«PROCEEDINGS OF THE 15TH ANNUAL CONFERENCE ON RESEARCH IN UNDERGRADUATE MATHEMATICS EDUCATION EDITORS STACY BROWN SEAN LARSEN KAREN MARRONGELLE ...»
Coe, 2007, p. 176). Coe (2007) reported that in more than one instance Peggy “did not use her thinking of a ratio as a comparison of values” to understand slope (p. 195). Considering slope as an index of slantiness allowed this teacher to correctly answer many questions without thinking about division. Coe (2007) concluded that none of the teachers “could clearly explain the use of division to calculate slope” and “there was no evidence of quantitative understanding of the ratio” (p. 237).
The transcripts of students in Castillo-Garsow’s (2010) and Carlson’s et al.’s (2002) studies suggests that the students thought about rates of change additively. In problems that would prompt multiplicative thinking, the students invoked “workaround” strategies including only considering rates of change on increments of equal size (usually 1), and thought of speed and slope as indices instead of as ratios. Since understanding division as relative size is an essential mathematical component in many problems identified as obstacles for students, we investigated our students’ meaning for division and rate to see if they had meanings for these topics that would allow them to understand derivatives.
Methodology To build models of students’ meanings for division we used Simon’s (1993) descriptions of partitive and quotative meanings for division. These two meanings for division do not require multiplicative reasoning. A third model for division, relative size, requires students to reason multiplicatively; the relative size model for division calls upon a comparison between the size of one quantity with respect to another quantity (Thompson & Saldanha, 2003). Division as 15TH Annual Conference on Research in Undergraduate Mathematics Education 359 relative size allows students to be able to reason about non-integer divisors. If division is viewed partitively, it only makes sense to divide a number into n equal parts if n is an whole number.
In order to investigate the understandings/meanings that calculus students might have for division and rate of change, we conducted exploratory interviews with seven undergraduate calculus 1 students, guided by the theoretical perspectives of Steffe and Thompson (2000b). Our interview protocol contained tasks and questions that had been used in class or in other research on understandings of division (See Ball, 1990; Simon, 1993). For example, “Describe a situation where you would need to divide 6 by 3/4ths.” or “How can you tell if your puppy is growing at a constant rate?” We conducted retrospective analysis to create models for students’ understandings of division and rate. In our exploratory interviews, we attended to the idea that phrases students used such as “constant rate” do not necessarily mean the same thing to them as to us.
Preliminary Results Preliminary results from our research confirm that individual students held various (and sometimes unproductive) meanings for division. Additionally, students with partitive meanings for division struggled to interpret answers to division problems involving decimals and struggled to provide a context where division by a fraction is needed to solve a problem.
Jack had strong quotative meanings for division but struggled to interpret the quotient as a measure of relative size. When asked to determine if a puppy was growing at a constant rate he explained that if it is measured on equally spaced intervals of time you can compare the changes in height using subtraction. He proposed if the changes in height are equal the puppy is growing at a constant rate. When asked what he would do if he had measurements corresponding to unequally spaced intervals of time, Jack could not use a multiplicative comparison to show the puppy was growing at a constant rate. Eventually he guessed that division might be an appropriate operation, but was unable to identify the expression “four units of height divided by two days” as a rate of growth. Jack’s definition for proportionality referred to quantity A growing by a units every time quantity B grows by b units, which was consistent with his additive thinking about rate of change but distinct from thinking that changes in A are a/b times as large as changes in B.
Another student, Arlene, had been successful on high school Calculus assessments but had additive and procedural meanings for division. Arlene saw division as a command to perform a calculation. She also struggled to explain how 29.66 related to 0.236 when given the statement 7 ÷ 0.236 = 29.66. Consistent with the findings of Ball (1990), Arlene’s quotative meaning for division broke down when prompted to give a scenario where one would need to divide six by three-fourths. When asked to explain what 6 ÷ ( 3 / 4 ) meant, she invoked the rule of “skip-flip-and-multiply”, explaining that this “is what we learned to do” and then gave a numerical answer instead of a meaning or a sensible scenario. Later on, Arlene could not explain why one divides in the slope formula, exclaiming, “I don’t really see it as division…I see that there is division but when I think of it in terms of slope I don’t, I don’t see that.” Like the teachers in Simon and Blume’s (1994) study, Arelene was inexperienced in representing a physical situation with a mathematical relationship.
Don, who planned to teach high school math, revealed a dominant partitive scheme for division. Don stated that he would emphasize using the long division algorithm to his future students. As a real world example for 37 divided by 3, Don suggested to partition 37 pencils into 3 groups, and later modified his example to each pencil being a bag of 10 M&M’s so that he 360 15TH Annual Conference on Research in Undergraduate Mathematics Education can divide the M&M’s into three equal groups. (Don didn’t notice that multiplication by 10 doesn’t make 37 divisible by three.) Another mathematics education student, Cindy, possessed strong quotative meanings for division. She was able to correctly determine when division was an appropriate operation and construct situations where division by fractions was necessary. However, when explaining what an idea like proportional meant she used additive descriptions and struggled to explain why we divide when we find a slope. This strong student was able to correctly solve many problems but still offered primarily additive explanations.
Early ConclusionsGiven our preliminary interviews we believe that it is possible that many Calculus students do not understand quotient as a measure of relative size and will be unable to make sense of average and instantaneous rate in the ways needed to understand derivatives. For example if one thinks of rate as an amount added, common explanations of the derivative which ask students to envision the numerator and denominator of a difference quotient becoming arbitrarily small do not make sense. If a student believes a rate is the amount added to the output instead of a multiplicative comparison, the rate is getting smaller and smaller in the limiting process because the change in y values is getting smaller and smaller. If they understand rates as an index of slantiness of a line, then the derivative is a way to measure a geometric property of a graph and they might not attend to the changing quantities being compared. We plan to conduct individual teaching experiments with pre-service secondary teachers to build models of how they understand division and associated concepts such as multiplication, rate, measure and fractions.
We aim to understand why thinking of quotients as a measure of relative size appears to be so challenging.
Questions for the Audience How can we promote understandings of division as relative size?
In the research that you do, are there any concepts related to division that students struggle with?
Can you think of any alternative explanations/models for our data?
Why do you suppose articulating meanings for seemingly elementary topics is so difficult?
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.
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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.
Castillo-Garsow, C. (2010). Teaching the Verhulst Model: A Teaching Experiment in Covariational Reasoning and Exponential Growth. ARIZONA STATE UNIVERSITY.
Coe, E. E. (2007). Modeling Teachers’ Ways of Thinking About Rate of Change. Arizona State
Orton, A. (1983). Students’ Understanding of Differentiation. Educational Studies in Mathematics, 14(3), 235-250.
Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers. The Journal of Mathematical Behavior, 13(2), 183-197. doi:10.1016/0732-3123(94)90022-1 Simon, M.A. (1993). Prospective elementary teachers’ knowledge of division. Journal for Research in Mathematics Education, 233–254.
Steffe, L. P., & Thompson, P. W. (2000a). Radical constructivism in action: building on the pioneering work of Ernst von Glasersfeld. Psychology Press.
362 15TH Annual Conference on Research in Undergraduate Mathematics Education Steffe, L. P., & Thompson, P. W. (2000b). Teaching experiment methodology: Underlying principles and essential elements. Handbook of research design in mathematics and science education, 267–306.
Thompson, P.W. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. Proceedings of the Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 45–64).
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.
Abstract: This study will use reader-oriented theory and the analysis of example spaces to understand abstract algebra textbooks. Textbooks can lay the foundation for a course, and greatly influence student understanding of the material. Multiple undergraduate abstract algebra texts were studied to investigate potential audiences of the books, the level of detail in explanations, examples, and proofs, and the overall material included in the book. Conclusions were drawn regarding some discrepancies between the intended reader and the actual reader and the appropriateness and differences among example spaces.
Keywords: Textbooks, Abstract Algebra, Reader-Oriented Theory, Example Spaces
Theory: Although there has been significant research on mathematics textbooks, much of it has focused on the K-12 level (K-12 Mathematics Curriculum Center, 2005). The calculus reform movement motivated an extension of the study of textbooks into the collegiate level, but still the focus remained on lower-level mathematics or calculus books. Little work has been done to investigate the use, purpose, strengths, and disadvantages of upper-level mathematics textbooks, especially for an abstract algebra course. Many teachers, even in abstract algebra, use the textbook as a foundation, if not an outline, of the course material. As Robitaille and Travers (1992) stated, “Teachers of mathematics in all countries rely heavily on textbooks in their dayto-day teaching, and this is perhaps more characteristic of the teaching of mathematics than of any other subject in the curriculum. Teachers decide what to teach, how to teach it, and what sorts of exercises to assign to their students largely on the basis of what is contained in the textbook authorized for their course.” Authors, even within the field of undergraduate abstract algebra textbooks, have different intentions for the content and use of their texts. Also, generational differences on how mathematics should be presented and learned can affect the language and style of the text.
Modern theories of learning indicate the need for student-oriented teaching methods and readeroriented textbook methods (Weinberg & Wiesner, 2011). Teachers, and textbooks, are no longer meant to simply “cover” material, but should facilitate a learning environment that inspires curiosity, speculation, inference, and quantitative literacy. Student thinking, and the multiple strategies that it may involve, should be valued (Reys, B. J., Reys, R. E., & Chaves, O. (2004).
Reader-oriented theory, although not a new concept in general, was recently applied to the specific area of mathematics textbooks by Weinberg and Wiesner (2011). Within this theory, the use of textbooks moves beyond considering them as a static collection of ideas from which meaning is extracted, and instead considers a student’s active engagement with the material and the processes of reading and understanding. In other words, “the meaning of a text does not reside in the text itself, but rather is generated through a transaction between the text and the reader…” (Weinberg & Wiesner, 2011). This theory takes into consideration the intended, implied, and empirical reader. In other words, the author’s intended audience, the audience that would truly understand the text, and the actual audience. When the three readers do not match, or even when just the implied and empirical readers do not correspond, the success of the book in terms of student comprehension and engagement is lessened.
Another aspect of textbooks that can influence reader understanding, and which also can illustrate the intended, implied, and empirical reader, is that of example spaces. The creation of 364 15TH Annual Conference on Research in Undergraduate Mathematics Education examples is essential in the teaching and learning of mathematics. They are used for reference and as a means to generate other examples, conjectures, and perceptions (Bills & Watson 2008;