Modeling and Quantitative Reasoning in the Functions and Modeling Pathway

Modeling is the process of using mathematics to describe, analyze, and gain insight into real life phenomena. It entails identifying and representing quantities and determining relationships among relevant quantities. Modeling requires careful recognition of, and attention to, the relevant quantities involved in the situation and use of either (1) patterns of covariation and/or rate of change to determine a class of functions (e.g. linear, exponential) that best model a relationship, and/or (2) prior knowledge of the relationship (e.g. physical, geometric) between these quantities to devise a model (e.g. recognizing that the volume of a box is a function of its height). Careful attention to the quantities involved is the heart of what is called quantitative reasoning (e.g. Thompson, 2011), and it is an indispensable component of modeling. Three key questions lie at the heart of quantitative reasoning and are instrumental in guiding the integration of quantitative reasoning into the design of instructional tasks: (1) what object is being measured?, (2) what attribute of that object are we measuring?, and (3) what is the unit of measurement?

Faculty at the MIP Functions and Modeling Initiation Workshop in June, 2019 identified the following aspects of Modeling and Quantitative Reasoning as critical for success in the Oklahoma Functions and Modeling Math Pathway.

Quantitative Modeling: Activities should encourage students to develop clear (mental and physical) images of a problem scenario to identify relevant quantities and relationships among them (Moore & Carlson, 2010). We see such imagery as one piece of a larger effort to work with function relationships flexibly across the various function representations (Oehrtman, Carlson, & Thompson, 2008). For example, the change in a quantity can be represented in a diagram, as an expression in function notation (formula), the length of a line segment in a graph, an extra column in a table, and through a verbal description. Making connections between a clear and detailed mental image of the situation and other function representations supports the development of flexible quantitative understandings that are not specific to any single representation and increases the scope of situations to which a student can apply these quantitative understandings. Attending to quantities across representations also supports the development of quantitative habits of mind in which identifying and describing quantities in various forms becomes an essential way a student approaches any new problem. Such habits and skills will serve them well later in this course and in future mathematics courses.

Multiple Approaches to Modeling: Activities should prompt students to reason with models in various ways, including identifying relationships amongst quantities to construct their own model as well as analyzing a situation with a predetermined model. In addition to explicitly asking for measurements of specific quantities (e.g. what is the average rate of change on this interval?), which can promote overly procedural understandings if relied upon too frequently, tasks can also phrase quantities in everyday language (e.g. what is the average daily increase in your credit card balance during this time?) to emphasize that mathematics is a tool we can use to gain insight into real-world phenomena.

Quantitative Reasoning: Activities should require quantitative reasoning. Students are adept at procedural ‘shortcuts’ that substantially decrease the cognitive demand of a task and hence, its pedagogical effect. Nonroutine problems (e.g. Moore & Carlson, 2012) whose solutions require attending to the 3 questions at the heart of quantitative reasoning (above) encourage more meaningful attention to patterns of covariation and rate of change (e.g. Carlson et al., 2002), which are key relationships students can leverage to develop quantitative meanings for function concepts.

Technology: Activities should leverage technology for modeling, particularly involving regression. A graphing calculator can display multiple representations of a model and quickly compute regressions. The ability to efficiently generate regression equations and their corresponding graphs, scatterplots, and tables is particularly helpful in promoting the use of multiple representations. Such use of technology replaces the need for tedious calculations and procedures, which also affords additional opportunities to emphasize quantitative meanings for the various components of a regression (e.g. what do various components of a regression output measure, how are they being measured, and how do they manifest in various function representations?).

 

Addressing Components of Inquiry

 

Participants of the MIP Workshop on Functions and Modeling suggested the following ways modules about Modeling and Quantitative Reasoning could address the three MIP components of mathematical inquiry:

Active Learning: Activities 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 involves coordinating meanings between real world contexts and mathematical representations, not all modeling activity productively develops conceptual understanding. 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: Activities 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 agency in creating these meanings and reinforce their ability to learn through persistence.

 

References

 

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, 352-378.

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.

Oehrtman, M., Carlson, M., & Thompson, P. W. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. Making the connection: Research and teaching in undergraduate mathematics education, 27, 42.

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 Mongraphs (Vol. 1, pp. 33- 57). Laramie, WY: University of Wyoming.

Additional Resources

Carlson, M., Oehrtman, M., & Moore, K. (2010). Precalculus: Pathways to calculus: A problem solving approach. Rational Reasoning.

Crauder, B., Evans, B., & Noell, A. (2013). Functions and change: A modeling approach to college algebra. Cengage Publishing.