«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
Methods The research took place at a medium-sized public university in the southeast. To address the research question, we met for a single 75-minute semi-structured, task-based interview with Charlotte Knight. We purposefully identified Charlotte, an undergraduate mathematics major with a concentration in teacher licensure, as a participant because of a “colored” proof she provided on an in-class exam. Very similar to the proofs Oliver Byrne presented in his reworking of Euclid’s Elements, we were curious about Charlotte’s reasoning. The audio-recorded interview focused on a discussion of Charlotte’s original proof and the construction of a new “colored” proof.
We are using the constant-comparative method of analysis as outlined by Corbin and Strauss (2008). That is, using the transcription of the interview, we are systematically open and axial coding the data to identify emergent themes in Charlotte’s interview, while regularly revisiting the theory identified in Gibson (1998) and supported by Yestness and Soto (2008).
Results and Future Work Preliminary results indicate that all four aspects of diagramming offered by Gibson (1998) and supported by Yestness & Soto (2008) are apparent in Charlotte’s colored proof.
Additionally, she appears to use color (1) as an organizational tool to connect her diagrams to the content of her proofs, (2) to enhance her understanding of the proof she is writing, and (3) to illustrate relationships within her diagrams and proofs. It may be the case that these are, in fact, embedded within Gibson’s categories.
We feel that this small study will have some particularly interesting pedagogical implications. Byrne (1847) asserted that using color-coded proofs allows a whole class to see the key parts of the argument rather than having to mentally connect what the letters refer too, and thus reducing opportunities for confusion. Early passes through Charlotte’s interview transcription support and expand upon this argument. Extending this research to include other participants who utilize diagrams (and, in particular, utilize colored diagrams) may shed light onto how to reform instruction accordingly.
15TH Annual Conference on Research in Undergraduate Mathematics Education 577 Necessary next steps for this research study include identifying additional participants who employ color in their proof-writing techniques. This will enable us to further investigate any conjectures that emerge as a result of this research. By the time of the RUME conference, we will be ready to report on our constant-comparative analysis of Charlotte’s interview. Questions
we intend to pose to the audience include the following:
• Charlotte was selected for an interview because of an isolated proof she provided on an exam. How might we go about identifying additional participants without creating an artificial environment?
• The aforementioned issue is a limitation to this research. What steps might we take to “beef up” the validity and reliability of our small study?
References Byrne, O. (1847). The first six books of the elements of Euclid in which colored diagrams and symbols are used instead of letters for the greater ease of learning. London: William Pickering.
Cajori, F. (1928). A history of mathematical notations, volume 1: Notations in elementary mathematics. London: Open Court Company.
Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. In M. J.Hoines & A. B. Fuglestad (Eds.) Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education, Bergen, Norway, July 11 - 14, 2004.
Corbin, J. & Strauss, A. C. (2008). Basics of qualitative research. Thousand Oaks, CA: Sage.
Gibson, D. (1998). Students’ use of diagrams to develop proofs in an introductory analysis course. In A.H. Shoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education. III (pp. 284-307). Providence, RI: American Mathematical Society.
Hallet, D. H. (1991). Visualization and calculus reform. In W. Zimmerman & S. Cunningham (Eds.) Visualization in teaching and learning mathematics: MAA notes number 19 (pp.
121-126). Washington, DC: Mathematical Association of America.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics, 44(1), 5-23.
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Selden, J. & Selden, A. (2003). Validation of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
Tall, D. (1991). Intuition and rigor: The role of visualization in the calculus. In W. Zimmerman & S. Cunningham (Eds.) Visualization in teaching and learning mathematics: MAA notes number 19 (pp. 105-119). Washington, DC: Mathematical Association of America.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge.
Educational Studies in Mathematics, 48(1), 101-119.
Ye, Z., Chou, S. C., Gao, X. S. (2010). Visually dynamic presentation of proofs in plane geometry. Journal of Automated Reasoning, 45, 213-241.
578 15TH Annual Conference on Research in Undergraduate Mathematics Education Yestness, N. & Soto, H. (2008). Student’s use of diagrams in understanding and explaining groups and subgroups in abstract algebra. In Zandieh, M. (Ed.) Proceedings of the 11th Annual Conference on Research in Undergraduate Mathematics Education, San Diego, CA, February 28 - March 2, 2008.
Zimmerman, W. (1991). Visual thinking in calculus. In W. Zimmerman & S. Cunningham (Eds.) Visualization in teaching and learning mathematics: MAA notes number 19 (pp. 127Washington, DC: Mathematical Association of America.
Past research has shown that students struggle when solving definite integral application problems, but little has been done to examine the sources of their difficulties. This study aims to more thoroughly examine student misconceptions about definite integrals and develop new curricula to address these issues. Participants were second-semester calculus students enrolled in a large, public university. Exam problems required students to sketch approximating slices of given solids, and set up a corresponding volume integral. Students’ written work was analyzed for common mistakes and misconceptions. Although some students solved the problems correctly, a majority exhibited major deficiencies in their understanding of how to apply the definite integral. Most surprising was students’ widespread failure to make a connection between the sketch and the set up of the integral. Further research is currently under way that aims to expose sources of students’ faulty thought processes when using definite integrals to solve volume problems.
Keywords: Calculus, definite integral, visualization, conceptual understanding
Finding volumes of solids is an application of the definite integral that is routinely covered in a second-semester calculus course, but very little research has been conducted with the aim of understanding how students conceptualize these problems. The definite integral is typically introduced to students as a tool for determining the area of a region contained between a continuous, positive curve y = f ( x) and the x-axis on a closed interval. Since students generally encounter the area conception of definite integral first, this can lead to the definite integral being tied only to the physical quantity of area in students’ minds (Bezuidenhout & Olivier, 2000; Gonzalez-Martin & Camacho, 2004; Sealey, 2006). This rigid association would € almost definitely lead to difficulties in applying the definite integral in other applicable physical situations.
Previous research has found that when solving definite integral application problems, students often rely on previously encountered methods for setting up and evaluating integrals (i.e., mimicking methods encountered in class) (Grundmeier, Hansen, & Sousa, 2006; Huang, 2010). Yeatts & Hundhausen (1992) examined student difficulties in applying calculus concepts to physics problems and found that students relied “heavily upon memory and pattern to establish the integrals prior to routine manipulation.” A key component in successfully solving volume problems is visualization of the solid.
Optimally, visualization of the solid and its constituent parts guides and dictates the construction of the corresponding volume integral. Unfortunately, it is possible to correctly solve many routine volume problems without the aid of visualization (we consider routine volume problems to be those in which the required function formulas are stated explicitly). In an early study on student understanding of integration (Orton, 1983), students were asked to give detailed explanations of their reasoning when solving integration problems. Orton observed that students had very little idea of the dissecting, summing, and limiting processes involved in integration.
580 15TH Annual Conference on Research in Undergraduate Mathematics Education Huang (2010) observed students focusing on “calculating correctly, while ignoring the true meaning of the concepts behind the calculations.” Current Research Aims and Questions The goal of this study is to more deeply explore student understanding of applications of the definite integral. The first phase involves identification and classification of common mistakes students make when setting up and solving volume problems. The second phase consists of a more in-depth analysis of student thinking via data collected from one-on-one interviews concerning past written work and novel problems, and small-group problem-solving sessions concerning novel problems, with the goal being identification of the sources of students’ misconceptions. The third phase involves development and implementation of new teaching techniques and materials that will aid in greater student understanding of the definite integral.
Although area is involved in certain volume calculations, it is not the physical quantity that is being determined by the integration problems considered in this study. Because of this, we believe that volume problems can expose any underlying deficiencies students may have that may otherwise be concealed due to the relative simplicity of integral-as-area problems.
Visualization is an important aspect of integrating to find volumes, so we want to examine the connections (or lack thereof) between students’ visualizations of solids and their set-up of the volume integral.
Conceptual Framework Our research is built on the foundation of the constructivist learning theory (Piaget, 1970). We believe that students construct their own understandings of mathematical concepts given the information that is presented to them and the information that they extract from the learning materials. The theoretical perspective guiding our analysis of student thinking in this study is based on Dubinsky’s (1991) Action-Process-Object-Schema framework. The portion of our study where we analyze student written work and subsequently interview students about their work will also be guided by Vinner’s (1997) conceptual framework that describes and analyzes verbal (oral/written) mathematical behaviors of students. Vinner classified student mathematical behaviors as occurring within two different contexts – a conceptual context, which involves understanding of mathematical symbols, notation, and meanings of words; and an analytical context, which involves problem solving. Conceptual behaviors are a result of conceptual thinking, which arises from meaningful learning and correct conceptual understanding.
Analytical behaviors are a result of analytical thinking, which involves accurate analysis of the type and structure of a problem, and selection of a valid solution procedure. When students act in ways that superficially resemble these types of behaviors, but lack the deep, proactive “thinking” aspects of each, they are exhibiting what Vinner calls pseudo-conceptual behaviors or pseudoanalytical behaviors. He explains that, in “mental processes that produce conceptual behaviors, words are associated with ideas, whereas in mental processes that produce pseudo-conceptual behaviors, words are associated with words; ideas are not involved” (p. 101). Similarly, in analytical mental processes that produce analytical behaviors, problem-solving strategies are associated with ideas, whereas in mental processes that produce pseudo-analytical behaviors, problem-solving strategies are associated with methods that have been previously encountered.
Application problems come in a variety of types and forms, and require a solid understanding of the underlying mathematical concepts. Optimally, when students begin solving application problems, they are familiar and comfortable with the relevant mathematical concepts – in other words, the concepts have been encapsulated into objects that can be used as problemsolving tools (Dubinsky, 1991). Incomplete or insufficient understanding of these concepts can 15TH Annual Conference on Research in Undergraduate Mathematics Education 581 lead to pseudo-conceptual and pseudo-analytical behaviors in the classroom. It is the aim of the researchers to identify and examine these pseudo-behaviors for definite integral problems, and determine where and how students’ misunderstandings occur. We believe that in the interview and problem-solving sessions, we will be able to uncover where in the action-process-object procedure students become stuck that requires them to resort to pseudo-strategies. We hope to create problems that better expose these inconsistencies, and develop teaching methods that discourage these types of student actions and foster more meaningful learning.