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

The category doing mathematics includes many different types of tasks that have the shared characteristic of having no pathway for solving the task explicitly or implicitly suggested and therefore requiring nonalgorithmic thinking. This category includes tasks that are nonroutine in nature, are intended to explore a mathematical concept in depth, embody the complexities of real-life situations, or represent mathematical abstractions (p. 23).

Taking this a step further, Cuoco, Goldenberg, and Mark (1996) suggest that particular habits of mind, developed through a variety of tasks, should be an organizing principle of mathematics curricula. They argue that students should learn mathematics by engaging in activities similar to the activities mathematicians do. These include searching for patterns, experimenting, communicating, exploring ideas, inventing notation, visualizing relationships, and making conjectures.

From this perspective, the overriding goal of the curriculum is to help students develop habits that enable them to be mathematically proﬁcient, blending strands such as conceptual understanding, procedural ﬂuency, strategic competence, adaptive reasoning, and productive disposition (National Research Council, 2001).

3 Methods Sixteen university mathematics and mathematics education faculty members participated in the study. Faculty from both public and private liberal arts colleges and research institutions in the western United States were included. All participants had PhD’s and had taught full time in a university setting for between one and 39 years. Participation in the study involved completing a written survey with some background information as well as a phone interview discussing their perspectives on “doing mathematics.” The phone interview questions are listed below; the ﬁrst four questions involve their expectations of students while the last three questions involve their own experiences with mathematics.

1. You are teaching a university level calculus course. What does it mean for one of your students to do mathematics in that setting?

2. What role do you think applications have in doing mathematics?

3. You are teaching an upper division proof-oriented mathematics course. What does it mean for one of your students to do mathematics in that setting?

4. What role do you think group work has in doing mathematics?

5. What does it mean for you to do mathematics?

6. What kind of activities do you do when you do mathematics?

7. What role do other people play in your doing mathematics?

15TH Annual Conference on Research in Undergraduate Mathematics Education 355 Phone interviews were recorded and transcribed. Participant responses were reviewed and similar responses were placed together in some initial categories. A more thorough review of the transcripts with a corresponding revision of categories is currently underway.

4 Results In this preliminary report, we will only discuss participants’ responses to the questions regarding “doing mathematics” within the context of a calculus course, a proof-oriented course, and their own experience. While there was a great deal of variability in participants’ responses, common phrases given in response to these questions, in decreasing order of frequency, are outlined below.

1. Mathematicians

• Calculus: computation, application, conceptual understanding, problem solving

• Proof-oriented: conceptual understanding, recognize logical structure, communicate

• Own experience: developing content knowledge, original research, communicate

2. Mathematics educators

• Calculus: conceptual understanding, making connections, reasoning, application

• Proof-oriented: proving, conceptual understanding, making connections

• Own experience: exploring concepts, proving, problem solving For the majority of the mathematicians, there was a deﬁnite progression in terms of expectations between the different contexts, moving from a computational focus to a conceptual focus. Their own experience often involved learning new content (through papers, presentations, and communication with colleagues) in order to do original research. Interestingly, while proof and conceptual understanding are key for students and are essential in original research, mathematicians did not mention these terms when describing their own work. On the other hand, the responses from mathematics educators were much more consistent across the different contexts. Exploring and understanding concepts, making connections, and logical reasoning were common responses to all three questions. In fact, several mathematics education faculty indicated that doing mathematics was essentially the same at any level. It is important to note that mathematics education faculty reported rarely if ever teaching either calculus or a proof-oriented course.

5 Discussion While there was some overlap and similarities between individual responses, our data indicates that there is not a general consensus among mathematicians and mathematics educators regarding the meaning of the phrase “doing mathematics.” Further, for many individuals the meaning was highly dependent on the context; for others the meaning was quite consistent with small changes in focus. Returning to the broader context of beliefs about mathematics teaching and learning, Sfard (1998) distinguishes between two metaphors for learning. In the acquisition metaphor, learning is viewed as acquiring or accumulating conceptual knowledge. This contrasts with the participation metaphor, where learning is conceived as a process of becoming a member of certain community in which individuals use a common language and act according to certain social norms. One way of interpreting our data is that mathematicians tend to have a more aquisititionist perspective, expecting students to acquire speciﬁc knowledge and skills as they progress, while mathematics educators lean towards a participationist perspective, expecting students to participate in increasingly sophisticated ways of exploring and reasoning about mathematical ideas. In discussing these metaphors for learning, Sfard (1998) argues that, “Naturally, the discussion between the participationist and acquisitionist is bound to be futile... It takes a common language to make one’s 356 15TH Annual Conference on Research in Undergraduate Mathematics Education position acceptable - or even just comprehensible - to another person” (p. 9). Similarly, when discussing perceptions about “doing mathematics” there is a danger that individuals might use the same words to mean different things. This potential communication issue may interfere with attaining our common goal of improving mathematics teaching and learning.

6 Discussion questions

• Are you aware of previous research regarding university mathematics teachers’ beliefs?

• Are there other questions or types of questions that we should ask to get a better sense of faculty perspectives on “doing mathematics”?

• What are the implications of this research?

References Bressoud, D. (2011). The best way to learn [MAA Launchings online column]. Retrieved September 30, 2011 from http://launchings.blogspot.com/2011/08/best-way-to-learn.html.

Cuoco, A., Goldenberg, E. P., & Mark, J. (1996). Habits of Mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior, 15, 375-402.

Hassi, M., Kogan, M., & Laursen, S. (2011). Student outcomes from inquiry-based college mathematics courses: Beneﬁts of IBL for students from under-served groups. In Proceedings of the 14th Annual Conference on Research in Undergraduate Mathematics Education (vol. 3, pp.

73-77). Portland, OR. Retrieved September 30, 2011 from http://sigmaa.maa.org/rume/RUME_XIV_Proceedings_Volume_3.pdf.

National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Washington, DC: National Academy Press.

Philipp, R. A. (2007). Mathematics teachers’ belief and affect. In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, National Council of Teachers of Mathematics.

Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4-13.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing Standardsbased mathematics instruction: A casebook for professional development. New York: Teachers College Press.

Abstract: We have conducted a preliminary investigation of university Calculus students’ conceptions of division and rate of change because these ideas are used to define the derivative.

We conducted exploratory interviews focused on building models of student understandings of division and rate. Retrospective analysis revealed the students interviewed had a variety of meanings for these concepts. Difficulty thinking about division as multiplicative comparisons of relative size was observed in multiple students. Additionally a student who explained rate as an amount added in equivalent x-intervals struggled to determine if a quantity was changing at a constant rate over unequally spaced x intervals. We hypothesize that difficulty conceptualizing division as quotient, and quotient as a measure of relative size2 of two quantities, obstructs students’ understandings of average and instantaneous rate of change. This research will further our goal of understanding student difficulties with derivatives.

Key words: calculus, derivative, rate of change, division, student thinking, multiplicative thinking

## Introduction and Background

As Thompson and Saldanha urged, we take seriously the idea that “how students understand a concept has important implications for what they can do and learn subsequently” (Thompson & Saldanha, 2003, p. 1). Understanding is “what results from a person’s interpreting signs, symbols, interchanges or conversations-assigning meanings according to a web of connections the person builds over time through interactions with his or her own interpretations of settings and through interactions with other people as they attempt to do the same” (Thompson & Saldanha, 2003, p. 12). We believe students build particular meanings for mathematical ideas by building on preexisting understandings (Steffe & Thompson, 2000a).Based on a conceptual analysis (Thompson, 2008) of the concepts of constant and average rate of change, we believe that conceptualizing division and rates as a multiplicative comparison of relative size is essential to understanding the derivative as a rate of change function. We interviewed university Calculus students to create models of their meanings for division and rate so that we can address the question “How do Calculus students understand division and rate?” Our inquiry into Calculus students’ meanings for division and rates of change emerged from observations of our own Calculus students and research on rates of change, division and derivatives. Asiala et al. (1997) summarizes a variety of studies that show that most Calculus students do not have a strong conceptual understanding of the derivative and struggle to solve non-routine problems. In Orton’s (1983) study of student understanding of the derivative, he found that the rule where one divides the difference in y by the difference in x to obtain a rate Research reported in this article was supported by NSF Grant No. MSP-1050595. Any recommendations or conclusions stated here are the authors and do not necessarily reflect official positions of the NSF.

It is more appropriate to say “relative magnitude” instead of “relative size” to account for comparisons of quantities of different physical dimensions (e.g., distance, time) but space is insufficient to explain this fully.

358 15TH Annual Conference on Research in Undergraduate Mathematics Education was not elementary for a large number of students. Orton (1983) alluded to the possibility that “one of the problems of learning about rate of change is that the ideas are basically concerned with ratio and proportion” (p. 243).

Carlson et al.’s (2002) study of 20 high-performing Calculus students revealed that most students struggled on tasks involving average and instantaneous rate of change. Although most students “were frequently able to coordinate images of the amount of change of the output variable while considering changes in the input variable”, students were typically unable to coordinate changes in a function’s average rate of change with uniform changes in the input (Carlson et al., 2002, p. 372). Most students did not understand situations where rates must be considered as multiplicative comparisons of changes in two variables. They were successful in describing rates of change as additive changes in the output.

Castillo-Garsow (2010) provided a model of one high performing secondary student’s meaning for rate that could explain why students find understanding rates of change in Calculus challenging. For this student, an interest rate told her how much money to add to a bank account each year. Thinking of a rate as an amount added results in correct interpretations of situations as long as one always considers uniform changes in the independent variable. The student reworked problems with fractional amounts of one year into whole numbers of months so that the denominator of her division problem (change in money)/(change in time) was one unit. This allowed her to ignore division and consider additive changes in account balances. Simon and Blume (1994) cite studies indicating that many other students think additively when multiplicative thinking is more appropriate.

Coe (2007) conducted an in-depth study of three secondary math teachers’ understandings of rates of change and revealed experienced teachers were not always able to articulate coherent connections between ideas of division, rate, and slope. For one teacher, Peggy, "the slope of a tangent gives a steepness that connects to speed in some contexts” (E. E.