Modeling in the Calculus I Pathway
Creating and interpreting mathematical models is a critical path for students to both better understand the underlying mathematics and to be prepared to apply that mathematics in other disciplines. In calculus, students have an opportunity to reason about new types of quantities and quantitative relationships (such as instantaneous rates) and to distinguish them from previous non-calculus quantities (such as constant or average rates). Students may represent quantities and quantitative relationships as ways to mathematize a context or, conversely, to give contextual meaning to mathematical symbols. They may strategically manipulate or interpret these representations to draw inferences about a context or use the context to construct conjectures or arguments about the mathematics.
Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Modeling for success in the Oklahoma Calculus I Pathway should:
1. Help students conceive and describe real-world quantities through appropriate mathematical representations. Contexts should be chosen to make the mathematics amenable to students’ intuitive reasoning that can subsequently be represented by mathematical variables, expressions, diagrams, and graphs.
2. Help students conceive and describe relationships between quantities through appropriate mathematical representations. Again, contexts should be chosen to enable students to more intuitively state, justify, or question relationships between quantities, before expressing them through mathematical representations.
3. Help students generalize context-specific reasoning by exploring the same underlying mathematical structure in multiple contexts, then reflecting on the similarities and differences across the resulting models (e.g., see the description of a learning trajectory across calculus leveraging abstraction across multiple contexts in Oehrtman, 2008).
4. Help students abstract mathematical structure by applying concepts developed earlier tasks as tools for making sense of new situations in later tasks (e.g., see the description of levels of emergent models in Gravemeijer, Cobb, Bowers, & Whitenack, 2000).
5. Develop working with quantities as a central habit of mind for students. This includes approaching any modeling situation with the initial aim to identify the relevant quantities for the given goal (e.g., see the discussion of extensive quantification in Thompson, 1994). Students should then distinguish between constant and variable quantities and identify relationships between these quantities determined by the situation. Many students will need help articulating these relationships initially using concrete numerical values for specific variable quantities, then seeing the algebra as a generalization of the multiple arithmetic expressions generated by choosing different values.
6. Help students draw effective diagrams of situations with the appropriate information and level of detail to support mathematical modeling.
7. Help students model changes in quantities and rates of change of one quantity with respect to another. This modeling should i) reinforce a concept of changes in quantities as meaningful quantities in their own right, ii) develop a quantitative conception of rate of change, and iii) help students identify rate of change features in contexts that correspond to particular function types to choose an appropriate algebraic form of a model (e.g., see examples of tasks involving modeling with changes and rates of change in Carlson, Oehrtman, & Moore, 2016).
8. Emphasize linear, exponential, and quadratic models that reinforce key quantitative concepts of constant rate of change, rate proportional to amount, and constant acceleration, respectively.
9. Provide opportunities for students to modeling situations with the full range of central constructs in calculus: linear relationships, limits, various types of rates, and accumulation.
Addressing Components of Inquiry
Participants of the MIP Workshop on Calculus I suggested the following ways resources about Modeling could address the three MIP components of mathematical inquiry:
Active Learning: Modules should engage students as the primary actors in creating and interpreting mathematical models. This engagement may focus on certain parts of a broader problem-solving process, but throughout the collection of CoRD resources should extend through developing, applying, and interpreting models at all stages. In doing so, they should support students in representing the products of their modeling activity using increasingly appropriate mathematical representations (terminology, symbols, expressions, graphs, procedures, etc.) and in interpreting the results of their mathematical computations in terms of the context.
Meaningful Applications: Although modeling essentially involves coordinating meanings between real-world contexts and mathematical objects and relationships, not all modeling activity productively develops conceptual understanding. In particular modules should focus students on identifying common structure across multiple modeling activities with different contexts as the source of abstracting the particular mathematical concept(s) common to them all.
Academic Success Skills: Modules should help students develop a view that mathematics is meaningful, both as a set of tools to model real-world situations, but also in the abstract, as generalizations of structures present across a wide variety of contexts. Students’ engagement in this process should develop their own agency in creating these meanings and reinforce their ability to learn through persistence.
References
Council of Chief State School Officers & National Governors Association Center for Best Practices (2010). Common core state standards for mathematics. Common Core State Standards Initiative. http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf.
Carlson, M., Oehrtman, M., & Moore, K. (2016). Precalculus, Pathways to Calculus: A Problem Solving Approach, Sixth Edition. Phoenix, AZ: Rational Reasoning.
Gravemeijer, K., Cobb, P., Bowers, J., & Whitenack, J. (2000). Symbolizing, Modeling, and Instructional Design. In Paul Cobb, Erna Yackel, & Kay McClain (Eds.) Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design. Mahwah, NJ: Erlbaum and Associates. 225-273.
Jones, S. R. (2017). An exploratory study on student understandings of derivatives in real-world, non-kinematics contexts. The Journal of Mathematical Behavior, 45, 95-110.
Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. In M. P. Carlson & C. Rasmussen (Eds.), Making the Connection: Research and Teaching in Undergraduate Mathematics Education, (MAA Notes, Vol. 73, pp. 65-80). Washington, DC: Mathematical Association of America.
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, NY: SUNY Press.