Investigating graphical representations of slope and derivative without a physics context

Christensen, Warren M.; Thompson, John R.

doi:10.1103/physrevstper.8.023101

Cited by 67 publications

(54 citation statements)

References 12 publications

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“…Other researchers have investigated student understanding of P-V (pressure versus volume) diagrams both in upper-level thermodynamics courses [34] as well as in introductory calculus-based physics courses [35]. In a later study, Christensen and Thompson [36] investigated student difficulties with the concept of slope and derivative in a mathematical (graphical) context. Student difficulties with the concept of a function have been researched by mathematics education researchers [37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Challenges in designing appropriate scaffolding to improve students’ representational consistency: The case of a Gauss’s law problem

Maries

Lin

Singh

2017

Phys. Rev. Phys. Educ. Res.

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Prior research suggests that introductory physics students have difficulty with graphing and interpreting graphs. Here, we discuss an investigation of student difficulties in translating between mathematical and graphical representations for a problem in electrostatics and the effect of increasing levels of scaffolding on students' representational consistency. Students in calculus-based introductory physics were given a typical problem that can be solved using Gauss's law involving a spherically symmetric charge distribution in which they were asked to write a mathematical expression for the electric field in various regions and then plot the electric field. In study 1, we found that students had great difficulty in plotting the electric field as a function of the distance from the center of the sphere consistent with the mathematical expressions in various regions, and interviews with students suggested possible reasons which may account for this difficulty. Therefore, in study 2, we designed two scaffolding interventions with levels of support which built on each other (i.e., the second scaffolding level built on the first) in order to help students plot their expressions consistently and compared the performance of students provided with scaffolding with a comparison group which was not given any scaffolding support. Analysis of student performance with different levels of scaffolding reveals that scaffolding from an expert perspective beyond a certain level may sometimes hinder student performance and students may not even discern the relevance of the additional support. We provide possible interpretations for these findings based on in-depth, think-aloud student interviews.

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Section: Introductionmentioning

confidence: 99%

Challenges in designing appropriate scaffolding to improve students’ representational consistency: The case of a Gauss’s law problem

Maries

Lin

Singh

2017

Phys. Rev. Phys. Educ. Res.

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“…As another example, students often use a tangent line and rely on visual judgments to sketch the derivative function [19]. In yet another study, students who were able to correctly rank the slope at points on a graph were less able to find the sign of the derivative at those points [20]. While this approach is not necessarily problematic, thinking about sliding tangent lines does not necessitate images of variation.…”

Section: A Students' Ways Of Thinking About Derivativementioning

confidence: 99%

Experts’ understanding of partial derivatives using the partial derivative machine

Roundy

Weber

Dray

et al. 2015

Phys. Rev. ST Phys. Educ. Res.

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[This paper is part of the Focused Collection on Upper Division Physics Courses.] Partial derivatives are used in a variety of different ways within physics. Thermodynamics, in particular, uses partial derivatives in ways that students often find especially confusing. We are at the beginning of a study of the teaching of partial derivatives, with a goal of better aligning the teaching of multivariable calculus with the needs of students in STEM disciplines. In this paper, we report on an initial study of expert understanding of partial derivatives across three disciplines: physics, engineering, and mathematics. We report on the central research question of how disciplinary experts understand partial derivatives, and how their concept images of partial derivatives differ, with a focus on experimentally measured quantities. Using the partial derivative machine (PDM), we probed expert understanding of partial derivatives in an experimental context without a known functional form. In particular, we investigated which representations were cued by the experts' interactions with the PDM. Whereas the physicists and engineers were quick to use measurements to find a numeric approximation for a derivative, the mathematicians repeatedly returned to speculation as to the functional form; although they were comfortable drawing qualitative conclusions about the system from measurements, they were reluctant to approximate the derivative through measurement. On a theoretical front, we found ways in which existing frameworks for the concept of derivative could be expanded to include numerical approximation.

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“…Planinic, Milin-Sipus, Katic, Susac, & Ivanjek, 2012;Planinic, Ivanjek & Susac, 2013), while some of them use other contexts (Woolnough, 2000;Planinic, Ivanjek & Susac, 2013). Several studies also analyze this understanding in the context of mathematics (Leinhardt et al 1990;Hadjidemetriou & Williams, 2002;Christensen & Thompson, 2012;Planinic, Milin-Sipus, Katic, Susac, & Ivanjek, 2012;Planinic, Ivanjek & Susac, 2013;Epstein, 2013).…”

Section: Previous Researchmentioning

confidence: 99%

“…Many researchers have analyzed students' understanding of the concepts of slope and area under the curve in the context of science, specifically in physics (McDermott et al, 1987;Beichner, 1994;Woolnough, 2000;Meltzer, 2004;Pollock, Thompson & Mountcastle, 2007;Nguyen and Rebello, 2011), while others have studied this understanding in the context of mathematics (Orton, 1983;Leinhardt et al 1990;Hadjidemetriou & Williams, 2002;Bajracharya et al 2012;Christensen & Thompson, 2012;Epstein, 2013). However, to date, no study has presented a multiple-choice test that evaluates students' understanding of these concepts in the context of calculus, with a design that follows the steps recommended by mathematics and science education researchers (Beichner, 1994;Ding et al 2006, Engelhardt, 2009).…”

Section: Introductionmentioning

confidence: 99%

Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs

Domínguez

Barniol

Zavala

2017

EURASIA J MATH SCI T

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Studies show that students, within the context of mathematics and science, have difficulties understanding the concepts of the derivative as the slope and the concept of the antiderivative as the area under the curve. In this article, we present the Test of Understanding Graphs in Calculus (TUG-C), an assessment tool that will help to evaluate students' understanding of these two concepts by a graphical representation. Data from 144 students of introductory courses of physics and mathematics at a university was collected and analyzed. To evaluate the reliability and discriminatory power of this test, we used statistical techniques for individual items and the test as a whole, and proved that the test's results are satisfactory within the standard requirements. We present the design process in this paper and the test in the appendix. We discuss the findings of our research, students' understanding of the relations between these two concepts, using this new multiple-choice test. Finally, we outline specific recommendations. The analysis and recommendations can be used by mathematics or science education researchers, and by teachers that teach these concepts.

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Investigating graphical representations of slope and derivative without a physics context

Cited by 67 publications

References 12 publications

Challenges in designing appropriate scaffolding to improve students’ representational consistency: The case of a Gauss’s law problem

Challenges in designing appropriate scaffolding to improve students’ representational consistency: The case of a Gauss’s law problem

Experts’ understanding of partial derivatives using the partial derivative machine

Test of Understanding Graphs in Calculus: Test of Students’ Interpretation of Calculus Graphs

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