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Abstract Title: Research on student conceptions of integration in math and physics
Abstract: Integration is a complex mathematical idea that has multiple meanings and layers. An in-depth knowledge of definite and indefinite integrals is extremely important to understanding various physics concepts and being able to solve relevant problems. This session includes researchers in both mathematics and physics education who have conducted investigations into the teaching and learning of integrals, primarily definite integrals.  Common themes in the presented work are student understanding of the Riemann sum, the interplay between the mathematics and the physics across multiple contexts, and indeed the role of context itself.
Abstract Type: Parallel session: Talk Symposium

Author/Organizer Information

Primary Contact: Michael Loverude
California State University - Fullerton
Dept of Physics
Fullerton, CA 92834
and Co-Presenter(s)
Warren M. Christensen, North Dakota State University
John Thompson, University of Maine

Symposium Specific Information

Discussant: Michael Loverude
Moderator: Warren Christensen
Presentation 1 Title: Using physics to highlight the underlying structure of Riemann sums in learning definite integrals
Presentation 1 Authors: Vicki L. Sealey, West Virginia University
Presentation 1 Abstract: Several studies have shown that students are most successful solving definite integral application problems when they are able to see the definite integral as the sum of infinitely small products, or in other words when they understand the underlying structure of Riemann sums. This research involves classroom teaching experiments and clinical interviews with first-semester calculus students where initial instruction on definite integrals is centered on physics applications that highlight the underlying structure of Riemann sums. For example, when students approximate the force of water exerted on a dam, they attend to the underlying structure of force as the product of pressure and area. This research aims to identify strengths and weaknesses of the students as they develop an initial understanding of definite integrals, and how they use this understanding to build more formal ideas, such as the Fundamental Theorem of Calculus and the relationship to area under a curve.
Presentation 2 Title: What the integral does: Physics students' efforts at making sense of integration
Presentation 2 Authors: Joseph F. Wagner, Xavier University
Presentation 2 Abstract: Students use of a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts.  This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discomforts and skepticism some students maintain even after constructing useful conceptions of the integral.  In particular, many students attempt to explain what integration does by trying to interpret the algebraic manipulations and computations involved in finding antiderivatives.  This tendency, perhaps arising from their past experience of making sense of algebraic expressions and equations, suggests a reluctance to use their understanding of "what a Riemann sum does" to interpret "what an integral does."
Presentation 3 Title: Physics and calculus students' understanding of the definite integral using graphical representations
Presentation 3 Authors: John R. Thompson, University of Maine
Rabindra R. Bajracharya, Oregon State University
Presentation 3 Abstract: Using written surveys and individual interviews administered in introductory calculus-based physics and multivariable calculus classes, we studied the extent to which the conceptual understanding of definite integrals affects the understanding of physics concepts that involve definite integrals. We also elicited specific difficulties that students have with definite integrals, particularly with graphical representations, including applying the Fundamental Theorem of Calculus to find the integral of a graphed function. One strong focus of this work was how students reasoned about integrals that yield a negative result.  We noticed student success invoking physical context to interpret certain aspects of definite integrals. Furthermore, we found that although students dominantly used area-under-the-curve reasoning, including unprompted invocation of the Riemann sum, when contemplating definite integrals, their reasoning was often not sufficiently deep to help think about negative definite integrals.
Presentation 4 Title: What integration cues, and what cues integration in intermediate electromagnetism
Presentation 4 Authors: Leanne Doughty, Michigan State University
Eilish McLoughlin, Dublin City University
Paul van Kampen, Dublin City University
Presentation 4 Abstract: The use of sophisticated math in physical models and problem solving has been identified as a major challenge for students in their intermediate and upper-division courses. Integration is a math tool widely used across physics contexts. It is perhaps the foremost mathematical technique in intermediate level electromagnetism courses. We present semi-quantitative research into students' difficulties with integration in such an electromagnetism course with cohorts of about 50 students. We have investigated what students' views on integration are, before they enter the electromagnetism course, and interpret the results in terms of their concept images of integration. We have found that students primarily see integration as a process of evaluation, and that the majority of students have no conceptual aspect in their concept image of integration. We confirm and quantify earlier results that recognizing dependency on a variable is a strong cue that prompts students to integrate. In addition, various technical difficulties with integration prevent almost all students from getting a completely correct answer to a typical electromagnetism problem involving integration.
Presentation 5 Title: Framework for Understanding Introductory Students' Application of Integrals in Physics Problem Solving
Presentation 5 Authors: Dehui Hu, Rochester Institute of Technology
N. Sanjay Rebello, Kansas State University
Presentation 5 Abstract: The concepts of differentiation and integration are important tools for solving physics problems. Research in physics education has reported students' lack of ability to transfer their calculus knowledge to physics problem solving.  We conducted group teaching/learning interviews with 13 students to investigate their reasoning in setting up integrals in the context of E&M.  We proposed a conceptual framework by integrating aspects of several theoretical constructs-mathematical resources, conceptual metaphors, and conceptual blending to help us understand student use of calculus concept in physics. The main contributions of this research include (1)  providing evidence for the existence of symbolic forms in students' reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems.