1 Calculus Student Understandings of Quotient and Rate Cameron Byerley Neil Hatfield Pat Thompson

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RUME XV/ Feb. 23 rd, 2012 The Fundamental Theorem of Calculus Guides Our Instruction Calculus exists to solve two basic problems: 1) You know how fast something is changing and you want to know how much of it you have. 2) You know how much of something you have and you want to know how fast it is changing. (Thompson, Byerley and Hatfield, in press)

RUME XV/ Feb. 23 rd, 2012 Our Students’ Understandings of Rate of Change Taking our instructional goals seriously demanded we build students’ understanding of rate of change. Many of our students struggled to explain average rate of change after explicit, repeated instruction. Our students struggled to create graphs of rate of change functions given an accumulation function.

RUME XV/ Feb. 23 rd, 2012 What Do These Issues Have in Common? Ideas of speed and velocity are non-trivial for calculus students. (Gravemeijer & Doorman, 1999) Students struggle with graphical understandings of the derivative. (Asiala et al., 1997) Study of 110 students found items requiring “elementary” idea of finding a rate by dividing changes in y by changes in x to be the most difficult in survey of understanding of differentiation. (Orton, 1983) High-performing Calculus students struggle to describe multiplicative relationships between two variables. (Carlson et al., 2002)

RUME XV/ Feb. 23 rd, 2012 They involve interpreting quotients as a measure of relative magnitude. What would it mean for instruction if undergraduates had weak meanings for quotient?

RUME XV/ Feb. 23 rd, 2012 Research Questions What meanings do undergraduate Calculus students have for quotient? What work do particular meanings for quotient do in understand slope, rate of change and derivative?

RUME XV/ Feb. 23 rd, 2012 Understandings of Quotient Relative Magnitude (Thompson & Saldanha, 2003) The individual understands the quotient as a multiplicative comparison of two quantities’ magnitudes as a measure of the magnitudes’ relative size. She sees that a measure induces a partition and a partition induces a measure. Partitive (Simon, 1993) The individual anticipates the quotient as a number of items in each group after an “equal sharing” process. Quotitive (Simon, 1993) The individual anticipates the quotient, A/B as representing number of times that a B-sized ruler measures the quantity A. 8

RUME XV/ Feb. 23 rd, 2012 Research on Understandings of Quotients Ball and Simon found that elementary and secondary school teachers struggled to create situations that require division by a fraction. (Ball, 1990; Simon, 1993; Ma, 1999) Secondary teachers have weak meanings for average rate of change and can’t explain use of division in slope formula. To them rates are amounts added in unit intervals or a measure of slantiness. (Coe, 2007)

RUME XV/ Feb. 23 rd, 2012 General Findings The seven Calculus students we interviewed had various meanings for quotient and rate. There were relationships between what they said when asked what quotient meant to them and which types of problems they understood.

RUME XV/ Feb. 23 rd, 2012 Arlene: Math Is Something You Do With Symbols Arlene, an AP calculus student in high school, struggles to interpret the meaning for the in the statement “7 divided by.236 = 29.66” outside of a computation. She had never considered why you divide to find slope.

RUME XV/ Feb. 23 rd, 2012 Jack: Rates as Amount Added Jack has a strong quotitive scheme but does not appear to have a way to think about relative magnitude. Jack thinks of rate as amounts added and struggles to determine how to characterize constant rate on unequally sized intervals. He is unlikely to create situations about rate that have differently spaced intervals because his definition depends on the equal spacing.

RUME XV/ Feb. 23 rd, 2012 Secondary Mathematics Education Major Survey Findings When asked to use a picture to explain what means, six out of seventeen were able to draw a picture that expressed a meaning for quotient. When asked to provide a real world example of dividing by a fraction, ten out of 18 responses gave a scenario involving division. When asked why division is present in the slope formula, one out of 17 explained using a meaning for quotient.

RUME XV/ Feb. 23 rd, 2012 Conclusions Thinking about rates as amounts added and slope as an index of slantiness could be coping mechanisms for students with weak meanings for quotient. If our findings about undergraduates meanings for quotient generalize it could partially explain observed difficulties with secondary topics involving division.

RUME XV/ Feb. 23 rd, 2012 Questions for Audience In the research or teaching that you do, are there any concepts requiring meanings for quotient that students struggle with? Contact Us

RUME XV/ Feb. 23 rd, 2012 References Asiala, M., Dubinsky, E., Cottrill, J., & Schwingendorf, K. (1997). The development of students’ graphical understanding of the derivative. Journal of Mathematical Behavior, 16(4), 399–431. Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132–44. Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352– 378. Coe, E. E. (2007). Modeling teachers’ ways of thinking about rate of change. (Unpublished doctoral dissertation). Arizona State University, Tempe, AZ. Gravemeijer, K., & Doorman, M. (1999). Context problems in realistic mathematics education: A calculus course as an example. Educational Studies in Mathematics, 39(1), 111–129. Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understandings of fundamental mathematics in China and the United States. Routledge. Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14(3), 235–250. Simon, M. A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 233–254. Thompson, P. W., Byerley, C., & Hatfield, N. (In Press). A conceptual approach to calculus made possible by technology. Computers in Schools. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. A Research Companion to Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. 16