Analysis and natural philosophy owe their most important discoveries to this fruitful means, which is called induction. Newton was indebted to it for his theorems of the binomial and the principle of universal gravity. (Laplace, 1902, p. 176)
I have had my results for a long time, but I do not yet know how I am to arrive at them. (Gauss, quoted by Lakatos, 1976, p. 9)
Much of the empirical work reported in this book is set in contexts where students of various ages are carrying out and talking about mathematical tasks. The precise mathematical content of the tasks is of less importance, for my purposes, than the mathematical processes in which the students are engaged. In order to understand what students say in such circumstances, and why they say things the way they do, it is important to understand the nature of the processes themselves. Yet, it can be dangerous and unhelpful to draw a sharp distinction between process and content in mathematics. Arguably it is the content-numbers, shapes, groups, topological spaces, and so on-that most clearly distinguishes mathematics as mathematics, that marks it out from other domains of knowledge, for example science or history-whilst in both these cases there are content overlaps with mathematics. Time, for example, is a concern for all three. On the other hand, without the processes there would be no mathematics, or at least mathematics would have no products, no propositional content, no truths (theorems) about the objects of study-numbers, groups, and so on. Bell et al. (1983, p. 206) describe the process dimension of mathematics in terms of:
the style and atmosphere of the activity in the mathematics classroom […] whether [pupils] see mathematics as a field of enquiry, or a deductive system, or a set of methods to be learnt from the teacher.