How do you introduce and teach calculus?
Do you start fairly informally exploring rates of change and gradients of tangents/curves and then move on to the rules of differentiation, then integration both as the inverse operator and as a way of calculating areas under curves, while always focussing on knitting together all the required prior knowledge from algebra, functions and geometry with the new material?
Or do you extend this informal approach to a more formal approach to differentiation linking the tangent as a limiting secant (geometrical representation), the gradient of the tangent as the limit of a sequence of gradients of secants (numerical representation) and the derivative as the limit of a difference quotient (algebraic representation)?
Do you also develop a more formal approach to integration linking the area under a curve to the limiting sum of areas of rectangles (geometric representation), the limit of a series of products (numerical representation) and the integral as the limit of a sum of products (algebraic representation)?
How much do you state, how much do you demonstrate and how much do you prove, including the Fundamental Theorem of Calculus?
Do you go as far as the formal ‘epsilon-delta’ type analysis normally first encountered in undergraduate Mathematics courses in the UK?
Whichever approach you take and however you develop it, the likelihood is that you will support your approach with the use of technology both to improve and increase the speed of understanding and to enhance visually the links between the geometric, numerical and algebraic representations that pupils need to make if they are to develop conceptual understanding as well as procedural fluency.
In Teaching and Learning Calculus with Free Dynamic Mathematics Software Geogebra by M Hohenwarter, J Hohenwarter, Y Kreis and Z Lavicka and presented at ICME 11 in 2008, the creator of Geogebra, now based at the University of Linz in Austria, and his colleagues consider in depth six different approaches to the use of Geogebra in teaching calculus, as well as reviewing the teacher-centred and pupil-centred approaches to using technology in the classroom. The paper emphasises the collaborative potential of the use of free, open-source resources such as Geogebra as well as the more obvious financial benefits.
Further reading on approaches to calculus teaching to develop conceptual understanding rather than simple reliance on learning a set of procedural rules, and an extensive bibliography, can be found in Key Ideas in Teaching Mathematics: Research Based Guidance for Ages 9-19 (Anne Watson, Keith Jones and Dave Pratt) 2013 OUP, in particular, Chapter 9 Moving To Mathematics Beyond Age 16. Indeed, the whole book is recommended reading for all mathematics teachers in its entirety.
As a final thought, it is worth noting that that the proposed content for the revised A Level Mathematics from September 2017 specifically includes both differentiation and integration from “first principles”. This will possibly require a more formal approach than that sometimes currently adopted in AS Level and A Level Mathematics teaching, but one that will be equally well supported by technology as discussed in the research paper above.
Footnote: in earlier issues of the Secondary Magazine, we featured a set of six articles by the authors of Key Ideas in Teaching Mathematics, following the themes of each chapter. They appear in Issues 105, 106, 107, 108, 109 and 110.