Respecto a los resultados pre prueba y pos prueba del grupo experimental.

This last paper is typical for Butters in several ways. It fits in nicely with overarching themes, such as generalisation and simplification. He is developing a more general method of proof with a wide applicability, as he is also doing in paper 2.3.3. In doing so, he is attempting to simplify the teaching of the subject. Simplification is also a key goal in paper 2.3.5, where he argues for a simpler way of dealing with sums of money. In this fifth paper, he also touches upon another recurring topic, that of speed. He remarks how much time can be wasted by poor methods, and suggests better ones, as he also does in 2.3.2.

The final paper is typical in choice of readers as well, targeted as it is at schoolteach- ers. Butters is usually writing with this particular audience in mind; 6 of 7 papers deal with educational matters. He is also expecting his readers to hold such knowledge as would be typical for teachers, such as a solid grounding in Euclidean geometry. Most of Euclid’s propositions in 2.3.7 are for instance given by number only, the actual wording being left out. In 2.3.3, he is using what may be presumed to be conventional names for certain lines and points without any explanation, not even marking them on the relevant figure. A teacher would know these things by heart, encountering them on a daily basis.

Butters’s choice of audience reveals itself in other ways, too. His papers are inter- spersed with comments on how best to teach the subject. In his fifth paper (2.3.5), he

argues that the conversion between coinage and decimals should never be written down, but should always be taught orally. In the final paper, he suggests teaching reflection symmetry ‘experimentally’, by folding a piece of paper, letting the fold be the axis, and pricking holes through it. Many more such examples can be found.

Butters can safely make such assumptions on behalf of the teachers, being a teacher himself. These papers can therefore provide some insight into the knowledge of his colleagues. Negative geometrical magnitudes are for instance standard. He also expects them to be familiar with projections. The other side of the coin is perhaps even more interesting. Butters does not expect the teachers to be familiar with modular arithmetic, and it looks as if he does not expect them to be too familiar with decimal arithmetic either. This latter supposition is perhaps more surprising, considering how he refers to school textbooks on decimal arithmetic dating from as early as 1685.59 _{It could of}

course be that he does expect them to know this, and that he is just emphasising which parts are necessary to convey to pupils. However, his treatment of decimal arithmetic seems a little too thorough for that interpretation to hold.

Whether he is writing for teachers only, or simply taking for granted that all non- teachers know the subject equally well is not quite clear. It is clear, however, that the papers would be of less value to an academic. His tendency to leave things unsaid would perhaps not be up to the academic standard, as illustrated by his third paper (2.3.3). As explained in that section, when compared to the academics’ treatment of the same topic, Butters’s exposition falls short.

It can also be argued that Butters’s proofs are a bit unsound. This is particularly relevant for his proof in the sixth paper (2.3.6). In this, he sets out to prove a general theorem, but does so by showing that it holds for a special case only. This procedure gives only a sketch of a proof, albeit an accurate sketch, that can easily be adapted to the general case. He argues that he does so for simplicity, which is plausible enough. Not only is the general case harder to follow, it also places higher demands on the printers, an expensive procedure in 1904.

Although his style may be slightly flawed from an academic point of view, his topics are far from uninteresting. As mentioned, the topic from his third paper is covered by academics, though in different forms, in 1900 [59] and 1904 [60], and by yet again in the American Mathematical Monthly in 1946 [45]. Butters’s presentation of Gauss’s work would no doubt be of interest as well, considering how the topic was still subject

CHAPTER 2: THE MATHEMATICS OF THE TEACHERS of research as late as 1909 [65].

His final paper stands out in this regard. Parallels can be drawn between his sym- metries and the transformations in Klein’s Erlangen Programm. This was published in 1872, but was not translated into English until 1892, and its contents were still impor- tant topics around the turn of the century. Whether or not Butters is aware of this, it still shows that teachers could benefit directly from learning of contemporary research. What separates Butters from the academics more than anything else is mode of presentation. The choice of topics is obviously influenced by his profession, but most of his papers would be of interest outside the schools as well.

2.4

Conclusion

Most of what the teachers published qualifies as research, as it was, at least as far as they were aware, original in content. Their topics somewhat reflect the trends in UK at the time, with gradually fewer papers on Euclidean Geometry, and somewhat more analysis. Euclidean Geometry was, and remained, their area of expertise, so it would be only natural to expect a decline in contributions as the current areas of research moved away from this. As they did take an interest in other areas as well, there would be no reason to expect it to stop entirely.

The contributions from the teachers to the Proceedings begin declining towards the end of the First World War. If this had been purely a result of the rising levels of mathematics, there would have been no reason for the same to happen in the Notes, but it does. It is therefore very likely that this is connected to the First World War. It is for instance not unlikely that the two groups were called in for different kinds of war-service, which would affect them differently. As will be seen in chapter 4, there were other reasons for why this decline became permanent and not just a passing phase.

The Enumeration of Rhyme

Schemes

3.1

Introduction

The current chapter deals with correspondence between A. C. Aitken, Sir D’Arcy Thompson and G. T. Bennett in December 1938 and January 1939. The correspondence begins with the counting of rhyme schemes and quickly leads to Bennett rediscovering certain properties of the Bell numbers, most of which Aitken has already rediscovered in his paper ‘On a problem in combinations’ from 1933. The chapter will go through the 13 letters and place them into context, especially regarding Aitken’s article. Before this can be done, however, a few preliminaries are in order.