Coherence is a term often used by mathematics educators to describe desired learning outcomes. Numerous policy documents [e.g., 1-4] stress the necessity of creating coherent curricula and mathematical experiences for students. Early policy documents [e.g., 1] mainly focused on identifying logical progressions of topics, while more recent documents [e.g., 2-4] provide increased efforts in explicating meanings for specific mathematical topics and articulating mathematical processes that extend across topics. As an example, the Common Core State Standards Initiative (CCSSI) [4] built on the Process Standards [2], specifically addressing quantitative reasoning (QR) as a mathematical practice. In this symposium we explore the role of QR in creating coherent mathematical experiences for students, where we define coherence as how well the components of a system of meanings fit together.

Nearly two decades ago, Kaput [5] suggested that quantitative reasoning could support students in constructing a coherent system of mathematical meanings. Additionally, multiple studies [e.g., 6-8] have emphasized the important role that quantitative reasoning plays in students’ development of deeper ways of reasoning and thinking. With the recent developments in educational policy (e.g., CCSSI), it is of great importance to identify and articulate ways of reasoning that support student learning to researchers, curricula developers, and practitioners.

The proposed symposium consists of a brief introduction (4 minutes), three presentations that analyze QR in secondary mathematics (36 minutes), responses by two discussants (20 minutes), followed by audience participation (30 minutes). Together, the three presentations illustrate how a focus on QR can build coherence in secondary mathematics by providing examples of how students’ emerging QR can impact their understanding of mathematical concepts. A focus on QR has the potential to inform instructional interventions that target students’ ways of reasoning and help students to develop robust ways of understanding particular mathematical topics. The first discussant offers a broader perspective of QR in mathematics education that includes discussing QR in elementary and middle grades. The second discussant raises issues of QR in science education and how these issues connect to mathematics education. The following questions focus audience participation:

1. How might students’ lived experience outside of the classroom impact and create foundations for their QR?
2. How might students' different conceptions of ratio and variable impact their understandings of concepts central to calculus (e.g. function and rate of change)?
3. What kind of evidence would indicate a student conceives of quantities as intensive, extensive or covarying?
4. What imagistic and operational aspects could be involved in students’ reasoning about varying quantities?
5. How might the emergent nature of students' QR be applicable to reasoning about advanced mathematical concepts (e.g., derivative and uniform continuity)? 

Presentation 1 - Articulating two forms of covariational reasoning about quantities involved in rate of change: Expanding an existing framework

This presentation examines students’ quantitative and covariational reasoning related to the mathematical concept, rate of change. Based on a study of the reasoning of 6 secondary students who had not taken calculus, two forms of covariational reasoning about quantities involved in rate of change are proposed. The first form—comparison of numeric or nonnumeric amounts of change in covarying quantities in particular intervals—affords the construction of an extensive or intensive [9] quantity measuring that comparison. The second form—coordination of change in one quantity with change in a related quantity—affords the construction of an intensive quantity, a rate of change, measuring that coordination. These forms of reasoning provide finer-grained markers between the Quantitative Coordination and Average Rate levels of the Covariation Framework [10]. Students’ use of comparison-based or coordination-based forms of covariational reasoning about rate of change has implications for students’ developing understanding of average and instantaneous rate of change.

Presentation 2 – Students’ interpretations of financial models and thinking about change

Small differences in fundamental concepts can lead to large differences in more advanced mathematical understandings. This presentation focuses on a teaching experiment [11] with two high school Algebra II students. In this experiment, the students built financial models from small variations in the description of a bank’s savings account policy. The ultimate goal of this sequence of policies was to have the students develop an understanding of exponential growth from a differential equations perspective.

Using retrospective analysis, two different ways of thinking about change were identified: change in chunks that have been completed, and a smooth change in progress. From these two different ways of thinking about change, the four teaching experiment participants (students, researcher, and observer) developed different meanings of rate. Consequently all four participants had different interpretations of the bank account policies than those originally intended. Participants’ different meanings of change and rate led to different interpretations of the policies without the participants being aware of the extent to which they disagreed. These different meanings of rate could impact students’ understanding exponential growth in two contexts: compound interest, and the differential equation for the exponential.

Presentation 3 – The emergent nature of students’ quantitative reasoning

This presentation addresses common themes that emerged from a series of studies investigating precalculus students’ QR. Theories of quantitative and covariational reasoning [10, 12] informed the series of studies of student thinking in the context of their solving novel precalculus problems. Each study utilized clinical interview [13] or teaching experiment [11] methodologies. The subjects of the studies were gathered from two large public universities in the United States. Data analysis involved conceptual analysis to characterize, compare, and contrast the students’ thinking. 

Findings emphasize the emergent nature of students’ QR. Over the course of solving a problem, students engage in a process of refining and modifying their understanding of a problem’s context, where these modifications influence their ability to provide correct solutions. Students often initially provide incorrect solutions, but students correct their solutions by continuing to reflect on a problem’s context in ways that support their identifying quantities and relationships between quantities. The students’ actions illustrate the impact of their quantitative structures on their solutions (e.g., graphs and formulas) while implying that instruction must account for the emergent nature of students’ QR.  

References

1. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
2. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
3. National Mathematics Advisory Panel. (2008). Foundations for success: Final report of the National Mathematics Advisory Panel. Washington, D.C.: U.S. Department of Education.
4. Council of Chief State School Officers and National Governors Association. (2010, March). Common Core Standards for Mathematics. Retrieved on July 25, 2011 from www.corestandards.org.
5. Kaput, J. (1995). Long term algebra reform: Democratizing access to big ideas. In C. Lacampagne, W. Blair & J. Kaput (Eds.), The algebra initiative colloquium (Vol. 1, pp. 33-49). Washington, D.C.: U.S. Department of Education.
6. Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165-208.
7. 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. Albany, NY: SUNY Press.
8. Steffe, L., & Izák, A. (2002). Pre-service middle-school teachers' construction of linear equation concepts through quantitative reasoning. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant & K. Nooney (Eds.), Proceedings of the twenty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 1163-1172). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.
9. 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 (Vol. 2, pp. 41-52). Reston, VA: National Council of Teachers of Mathematics.
10. 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.
11. 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 (pp. 267-307). Hillside, NJ: Erlbaum.
12. Smith III, J., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher & M. L. Blanton (Eds.), Algebra in the Early Grades (pp. 95-132). New York, NY: Lawrence Erlbaum Associates.
13. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Handbook of Research Design in Mathematics and Science Education (pp. 547-589). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.