Local Linearity, Differentials, Infinity, and Infinitesimals in the Calculus I Pathway
Researchers (e.g., Thompson et al, 2015) have demonstrated that many calculus students’ understanding of derivatives is not sufficiently grounded in robust meanings of rate of change. When prompted to explain what the derivative at a point represents, students often reply, “The slope of the tangent line.” Limited to such geometric interpretations, students struggle to apply derivatives in novel contexts and to understand more advanced topics in calculus, such as linear approximation, L’Hopital’s rule, implicit differentiation, related rates, Riemann sums, definite integrals, and the fundamental theorem of calculus. To address this problem, several mathematics educators (e.g., Ely, 2021) have documented the affordances of students’ conceptualizing differentials as linear functions (i.e., as infinitesimal changes that vary proportionally). Related recommendations include supporting students’ interpretation of “instantaneous rate of change” as “average rate of change over infinitesimally small intervals where the corresponding changes in the measures of the input and output quantities are proportional.”
Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Local linearity, differentials, infinity, and infinitesimals for success in the Oklahoma Calculus I Pathway should:
1. Engage students with applied contexts (other than distance-velocity-time and area) that require them to reason quantitatively about derivatives and integrals.
2. Use derivatives for linear approximations in instances where an instantaneous rate is known but a derivative function is unknown (as in applying Euler’s method or Newton’s method).
3. Conceptualizing dx not just as a trivial part of integral notation, or as just an indicator of the independent variable for antidifferentiation, but as an infinitesimal change that when multiplied by the quantity represented by the expression in the integrand yields an incremental accumulation of some quantity.
4. Understand dy/dx not just as a fraction but as the constant of proportionality that relates corresponding infinitesimal changes of covarying quantities.
Addressing Components of Inquiry
Participants of the MIP Workshop on Calculus I suggested the following ways resources about Local linearity, differentials, infinity, and infinitesimals could address the three MIP components of mathematical inquiry:
Active Learning: Engage students in tasks that require them to leverage their understanding of the invariant multiplicative relationship between corresponding infinitesimal changes in the input and output quantities of a differentiable function to make inferences about the function, and to use information about the function to make inferences about its derivative at a point.
Meaningful Applications: Meaningful applications should support students’ abstraction of local linearity, or a proportional relationship between differentials, conceptualized as corresponding infinitesimal changes in the input and output quantities of a differentiable function. Meaningful applications should support the need for the local linearity. It is important to remain aware that just because an application is a practical application does not automatically make it meaningful. Some criteria for meaningful applications that might support students’ learning of the targeted concept include (1) contexts that require students to solve for changes (or nearby points) given a rate function they cannot simply antidifferentiate, (2) reference to quantities other than speed that are defined in terms of a variable other than elapsed time can support generalization and abstraction.
Academic Success Skills: Students are often overwhelmed by their perceived expectation to become proficient in applying a variety of skills and strategies required to solve different classes of disconnected problem types. Engaging students in experiences that enable them to recognize the broad applicability of conceptualizing differentials as linear functions, and the derivative as a linear map, reduces the cognitive load of memorizing an assortment of procedures for solving routine problems, and fosters students’ positive affect and productive mathematical engagement.
References
Bos, H. J. M. (1974). Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Archive for History of Exact Sciences, 14, 1–90.
Dray, T., & Manogue, C. (2003). Using differentials to bridge the vector calculus gap. College Mathematics Journal, 34, 283–290.
Dray, T., & Manogue, C. (2010). Putting differentials back into calculus. College Mathematics Journal, 41, 90–100.
Ely, R. (2010). Nonstandard student conceptions about infinitesimal and infinite numbers. Journal for Research in Mathematics Education, 41, 117–146.
Ely, R. (2017). Reasoning with definite integrals using infinitesimals. Journal of Mathematical Behavior, 48, 158–167.
Ely, R. (2021). Teaching calculus with infinitesimals and differentials? ZDM Mathematics Education, 53(3), 591-604.
Keisler, H. J. (2011). Elementary calculus: an infinitesimal approach (2nd ed.). New York: Dover Publications. (ISBN 978-0-486-48452-5).
Ransom, W. R. (1951). Bringing in differentials earlier. The American Mathematical Monthly, 58, 336–337.
Tall, D., (1980). Intuitive infinitesimals in the calculus. Abstracts of short communications, Fourth International Congress on Mathematical Education, Berkeley, p. C5.
Tall, D. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48, 199–238.
Tall, D. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM, 41(4), 481–492.
Thompson, P. W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics, 26(2–3), 229–274.
Thompson, P. W., & Ashbrook, M. (2019). Calculus: Newton, Leibniz, and Robinson meet technology.