Mamikon\'s Approach to Teaching Calculus

Mamikon's Approach To Teaching Calculus

Mamikon's A. Mnatsakanian, often along with his colleague Tom M. Apostol, has published many papers detailing new instructional methods for explaining otherwise complex concepts in the realm of calculus, as well as new ways of understanding these concepts. His emphasis is on a visual understanding of calculus, which is more easily observed and intuited by students -- and at a younger age, it seems increasingly evident -- than traditional textual and purely mathematic explanations and understandings. For years, a website has been available with several puzzles and games that help to visually express many of the mathematical measurements and principles of calculus. Several brief examples of Mamikon's teaching style make it clear how the principles of calculus build on lower mathematic understanding, and are in fact easily understood themselves.

Measuring the area of a curved space is essential for many applications of calculus, yet can be one of the more difficult among the basic principles and practices of the average calculus student. Mamikon's illustration of the curving bicycle, and subsequent related illustrations, show quickly and easily how the area described by such a curve is the same as the area -- or partial area -- of a circle (CalTech). The preceding sentence is proof of how difficult such concepts can be to clearly and efficiently explain, but accompanied by Mamikon's illustrations the principle is instantly observed and far more easily remembered and recognized.

A more thorough and elegant explanation of the same concept is provided on Mamikon;s paper (with Apostol) entitled "Subtangents -- An Aid to Visual Calculus." Again, Mamikon starts with a visual explanation of the principle, but goes on to detail this principles work in calculus (Apostol & Mamikon 2002). Thus, his method of teaching calculus visually creates at least a rudimentary understanding of a principle or practice before any theorem or even a simple equation is introduced. This is the opposite of the way calculus -- and mathematics in general -- is often taught, where understanding comes only after the working our of many illustrative problems. Mamikon even takes this simple observation about curves to establish a new relationship between the tractrix and exponential curves (Apostol & Mamikon 2002).

Mamikon's visual understanding and explanation of calculus is not limited to two-diemnsional curves, nor does he concern himself only with new insights into mathematical relationships. In another paper, again published with Apostol, Mamikon established new proofs for Archimedes' discoveries concerning polyhedrons and their circumscribing prisms (Apostol & Mamikon 2004). Again, his explanation abounds with visual examples, clearly shaded in various tones to correlate areas and volumes for an easy understanding of the relationships Mamikon is describing. The mathematical formula are present too, of course, but they are far more easily understood for most students when accompanied with visual examples.