Teaching the Fundamental Theorem of Calculus: A Historical Reflection - Concluding Remarks: The Relation between the History of Mathematics and Mathematics Education

It is generally believed that the history of mathematics is relevant to its teaching and learning in so far as it shows some of the workings of the human mind as it grapples in its quest to organize collections of problems and other mathematical data into coherent bodies of knowledge. While this statement is remarkable for what it says, it is perhaps more so for what it stops short of saying. Surely, history does suggest specific ways to approach the teaching of particular areas of mathematics. But in far too many instances, it is clear that we cannot limit mathematics teaching to the ideas nurtured in a particular epoch of its development, without sacrificing important ideas in the evolution of mathematical knowledge. A case in point is our reservations concerning the pedagogical wisdom of basing the theory of the elementary integral on Cauchy's idea of the limit of Riemann sums. The historical evidence available shows (and in no ambivalent terms) that during the second half of the seventeenth century, when the elementary integral was actually invented, there was no notion resembling limits of Riemann sums lurking in the air. In fact, the notion of “area" was based on the concepts of indivisibles and infinitesimals. As vague and controversial as such conceptions might have turned out to be in the perspective of history, a close analysis of the mathematical ideas of the second half of the seventeenth century convincingly shows that, apart from the notions just mentioned, the core idea of the elementary integral is based on the leitmotif of areas being generated by moving abscissas on curves (area functions). It is truly amazing the extent to which the properties of the integral are dependent on just this simple idea. An added virtue of our teaching proposal is that it leads naturally to more sophisticated ideas like those contained in Cauchy's limit-sum definition of the integral.

An ever increasing amount of evidence in the mathematics education literature documents what every mathematics professor knows: the majority of students, even high performance students, have serious difficulties with the epsilon-delta definition of limit (and also with the so-called sequential definition of the limit) [42, p. 4; 39]. In [11] evidence is presented that documents specific challenges students have with limits, and in particular with the limit involved in Cauchy's definition of the integral. Similar problems are described and data presented in [1, p. 196], [10, p. 153], and [43, p. 11]. Other interesting articles related specifically to the teaching of calculus include [12], [13], [30], [36], [40], and [45]. Mathematics teaching seems to be full of examples of the fact that mixing mathematical “notations” and “meanings” from different mathematical eras can cause difficulties for the student who is trying to understand the corresponding mathematics. This last statement should not necessarily be construed as an argument to avoid the mixing of ideas from different historical times for the sake of pedagogy, but rather as a warning that, in doing so, troubles may ensue. The case at hand, related to the use of Riemann sums in elementary integration, is one such example.

Another example is the teaching of the chain rule; see [25] and [19]. What is the chain rule? The standard answer: it is an algorithm to find the derivative of the composition of two differentiable functions. It is curious that in L'Hospital's Analyse des infiniment petits [34] or in any of Euler's famous calculus books [14, 15, 16], there is no “proof” or justifying argument for the chain rule. Of course, in L'Hospital’s time there was no notion of function, nor any idea that functions could be composed (see note 8.1). It is interesting that the notion of composition is what seems to be the main source of difficulty for student understanding of the chain rule (see note 8.2). In L'Hospital's and Euler's calculus, the use of infinitesimals turns the chain rule into a virtually trivial statement having to do with substitutions rather than compositions.

Indeed, after Abraham Robinson's vindication of Leibniz, we can successfully write most of Euler's beautiful arguments in correct mathematical language, thus giving back to calculus its dynamic character. Calculus was originally developed as a study of change and motion (see note 8.3) in a numerical setting that included infinitesimals and infinite numbers. With the construction of the standard real number system, the theory of limits was invented, and this presupposed a change of emphasis biased towards logic and quantification, and the handling of inequalities. Whether this is adequate language for the pedagogy of the calculus is an interesting topic that, unfortunately, has not received sufficient attention in the literature. Surely there must be a compromise between the extremes of teaching mathematics using the ideas of previous times as if nothing important happened afterwards, and doing so by using the cold implicit esthetics that shifts around theorems, corollaries, and definitions in order to maximize logical coherence and minimize redundancy, and without regard to the historical development of ideas. Whatever the solution to this intriguing dilemma, mathematics departments must begin to think about such issues if they are to improve on the teaching of the calculus, an area that leaves much to be desired.

Notes for page 8:

8.1. The first reference to the chain rule as a rule for differentiation of a composition of two functions occurred in [27, Section 31, p. 29] almost a century after the chain rule was first used; it was also recorded in [8, p. 25].

8.2. This information was gathered in several non-standard calculus courses given at the University of Puerto Rico by one of the authors.

8.3. This was certainly true from Barrow all the way to Leibniz (including Newton and Gregory). For instance, Barrow's Lectures II and III were devoted to the generation of quantities by “local motions” [9, p. 42]. Leibniz, in his proof of the FTC [41, p. 282], said, “Then we shall show how this line can be described by a motion that I have invented.”