Functions in the Calculus I Pathway

Functions serve as the basic language and notation for students’ experience in Calculus I. A robust understanding of functions is therefore critical for students’ success in the course. Many difficulties students experience while reasoning with functions are based in a static “action view” of evaluating a function for one input at a time, typically based on an algebraic formula. In contrast, a “process view” of function in which a student can conceive of the entire process happening to all input values at once, enables them 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). This way of thinking about the covariation of input and output values is foundational for constructing meaningful formulas and graphs when modeling relationships in applied contexts, interpreting limits conceptually or formally, and thus reasoning about all concepts defined in terms of limits (Carlson et al., 2002; Moore & Carlson, 2012; Oehrtman, Carlson, & Thompson, 2008).

Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Functions for success in the Oklahoma Calculus I Pathway should:

1. Connect/distinguish continuity at a point with the limit of a function at that point.

2. Understanding the relationship between differentiability and continuity, namely that differentiability implies continuity.

3. Relate points on non-differentiability (e.g., “corners,” “cusps,” “vertical tangents”) to particular discontinuities of the derivative function.

4. Appreciate the role of continuity as a requirement in a variety of important theorems (e.g., intermediate value theorem, mean value theorem, extreme value theorem, integrability, the fundamental theorem of calculus).

5. Understand the relationship between one-sided limits, two-sided limits, the value of a function at a point, and continuity.

6. Interpret various types of discontinuity (e.g., “jump,” “removable,” “oscillating,” “infinite”) in terms of limit expressions.

7. Connect the limit definition of continuity to a function’s graph and to intuitive notions of continuity (i.e., “you can sketch the graph without lifting your pencil”).

8. Understand of continuity as an extension of limits.

9. Understanding the role of continuity in differentiability conditions.

Addressing Components of Inquiry

Participants of the MIP Workshop on Calculus I suggested the following ways resources about Functions could address the three MIP components of mathematical inquiry:

Meaningful Applications: An instructor might engage students in active learning to support their understanding of continuity and its implications by presenting tasks that require students to find the value of a parameter to make a piecewise defined function continuous where at least one of the expressions of the function involves computing a limit using a method other than direct evaluation. Such tasks require students to apply their understanding that the limit of one expression must be equal to the value of the other at a particular input. Other tasks that might support students’ active learning include asking them to sketch a graph of piecewise function given information about its one-sided limits and value at a point. Crucially, these tasks require students to coordinate the limiting value of a function with its value to assess whether the function is continuous at a particular point.

Active Learning: Students’ understanding of continuity can be reinforced and extended by engaging in applied tasks that require them to use continuity to compute the limit of a function and to interpret computational strategies for limit evaluation as an instance of leveraging the definition of continuity to evaluate a limit. Additionally, applied contexts can support students’ recognition of the importance of continuity as a hypothesis in the intermediate value theorem, mean value theorem, and extreme value theorem by exploring counterexamples.

Academic Success Skills: Supporting students’ understanding of why the conclusions of important theorems in calculus depend on a function being continuous on a closed interval enables students to interpret these conclusions as intuitive implications of continuity, not as a list of facts to be memorized and applied to solve routine exercises. Interpreting the conclusions of these theorems as intuitive implications challenges the common assumption that mathematical proficiency is based principally on one’s ability to efficiently recall declarative knowledge—a perspective that increases students’ uncertainty as to whether they can successfully participate in mathematics. Constructing meaning for continuity and its implications allows students to recognize that a small number of essential ways of reasoning are sufficient for engaging productively in a variety of tasks, and if students have experienced their capacity to engage in these ways of reasoning in several mathematical contexts, then they are more likely to appraise task demands as manageable, thus encouraging perseverance in problem solving and a growth mindset about mathematical ability.

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, 33(5), 352–378.

Jayakody, G., & Zazkis, R. (2015). Continuous problem of function continuity. For the Learning of Mathematics, 35(1), 8-14.

Maharajh, N., Brijlall, D., & Govender, N. (2008). Preservice mathematics students’ notions of the concept definition of continuity in calculus through collaborative instructional design worksheets. African Journal of Research in Mathematics, Science and Technology Education, 12(1), 93-106.

Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. Journal of Mathematical Behavior, 31, 48–59.

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.

Patenaude, R. E. (2013). The use of applets for developing understanding in mathematics: A case study using Maplets for Calculus with continuity concepts (Doctoral dissertation, University of South Carolina).

Tall, D., & Katz, M. (2014). A cognitive analysis of Cauchy’s conceptions of function, continuity, limit and infinitesimal, with implications for teaching the calculus. Educational Studies in Mathematics, 86(1), 97-124.