Research-based guidance and classroom activities for teachers of mathematics

Calculus and analysis

Post-16, calculus entails the study of change and is commonly divided into two major branches: differential calculus (or differentiation) and integral calculus (or integration). The two branches are linked by the remarkable fact that differentiation and integration are inverses (as encapsulated by the Fundamental Theorem of Calculus).

Mathematical analysis

Mathematical analysis, usually referred to simply as ‘analysis’, is the branch of pure mathematics that encompasses differential and integral calculus but also extends to cover measure, limits, infinite series, and analytic functions. When students begin to study the calculus, their success depends on their previous experience and current knowledge of both algebra and geometry.

Students’ knowledge and ideas about functions and their capability to manipulate algebraic expressions are particularly important here, as are their ideas of ratio, similarity, measure, slope/rate/gradient, right-angled triangles, and circle geometry as it relates to radians. This entails students being able to interpret the graphs of functions such as simple polynomials, but also knowing about rational functions, trigonometric functions, and the relationship between powers and logarithms.

Different teaching approaches

Students who are taught the calculus beginning with the epsilon–delta definition of limits can encounter difficulties, and this has led to other teaching approaches being developed. One approach to differentiation, known as the ‘locally-straight approach’, is based around the idea of magnifying (or zooming in on) a part of a graph of a function to see it approximate to a straight line whose slope can be measured. The idea of ‘accumulation’ (of a quantity described by its rate of change) is being suggested for use in the teaching of integral calculus.

Calculus pre-16?

These teaching approaches (including ‘local straightness’ and ‘accumulation’) exploit the potential of computer environments to handle multiple representations (graphical, numeric, symbolic) of mathematical functions. This raises the question of whether students younger than 16, given appropriate technologies, might be able to engage with the mathematics of change and variation that is the focus of the calculus. This could lead to a new strand of the mathematics curriculum that is neither a simplified version of symbolic calculus, nor an exploration of linear functions and related notions of rate and ratio.