Calculus: Seeing is Believing - Pedagogical Motivations

Reform of post-secondary mathematics education, particularly introductory calculus, is becoming commonplace across North America. Although there are many varieties of reform, most can be placed within the philosophical camp of social constructivism. According to this movement, mathematical knowledge is constructed in an interactive way through instructor-student and inter-student dialogue, rather than built in an axiomatic sense such as the "new math" of 20 years ago, or in the reductionistic, algorithmic sense dominant in secondary and introductory college mathematics. While I hold serious concerns about the relativizing of mathematical knowledge that occurs when social constructivism is adopted as a philosophy of mathematics, I believe that it sheds light on the educational process, and some of its tenets can be used to enhance pedagogy. Reform has the potential to allow us to teach mathematics with integrity, by presenting our subject in a way that reflects its historical, cultural, and cognitive nature.

The social constructivist view of knowledge is dealt with in some detail in another paper at this conference (by Ted Watanabe). Aspects of this philosophy that can enhance mathematics teaching include the following:

• A holistic, as opposed to reductionistic, view of knowledge. Traditionally-trained students are inclined to think of mathematics as the foundational basis of science: a timeless edifice to be mastered and from which to build other forms of knowledge. Social constructivism shifts the emphasis from mastery to interaction between students and disciplines, transforming mathematics from knowing to doing.
• Motivation from physical context, rather than theoretical construct. Mathematics often emerges from empirical circumstances, rather than from theoretical concerns. This is particularly true for the curriculum in introductory courses. The order of presentation is reversed from the traditional theoretical discussion, followed by one or more applications.
• Higher-level reasoning as opposed to skill mastery. The primary emphasis is on the ability to address a wide variety of problems, rather than those that fit simple templates. The advent of computer algebra systems will force this innovation upon us, to some extent. Nevertheless skill mastery must remain a concern, although secondary.
• Quasi-empirical exploration leading toward abstraction, rather than axiomatic knowledge construction. This is one aspect of the Archimedean distinction between "analysis" and "synthesis". The former refers to the unpublished, less structured patterns of thought that produce the original path to the solution; the latter is the polished and published, yet often unmotivated formal proof. Constructivists propose to bring the analysis to light, de-emphasizing synthesis.

These goals are common to many reform projects. I propose to add another, in keeping with the aims of social constructivism:

• A continuous honing of native language skills, leading toward mathematical language. Mathematical thought remains an alien, separated skill in the public perception, partially because the synthesis discussed above does not often truly depict the thought process leading to a mathematical conclusion. The student’s inherent mathematical ability must be brought forth from his/her existing reasoning and communication skills, not built from scratch in a new thought category.

The realization of these goals occurs at The King’s University College in a number of ways, including (i) the lecture period (which becomes an interactive problem-solving time, leading to the development of theory), (ii) the assigned homework (emphasizing grasp of concepts already taught and skill acquisition), and (iii) the laboratories (which model mathematical behaviour in a variety of settings). We believe the innovative nature of our program manifests itself most strongly in the laboratories. One of our labs, on L’Hôpital’s Rule, is provided at the end of this paper as a sample of our efforts.

Many of the weekly laboratory periods begin with the presentation of some empirical setting. A qualitative exploration of the relations involved, using pencil/paper and the computer, leads to an anomaly, problem, or need for an idea. The concept is developed primarily in a visual manner at first (since the main ideas of calculus are essentially geometric), with some symbolic assistance as necessary. The solution of the empirical problem is the climax of the laboratory. If appropriate, the solution is extended to other contexts.

Presented in this way, the students are led to expect a dynamic interplay to occur between mathematical idea and context: each informs the other. The motivation for the theory is easily grasped from the context, and the subsequent power of the theory leads to a recognition of the need for (and respect for) "pure" mathematical reasoning. The unity between exploration and theory is restored.

While empirical settings are a primary motivation for the study of calculus, they should not be the only path. A number of our laboratories emphasize other sources of mathematical inspiration. These include the aesthetic feel for a beautiful theorem or image, such as Fermat’s integration of polynomial curves (Lab 7) or the ubiquitous lab on fractals (Lab 11). Some relate historical episodes, revealing that calculus grew through the interplay of personalities and cultural ideas. This variety helps to illustrate the diversity of styles, approaches and philosophies that exist within mathematics.

Perhaps the most unusual feature of the laboratories is their conversational and visual approach. Difficulties with mathematical communication are insurmountable to many students. We attempt to construct mathematics from these students’ native skills primarily through a careful use of modes of communication. This includes:

• visualization: The ideas of calculus are primarily visual; thus pictures are the most natural method of conveying them. Furthermore, visualization plays a greater role in the learning process for students of the video age than has been the case in the past.
• conversational language: This provides an accessible bridge between native and formal reasoning. A typical lab begins with natural conversation, and is continuously honed as required by the greater precision necessitated by the interplay between theory and context.

Finally, the use of collaborative learning techniques is essential to the social aspects of the students’ construction of mathematical skills. Conflict resolution for problems involving multiple approaches provides unique learning opportunities. A broader knowledge base and problem-solving competence is established, and precision of language improves markedly when students are required to justify their reasoning to their peers.

Conclusion: Ours is only one vision of calculus education among many, informed by but not limited to social constructivism. It is presented to illustrate that a careful consideration of one’s pedagogical goals, and of the alternative methods of attaining them, can be a valuable exercise. Calculus education can become a lesson not merely in symbolic and graphical techniques, but in the endless varieties of reason itself.

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Last updated: August 25, 1995