A number of introductory statistics books published during recent years have presented the basic theory of point estimation (see, for example, [1]–[4]). The time now seems ripe for a few cautionary tales to be told, or retold, for there is more to the estimation of a parameter than searching for an unbiased estimator or maximising the likelihood of a random sample. Having said that, I need at once to emphasise the distinction between methods of estimation on the one hand, and properties of estimators on the other. For a particular problem, the method of maximum likelihood, fcr example, provides a possible set of estimators for the parameters involved. Other methods such as least squares or minimum-χ2 might provide alternatives; but no estimator should ever be put to use until its properties have been investigated in some depth. And here lies the rub; for, as I intend to show in the following paragraphs, the choice of criteria by which one estimator is judged superior to another rests, ultimately, with the individual statistician or researcher responsible for the estimate produced. In other words, no completely objective solution to the estimation problem is tenable.

Those who have the privilege of contributing new proofs of old theorems to the columns of the Gazette do so, normally, with complete confidence in the validity of the new proofs but possibly with some doubts as to whether the new proofs will be regarded as better than the old ones. The following attempt to set out new proofs for certain properties of the triangle differs from the norm in that there is no doubt whatever that the new proofs will be regarded as worse than the old ones and the only doubt is whether they will be regarded as valid. The doubt may arise because the aim is to deduce the truth of certain general theorems from the proved truth of one or two particular cases of the theorems. This is far from being the usual procedure in mathematics, though it is not without precedent; for example, if we show that ax2 + bx + c is zero for three values of x, it follows that it is zero for any and all values of x.

In the years 1603–1867, known as the Edo period, when Japan isolated itself from the western world, the country developed its own style of mathematics, especially geometry. Results and theorems of traditional Japanese mathematics, known as ‘Wasan’, were usually stated in the form of problems; these were originally displayed on wooden boards (‘sangaku’) hung in shrines and temples, but many later appeared in books, either handwritten with a brush or printed from wood blocks. (See [1, 2, 3] for more details.) Solutions to the problems were not given, but numerical answers were sometimes provided. The problems were written in a form of language called ‘Kanbun’, based on Chinese, which cannot be readily understood by modern Japanese readers even though written Japanese makes extensive use of Chinese characters.

This article begins with the consideration of a short program designed to produce random values from a normal distribution and goes on to suggest how a similar program might be used to introduce confidence intervals to a group of A-level students. The programs are written in BASIC for the BBC microcomputer.

One of the difficulties encountered by students when they are first introduced to axiomatic algebra is that the axioms, such as commutativity, associativity, and distributivity, seem self-evident in the algebraic situations with which they are familiar. Indeed, the axioms are so useful, of course, because of their wide applicability. However, it can be instructive to study situations in which these axioms break down, particularly if the situations are not too contrived.

When students first meet Simpson’s rule, we look for simple applications to show them what it can do. I don’t imagine I am the only teacher who has suggested using it to find a numerical value for π - which up until then they have always had to take on trust.

A series of investigations is being planned by a section of the Board of Studies for Mathematics in the Institute of Education of the University of Birmingham, with the object of gaining information about the types of difficulty experienced by children in various branches of grammar school mathematics, and the types of error made. An analysis of these errors, both from the mathematical and psychological points of view, is highly informative and relevant to a discussion of teaching methods in the subject. It is hoped that later it will be possible to prepare standardised tests which would be of diagnostic value to teachers. The co-operation of other teachers would be welcomed in furthering this investigation, either in the form of comments or criticisms in the light of their own experience, or in helping to widen the sample of children tested in future work.

The difficulty of a good calendar lies in the fact that the earth travels round the sun in 365 days, or to be precise in 365 days 5 hours, 48 mins., 46 secs, which, we should note, is less than 365 days by just over 11 minutes.