicists to work closely with biologists to learn not only what physics topics and habits of mind that are useful to biologists, but how biologists work is fundamentally different from ours, and how to bridge that gap. In this article, we discussed what we have learned about these issues from years of conversations with colleagues in biology.

A common feature of the recent calls for reform of the undergraduate biology curriculum has been for better coordination between biology and the courses from the allied disciplines of mathematics, chemistry, and physics. Physics has lagged math and chemistry in creating new biologically oriented curricula, though much activity is now taking place and significant progress is being made. In this article we consider a case study: a multi-year conversation between a physicist interested in adapting his physics course for biologists (Redish) and a biologist interested in including more physics in his biology course (Cooke). These extended discussions have led us both to a deeper understanding of each others' discipline and to significant changes in the way we each think about and present our classes. We discuss two examples in detail: the creation of a physics problem for a biology class on fluid flow, and the creation of a biologically authentic physics problem on scaling and dimensional analysis. In each case, we see differences in how the two disciplines frame and see value in the tasks. We conclude with some generalizations about how biology and physics look at the world differently that help us navigate the minefield of counterproductive stereotypical responses., Comment: 19 pages, 5 figures

Students' sometimes contradictory ideas about ATP (adenosine triphosphate) and the nature of chemical bonds have been studied in the biology and chemistry education literatures, but these topics are rarely part of the introductory physics curriculum. We present qualitative data from an introductory physics course for undergraduate biology majors that seeks to build greater interdisciplinary coherence and therefore includes these topics. In these data, students grapple with the apparent contradiction between the energy released when the phosphate bond in ATP is broken and the idea that an energy input is required to break a bond. We see that students' perceptions of how each scientific discipline bounds the system of interest can influence how they justify their reasoning about a topic that crosses disciplines. This has consequences for a vision of interdisciplinary education that respects disciplinary perspectives while bringing them into interaction in ways that demonstrate consistency amongst the perspectives., Comment: 4 pages, submitted to PERC 2012

The theme of the World Conference on Physics Education 2012 is "Context, Culture, and Representations." In this talk I present a brief outline of a theoretical framework that allows us to discuss these issues using a model based in psychology and sociology: the resources framework. The framework brings together a model of individual behavior based on brain function with a model of how the behavior of an individual is controlled by the individual's perception of the social context they find themselves in. This control process is the process I refer to as "framing". In the paper I give three experiments that the reader can carry out for themselves that illustrate the basic principles of the framework. I then discuss a number of specific examples showing how framing can have powerful effects leading to context dependence and cultural responses at a variety of levels and grain sizes. One such is the impact of differences between the epistemological stances of physics and biology on the creation of a reformed physics class for biology students., Comment: 22 pages, 14 figures; paper based on keynote address given at World Conference on Physics Education, Istanbul, July 2012

Educators and policy-makers have advocated for reform of undergraduate biology education, calling for greater integration of mathematics and physics in the biology curriculum. While these calls reflect the increasingly interdisciplinary nature of biology research, crossing disciplinary boundaries in the classroom carries epistemological challenges for both instructors and students. In this paper we expand on the construct of authenticity to better describe and understand disciplinary practices, in particular to examine those used in physics and biology courses. We then apply these ideas to examine an introductory biology course that incorporates physics and mathematics. We characterize the uses of interdisciplinary tools in this biology course and contrast them with the typical uses of these tools in physics courses. Finally, we examine student responses to the use of mathematics and physics in this course, to better understand the challenges and consequences of using interdisciplinary tools in introductory courses. We link these results to the reform initiatives of introductory physics courses for life-science students., Comment: 19 pages

Energy concepts are fundamental across the sciences, yet these concepts can be fragmented along disciplinary boundaries, rather than integrated into a coherent whole. To teach physics effectively to biology students, we need to understand students' disciplinary perspectives. We present interview data from an undergraduate student who displays multiple stances towards the concept of energy. At times he views energy in macroscopic contexts as a separate entity from energy in microscopic (particularly biological) contexts, while at other times he uses macroscopic physics phenomena as productive analogies for understanding energy in the microscopic biological context, and he reasons about energy transformations between the microscopic and macroscopic scales. This case study displays preliminary evidence for the context dependence of students' ability to translate energy concepts across scientific disciplines. This points to challenges that must be taken into account in developing curricula for biology students that integrate physics and biology concepts., Comment: 4 pages, submitted to PERC 2011

The past 10-15 years have seen numerous calls for curricular reform in undergraduate biology education, most of which focus on changes to curriculum or pedagogy. Data collected from students in a large introductory undergraduate biology course indicate that student expectations about the nature of the knowledge they were learning influence how they interacted with reform efforts in that class. Given that student expectations influence the ways in whichthey participate in course activities, this paper (the first in a series that looks at student expectations in biology) argues that curriculum reform initiatives should consider student expectations in order to increase the chance for effective implementation., Comment: 10 pages

Physics students can encounter difficulties in physics problem solving as a result of failing to use knowledge that they have but do not perceive as relevant or appropriate. In previous work the authors have demonstrated that some of these difficulties may be epistemological. Students may limit the kinds of knowledge that they use. For example, they may use formal manipulations and ignore physical sense making or vice versa. Both beginning (novice) and intermediate (journeymen) students demonstrate these difficulties. Learning both to switch one's epistemological lens on a problem and to integrate different kinds of knowledge is a critical component of learning to solve problems in physics effectively. In this paper, we present two case studies in which journeyman students (upper-division physics majors) demonstrate switching between epistemological resources in approaching a complex problem. We conjecture that mastering these epistemological skills is an essential component of learning complex problem solving in physics., Comment: 12 pages

The University of Maryland (UMD) Biology Education and Physics Education Research Groups are investigating students' views on the role of physics in introductory biology courses. This paper presents data from an introductory course that addresses the fundamental principles of organismal biology and that incorporates several topics directly related to physics, including thermodynamics, diffusion, and fluid flow. We examine how the instructors use mathematics and physics in this introductory biology course and look at two students' responses to this use. Our preliminary observations are intended to start a discussion about the epistemological issues resulting from the integration of the science disciplines and to motivate the need for further research., Comment: Physics Education Research Conference 2010, Portland OR, 4 pages

Education is a goal-oriented field. But if we want to treat education scientifically so we can accumulate, evaluate, and refine what we learn, then we must develop a theoretical framework that is strongly rooted in objective observations and through which different theoretical models of student thinking can be compared. Much that is known in the behavioral sciences is robust and observationally based. In this paper, I draw from a variety of fields ranging from neuroscience to sociolinguistics to propose an over-arching theoretical framework that allows us to both make sense of what we see in the classroom and to compare a variety of specific theoretical approaches. My synthesis is organized around an analysis of the individual's cognition and how it interacts with the environment. This leads to a two level system, a knowledge-structure level where associational patterns dominate, and a control-structure level where one can describe expectations and epistemology. For each level, I sketch some plausible starting models for student thinking and learning in physics and give examples of how a theoretical orientation can affect instruction and research. (Contains 26 figures and 2 tables.) [This document presents lectures given at the "Enrico Fermi" Summer School in Physics, Course CLVI, Varenna, Italy, 2003. The lectures were published in: "The Proceedings of the Enrico Fermi Summer School in Physics, Course CLVI" (Italian Physical Society, 2004).]

When we teach physics to prospective scientists and engineers we are teaching more than the "facts" of physics - more, even, than the methods and concepts of physics. We are introducing them to a complex culture - a mode of thinking and the cultural code of behavior of a community of practicing scientists. This culture has components that are often part of our hidden curriculum: epistemology - how we decide that we know something; ontology - how we parse the observable world into categories, objects, and concepts; and discourse - how we hold a conversation in order to generate new knowledge and understanding. Underlying all of this is intuition - a culturally created sense of meaning. To explicitly identify teach our hidden curriculum we must pay attention to students' intuition and perception of physics, not just to their reasoning., Comment: 4 pages, Physics Education Research Conference 2010 Plenary talk

Physics makes powerful use of mathematics, yet the way this use is made is often poorly understood. Professionals closely integrate their mathematical symbology with physical meaning, resulting in a powerful and productive structure. But because of the way the cognitive system builds expertise through binding, experts may have difficulty in unpacking their well established knowledge in order to understand the difficulties novice students have in learning their subject. This is particularly evident in subjects in which the students are learning to use mathematics to which they have previously been exposed in math classes in complex new ways. In this paper, we propose that some of this unpacking can be facilitated by adopting ideas and methods developed in the field of cognitive semantics, a sub-branch of linguistics devoted to understanding how meaning is associated with language., Comment: GIREP Conference, Leicester, UK, 2009, 15 pages

Developing expertise in physics entails learning to use mathematics effectively and efficiently as applied to the context of physical situations. Doing so involves coordinating a variety of concepts and skills including mathematical processing, computation, blending ancillary information with the math, and reading out physical implications from the math and vice versa. From videotaped observations of intermediate level students solving problems in groups, we note that students often "get stuck" using a limited group of skills or reasoning and fail to notice that a different set of tools (which they possess and know how to use effectively) could quickly and easily solve their problem. We refer to a student's perception/judgment of the kind of knowledge that is appropriate to bring to bear in a particular situation as epistemological framing. Although epistemological framing is often unstated (and even unconscious), in group problem solving situations students sometimes get into disagreements about how to progress. During these disagreements, they bring forth explicit reasons or warrants in support of their point of view. For the context of mathematics use in physics problem solving, we present a system for classifying physics students' warrants. This warrant analysis offers tangible evidence of their epistemological framing., Comment: 23 pages

Mathematics can serve many functions in physics. It can provide a computational system, reflect a physical idea, conveniently encode a rule, and so forth. A physics student thus has many different options for using mathematics in his physics problem solving. We present a short example from the problem solving work of upper level physics students and use it to illustrate the epistemic framing process: framing because these students are focusing on a subset of their total math knowledge, epistemic because their choice of subset relates to what they see (at that particular time) as the nature of the math knowledge in play. We illustrate how looking for the warrants students use, the often unspoken reasons they think their evidence supports their mathematical claims, serves as a window to their epistemic framing. These warrants provide a powerful, concise piece of evidence of their epistemic framing., Comment: accepted for publication in 2008 Physics Education Research Conference Proceedings

The University of Maryland Physics Education Research Group (UMd-PERG) carried out a five-year research project to rethink, observe, and reform introductory algebra-based (college) physics. This class is one of the Maryland Physics Department's large service courses, serving primarily life-science majors. After consultation with biologists, we re-focused the class on helping the students learn to think scientifically -- to build coherence, think in terms of mechanism, and to follow the implications of assumptions. We designed the course to tap into students' productive conceptual and epistemological resources, based on a theoretical framework from research on learning. The reformed class retains its traditional structure in terms of time and instructional personnel, but we modified existing best-practices curricular materials, including Peer Instruction, Interactive Lecture Demonstrations, and Tutorials. We provided class-controlled spaces for student collaboration, which allowed us to observe and record students learning directly. We also scanned all written homework and examinations, and we administered pre-post conceptual and epistemological surveys. The reformed class enhanced the strong gains on pre-post conceptual tests produced by the best-practices materials while obtaining unprecedented pre-post gains on epistemological surveys instead of the traditional losses., Comment: 35 pages including a 15 page appendix of supplementary materials

The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and elegance of the method are often missing, unlike the treatments frequently enjoyed by Fourier transforms. We review and modify the presentation of Legendre transforms in a way that explicates the formal mathematics, resulting in manifestly symmetric equations, thereby clarifying the structure of the transform algebraically and geometrically. Then we bring in the physics to motivate the transform as a way of choosing independent variables that are more easily controlled. We demonstrate how the Legendre transform arises naturally from statistical mechanics and show how the use of dimensionless thermodynamic potentials leads to more natural and symmetric relations., Comment: 11 pages, 3 figures

We investigate the dynamics of how graduate students coordinate their mathematics and physics knowledge within the context of solving a homework problem for a plasma physics survey course. Students were asked to obtain the complex dielectric function for a plasma with a specified distribution function and find the roots of that expression. While all the 16 participating students obtained the dielectric function correctly in one of two equivalent expressions, roughly half of them (7 of 16) failed to compute the roots correctly. All seven took the same initial step that led them to the incorrect answer. We note a perfect correlation between the specific expression of dielectric function obtained and the student's success in solving for the roots. We analyze student responses in terms of a resources framework and suggest routes for future research., Comment: 4 pages

In a series of well-known papers, Chi and Slotta (Chi, 1992; Chi & Slotta, 1993; Chi, Slotta & de Leeuw, 1994; Slotta, Chi & Joram, 1995; Chi, 2005; Slotta & Chi, 2006) have contended that a reason for students' difficulties in learning physics is that they think about concepts as things rather than as processes, and that there is a significant barrier between these two ontological categories. We contest this view, arguing that expert and novice reasoning often and productively traverses ontological categories. We cite examples from everyday, classroom, and professional contexts to illustrate this. We agree with Chi and Slotta that instruction should attend to learners' ontologies; but we find these ontologies are better understood as dynamic and context-dependent, rather than as static constraints. To promote one ontological description in physics instruction, as suggested by Slotta and Chi, could undermine novices' access to productive cognitive resources they bring to their studies and inhibit their transition to the dynamic ontological flexibility required of experts., Comment: The Journal of the Learning Sciences (In Press)

Current concerns over reforming engineering education have focused attention on helping students develop skills and an adaptive expertise. Phenomenological guidelines for instruction along these lines can be understood as arising out of an emerging theory of thinking and learning built on results in the neural, cognitive, and behavioral sciences. We outline this framework and consider some of its implications for one example: developing a more detailed understanding of the specific skill of using mathematics in modeling physical situations. This approach provides theoretical underpinnings for some best-practice instructional methods designed to help students develop this skill and providesguidance for further research in the area., Comment: 20 pages

Multiple-choice tests such as the Force Concept Inventory (FCI) provide useful instruments to probe the distribution of student difficulties on a large scale. However, traditional analysis often relies solely on scores (number of students giving the correct answer). This ignores what can be significant and important information: the distribution of wrong answers given by the class. In this paper a new method is introduced called "concentration analysis" to measure how students' responses on multiple-choice questions are distributed. This information can be used to study if the students have common incorrect models or if the question is effective in detecting student models. When combined with information obtained from qualitative research, the method can be used to identify cleanly what FCI results are saying about student knowledge. (Contains 19 references, 9 figures, and 8 tables.) (Author/YDS)

Symbolic calculators like Mathematica are becoming more commonplace among upper level physics students. The presence of such a powerful calculator can couple strongly to the type of mathematical reasoning students employ. It does not merely offer a convenient way to perform the computations students would have otherwise wanted to do by hand. This paper presents examples from the work of upper level physics majors where Mathematica plays an active role in focusing and sustaining their thought around calculation. These students still engage in powerful mathematical reasoning while they calculate but struggle because of the narrowed breadth of their thinking. Their reasoning is drawn into local attractors where they look to calculation schemes to resolve questions instead of, for example, mapping the mathematics to the physical system at hand. We model the influence of Mathematica as an integral part of the constant feedback that occurs in how students frame, and hence focus, their work.

Since 1983, the Maryland University Project in Physics and Educational Technology (M.U.P.P.E.T.) has been investigating the implication of including student programming in an introductory physics course for physics majors. Many significant changes can result. One can rearrange some content to be more physically appropriate, include more realistic problems, and introduce some contemporary topics. One can begin training the student in professional research-related skills at an earlier stage than is traditional. The inclusion of carefully considered computer content requires an increased emphasis on qualitative and analytic thinking. (Contains 20 notes and 5 figures.) (Author/NB)

Students' understanding of what science is about and how it is done and their expectations as to what goes on in a science course play a powerful role in what they can get out of introductory college physics. This is particularly true when there is a large gap between what the students expect to do and what the instructor expects them to do. This paper describes the Maryland Physics Expectations (MPEX) Survey, a 34-item Likert-scale survey that probes student attitudes, beliefs, and assumptions about physics. The results of pre- and post-instruction delivery of this survey to 1500 students in introductory calculus-based physics at 6 colleges and universities are presented. Findings indicate a large gap between the expectations of experts and novices and a tendency for student expectations to deteriorate rather than improve as a result of a semester of introductory physics. (Contains 36 references.) (Author/WRM)

During the past 15 years, physics education research has revealed many surprising things about the difficulties introductory physics students have in learning physics. At the same time, the ongoing revolution in information technology has led to new tools for creating innovative educational environments. In response to these two developments, a wide variety of new models of physics instruction are beginning to appear. This paper reviews some of the findings of physics education research, putting them into the context of a theory of thinking and learning. Some of the most promising instructional models currently being developed in the U.S. are discussed. (Contains 20 references and 12 figures.) (Author/WRM)