Limits in the Calculus I Pathway

Limits are often the first mathematical operation students encounter that cannot be conceived through finite computation, leaving them to negotiate spontaneous concepts of an actual infinity with infinite processes that do not end. Covariational reasoning about functional dependence is required to first conceptualize, then to coordinate, two such infinite processes quantitatively (e.g., a domain process in which $x\to a$ and a codomain process in which $f(x)\to L$ or a domain process in which $n\to\infty$ and a codomain process in which $a_n\to L$ ). Such dynamic reasoning about functions is especially important in calculus, as the argument of a limit becomes more quantitatively complex, such as a rate of change or an accumulation.

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

1. Operating in small neighborhoods then extending inferences beyond them.

2. Treating limits conceptually in terms of approximating and refining approximations to achieve a desired level of accuracy. These ideas can be initially developed in terms of approximating instantaneous rates, such as speed, and accumulation of quantities with continuously varying rates. Subsequently, they should generalize to other contexts with the same limiting structure.

3. Computer-based methods to experience and visualize the limit process.

4. Repeated refinement of approximations as an experience of the limiting process

Addressing Components of Inquiry

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

Active Learning: By actively computing several values of a difference quotient or Riemann sums for a derivative or definite integral, respectively, students can experience the limit process. Furthermore, by asking students to find such approximations to given degrees of accuracy, they must then reason in a way consistent with the formal ε-δ definition (just in a different language).

Meaningful Applications: Applications can play a particularly important role in students’ experiences about limits. Specifically, when derivatives and integrals are introduced by engaging in the same problem-solving process across multiple contexts, students may recognize the common structure across all of their activity as the mathematical concept. While distance-velocity-time examples make excellent first examples, it is too often the only example students see. Such students are then likely limited to a context-dependent understanding of derivatives and integrals. Other contexts such as area and volume, mass and density, pressure and force, etc. provide good opportunities to help students generalize their understanding.

Academic Success Skills: We often see students stagnate by the fact that they think they develop enough procedural fluency to solve computational problems, yet do not understand the meanings of the limit values they are finding. Communicate to students that they need a richer understanding in a way that doesn’t discourage them. Help motivate digging in to the deeper meanings of calculus rather than just the procedural “shortcuts.” Encourage them to value understanding why they are working with the particular derivatives or integrals that appear in a situation.

References

Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K. & Vidakovic, D. (1996). Understanding the limit concept: Beginning with a coordinated process schema, Journal of Mathematical Behavior, 15(2), 167-192.

Cornu, B. (1991). Limits. In D. Tall (Ed.). Advanced Mathematical Thinking, pp. 153-166. Boston: Kluwer.

Oehrtman, M. (2008). Layers of abstraction: Theory and design for the instruction of limit concepts. In Making the connection: Research and teaching in undergraduate mathematics education (Vol. 73, pp. 65-80). Mathematical Association of America Washington, D.C.

Oehrtman, M. (2009). Collapsing dimensions, physical limitation, and other student metaphors for limit concepts. Journal for Research in Mathematics Education, 40(4), 396-426.

Szydlik, J. (2000). Mathematical beliefs and conceptual understanding of the limit of a function. Journal for Research in Mathematics Education, 31, 258-276.

Tall, D. (1992). The Transition to Advanced Mathematical Thinking: Function, Limits, Infinity, and Proof. In D.A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning. MacMillan Publishing Company, New York, 495-511.

Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22, 219-236.