This assignment: A Prelude to Calculus – History and Pedagogy was part of my Masters of Teaching. It was a way of introducing students to the concept of a limit based on an historical account of the way the ancient Greeks reasoned about the concept in order to derive equations for the volumes of a sphere, cone, and cylinder. This occurred long before the development of calculus, and even of algebra. Central to the solution was a focus on Archimedes’ Method of Equilibrium so that the idea of limits would be seen to solve a real problem within a real historical context.
The concept of a limit which is fundamental to calculus is a radically new concept for students who are studying it for the first time. The problem I was exploring was how to introduce this new concept to a student in such a way that they had time to ponder and eventually assimilate it. The standard method for introducing the limit is to make reference to graphs of curves and show how in the limit their secants become tangents. However, this method fails to demonstrate to students how this can be used and why it is important, and proceeds almost immediately to finding the limit of algebraic expressions.
Language of Mathematics
This action research project was coupled with a ten-week internship and was my final assignment for my Masters of Teaching. The question upon which it was based was whether an increased understanding and use of mathematical terminology would help students to better understand the concepts and procedures to which those terms relate, and if so, whether that increased understanding would result in an increased ability to actually do math.
A class dictionary of relevant mathematical terms was made and each student kept their own version in a small booklet. Written records of students’ work were collected over a three week period and audio tape was used to record classroom conversations. The hope was that there would be an improvement in both the students’ understanding of mathematical concepts and their ability to apply those concepts to solve problems.
The students responded well to the math dictionary but many struggled to use the words to express their understanding. Severe time constraints placed significant limitations on the conclusions which could be drawn from this research project. Nevertheless, the results suggested that while there was a distinct increase in students’ understanding of the concepts associated with the terminology studied, more time was needed to demonstrate a clear link between an understanding of the terminology and an increased ability to do mathematics.