Functions and their Fundamental Characteristics in the College Algebra and Precalculus Pathway
Many student difficulties in reasoning with functions are based in a static conception tied to evaluating a function one step at a time, typically tied to the formula. This is often called an “action view,” and renders reasoning dynamically or about multiple values at a time nearly impossible. In contrast, a “process view” of function in which a student can conceive of the entire process happening to all input values at once, and is thus able to conceptually run through a continuum of input values while attending to the resulting impact on output (e.g., see the discussion of action and process views in Oehrtman, Carlson, & Thompson, 2008).
Faculty at the MIP College Algebra and Precalculus Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Functions and their Fundamental Characteristics for success in the Oklahoma Functions and Modeling Math Pathway should:
1. Ask students to coordinate multiple function processes (e.g., through composition, addition, or in defining an increasing function).
2. Ask students about the behavior of functions on entire intervals in addition to single points (e.g., describing a function’s behavior as input values increase continuously through the domain or finding the image of an interval).
3. Ask students to reverse a function processes (e.g., finding the preimage of a specified output value or interval).
4. Ask students to make and compare judgments about functions across multiple representations.
5. Help students understand, explore, apply, and contrast the fundamental features of various classes of functions including linear, exponential, quadratic, polynomial, logarithmic, rational, trigonometric, and radical functions. Important characteristics to emphasize include, but are not limited to, domain, range, zeroes, extrema, limiting behavior, increasing, decreasing, periodic behavior.
Addressing Components of Inquiry
Participants of the MIP Workshop on College Algebra and Precalculus suggested the following ways resources about Functions and their Fundamental Characteristics could address the three MIP components of mathematical inquiry:
Active Learning: Students should be engaged in tasks that go beyond treating functions as equations and provides opportunities for them to create functions appropriate to solve novel problems and invoke function notation in ways responsive to that problem-solving activity.
Meaningful Applications: Activities could emphasize modeling and interpretation to reinforce functions as a tool to describe the world. The coordination of two quantities and univalence built into the mathematical structure of functions can gain compelling meaning from natural relationships and constraints between quantities in real world situations. One may ask students to contrast the domain and range of functions based on the problem context with the domain and range derived from algebraic constraints alone. The concept of functions and function notation can be motivated and reinforced by engaging students in reasoning with and expressing quantities determined through correspondence, such as, Δh = h(V+ΔV) – h(V). Students should also identify and interpret key parameters in each function class in terms of the context in which it is being applied and in its various mathematical representations.
Academic Success Skills: 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
Oehrtman, M., Carlson, M., & Thompson, P. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the Connection: Research and Practice in Undergraduate Mathematics, MAA Notes, Volume 73, 27-41. Washington, DC: Mathematical Association of America.