Augustus De Morgan (1806-1871) was a prolific writer, a notable mathematician and logician, and an excellent teacher. His work in logic and his invention of the logic of relations are well-known. However he also thought and wrote extensively about mathematics teaching pedagogy, and his approach is still relevant to mathematics teachers today. In this short note I offer selected quotations from De Morgan's writings on mathematics education, preceded by a short biography taken from.

Matches of a certain popular brand are contained in a box designed to hold 100 matches. This box has two compartments so that if you open the box casually and without thought as I do, then you will select a match at random from one compartment or the other with equal probability. At some stage you will empty one of the compartments; and having more statistical insight, perhaps you will be less surprised than me by the large number of matches often left unused in the other compartment.

There are two well known constructions of the real numbers from the rationals—namely the Dedekind cuts method in which a real number is defined as a class of rationals, and the Cantor–Cauchy completion method in which a real number is defined as an equivalence class of Cauchy sequences of rational numbers. In this article we give a new construction of the reals from the rationals in which a real number is defined as an equivalence class of sets of natural numbers.

Teaching mathematics has never been easy. There is no reason to imagine that it ever will be. Yet it is still worth asking whether there might not be relatively simple ways of improving our teaching of particular topics and of particular groups of pupils. History of mathematics has much to offer on both counts, and can enrich ordinary teaching in many ways and on many levels. But it also has its problems and its pitfalls. This article focuses on two issues: the temptation to enlist the support of ‘history’ when trying to change social attitudes; and the uncritical way in which intelligent students respond to such pseudohistory.

It has been customary when Euclid, considered as a text-book, is attacked for his verbosity or his obscurity or his pedantry, to defend him on the ground that his logical excellence is transcendent, and affords an invaluable training to the youthful powers of reasoning. This claim, however, vanishes on a close inspection. His definitions do not alwaysdefine, his axioms are not always indemonstrable, his demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative force when no figure is drawn, but very many of Euclid’s earlier proofs fail before this test.

Most mathematical students are familiar with Pascal’s statement, made in a communication to a learned society in Paris in 1653, that he had a number of mathematical treatises, chiefly geometrical, ready for publication, amongst them being the following : Conicorum opus completum, et conica Apollonii et alia innumera unica fere propositione amplectens ; quod quidem nondum sex decimum aetatis annum assecutus excogitavi ; et deinde in ordinem congessi.

A famous unsolved problem in mathematics is the 3x + 1 conjecture. An excellent discussion of its origin and history is found in [2]. It is deceptively simple to state. Many are tempted to try it because of its enticing appearance. One aim of this note is to warn the uninitiated that a person can grow old and infirm by trying to solve the problem. There are probably fine mathematicians who have frittered away what might have been productive periods of their lives working on this devilish puzzle. The other aim is to say something at least mildly interesting about a question that I can’t answer. These are the writings of a person who has been frustrated, but not defeated, by the 3x + 1 conjecture.

One of the oldest and best known of chessboard problems is to place the largest possible number of similar pieces on the board such that no two of these pieces are attacking each other. In his book Amusements in mathematics [1], Dudeney considered this problem on a generalised square chessboard containing n 2 cells, and proved that for rooks, queens and bishops this maximum number is equal to n, n and 2n – 2 respectively.

The discussion of symmetry in modern mathematics syllabuses can be taken a stage further in the sixth form to show how symmetry can often be used to simplify the solution of certain types of problems. This article gives just one example, that of determining the eigenvalues and eigenvectors of an n × n matrix. The theory involves two steps. In the first we show how the eigenvalues of a matrix M can under certain circumstances be found from those of a commuting matrix P. In the second step we show how to construct, from the symmetry properties of matrix M, commuting matrices P whose eigenvalues are readily found.