Prologue. This article began in an attempt, for purposes which shall be nameless to find the result of inverting a triangle and associated lines and circles with respect to its nine-points circle. The work outran its aim, and I now present it from a quite different starting-point.

A very simple way of generating a sequence is to take two starting values, a1 and a2, and construct a3 as their arithmetic mean, a4 as the mean of a2 and a3 and so on. This sequence always converges to the limit and seems to be fairly widely known as a topic for investigation in schools and colleges. It can immediately be generalised to the case of k starting values, and this forms a good subject for investigation using a micro-computer. It seems likely that such a simple set-up has arisen independently many times. One emergence occurred about ten years ago at a conference of the Association of Teachers of Mathematics, where the originator introduced it under the fictitious name ‘Littov sequences’. It has recently resurfaced at a Department of Education and Science conference.

In the literature we find several different ways of introducing elementary functions. For the exponential function, we mention the following ways of characterising the exponential function:

(a)

(b) , also for complex values of x;

(c) x → exp (x) is the unique solution to the initial value problem [4]

(d) x → exp (x) is the inverse of

(e)x → exp (x) is the unique continuous function satisfying the

functional equation f (x + y) = f (x) f (y) and f(0) = 1 [6]; the corresponding definition is done for the logarithm in [7];

(f) Define dr for rational r, and then use a continuity/density argument [8].

All of them have their advantages and disadvantages. We like (a) and (c), mostly because they have natural interpretations, (a) in the setting of compound interest and (c) being a simple model of many processes in physics and other sciences, but also because they are related to methods and ideas that are (usually) introduced rather early to the students.

In this paper I look at different aspects of mathematical reasoning, and argue that we need to make sure students of all ages engage in a range of mathematical reasoning, particularly given the evidence that teaching for reasoning is a powerful complement to teaching that is more focused on skills and procedures.

If P is a point of intersection of two conics which have one common focus, then e1 |PM1| =e2 |PM2|. It follows that if the two directrices cut in O, then OP passes through a second point of intersection, and that if directions of measurement perpendicular to the directrices are assigned, the equations e1 . M1P = ±e2M2P give two of the chords of intersection, we may say that these are the chords associated with the common focus.