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.

A driver’s estimate of the average velocity of vehicles on a highway is biased by the driver’s sampling of those vehicles that pass the driver or are passed by the driver. Surprisingly, the magnitude of the bias does not head towards infinity as the driver’s speed heads towards infinity. Here, we explore these interesting facts.

For n ≥ 1 an integer, we shall denote by σ(n) the sum of the (positive) divisors of n.

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analogue to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley-Hamilton theorem.

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

We tackle an unusual problem that, as far as I know, is not in the standard literature. To state it concisely I use what I call ‘valid sets’. We know that in any three or more positive quantities, only the largest can be half or more of the total value; and is then obviously greater than the sum of all the others. But if the largest is less than half the total of the set, it must be less than the sum of the others; and this is true for every element in this set. I call such a set ‘valid’. For example, the sides of a triangle are valid, as are the face areas of a tetrahedron. Our problem relates to the converse for a tetrahedron: given any four valid quantities, is there always a tetrahedron with those face areas? In this article I answer this by showing that, for any valid set, we can always construct the net of a corresponding tetrahedron. In fact, for any given valid set there is always an infinity of non-congruent tetrahedra with the given face areas. Although in general there are no formulae that give the exact edge lengths of these solids, I show that there are always several distinct infinite subsets that are constructible and whose edge lengths can be determined exactly.

Given a parallelogram, construct squares outwardly on its sides; hereafter we will call this the ‘Thébault configuration’. Our name derives from Thébault’s celebrated first problem, which states that the centres of these newly constructed squares also form a square.

Certain ideas recur in many areas of mathematics. One example is groups of symmetries, which appear in the Galois theory of equations and in Lie groups. Lie groups are of great value in physics, where Noether’s theorem enables us to derive a conservation law for every case in which a function known as the Lagrangian is invariant under a one-parameter Lie group. The importance of this approach can be seen from the fact that the laws of the conservation of energy, linear momentum and angular momentum are all outcomes of Noether’s theorem, though they can of course be derived by simpler methods. The full power of Noether’s approach is shown in its applications to quantum field theory, where it can be used to find conserved currents and charges.