During this lecture I am working from the premise that all the world is a mathematical classroom. By the time children begin formal education they have spent up to five years in this classroom [1] and during their years of schooling the fraction of time spent in a formal classroom is minimal compared to the time spent out of it. I wish to argue that this broader definition of a classroom impacts in many strong ways on our conceptions of mathematics and helps to construct the misconceptions about the nature of the subject that impinge on the learning of mathematics and the value which we place on it. It is within the formal classroom that many of the misconceptions about mathematical knowledge and skills develop. Although a great deal of writing exists about these, and while teachers use this knowledge to inform their teaching I wish to place more emphasis on the less familiar conceptions and misconceptions of mathematics. I will be challenging you to examine your own conceptions by showing you examples from television, radio, advertising, newspapers, film and literature. I will ask you to reflect on how a consideration of these aspects of pupils' experiences of mathematics might affect our teaching. I am also going to ask you to think about whether influencing young people's conceptions of mathematics might affect their decisions as to stop or continue studying the subject.

C. A. B. Smith (c.1918-2002) holder of the Weldon Chair of Biometry at the Galton Laboratory was a member of the Mathematical Association for over sixty years. Amongst his scientific work, he developed the standard statistical methods now used to map genes onto chromosomes and he wrote a text book Biomathematics. He recently contributed a poem written by the mysterious ‘Blanche Descartes’ to the Gazette and it was published posthumously.

Recently a poem, written by Blanche Descartes in the 1930s, was republished in the Gazette, with the additional attribution ‘communicated via Cedric A. B. Smith’. Cedric had reported ‘it was written after Blanche heard that Hector Pétard was about to be married. She composed the verse as a wedding present. Hector is still active mathematically, though I suspect that he may be in his mid nineties’. We felt that readers might be interested to know something about the mysterious Blanche Descartes, and the origins of the poem. It makes quite a story. At the risk of boring you, we will start at the beginning.

Candidates are graded by their examination results in some such way as 1st B; 2nd equal A, D; 4th equal C, E; and so on. In how many ways gncan n candidates be graded? This problem was solved by Cayley [1, pp. 374-378] and later by others. It does not seem to be well known, and the following result may be of interest.

Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.

In this paper we shall consider two situations in triangle geometry when equilateral triangles appear and then show that they are closely related.

In the first (known as the Napoleon theorem) equilateral triangles BCAT, CABT, and ABCT, are built on the sides of an arbitrary triangle ABC and their centroids are (almost always) vertices of an equilateral triangle ANBNCN (known as a Napoleon triangle of ABC; see Figure 1).

The familiar fact that we may cut any triangle into three polygonal pieces which can be rearranged to form a rectangle is one of the most elementary problems of polygonal dissection. This fact is shown in two ways in Figure 1 where both sides of the rectangle are predetermined after setting up the triangle: one side of the triangle and half of the altitude to this side, or one altitude and half of the side at right angles to it.

Given a triangle with vertices P1, P2, P3 its Fermat point is defined as that point F which minimises the sum P1F + P2F + P3F. It is known that if each of the angles P1, P2, P3 is less than 120°, then F will be the point inside the triangle at which each pair of P1, P2, P3 subtend an angle of 120°, while if any of the angles at P1, P2, P3 exceed 120° then F will coincide with that vertex – see Figure (1).

One of the long-standing problems of number theory, appealing to professional and recreational mathematicians alike, is the existence of Mersenne primes. These puzzling primes, for example 7, 31, 127 and 8191, are of the form 2P - 1, where p is itself a prime. The problem of their existence originated some 2400 years ago with the early Greek mathematicians' quest for the so-called perfect numbers, those like 6 and 28 which are the sum of their proper divisors. The connection was given in Euclid's Elements in 300 BC: if 2P - 1 is prime, then 2P-1(2P - 1) is a perfect number. Much later, Euler proved that all the even perfect numbers correspond to Mersenne primes. So the interest for many years has been in finding Mersenne primes. Only 39 are known, including several monsters discovered in recent years using thousands of PCs coordinated via the internet (see [1] for information on the project). Many of us would like to know if there is any way of predicting which exponents yield Mersenne primes and whether there is an infinite number of them.

The roots of a polynomial can be represented as points in the complex plane. The time value of money (TVM) equation that is commonly used in finance is a polynomial equation. (See the appendix for a short description of the TVM equation and an example of its use in finance.) In [1] and [2] it is shown that concepts from financial mathematics can be obtained from the pattern of the roots of the TVM equation. The concepts are given in terms of distances between the roots and other salient points in the plane. This note shows that this particular polynomial, and the technique, can be applied more generally. When a series of data is fed into the coefficients of the polynomial, the mean and standard deviation of the data are seen in the complex plane as combinations of distances between the roots and other salient points. The results are aesthetically pleasing as well as mathematically interesting.

We have developed a novel diagrammatic approach for understanding and teaching probability theory — Probability Space diagrams [1]. Our studies of learning and instruction with Probability Space (PS) diagrams have demonstrated that they can significantly enhance students' conceptual understanding. This article illustrates the utility of PS diagrams by applying them to the explanation of some difficult concepts and notoriously counterintuitive problems in probability. We first outline the nature of the system.