Bulletin of Science, Technology & Society, 1988

The Differential Calculus provides a precise, standardized language in which to learn about and discuss specific "changes" other than just in mathematics. Unfortunately, rather than to disseminate this language widely, the mathematical community has mostly considered the Differential Calculus as just a part of mathematics, if a fairly central one, and has taught it accordingly. We argue that a major rethinking of whom we want to teach it to and of the way we should then teach it is necessary and would have some far reaching consequences. One might want to learn the language of the Differential Calculus without necessarily wanting to learn its technique as do engineers and scientists or its theory as do mathematicians. To replace the usual Arithmetic and Algebra courses followed by the usual three semesters of Precalculus I-II and Calculus I, we propose, for students who passed or placed out of an integrated Arithmetic-Algebra I sequence with an A, a two four-hour semesters Integrated Precalculus I-II & Calculus I Sequence whose first semester is designed to impart Calculus Literacy and whose second semester is designed as bridge to the mainstream integral calculus. The characteristics of this sequence are that: i. it takes a Lagrangian viewpoint and ii. it proceeds through classes of functions of increasing complexity. This, together with a taskcard format gives realistic access to the Calculus to large numbers of students who never even dreamt of coming anywhere near it.

This syllabus is for a course I teach on Calculus to high school students. Its introduction is helpful for those who wish to better understand the significance of mathematics in a liberal arts education.

K. Lee. Lerner. "The Elaboration of the Calculus." (Preprint) Originally published in Schlager, N. Science and Its Times: Understanding the Social Significance of Scientific Discovery. Thomson Gale, 2001

Many of the most influential advances in mathematics during the 18th century involved the elaboration of the calculus, a branch of mathematical analysis which describes properties of functions (curves) associated with a limit process. Although the evolution of the techniques included in the calculus spanned the history of mathematics, calculus was formally developed during the last decades of the 17th century by English mathematician and physicist Sir Isaac Newton (1643-1727) and, independently, by German mathematician Gottfried Wilhelm von Leibniz (1646-1716). Although the logical underpinnings of calculus were hotly debated, the techniques of calculus were immediately applied to a variety of problems in physics, astronomy, and engineering. By the end of the 18th century, calculus had proved a powerful tool that allowed mathematicians and scientists to construct accurate mathematical models of physical phenomena ranging from orbital mechanics to particle dynamics. Although it is clear that Newton made his discoveries regarding calculus years before Leibniz, most historians of mathematics assert that Leibniz independently developed the techniques, symbolism, and nomenclature reflected in his preemptory publications of the calculus in 1684 and 1686. The controversy regarding credit for the origin of calculus quickly became more than a simple dispute between mathematicians. Supporters of Newton and Leibniz often arguing along bitter and blatantly nationalistic lines and the feud itself had a profound influence on the subsequent development of calculus and other branches of mathematical analysis in England and in Continental Europe.

Journal of Macrodynamic Analysis, 2002

I will discuss some of the difficulties that I have encountered in teaching Calculus. I will follow this, in Part I, with certain examples that my students have been finding helpful in reaching a preliminary notion of derivative. The focus in Part II is the genesis of the Fundamental Theorem of Calculus.

Journal für Mathematik-Didaktik, 2016

This paper discusses aspects and Grundvorstellungen in the development of concepts of derivative and integral, which are considered central to the teaching of calculus in senior high school. We will focus on perspectives that are relevant when these concepts are first introduced. In the context of a subject matter didactical debate, the ideas are separated into two classes: firstly, more mathematically motivated aspects such as the limit of difference quotients or local linearization within the concept of derivative, as well as the product sum, antiderivative, and measure aspects of integration; secondly, the Grundvorstellungen associated with the concepts of derivative and integral. We consider finding a comprehensive description of aspects and Grundvorstellungen to be an important objective of subject matter didactics. This description should clarify both the differences and the relationships between these perspectives, including both mathematically motivated aspects and Grundvorstellungen which are central to the students' perspective. The primary objectives of this paper include a specification of the concepts of aspects and Grundvorstellungen in the context of differentiation and integration and a discussion of the relationships between the aspects and Grundvorstellungen associated with the concepts of derivative and integral. We begin by presenting the characteristic properties of aspects and Grundvorstellungen, including an account of related concepts and the current state of research. These two concepts are then analyzed, based on a subject matter didactical analysis of the concepts of derivative and integral. We conclude with an account of how these insights can be beneficially exploited for introducing differentiation and integration in real-life environments, within the framework of a theory of concept understanding and subject matter didactics.

Proceedings of the Working Group on Mathematical Modelling and Applications at the 5th Conference on European Research in Mathematics Education (CERME-5). Nicosia, Cyprus: University of Cyprus, 2007

This paper reports on a longitudinal observation study characterising student's development in their understanding of derivatives. Through the Dutch context-based curriculum, students learn this concept in relation to applications. In our study, we assess student's understanding. We used a framework for data analysis, which focuses on representations and their connections as part of understanding derivatives, and it includes applications as well. We followed students from grade 10 to grade 12, and in these years we administered four task-based interviews. In this paper we report on the development of one 'average' student Otto. His growth consists of an increasing variety of relations, both between and within representations and also between a physical application and mathematical representations. We also find a continuity in his preferences for and avoidances of certain relations.

2009

I explain a direct approach to differentiation and integration. Instead of relying on the general notions of real numbers, limits and continuity, we treat functions as the primary objects of our theory, and view differentiation as division of f (x) − f (a) by x − a in a certain class of functions. When f is a polynomial, the division can be carried out explicitly. To see why a polynomial with a positive derivative is increasing (the monotonicity theorem), we use the estimate |f (x) − f (a) − f ′ (a)(x − a)| K(x − a) 2. By making it into a definition we arrive at the notion of uniform Lipschitz differentiability (ULD), and see that the derivative of a ULD function is Lipschitz. Taking different moduli of continuity instead of the absolute value, we get different flavors of calculus, each rather elementary, but all together covering the total range of uniformly differentiable functions. Using the class of functions continuous at a, we recapture the classical notion of pointwise differentiability. It turns out that uniform Lipschitz differentiability is equivalent to divisibility of f (x) − f (a) by x − a in the class of Lipschitz functions of two variables, x and a. The same is true for any subadditive modulus of continuity. In this bottom-up, computational, one modulus of continuity at a time approach to calculus, the monotonicity theorem takes the central stage and provides the aspects of the subject that are important for practical applications. The weighty ontological issues of compactness and completeness can be treated lightly or postponed, since they are hardly used this streamlined approach that pretty much follows the Vladimir Arnold's "principle of minimal generality, according to which every idea should first be understood in the simplest situation; only then can the method developed be applied to more complicated cases." I discuss a generalization to many variables briefly.

Educational Studies in Mathematics, 2012

In this article, we explore the responses of a group of undergraduate mathematics students to tasks that deal with areas, perimeters, volumes, and derivatives. The tasks challenge the conventional representations of formulas that students are used to from their schooling. Our analysis attends to the specific mathematical ideas and ways of reasoning raised by students, which supported or hindered their appreciation of an unconventional representation. We identify themes that emerged in these responses and analyze those via different theoretical lenses-the lens of transfer and the lens of aesthetics. We conclude with pedagogical recommendations to help learners appreciate the structure of mathematics and challenge the resilience of certain conventions.

SSRN Electronic Journal, 2021

This relatively short dialogue is intended for beginners in calculus who are not satisfied with incomplete stories or ``sound right'' overview on calculus, but look for comprehensive coverage of the foundation and ``logically right" understanding. All covered concepts are clearly defined without ambiguity and theorems are proved vigorously so the readers are trained to get used to the analytical technics that are typically used in the development of the foundation. Readers are expected to dependent on themselves for the rest of the journey in their study of this subject.