The Function Concept in the Functions and Modeling Pathway

Function is the foundational topic in Functions and Modeling. The function concept enables us to identify, analyze, and gain insight into relationships between real-world quantities that vary in tandem, and is a key prerequisite to learning subsequent ideas in this course. Accordingly, students in Functions and Modeling should develop productive understandings of function (both single- and multi-variable) that can be used flexibly amongst various real-world contexts and representations. This involves awareness and use of appropriate conventions like function notation as well as aspects of quantitative reasoning and covariational reasoning.

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

Multiple Representations: Activities should engage students in analyzing function relationships and concepts through multiple representations. Being able to work proficiently with each of the major function representations (e.g. formula, table, graph, words) also promotes the dynamic view that a function is much more than a way to relate specific inputs to specific outputs (i.e. instructions for how to ‘convert’ an input value to an output value) and reinforces the view that each representation is a different manifestation of the same relationship between quantities that are changing together (see Oehrtman, Carlson, and Thompson, 2008). Working flexibly across multiple functions representations is also valuable for understanding function concepts because each representation can highlight various aspects of the concept. For example, examining function composition in table and graph form might enable a student to imagine how changes in the input of one function correspond to changes in the output of the other (which students possessing only a formula-based understanding of composition would be unlikely to achieve).

Covariational Reasoning: Activities should emphasize the concept of function as a relationship between quantities and design tasks that encourage students to reason explicitly about how a function’s quantities are changing in relation to each other. Carlson et al.’s (2002) covariation framework provides details of the patterns of mental actions that support reasoning covariationally. A covariational emphasis promotes a dynamic view of function as a relationship between two  changing quantities (as opposed to a static, input-output correspondence view). This emphasis also entails aspects of quantitative reasoning (which includes carefully attending to the following questions for each quantity: what is being measured, what is the measurement unit, and what does the value of the measurement?). Reasoning in this way is key for understanding the relationship between the original quantities (e.g. Moore and Carlson, 2010) and also foundational for understanding key ideas like constant and average rate of change (e.g. Thompson, 2008).

Function Notation: Activities should have students represent the various quantities associated with a function using function notation. Note that this includes not only a proficiency with basic conventions of expressing input-output pairs in function notation, but also extends to expressions of other related quantities like change and rates of change in function notation. This representational activity can be productive because it emphasizes the common structure held by all quantities of the same type (e.g. that changes in the output quantity are all of the form f(b)-f(a)) and provides students with an opportunity to develop meaningful understandings of what might otherwise be rote formulas. Participants of the Initiation Workshop stressed that students should come to see function notation as an efficient and useful tool that does work for us; that is, the CoRD should design activities that enable students to see function notation as necessary for expressing mathematical ideas.

Technology: Activities should leverage technology as a tool to advance students’ understanding of function and related function concepts. Technology should be used to enable students to better focus on ideas and concepts, instead of only procedures and algebraic manipulation. For example, a graphing calculator (or any graphing technology) makes it easier to shift between function representations because, having entered an equation, one can view a graph or a table without getting bogged down in procedures carried out by hand (promoting the recommendation regarding the benefits of viewing functions and related concepts in multiple representations).

Addressing Components of Inquiry

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

Active Learning: Supporting students’ quantitative reasoning with functions promotes insight into relationships between quantities (for example, a quantitative understanding for ‘increasing’ might involve the observation that the changes in output along the interval in question are all positive). Such meanings for function concepts provide rich opportunities for the MIP characterization of active learning (which includes students’ selecting, performing, and evaluating actions equivalent to the concept to be learned). Tasks can pose problems about the behavior of a function’s quantities in which the resolution requires attention to the desired quantitative understanding. In this way, the students have opportunities to intuitively develop function concepts as they devise their own solutions to nonroutine problems (for example, concavity can emerge in students’ reasoning as they use trends they notice in the average rate of change to make predictions about the behavior of quantities).

Meaningful Applications: Though examples of functions abound in everyday life, function is often seen by students as existing only with the confines of a mathematics class. Part of the philosophy behind Functions and Modeling is that all problems are based in real-world experiences. There are many  examples of functions that students are exposed to in classes, but unless the function concept does real work in students’ reasoning, they are likely to continue to confine notions of function to the classroom. Participants in the Initiation The MIP characterization of meaningful applications states that an application problem is meaningful only to the extent that it supports students in identifying mathematical relationships, justifying their reasoning, and generalizing key concepts across various contexts. Through careful instructional design, real-world applications that leverage students’ real-world knowledge can become key tools for students’ reasoning. For example, students can employ an analysis of a profit graph to reason about how many items yields maximal profits, break-even points, and so on.

Academic Success Skills: As function is such an integral idea upon which many future ideas depend, developing a robust, quantitative understanding of function can go a long way towards fostering students’ willingness to persevere in problem solving and their identities as capable of doing mathematics. When improperly motivated, introduction of functions can seem arbitrary and unnecessarily complicated, raising a barrier to many students. Modules should help students become confident in their use of functions as a foundation of the language of mathematics and science.

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. (2008). Conceptual analysis of mathematical ideas: Some spadework at the foundations of mathematics education. In Proceedings of the annual meeting of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 31-49). PME Morelia, Mexico.

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.

Musgrave, S., & Thompson, P. W. (2014). Function Notation as Idiom. Proceedings of the 38th Meeting of the International Group for the Psychology of Mathematics Education, (Vol 4, pp. 281-288). Vancouver, BC: PME.

Thompson, P. W. (2013, October). “Why use f(x) when all we really mean is y?”. OnCore, The Online Journal of the AAMT.