Quantitative Reasoning and Modeling in the College Algebra and Precalculus Pathway
The MIP aims to support modeling in College Algebra and Precalculus primarily through student activity to mathematically represent quantities and quantitative relationships and to manipulate or interpret these representations to draw inferences about the context. Modeling involves the ability to “decontextualize (to abstract a given situation, represent it symbolically, and manipulate the resultant symbols as if they have a life of their own) and to contextualize (to pause as needed during the manipulation process in order to probe into the referents for the symbols at hand)” (Common Core Practice Standards in the Common Core State Standards Initiative, 2010). Contextualizing also involves “interpreting the results in the context of a situation and reflecting on whether the results make sense, possibly improving the model if it has not served its purpose” (Common Core Practice Standards in the Common Core State Standards Initiative, 2010).
Faculty at the MIP College Algebra and Precalculus Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Quantitative Reasoning and Modeling for success in the Oklahoma Functions and Modeling Math 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 more mathematically by 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.
Addressing Components of Inquiry
Participants of the MIP Workshop on College Algebra and Precalculus suggested the following ways resources about Quantitative Reasoning and Modeling could address the three MIP components of mathematical inquiry:
Active Learning: Modules should engage students in developing, applying, and interpreting models at all stages. In doing so, they must transfer meaning both from context to mathematical representations and vice-versa.
Meaningful Applications: Although modeling essentially involves coordinating meanings between realworld 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.
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.