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

Interviews were used as a research tool in this case study. The purpose of the interviews was to understand student cognitive processes when discussing functions and function composition in relation to setting up the difference quotients. Over the course of two interviews, students were asked performance questions, unexpected “why” questions, and reﬂection questions (Zazkis & Hazzan, 1999). They were also asked to complete “give an example” tasks and construction tasks (Zazkis & Hazzan, 1999). For example, students were asked to construct a g(x) such that f (x + h) = f (g(x)).

Some of the questions were developed in an interview guide beforehand, while others came up during the interview as a result of individual student response. These were generally the unexpected why questions or questions posing counter-examples to student claims.

Students were also shown samples of other students’ work and asked to comment on the procedures the students took in the samples and why s/he believed those students performed those procedures. After the culmination of the interviews, data analysis began. Analysis of this data is still underway.

Data Analysis Plan Analysis of student interviews will be focused on student thinking of functions and function composition as revealed by interviews and the errors when setting up the difference quotients. I will analyze the instances of errors made by the interviewees, answers to pre-developed interview questions, answers to particular topics of function and function composition, and interviewee descriptions and explanations of student sample work. One error will be chosen at a time and data from the two aforementioned groups of students will be analyzed for trends. Data will be analyzed in this way to make claims about why these errors are occurring.

The framework for the data analysis tool will be developed iteratively. That is, data from the interviews will be used to inform the framework and in turn the framework will help classify students into a conception category. The conception categories will be correlated with instances of error within the interviewees, interviewee descriptions and explanations of student sample work, 566 15TH Annual Conference on Research in Undergraduate Mathematics Education and interviewee conceptions of function composition.

Preliminary ﬁndings on the effectiveness of using function composition to evaluate f (x + h) shows the concept can be a valuable learning tool to help students transition out of an action conception into a process conception; that is, they no longer view evaluation of f (x + h) as an action of replacing or substituting x for x + h. For instance, the student who took the longest (the times ranged from 6 seconds to 9.5 minutes) to complete the construction task described above said, “[Before] I just said you just replaced [x with x + h]...Because I didn’t know–or you just substitute in. It’s actually a composition. I get it.” In addition, ﬁndings from this study on errors can be used to inform teaching practices, curriculum development, or further research. Presenting errors found in this study to students learning composition can be used as a teaching method or part of the curriculum to open discussion with students. As observed during the interviews, this can lead to rich discussion on the concept of function itself.

**Questions for the Audience:**

1. What other implications can you see from this work?

2. What other criteria for function conception classiﬁcation have you seen or used?

## References

Ayers, T., Davis, G., Dubinsky, E., & Lewin, P. (1988). Computer experiences in learning composition of functions. Journal for Research in Mathematics Education, 19(3), pp. 246-259.Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23(3), pp. 247-285.

Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. Research in Collegiate Mathematics Education III. Issues in Mathematics Education, 7, 115-162.

Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment: A tool for assessing students reasoning abilities and understandings. Cognition and Instruction, 28(2), 113-145.

Cottrill, J. (1999). Students’ understanding of the concept of chain rule in ﬁrst year calculus and the relation to their understanding of composition of functions. Unpublished doctoral dissertation, Purdue University.

Dubinsky, E. (1991). Reﬂective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Dordrecht, The Netherlands: Kluwer.

Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 85–106). Washington, D. C.: Mathematical Association of America.

Eisenberg, T. (1991). Functions and associated learning difﬁculties. In D. Tall (Ed.), Advanced mathematical thinking (pp. 140–152). Dordrecht, The Netherlands: Kluwer.

Gooya, Z., & Javadi, M. (2011). Unversity students’ understanding of function is still a problem. In S. BROWN, S. LARSEN, K. MARRONGELLE, & M. OEHRTMAN (Eds.), Proceedings of the 14th annual conference on research in undergraduate mathematics education (Vol. 3, pp. 65–67). Portland, Oregon.

Harel, G., & Kaput, J. (1991). the role of conceptual entities and their symbols in building advanced mathematical concepts. In david tall (Ed.), Advanced mathematical thinking (p. 82).

Horvath, A. (2011). Treatment of composition at the secondary and early college mathematics curriculum.

In S. BROWN, S. LARSEN, K. MARRONGELLE, & M. OEHRTMAN (Eds.), Proceedings of the 14th annual conference on research in undergraduate mathematics education (Vol. 4, pp. 98–102). Portland, Oregon.

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**Oehrtman, M., Carlson, M., & Thompson, P. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the connection:**

Research and practice in undergraduate mathematics (p. 27-42). Washington, DC: Mathematical Association of America.

Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reiﬁcation - the case of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy.

Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Dordrecht, The Netherlands: Kluwer.

Thompson, P. (1994). Student, functions, and the undergraduate curriculum. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education I (pp. 21–43). Providence, RI: American Mathematical Society.

Yin, R. K. (2009). Case study research. Thousand Oaks, CA: Sage.

Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. Schoenfeld, & J. Kaput (Eds.), Research in collegiate mathematics education IV (pp. 103–127). Providence, RI: American Mathematical Society.

Zazkis, R., & Hazzan, O. (1999). interviewing in mathematics education research: Choosing the questions.

Journal of Mathematical Behavior, 17(4), 429–439.

Abstract The purpose of this study was to construct and apply a framework to examine opportunities to understand calculus deeply, as informed by prior research. I applied this framework to analyze opportunities to learn derivatives in two calculus texts: Hughes-Hallett et al. (2009) and Stewart (2012). These tests were chosen to represent different points on a continuum between conventional and reform calculus materials. An analysis of both texts suggests that they are more similar than might be expected with respect to the amount of context given in problems, their attention to position, velocity and acceleration, and opportunities to use multiple representations – algebraic, numeric, graphical, and descriptive. There were differences between graphical and descriptive problems between texts and opportunities to make connections between representations. The framework presented here illuminated degrees of variation and similarity between opportunities to understand calculus in these texts and could have further utility for examining additional calculus texts.

**Keywords: Calculus, Textbook analysis, Multiple representations, Derivatives**

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**Opportunities to Develop Understanding of Calculus:**

A Framework for Analyzing Homework Exercises What does it mean to learn and understand calculus, and how are these opportunities to learn represented in mathematical tasks in college curriculum materials? In this study, I have constructed a framework that integrates various perspectives on calculus learning, as informed by research literature. These perspectives were chosen because they represent what a range of scholars value about calculus understanding and opportunities to understand calculus deeply through multiple representations (Sofronas et al, 2011; Knuth, 2000; Aspinwall and Miller, 2001;

Porter and Masingila, 2000; Lithner, 2004; Cunningham, 2005; Roth and Bowen 2001; Bossé, 2010). The primary purpose of this study is to examine the degree to which curriculum materials support opportunities to learn calculus in these ways.

Why study textbooks? Outside of classroom instruction, students spend significant amounts of their time interacting with their text and solving problems and exercises in a typical college course. According to Rezat (2007), “The mathematics textbook is one of the most important resources for teaching and learning mathematics” (p. 1260). Additionally, textbook analysis is an area where research is lacking (Love and Pimm, 1996). In sum, this area is important because in mathematics learning in higher education, texts may play a large role.

Decisions about the content of mathematics curriculum, or what students should learn, must ultimately rest upon value judgments (Hiebert, 1999). Different value systems are likely to be represented in the design of particular curriculum materials. Comparing textbooks is one way to examine the opportunities that students may have to engage in various ways with calculus, but deciding which text is better from the analysis depends on the type of learning one would like to foster in calculus students. Using the framework I developed to examine opportunities to learn calculus, I compared opportunities to learn derivatives, a central idea in calculus (Sofronas et al, 2011), in these two sets of curriculum materials. Such an analysis could reveal whether the values of textbook authors differ, in what ways, and whether there is common ground. To this end, I investigated the following question: How do Stewart (2012) and Hughes-Hallett et al.

(2009) calculus texts compare in the opportunities that students have to engage with mathematics in the homework exercises?

Conceptual Framework The framework for this analysis was developed around components that could help students develop a robust understanding of calculus in which students see connections between calculus topics and connections between mathematics and the world outside of their classrooms (See Table1). Previous work by Sofranos et al. (2011) indicated that, with the emergence of many different ‘types’ of calculus classes for different audiences it may be important to agree on common elements of calculus that are vital. Three components identified in Sofranos et al. and other research include the importance of context, the importance of attending to the relationship between position, velocity, and acceleration, and the importance of including derivatives that are not the “typical” rates of change in regard to x, t, or. These components, such as context (Boaler, 1993), may be important for developing connections between classroom mathematics and real-world problems and the ability to use mathematics flexibly. These components may help to foster breadth and flexibility of students’ knowledge of calculus.

Drawing connections between and deeply understanding the meaning of different representations is also a worthy goal for calculus students. For this reason, texts ought to give students opportunities in homework exercises to deal with different mathematical representations, and both texts analyzed in this study explicitly state in the preface that 570 15TH Annual Conference on Research in Undergraduate Mathematics Education representing mathematics in multiple representations is one of their goals (Hughes-Hallett et al, 2009; Stewart, 2011). The framework divides representations into four types: 1) Algebraic representations consist of algebraic equations, algebraic proofs, and most symbolic notation. 2) Numeric representations consist of tables and numeric approximations. 3) Graphical representations consist of graphs on the Cartesian or polar coordinate plane. 4) Descriptive representations consist of describing mathematics in ‘plain language,’ in one’s own words, or integrating a real-world context into the problem.

This framework analyzes the representations that are present in the problems given to students, and the representations that are asked for in the expected student solutions. Both of these elements have merit. Research has shown that students do not typically form effective connections between multiple representations unless they have experience solving problems that ask them to transfer knowledge from one representation to another (Cunningham, 2005).

Additionally, calculus has traditionally focused on algebraic representation (Hughes-Hallett et al., 2006), but there are reasons to believe that other representations are important. In real-world mathematics, calculus problems will not always take this form, and other studies have shown the benefit of writing in calculus classes (Aspingwall and Miller, 2001) and in mathematics more generally (Bossé, 2010; Porter, 2000). Additionally, research has shown the benefit of understanding graphs in technical occupations (Roth and Bowen, 2001; Knuth, 2000). For these reasons, examiningthe degree to whichdifferent texts use and ask for representations is worthwhile.

Methods