# «PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»

**The two particular texts analyzed in this study were Hughes-Hallett et al.’s Calculus:**

Single Variable, 5th Edition (2009) and Stewart’s Single Variable Calculus, 7th Edition (2012).

The most recent editions of these texts were chosen to represent different standpoints on a continuum between conventional materials and the reform calculus materials which arose from the Tulane Conference of 1986 and the Harvard Calculus Consortium. Calculus instruction has undergone changes and criticisms in the last 25 years, in response generating interest in developing new reform textbooks (like Hughes-Hallett et al.) and sparking changes in existing texts (such as Stewart). The criticisms varied: not enough students were involved in higher mathematics, technology was not being implemented in ways that maximized its potential benefits, and procedures trumped problem solving and modeling. The biggest concern, however, was in developing a conceptual understanding of calculus which would allow students to use what they had learned in class in ways in unfamiliar territory (Hughes-Hallett, 2006).

The inclusion of Stewart’s textbook in this analysis is justified because of the popularity of Stewart’s texts. These textbooks are in many ways canonical across university introductory calculus classrooms. In 2009, Stewart’s textbook outside all other curriculum combined in the North American market of calculus texts (Peterson, 2009). Its widespread use and popularity make it a strong sign-post for comparison to other works. Though Stewart’s text may be labeled ‘conventional’ because of its widespread popularity before the calculus reform, Stewart has been influenced by reform thought, and he describes in the beginning of his textbook ways in which the reform movement has influenced the direction of new editions of his work (Stewart, 2012).

In this way, Stewart may be more of a hybrid between conventional and reform texts than firmly instanced in either trend. The framework presented here helps to illuminate potential similarities and differencesthat exist between the two texts, as well as point out areas where both texts have relatively similar opportunities for students to learn through homework exercises.

15TH Annual Conference on Research in Undergraduate Mathematics Education 571 Because of the importance given to derivatives in first-year calculus, I chose to analyze the homework problems in chapters of Hughes-Hallett et al. and Stewart that introduced this topic. This corresponds to three chapters in Hughes-Hallett et al. and two chapters in Stewart. In these chapters, I analyzed 1111 problems in Hughes-Hallett et al. and 1072 problems in Stewart for each of the components listed in the conceptual framework. In order to check for the consistency coding, I randomly selected 99 problems from Hughes-Hallett and Stewart texts.

These problems were re-coded by one instructor and one graduate student from a mathematics education program. The reliability for all codes exceeded 85%, ranging from 86.9% to 97.0%.

Results The most surprising finding from the study was that Hughes-Hallett et al. and Stewart have very similar distributions along many of the properties for comparison (See Table 2). Both texts have a similar percentage of problems with context and a similar distribution of representations across problems. Both texts are unlikely to give numeric data in a problem (3.2% for Hughes-Hallett et al. and 1.6% for Stewart). Both texts attend to position, velocity and acceleration around 5% of the time, and both texts have similar distributions of representations in the expected student solutions with the exception of graphical and descriptive representations.

The clearest difference between tests happened among these representations. Whereas HughesHallett et al. expected students to describe or explain their solution 28.8% of the time and to form a graph for 15.3% of problems, Stewart expected students to describe only 17.8% of the time and to construct a graph for 26.5% of problems. Another difference between the two texts was the percentage of problems which called for students to convert information from one representation to another, such as from algebraic to graphical, etc. The Stewart text was more likely than the Hughes-Hallett et al. text to ask students to make this kind of transfer (46.9% to 36.1% of problems were transfer tasks in each text). Although instructors have influence students’ opportunities to engage with these components in the framework, the analysis helped to showcase the degree to which the textbooks provide instructors with opportunities to engage students that would not have been apparent without a systematic study.

Applying the conceptual framework to an analysis of these two textbooks provided the following insights. I found the textbooks to be more similar than I had anticipated, given that Hughes-Hallett was formed out of the reform movement and editions of Stewart existed before the movement began. The components of this framework more generally can help reveal a better picture of the ways in which calculus texts give opportunities for students to engage in homework problems. These components are based on recommendations from experts and from research, and while they are not an exhaustive list of the important aspects of calculus texts, this framework provides an illuminating method of examining mathematics problems; this analysis suggests that the framework can be applied with high reliability. This framework could be used to track some of the changes that have occurred within calculus textbooks since the call for reform in the 1980s and to notice general trends or exceptional texts if applied to a range of textbooks representing periods in time. The framework also has practical implications, in that it could be used by members of mathematics departments to compare and contrast textbooks when school districts or universities are selecting textbooks.

Questions: 1) During this analysis, I became concerned about the authenticity of contexts given in problems. How could I analyze whether contexts given are relevant for making sense of the mathematics in the problem or whether they are “psuedocontexts” (Boaler, 1993)? 2) How could I extend this study to examine ways that students use their textbooks, i.e. the degree to which students take up these opportunities to learn?

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## References

Aspinwall, L., & Miller, L. D. (2001). Diagnosing conflict factors in calculus through students’ writing: One teacher’s reflections. Journal of Mathematical Behavior, 20 p. 89-107.Boaler, J. (1993). The role of contexts in the mathematics classroom: Do they make mathematics more “real”? For the Learning of Mathematics, 13(2), p. 12-17.

Bossé, M. J. &Faulconer, Johna.(2010). Learning and assessing mathematics through reading and writing.School Science and Mathematics. 108(1), p. 8-19.

Cunningham, R. F. (2005). Algebra teachers’ utilization of problems requiring transfer between algebraic, numeric, and graphic representations. School Science and Mathematics.

105(2), p. 73-81.

Hiebert, J. (1999). Relationships between research and the NCTM Standards. Journal for Research in Mathematics Education, 30(1), p. 3-19.

Hughes-Hallett, D. (2006). What have we learned from calculus reform? The road to conceptual understanding.Mathematical Association of America, Committee on the Mathematical Science,. 69, p. 43-45.

Hughes-Hallett, D., Gleason, A. M., McCallum, W. G., et al. (2009). Calculus: Single Variable (5thed.). United States: John Wiley & Sons, Inc.

Knuth, E. J. (2000). Student understanding of the Cartesian connection: An exploratory study.

Journal for Research in Mathematics Education, 31(4) p. 500-508.

Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. Journal of Mathematical Behavior, 23 p. 405-427.

Love, E., &Pimm, D. (1996). ‘This is so’: a text on texts. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International Handbook of Mathematics Education.

Vol. 1 (pp. 371-409). Dordrecht: Kluwer.

Peterson, I. (2009). James Stewart and the house that calculus built.MAA Focus, 29(4), p. 4-6.

Porter, M. K. &Masingila, J. O. (2000).Examining the effects of writing on conceptual and procedural knowledge in calculus. Educational Studies in Mathematics, 42, p. 165-177.

Rezat, S. (2010, Jan-Feb). The utilization of mathematics textbooks as instruments for learning.

Paper presented at the Conference of European Research in Mathematics Education, Lyon, France.

Roth, W. & Bowen, G. M. (2001). Professionals read graphs: A semiotic analysis. Journal for Research in Mathematics Education, 32(2) p. 159-194.

Sofronas, K. S., DeGranco, T. C., Vinsonhaler, C., Gorgievski, N., Schroeder, L., & Hamelin, C.

(2011). What does it mean for a student to understand the first-year calculus?

Perspectives of 24 experts. The Journal of Mathematical Behavior, 30, p. 131-148.

Stewart, J. (2012). Single Variable Calculus (7thed.). Australia: Thomson Brookes/Cole.

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**Table 1:**

Components of the Framework

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**Table 2:**

Number (and Percent) of Problems with the Given Components

15TH Annual Conference on Research in Undergraduate Mathematics Education 575 For Educational Color Work: Diagrams in Geometry Proofs Preliminary Research Report Allison F. Toney University of North Carolina Wilmington Kelli M. Slaten University of North Carolina Wilmington Elisabeth F. Peters University of North Carolina Wilmington Abstract. Historically grounded in Oliver Byrne's reworking of Euclid's Elements, and based on a student-generated proof, we investigate the use of coloring to enhance geometry proofs.

Charlotte Knight, an undergraduate mathematics major enrolled in Modern Geometry, regularly employed coloring techniques as a tool in her proof-writing. We met for a single semi-structured, task-based interview to discuss Charlotte’s use of coloring in her organization and understanding of geometry proofs. Preliminary results indicate that Charlotte’s use of diagrams is closely related to her construction of a proof. In particular, her use of color serves several purposes: (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. We feel this small study has particularly interesting pedagogical implications.

**Keywords: modern geometry, proofs, diagrams, color**

Background &Theoretical Framework In 1847, Oliver Byrne published his reworking of Euclid’s Elements. He used colored diagrams so extensively that the visual representations were inseparable from the proofs they were intended to support. Published during a period when geometer’s had their attention focused on non-Euclidean investigations, Byrne’s work was not taken seriously, and was “regarded as a curiosity” (Cajori, 1928, p. 429). However, Byrne did not intend his work for mere entertainment. Instead, he proposed that the book enhanced pedagogy by appealing to the visual and encouraging retention of the ideas. He suggested that by communicating Euclid’s ideas through a colored, visual means, instruction time could be used more efficiently and student retention is more permanent (Byrne, 1847).

Students’ transition to formal proof is a well-covered area of research in mathematics education (e.g., Moore, 1994; Selden & Selden, 2003; Weber 2001). However, students’ use of representations to support their arguments is still an emerging field of research at the postsecondary level. Where there is considerable research available about calculus students’ use of visual representations (e.g., Hallet, 1991; Tall, 1991; Zimmerman, 1991), there is still little research available about students in advanced undergraduate mathematics. Additionally, the National Council of Teachers of Mathematics (NCTM, 2000) asserts that creating and using representations is an essential component to mathematical understanding. As a result, the use of visual representations in K-12 mathematics (and, in particular, K-12 geometry) is welldocumented (e.g., Christou, Mousoulides, Pittalis, Pitta-Pantazi, 2004; Hanna, 2000; Ye, Chou, & Gao, 2010).

576 15TH Annual Conference on Research in Undergraduate Mathematics Education In his research investigating students’ use of visual representations in an introductory analysis course, Gibson (1998) found that students implement diagrams to (1) understand information, (2) determine the truthfulness of a statement, (3) discover new ideas, and (4) verbalize ideas. Yestness and Soto (2008) used Gibson’s results to frame their study of 7 students who used diagrams in the development of their understanding of abstract algebra concepts. They found students most commonly employing (1) and (4) in their diagramming. In particular, they discussed students who explained that their drawings were merely for personal use and not for proof or explanation. However, when asked to explain their proof, many drew a diagram to support their explanation.

The primary goal of this small research study is to investigate how students in an undergraduate modern geometry class use diagrams as proof-writing tools. In particular, we noticed a growing number of students employing the use of color to support their diagrams in our advanced undergraduate mathematics classes. We used the framework proposed by Gibson (1998) and reinforced by Yestness and Soto (2008) to guide our small phenomenological research study into a single geometry student’s use of color-enhanced diagrams as a proofwriting tool. The question guiding our research is: What is the nature of students’ use of color as a proof-writing tool in college geometry?