In my professional life, I have repeatedly had cause to return to the question of

It's time I began to frame some responses! In doing so, I am very aware that membership of The Mathematical Association is broad church, ranging from those with an amateur interest in mathematics (and mathematics education), through non-specialist teachers, to professional mathematicians. As such it is a rich community, but one in which it is challenging to address such questions in a way which is meaningful to all. However, I shall attempt to do so and I hope that the balance will in a small way also prove illuminating to the reader. In particular, I have a hope that it will point to the breadth and depth of what we expect from our teachers of mathematics, at all levels, and to a valuing of that.

Tangential quadrilaterals are defined to be quadrilaterals in which a circle can be inscribed that is tangent to all four sides. It is well known and easy to prove that a convex quadrilateral is tangential if, and only if, the angle bisectors of all four vertex angles are concurrent at a point, which is the centre of the inscribed circle (incircle). The most well-known and in problem solving useful characterisation of tangential quadrilaterals is Pitot's theorem, which states that a convex quadrilateral is tangential if and only if its consecutive sides a, b, c, d satisfy the relation a + c = b + d [1, pp. 64-67]. If you want to have more background information about characterisations of tangential quadrilaterals, then we recommend you to check out the lovely papers [2, 3, 4], as well as our previous contributions on the subject [5, 6, 7]. These six papers together include about 30 characterisations that are either proved or reviewed there with references.

Optimisation problems are among the most practical applications of calculus to everyday life, and a survey of exercises in various calculus textbooks will provide a teacher with many interesting scenarios for framing intriguing questions on this topic. Whether it is finding a container's dimensions that yield the least surface area for a given volume, or finding that ideal movie ticket price which will maximise a theatre's revenue, students can usually relate to these problems. Pólya in his book Plausible reasoning makes the following remarks about the attraction of extrema problems:

The theory of quadratic equations (with real coefficients) is an important topic in the secondary school mathematics curriculum. Usually students are taught to solve a quadratic equation ax2 + bx + c = 0 (a ≠ 0) algebraically (by factorisation, completing the square, quadratic formula), graphically (by plotting the graph of the quadratic polynomial y = ax2 + bx + c to find the x-intercepts, if any), and numerically (by the bisection method or Newton-Raphson method). Less well-known is that we can indeed solve a quadratic equation geometrically (by geometric construction tools such as a ruler and compasses, R&C for short). In this article we describe this approach. A more comprehensive discussion on geometric approaches to quadratic equations can be found in [1]. We have also gained much insight from [2] to develop our methods. The tool we use is a set square rather than the more common R&C. But the methods to be presented here can also be carried out with R&C. We choose a set square because it is more convenient (one tool is used instead of two).

A continued fraction is an expression of the form

The expression can continue for ever, in which case it is called an infinite continued fraction, or it can stop after some term, when we call it a finite continued fraction. For irrational numbers, a continued fraction expansion often reveals beautiful number patterns which remain obscured in their decimal expansion. The interested reader is referred to [1] for a collection of many interesting continued fractions for famous mathematical constants.

This problem was posed in the 1930s by Paul Erdős [1]. It is the simplest case of a still unsolved conjecture.

If k2 + 1 squares with no interior point in common are drawn inside a unit square then the maximum possible sum of their side-lengths is k [2].

We shall use the notation S(n) to denote the maximum possible sum of the side-lengths for n squares drawn with no interior point in common inside a unit square. The main aim of this article will be to develop an approach to the study of the function S which will give surprisingly simple proofs of a number of known results. This approach will then be used to prove a new result about the asymptotic behaviour of S.

During the 19th century various criticisms of Euclid's geometry emerged and alternative axiom systems were constructed. That of David Hilbert ([1], 1899) paid particular attention to the independence of the axioms, and it is his insights which have shaped many of the further developments during the 20th century.

We can, from his insights, define an affine plane as a set of points, with distinguished subsets called lines such that

Axiom 1: Given two distinct points, there is a unique line containing them both.

Axiom 2: Given a line L and a point, p, not contained in L, there is a unique line containing p which does not intersect L.

Axiom 3: There exist at least three points, not belonging to the same line.