Abstract
Quantitative and covariational reasoning characterize essential ways of thinking for conceptualizing and representing growth patterns in pairs of quantities in dynamic situations. Despite the broad body of research documenting the central role of quantitative reasoning in students’ ability to construct meaningful function formulas and graphs relating pairs of quantities’ values, little progress has been made to operationalize this theory in precalculus curricula and instruction. This chapter describes conventions for supporting instructors transitioning their students to spontaneously engage in quantitative reasoning toward the goal of conceptualizing the quantitative structure described in an applied problem context. The Pathways conventions for supporting quantitative reasoning reinforce: (i) consistent patterns of referencing and speaking about quantities and how they relate to other quantities in the problem context (speaking with meaning); (ii) physical motion for conceptualizing quantities and considering how the values of pairs of quantities vary together (quantity tracking tool); (iii) use of consistent representational conventions when making a drawing to represent the quantitative structure of a problem context (quantitative drawing); and (iv) consistent expectations and patterns for defining variables, and constructing algebraic expressions and formulas (emergent symbolization). These conventions emerged in the context of a 15-year research and development project focused on supporting instructors in making their precalculus teaching more engaging, meaningful, and coherent. We provide a rationale for introducing each convention and explain how each convention supports students in constructing and representing quantitative relationships symbolically and graphically. We describe our approach to motivating and supporting instructors to adopt these conventions and conclude by illustrating how these instructional conventions might impact an instructor’s mathematical connections and instructional practices.
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Notes
- 1.
Thompson (2008b) describes these strands as follows. The mathematics of quantity refers to how individuals conceptualize measurable attributes of a situation, create measurement schemes to quantify the attributes’ magnitudes, represent the quantities in various ways, and generalize aspects of these attributes. The mathematics of variation refers to how individuals imagine quantities with magnitudes that can vary, how they represent this variation in different ways, and how they draw inferences from noticing what in a relationship remains invariant as two quantities change in tandem. The mathematics of representational equivalence refers to how individuals think of arithmetic and, eventually, algebraic expressions non-computationally as “segues into structural properties of numbers and quantitative relationship[s]” (p. 7). These three strands are interrelated, and opportunities always exist to discuss elements of one strand even within contexts emphasizing another strand. For example, while supporting students in conceptualizing quantitative relationships in some given scenario it can be very natural to explore how two or more of the conceptualized quantities co-vary in tandem. Thompson states that “The three strands in interaction, each receiving appropriate emphasis, and always with the other two in the background, builds a foundation for algebraic reasoning that simultaneously builds a foundation for schemes of meanings that are crucial for understanding the calculus” (pp. 8–9).
- 2.
Since our initial introduction of the term speaking with meaning, 11 new colleges/universities have participated in Pathways professional development.
- 3.
In universities where a commitment to speaking with meaning became a social mathematical norm and was highly valued by the local coordinator in our initial Pathways workshop, we have since documented the persistence of speaking with meaning in this local community of instructors.
- 4.
When initially using the quantity tracking tool students typically decide that a positive value of one quantity is represented by a distance upward from a starting point and that positive values of a second quantity are represented by a distance to the right of the same starting point, while negative values are downward and to the left respectively.
- 5.
The instructor selects a specific button to indicate which two quantities to isolate when exploring the concurrent variation in two quantities’ values.
- 6.
The convention of quantitative drawing, as explained later in this chapter, can further support students in conceptualizing the quantities we want them to coordinate with the quantity tracking tool. The conventions in this chapter all support each other to maximize students’ learning opportunities.
- 7.
The conceptualizations for enacting the quantity tracking tool parallels the thinking for constructing a graph of two quantities’ values as they vary together.
- 8.
We repeat that the conventions and ideas described throughout this chapter support each other. Speaking with meaning, the quantity tracking tool, and quantitative drawing all support students in using symbols meaningfully and making connections between different representations of the same relationship.
- 9.
An “additive comparison” is the answer to the question, “By how much does the magnitude (or value) of one quantity exceed the magnitude (or value) of another quantity?” In contrast, one category of “multiplicative comparisons” is the answer to the question, “One quantity is how many times as large as a second quantity?”.
- 10.
Note that O’Bryan (2020a) uses emergent symbol meaning to describe a set of meanings and expectations that may guide an individual’s algebraic symbolization activity and interpretation of the algebraic symbols that others generate. He uses emergent symbolization when describing the actions an individual engages in that are motivated by these meanings and expectations.
- 11.
Numbers 1 and 3 in this list might seem quite similar, but there is a different intent. The first item focuses on the expectation that all calculations or parts of expressions should represent an evaluation process for some quantity in the situation, and thus each calculation or part of an expression can be quantitatively justified (and, if the individual cannot justify it, then it provides a motivation to reconsider how she has conceptualized the situation). The third item is about how mathematically equivalent expressions with different orders of operations reflect different ways of understanding the situation and that manipulations, including “simplifying” or rewriting an equation to solve for a different variable, may require reconceptualizing the quantitative structure to make sense of the result.
References
American Mathematical Association of Two-Year Colleges (2018). IMPACT: Improving mathematical prowess and college teaching. Author.
Baş-Ader, S., & Carlson, M. (2021). Decentering framework: A characterization of graduate student instructors’ actions to understand and act on student thinking. Mathematical Thinking and Learning. https://doi.org/10.1080/10986065.2020.1844608
Bell, A., & Janvier, C. (1981). The interpretation of graphs representing situations. For the Learning of Mathematics, 2, 34–42.
Boston, M. D., & Wilhelm, A. G. (2017). Middle school mathematics instruction in instructionally focused urban districts. Urban Education, 52(7), 829–861.
Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23, 247–285.
Bressoud, D., Ghedamsi, I., Martinez-Luaces, V., & Törner, G. (2016). Teaching and learning of calculus. Springer Nature.
Byerley, C. (2019). Calculus students’ fraction and measure schemes and implications for teaching rate of change functions conceptually. The Journal of Mathematical Behavior, 55, 100694.
Byerley, C., & Thompson, P. W. (2017). Instructors’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168–193.
Carlson, M. (1998). A cross-sectional investigation of the development of the function concept. In E. Dubinsky, A. H. Schoenfeld, & J. J. Kaput (Eds.), Research in collegiate mathematics education, III (p. 7). Issues in Mathematics Education.
Carlson, M., & Bloom, I. (2005). The cyclic nature of problem solving: An emergent multidimensional problem solving framework. Educational Studies in Mathematics, 58, 45–75.
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. https://doi.org/10.2307/4149958
Carlson, M. P., & Ohertman, M. C. (2010). Precalculus: Pathways to calculus. Rational Reasoning, LLC.
Carlson, M. P., Ohertman, M. C., Moore, K. C., & O’Bryan, A. E. (2020). Precalculus: Pathways to calculus (8th ed.). Rational Reasoning, LLC.
Carlson, M. P., Oehrtman, M. C., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: A tool for assessing students’ reasoning patterns and understandings. Cognition and Instruction, 28(2), 113–145.
Carlson, M. P., Madison, B., & West, R. D. (2015). A study of students’ readiness to learn calculus. International Journal of Research in Undergraduate Mathematics Education, 1, 209–233.
Carlson, M. P., & Bas-Ader, S. (2019). The interaction between a instructor’s mathematical conceptions and instructional practices. In A. Weinberg, D. Moore-Russo, H. Soto, & M. Wawro (Eds.), Proceedings of the 22nd annual conference on research in undergraduate mathematics (pp. 101–110). Mathematical Association of America.
Clark, P. G., Moore, K. C., & Carlson, M. P. (2008). Documenting the emergence of “speaking with meaning” as a sociomathematical norm in professional learning community discourse. The Journal of Mathematical Behavior, 27(4), 297–310. https://doi.org/10.1016/j.jmathb.2009.01.001
David, E. J., Roh, K. H., & Sellers, M. E. (2019). Value-thinking and location-thinking: Two ways students visualize points and think about graphs. The Journal of Mathematical Behavior, 54, 100675. https://doi.org/10.1016/j.jmathb.2018.09.004
Engelke, N. (2007). Students’ understanding of related rate problems in calculus (Unpublished doctoral dissertation). Arizona State University.
Frank, K. M. (2017a). Examining the development of students’ covariational reasoning in the context of graphing (Ph.D. dissertation). Arizona State University.
Frank, K. M. (2017b). Tinker Bell’s pixie dust: The role of differentiation in emergent shape thinking. In Proceedings of the twenty-first annual conference on research in undergraduate mathematics education (pp. 596–604).
Harel, G. (2008). DNR perspective on mathematics curriculum and instruction, part II: with reference to teacher’s knowledge base. ZDM—Mathematics Education, 40, 893–907. https://doi.org/10.1007/s11858-008-0146-4
Hiebert, J., Stigler, J. W., Jacobs, J. K., Givvin, K. B., Garnier, H., Smith, M., Hollingsworth, H., Manaster, A., Wearne, D., & Gallimore, R. (2005). Mathematics teaching in the United States today (and tomorrow): Results from the TIMSS 1999 video study. Educational Evaluation and Policy Analysis, 27(2), 111–132.
Hill, J. A. T. (2021). Using logistic regression to examine the relationship between early alert systems and success in mathematics (Doctoral dissertation, Northern Illinois University).
Jackson, K., Cobb, P., Wilson, J., Webster, M., Dunlap, C., & Applegate, M. (2015). Investigating the development of mathematics leaders’ capacity to support teachers’ learning on a large scale. ZDM Mathematics Education, 47(1), 93–104.
Johnson, H. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in Covarying quantities. Mathematical Thinking and Learning, 17(1), 64–90. https://doi.org/10.1080/10986065.2015.981946
Kaput, J. J. (1992). Patterns in students’ formalization of quantitative patterns. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 290–318). Mathematical Association of America.
Laursen, S. L., & Rasmussen, C. (2019). I on the prize: Inquiry approaches in undergraduate mathematics. International Journal of Research in Undergraduate Mathematics Education, 5(1), 129–146.
Litke, E. (2020). Instructional practice in algebra: Building from existing practices to inform an incremental improvement approach. Teaching and Teacher Education, 91, 103030.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64. https://doi.org/10.3102/00346543060001001
Lozano, D. (1998). El Concepto de Variable: Evolución a lo largo de la Instrucción Matemática. B.Sc. thesis, Instituto Tecnologico Autonomo de Mexico.
Mkhatshwa, T. P. (2020). Calculus students’ quantitative reasoning in the context of solving related rates of change problems. Mathematical Thinking and Learning, 22(2), 139–161.
Monk, S. (1992). Students’ understanding of a function given by a physical model. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA notes (Vol. 25, pp. 175–193). Mathematical Association of America.
Moore, K. C. (2010). The role of quantitative reasoning in precalculus students learning central concepts of trigonometry (Order no. 3425753). Available from Dissertations & Theses @Arizona State University; ProQuest Dissertations & Theses Global (757614251).
Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. The Journal of Mathematical Behavior, 31(1), 48–59. https://doi.org/10.1016/j.jmathb.2011.09.001
Moore, K. C., & Thompson, P. W. (2015). Shape thinking and students’ graphing activity. In T. Fukawa-Connelly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the 18th meeting of the MAA special interest group on research in undergraduate mathematics education (pp. 782–789). RUME.
Musgrave, S., & Carlson, M. P. (2017). Understanding and advancing graduate teaching assistants’ mathematical knowledge for teaching. The Journal of Mathematical Behavior, 45, 137–149. https://doi.org/10.1016/j.jmathb.2016.12.011
National Governors Association Center for Best Practices, Council of Chief State School Officers (2010). Common core state standards for mathematics. National Governors Association Center for Best Practices, Council of Chief State School Officers.
Oehrtman, M. (2009). Collapsing dimensions physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education JRME, 40(4), 396–426.
O’Bryan, A. E. (2018). Conceptual analysis in cognitive research: Purpose, uses, and the need for clarity. In Proceedings of the 21st annual conference on research in undergraduate mathematics education.
O’Bryan, A. (2019). Exponential growth and related ideas: Examining students’ meanings and learning in an online Precalculus course.
O’Bryan, A. E. (2020a). Quantitative reasoning and symbolization activity: Do individuals expect calculations and expressions to have quantitative significance? In Proceedings of the 23rd annual conference on research in undergraduate mathematics education.
O’Bryan, A. E. (2020b). You can’t use what you don’t see: Quantitative reasoning in applied contexts. OnCore: Journal of the Arizona Association of Instructors of Mathematics, 66–74.
O’Bryan, A. E., & Carlson, M. P. (2016). Fostering instructor change through increased noticing: Creating authentic opportunities for instructors to reflect on student thinking. In T. Fukawa-Connelly, N. Infante, M. Wawro, & S. Brown (Eds.), Proceedings of the 19th annual conference on research in undergraduate mathematics education (pp. 1192–1200).
Piaget, J. (1955). The language and thought of the child. Meridian Books.
Piaget, J. (1968). Quantification, conservation, and nativism. Science, 162(3857), 976–979. https://doi.org/10.1126/science.162.3857.976
Polya, G. (1957). How to solve it; a new aspect of mathematical method (2nd ed.). Doubleday.
Rocha, A., & Carlson, M. (2020). The role of mathematical meanings for teaching and decentering actions in productive student-instructor interactions. In Proceedings of the twenty-third annual special interest group of the mathematical association of America conference on research in undergraduate mathematics education.
Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.), Proceedings of the annual meeting of the psychology of mathematics education. North Carolina State University.
Schmidt, W. H., Wang, H. C., & McKnight, C. C. (2005). Curriculum coherence: An examination of US mathematics and science content standards from an international perspective. Journal of Curriculum Studies, 37(5), 525–559.
Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (p. 41Y52). National Council of Instructors of Mathematics (NCTM) and LEA.
Sierpinska, A. (1992). On understanding the notion of function. In G. Harel & E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy, MAA notes (Vol. 25, pp. 59–84). Mathematical Association of America.
Silverman, J., & Thompson, P. W. (2008). Toward a framework for the development of mathematical knowledge for teaching. Journal of Mathematics Instructor Education.
Simon, M. A., Tzur, R., Heinz, K., Kinzel, M., & Smith, M. S. (2000). Characterizing a perspective underlying the practice of mathematics teachers in transition. Journal for Research in Mathematics Education, 31(5), 579–601.
Simon, M., & Placa, N. (2012). Reasoning about intensive quantities in whole-number multiplication? A possible basis for ratio understanding. For the Leaning of Mathematics, 32, 35–41.
Smith, J., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. Carraher, & M. Blanton (Eds.), Algebra in the early grades (pp. 95–132). Erlbaum.
Steffe, L. P., & Thompson, P. W. (2000). Teaching experiment methodology: Underlying principles and essential elements. In R. Lesh & A. E. Kelly (Eds.), Research design in mathematics and science education. Kluwer.
Stigler, J. W., Gonzales, P., Kwanaka, T., Knoll, S., & Serrano, A. (1999). The TIMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States. A research and development report.
Stigler, J. W., & Hiebert, J. (2009). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. Simon and Schuster.
Tallman, M. A., & Frank, K. M. (2018). Angle measure, quantitative reasoning, and instructional coherence: An examination of the role of mathematical ways of thinking as a component of instructors’ knowledge base. Journal of Mathematics Instructor Education. https://doi.org/10.1007/s10857-018-9409-3
Thompson, P. W. (1988). Quantitative concepts as a foundation for algebra. In M. Behr (Ed.), Proceedings of the annual meeting of the North American chapter of the international group for the psychology of mathematics education (Vol. 1, pp. 163–170). Dekalb.
Thompson, P. W. (1990). A theoretical model of quantity-based reasoning in arithmetic and algebraic. Center for Research in Mathematics & Science Education, San Diego State University.
Thompson, P. W. (1992). Notations, conventions, and constraints: Contributions to effective uses of concrete materials in elementary mathematics. Journal for Research in Mathematics Education, 23(2), 123–147.
Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165–208.
Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179–234). Albany.
Thompson, P. W. (1996). Imagery and the development of mathematical reasoning. In L. P. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.), Theories of mathematical learning (pp. 267–283). Erlbaum.
Thompson, P. W. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing and modeling in mathematics education. Kluwer.
Thompson, P. W. (2008a). Conceptual analysis of mathematical ideas: Some spadework at the foundation of mathematics education. Plenary paper delivered at the 32nd annual meeting of the international group for the psychology of mathematics education. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano, & A. Sèpulveda (Eds.), Proceedings of the annual meeting of the international group for the psychology of mathematics education (Vol. 1, pp. 45–64). PME.
Thompson, P. W. (2008b). One approach to a coherent K-12 mathematics. Or, it takes 12 years to learn calculus. Paper presented at the Pathways to Algebra Conference, June 22–25, Mayenne, France.
Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education WISDOMe monographs (Vol. 1, pp. 33–57). University of Wyoming Press.
Thompson, P. W. (2012). Advances in research on quantitative reasoning. In R. Mayes, R. Bonillia, L. L. Hatfield, & S. Belbase (Eds.), Quantitative reasoning: Current state of understanding WISDOMe Monographs (Vol. 2, pp. 143–148). University of Wyoming Press.
Thompson, P. (2013). In the absence of meaning. In Vital directions for mathematics education research (pp. 57–93). https://doi.org/10.1007/978-1-4614-6977-3_4
Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. D. English & D. Kirshner (Eds.), Handbook of international research in mathematics education (pp. 435–461). Taylor & Francis.
Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421–456). National Council of Instructors of Mathematics.
Thompson, P. W, Carlson, M. P., Byerley, C., & Hatfield, N. (2014). Schemes for thinking with magnitudes: A hypothesis about foundational reasoning abilities in algebra. In K. C. Moore, L. P. Steffe, & L. L. Hatfield (Eds.), Epistemic algebra students: Emerging models of students’ algebraic knowing. WISDOMe monographs (Vol. 4, pp. 1–24). University of Wyoming.
Thompson, P. W., & Harel, G. (2021). Ideas foundational to calculus learning and their links to students’ difficulties. ZDM–Mathematics Education, 1–13.
Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byereley, C. (2017). Covariational reasoning among U.S. and South Korean secondary mathematics instructors. Journal of Mathematical Behavior, 48, 95–111.
Teuscher, D., Moore, K. C., & Carlson, M. P. (2016). Decentering: A construct to analyze and explain instructor actions as they relate to student thinking. Journal of Mathematics Instructor Education, 19(5), 433–456. https://doi.org/10.1007/s10857-015-9304-0
Underwood, K., & Carlson, M. P. (2012). Understanding how precalculus instructors develop mathematic knowledge for teaching the idea of rate of change. In Proceedings of the 15th annual conference on research in undergraduate mathematics education.
Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20, 356.
Winsløw, C. (2021). Abell, Braddy, Ensley, Ludwig, Soto: MAA instructional practices guide. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-021-00141-0
Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458–477.
Yoon, H., Byerley, C., & Thompson, P. W. (2015). Instructors’ meanings for average rate of change in U.S.A. and Korea. In T. Fukawa-Connelly, N. E. Infante, K. Keene, & M. Zandieh (Eds.), Proceedings of the 18th meeting of the MAA special interest group on research in undergraduate mathematics education (pp. 335–348). RUME.
Yoon, H., & Thompson, P. W. (2020). Secondary instructors’ meanings for function notation in the United States and South Korea. The Journal of Mathematical Behavior, 60, 100804. https://doi.org/10.1016/j.jmathb.2020.100804
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 (Vol. 8, pp. 103–127). American Mathematical Society.
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Carlson, M.P., O’Bryan, A., Rocha, A. (2022). Instructional Conventions for Conceptualizing, Graphing and Symbolizing Quantitative Relationships. In: Karagöz Akar, G., Zembat, İ.Ö., Arslan, S., Thompson, P.W. (eds) Quantitative Reasoning in Mathematics and Science Education. Mathematics Education in the Digital Era, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_9
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