T.I. Smith, J.R. Thompson and D.B. Mountcastle

Student Understanding of Taylor Series Expansions in Statistical Mechanics

*Physical Review Special Topics - Physics Education Research* **9**, 020110 (2013). http://link.aps.org/doi/10.1103/PhysRevSTPER.9.020110

One goal of physics instruction is to have students learn to make physical meaning of specific mathematical expressions, concepts, and procedures in different physical settings. As part of research investigating student learning in statistical physics, we are developing curriculum materials that guide students through a derivation of the Boltzmann factor using a Taylor series expansion of entropy. Using results from written surveys, classroom observations, and both individual think-aloud and teaching interviews, we present evidence that many students can recognize and interpret series expansions, but they often lack fluency in creating and using a Taylor series appropriately, despite previous exposures in both calculus and physics courses.

## 2014-06-02

### Smith, Thompson, Mountcastle on Taylor Series Expansions in Statistical Mechanics

Labels: Boltzmann, mathematics, Mountcastle, Smith, statisticalmechanics, Taylorseries, Thompson

### Christensen and Thompson on graphical slope and derivative representations

W.M. Christensen and J.R. Thompson

Investigating graphical representations of slope and derivative without a physics context

*Phys. Rev. ST Physics Ed. Research* **8**, 023101 (2012).

http://link.aps.org/doi/10.1103/PhysRevSTPER.8.023101

By analysis of student use of mathematics in responses to conceptual physics questions, as well as analogous math questions stripped of physical meaning, we have previously found evidence that students often enter upper-level physics courses lacking the assumed prerequisite mathematics knowledge and/or the ability to apply it productively in a physics context. As an extension from this work on students’ mathematical competency at the upper level in physics, we report on a preliminary investigation of mathematical understanding of fundamental concepts of slope and derivative among students in a third- semester multivariable calculus course. Among the first published findings of physics education research are investigations on students’ understanding of kinematics, with particular attention to graphical representations of position-, velocity-, and acceleration-versus-time graphs. Underlying these physical quantities are relationships that depend on derivatives and slopes. We report on our findings as we attempt to isolate students’ understanding of these mathematical concepts.

Labels: calculus, Christensen, derivative, mathematics, slope, Thompson

## 2014-05-05

### Wittmann and Black on Consistency Plots

Michael C. Wittmann and Katrina E. Black

Visualizing changes in pretest and post-test student responses with consistency plots

Phys. Rev. ST Phys. Educ. Res 10, 010114 – Published 5 May 2014

Tabular presentations of student data often hide information about the switches in responses by individual students over the course of a semester. We extend unpublished work by Kanim on “escalator diagrams,” which show changes in student responses from correct to incorrect (and vice versa) while representing pre- and postinstruction results on questions. We introduce the representation of “consistency plots,” containing three pieces of information: each student’s method of solution and correctness of solution and the shift from before to after instruction. We present data from students in an intermediate mechanics class answering (nearly) identical midterm and final examination questions. These data serve as a proof of concept of the method; we suggest other possible uses of consistency plots in physics education research, as well.

Labels: Black, mathematics, Mechanics, Wittmann

## 2013-09-09

### Harrer, Flood, and Wittmann on productive resources about energy

Benedikt W. Harrer, Virginia J. Flood, and Michael C. Wittmann

Productive resources in students’ ideas about energy: An alternative analysis of Watts’ original interview transcripts

Phys. Rev. ST Phys. Educ. Res. 9, 023101 – Published 10 September 2013

For over 30 years, researchers have investigated students’ ideas about energy with the intent of reforming instructional practice. In this pursuit, Watts contributed an influential study with his 1983 paper “Some alternative views of energy” [Phys. Educ. 18, 213 (1983)]. Watts’ “alternative frameworks” continue to be used for categorizing students’ non-normative ideas about energy. Using a resources framework, we propose an alternate analysis of student responses from Watts’ interviews. In our analysis, we show how students’ activated resources about energy are disciplinarily productive. We suggest that fostering seeds of scientific understandings in students’ ideas about energy may play an important role in their development of scientific literacy.

## 2013-07-11

### Wittmann, Flood, and many others on negotiations of energy in a structured environment (Energy Theater)

Rachel E. Scherr*, Hunter G. Close†, Eleanor W. Close†, Virginia J. Flood‡, Sarah B. McKagan*, Amy D. Robertson*, Lane Seeley*, Michael C. Wittmann§‡, and Stamatis Vokos*

Negotiating energy dynamics through embodied action in a materially structured environment

Phys. Rev. ST Phys. Educ. Res. 9, 020105 – Published 11 July 2013

* Department of Physics, Seattle Pacific University, Seattle, WA

† Department of Physics, Texas State University, San Marcos, TX

‡ Maine Center for Research in STEM Education, Orono, ME

§ Department of Physics, University of Maine, Orono, ME

We provide evidence that a learning activity called Energy Theater engages learners with key conceptual issues in the learning of energy, including disambiguating matter flow and energy flow and theorizing mechanisms for energy transformation. A participationist theory of learning, in which learning is indicated by changes in speech and behavior, supports ethnographic analysis of learners’ embodied interaction with each other and the material setting. We conduct detailed analysis to build plausible causal links between specific features of Energy Theater and the conceptual engagement that we observe. Disambiguation of matter and energy appears to be promoted especially by the material structure of the Energy Theater environment, in which energy is represented by participants, while objects are represented by areas demarcated by loops of rope. Theorizing mechanisms of energy transformation is promoted especially by Energy Theater’s embodied action, which necessitates modeling the time ordering of energy transformations.

Labels: embodiment, energy, Flood, Wittmann

## 2013-02-01

### Wittmann, Flood, and Black on embodiment in separating variables

Algebraic manipulation as motion within a landscape

Michael C. Wittmann, Virginia J. Flood, Katrina E. Black

Educational Studies in Mathematics

February 2013, Volume 82, Issue 2, pp 169-181

We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment.

Labels: Black, embodiment, Flood, gesture, mathematics, Wittmann

## 2013-01-24

### Harrer, Flood, and Wittmann on Energy in middle school science

Students talk about energy in Project-Based Inquiry Science

Benedikt W. Harrer, Virginia J. Flood, and Michael C. Wittmann

AIP Conf. Proc. 1513, 162 (2013)

We examine the types of emergent language eighth grade students in rural Maine middle schools use when they discuss energy in their first experiences with Project-Based Inquiry Science: Energy, a research-based curriculum that uses a specific language for talking about energy. By comparative analysis of the language used by the curriculum materials to students’ language, we find that students’ talk is at times more aligned with a Stores and Transfer model of energy than the Forms model supported by the curriculum.

### Kaczynski and Wittmann on expectations about damped harmonic motion

Student expectations in a group learning activity on harmonic motion

Adam Kaczynski and Michael C. Wittmann

AIP Conf. Proc. 1513, 210 (2013)

Students in a sophomore-level mechanics course participated in a new group learning activity that was intended to support model-building and finding coherence between multiple representations in the context of an underdamped harmonic system. Not all of the student groups framed the activity in the same way, and many attempted tasks that existed outside of the prompts of the activity. For one group, this meant that instead of providing a rich verbal description, they framed the activity as finding a mathematical expression.

Labels: epistemology, harmonicmotion, Kaczynski, Mechanics, Wittmann

### Wittmann and Hawkins on new versions of FMCE questions

New ways of investigating the canonical coin toss acceleration problem

Michael C. Wittmann and Jeffrey M. Hawkins

AIP Conf. Proc. 1513, 422 (2013)

Asking students about the acceleration of a tossed object is a well-studied problem in physics education research. Students frequently respond using reasoning that describes the velocity of the object, in particular that acceleration is zero at the top. We created new versions of the canonical multiple-choice Force and Motion Conceptual Evaluation coin-toss questions to investigate what other reasoning students might use. Some students were asked “is the acceleration zero at the top?” Other students were told “the acceleration is not zero” and asked to explain. A third group answered the original multiple-choice version of the question. Our results suggest that some students give answers that they can explain are incorrect. We also find that some students’ responses about the acceleration at the turnaround point are affected by question format.

Labels: assessment, FMCE, Hawkins, kinematics, Wittmann

### Chase and Wittmann on Embodied Cognition

Evidence of embodied cognition via speech and gesture complementarity

Evan A. Chase and Michael C. Wittmann

AIP Conf. Proc. 1513, 94 (2013)

We are studying how students talk and gesture about physics problems involving directionality. Students discussing physics use more than words and equations; gestures are also a meaningful element of their thinking. Data come from one-on-one interviews in which students were asked to gesture about the sign and direction of velocity, acceleration, and other quantities. Specific contexts are a ball toss in the presence and absence of air resistance, including situations where the ball starts at greater than terminal velocity. Students show an aptitude for representing up to 6 characteristics of the ball with 2 hands. They switch quickly while talking about velocity, acceleration, and the different forces, frequently representing more than one quantity using a single hand. We believe that much of their thinking resides in their hands, and that their gestures complement their speech, as indicated by moments when speech and gesture represent different quantities.

Labels: Chase, embodiment, gesture, kinematics, Wittmann

## 2012-10-01

### Hawkins, Frank, Thompson, Wittmann, and Wemyss on alternative question strategies

Probing student understanding with alternative questioning strategies

Jeffrey M. Hawkins, Brian W. Frank, John R. Thompson, Michael C. Wittmann, and Thomas M. Weymss

AIP Conf. Proc. 1413, 207 (2012)

Common research tasks ask students to identify a correct answer and justify their answer choice. We propose expanding the array of research tasks to access different knowledge that students might have. By asking students to discuss answers they may not have chosen naturally, we can investigate students' abilities to explain something that is already established or to disprove an incorrect response. The results of these research tasks also provide us with information about how students' responses vary across the different tasks. We discuss three underused question types, their possible benefits, and some preliminary results from an electric circuits pretest utilizing these novel question types. We find that the answer students most commonly choose as correct is the same choice most commonly eliminated as incorrect. Also, given the correct answer, students can provide valuable reasoning to explain it, but they do not spontaneously identify it as the correct answer.

### Wittmann and Black on integration and separation of variables

When basic changes to a solution suggest meaningful differences in mathematics

Michael C. Wittmann and Katrina E. Black

AIP Conf. Proc. 1413, 93 (2012)

When solving two integrals arising from the separation of variables in a first order linear differential equation, students have multiple correct choices for how to proceed. They might set limits on both integrals or use integration constants on both or only one equation. In each case, the physical meaning of the mathematics is equivalent. But, how students choose to represent the mathematics can tell us much about what they are thinking. We observe students debating how to integrate the quantity dt. One student seeks a general function that works for everyone, and does not wish to specify the value of the integration constant. Another student seeks a function consistent with the specific physics problem. They compromise by using a constant, undefined in value for one student, zero in value for the other.

Labels: Black, mathematics, Mechanics, Wittmann

### Harrer, Scherr, Wittmann, Close and Frank on proximal formative assessment about energy

Elements of proximal formative assessment in learners' discourse about energy

Benedikt W. Harrer, Rachel E. Scherr, Michael C. Wittmann, Hunter G. Close, and Brian W. Frank

AIP Conf. Proc. 1413, 203 (2012)

Proximal formative assessment, the just-in-time elicitation of students' ideas that informs ongoing instruction, is usually associated with the instructor in a formal classroom setting. However, the elicitation, assessment, and subsequent instruction that characterize proximal formative assessment are also seen in discourse among peers. We present a case in which secondary teachers in a professional development course at SPU are discussing energy flow in refrigerators. In this episode, a peer is invited to share her thinking (elicitation). Her idea that refrigerators move heat from a relatively cold compartment to a hotter environment is inappropriately judged as incorrect (assessment). The "instruction" (peer explanation) that follows is based on the second law of thermodynamics, and acts as corrective rather than collaborative.

Labels: assessment, energy, Frank, Harrer, Wittmann

### Wittmann and Chase on embodied cognition in wave propagation

Evidence of embodied cognition about wave propagation

Michael C. Wittmann and Evan Chase

AIP Conf. Proc. 1413, 383 (2012)

That students think of wavepulses as if throwing balls down a long taut spring is well established. Typical questions involve students imagining the spring already pulled taut; a different scenario would imagine them pulling the spring tight first. This situation creates a different baseline of physical experience from which to reason. For example, it provides a physical experience in which tension is a relevant measure in the system. We investigated the effects of students pulling the spring (or not) in interviews after instruction. We also wrote two surveys, each giving a different physical description of a typical problem. From interviews, we find evidence that a different embodiment of the problem affects students' responses. In surveys, with students asked to imagine different situations, we found no such evidence.

Labels: Chase, embodiment, waves, Wittmann

## 2011-05-20

### Thompson Christensen Wittmann on graduate course in PER

John R. Thompson, Warren M. Christensen, Michael C. Wittmann

Preparing future teachers to anticipate student difficulties in physics in a graduate-level course in physics, pedagogy, and education research

We describe courses designed to help future teachers reflect on and discuss both physics content and student knowledge thereof. We use three kinds of activities: reading and discussing the literature, experiencing research-based curricular materials, and learning to use the basic research methods of physics education research. We present a general overview of the two courses we have designed as well as a framework for assessing student performance on physics content knowledge and one aspect of pedagogical content knowledge—knowledge of student ideas—about one particular content area: electric circuits. We find that the quality of future teachers’ responses, especially on questions dealing with knowledge of student ideas, can be successfully categorized and may be higher for those with a nonphysics background than those with a physics background.

URL: http://link.aps.org/doi/10.1103/PhysRevSTPER.7.010108

DOI: 10.1103/PhysRevSTPER.7.010108

PACS: 01.40.J-

Labels: Christensen, circuits, PCK, professionaldevelopment, Thompson, Wittmann