M J M Hill on the Teaching of Mathematics

M J M Hill gave his Presidential Address On the Teaching of Mathematics to the Mathematical Association in 1927. Below are the opening paragraphs of that Address which were published as M J M Hill, 'On the Teaching of Mathematics. Presidential Address to the Mathematical Association, 1927', The Mathematical Gazette 13 (187) (1927), 296-312. In 1928 Hill gave a second Presidential Address The Logical Eye and the Mathematical Eye. Their Outlook on Euclid's Theory of Proportion to the Mathematical Association. Below are also the opening paragraphs of that Address.

  1. On the Teaching of Mathematics - Presidential Address 1927.

    Ladies and Gentlemen, - I desire to begin by thanking you for the great honour you have conferred on me by i nviting me to occupy the Presidential Chair of your Association.

    Great as is the honour, I am thinking more of the responsibilities it entails and of the opportunities for useful work which it offers to me.

    The noble passage which stands at the head of the statement of the objects of this Association opens with the words: "I hold every man a debtor to his profession." The consciousness of that debt I carry with me whether at work or in repose.

    Thus it happened that when some years ago I was spending a holiday on the shore of the North Sea, I was thinking of my profession. And whilst I was so engaged there came and sat down beside me a little lad. I asked him if he went to school. "Yes!" he said indifferently. Though somewhat repelled, I ventured on another question: Did he like his school? "No!" he answered, pouring into the word a depth of indignation which stung me to the quick, for I felt its implied censure on my profession. And as I moved uneasily away over the stones, my eyes fell upon a fragment of paper which contained a paragraph headed: "She knew her multiplication table." It read something like this: Little Sue had come home from school. She ran up to her mother and said, "Mother, I know my multiplication table." Her mother said, "Hush, Sue, Uncle Bob is reading. Don't disturb him." But her excitement was not to be quenched, so she answered, "Did you hear, Mother, what I said? I know my multiplication table." This roused Uncle Bob, who said, "Did you say you knew your multiplication table, Sue?" "Yes! Uncle," she replied. "Very well, then," he said, "tell me what are thirteen times three?" Whereupon she looked at him scornfully and said, "Don't be so silly, Uncle Bob, there ain't no such thing as thirteen times three."

    Some time after this, when I was thinking what significance this had for those interested in the efficiency of education, the scene before my eyes seemed to change. It was a summer afternoon. There was great excitement in the schoolroom. The air was quivering with heat; a distinguished ecclesiastic had just entered the room to address the lads. I was listening intently, and I seemed to hear these words: "We owe a great debt to the Headmaster for the way in which he has met the Philistine attack of those who are attempting to put down Classics in favour of such plebeian pursuits as the study of Mathematics."

    Plebeian! How strange an epithet to apply to the study of any branch of human thought. Can you wonder that in my indignation there flashed through my mind a distortion of a well-known saying so that it read thus: "Omne ignotum pro malefico." But a gibe carries one no farther, and reflection brought a wiser counsel, so that I remembered: "Fas est et ab hoste doceri," and I was forced to put some questions to myself.

    Have we teachers of mathematics, you and I, failed to strike a responsive chord in the minds of our pupils, whether in the public or in the elementary schools? Is there anything beyond the inherent difficulty of the subject that stands between them and us?

    I think there is. Driven by the great mass of routine, we teach, I fear, too much by rule, too little by principle. Let me try to make my meaning quite clear. I use the word "principle" as something furnishing the pupil with a direction which commends itself immediately to his reason, something which he can link up at once with his previous experience. He welcomes it gladly. On the other hand a "rule" is of the nature of an order; very often it is a summary of reasons which may either have been clearly and fully explained, or which may have been inadequately explained, or which may have been delivered ex cathedra without any explanation at all.

    Possibly an objection may be made at this point. The teacher is obliged to teach the subject so that the pupil can acquire great facility in using it as a tool, and this can only be done by the formulation of rules. Unfortunately, facility is often acquired at the price of forgetfulness of the reasons on which these rules were founded. This price is too great to pay. My suggestion is that in every lesson a few moments should be spared to go back to first principles, for whenever these have been fully grasped the desired facility will follow by itself.

    And so the plan of my address from this point onwards is to choose out two or three rules or statements commonly made without explanation, or as it seems to me with inadequate explanation, and to suggest some line of action which has helped me in the past in the hope that it may help some of you in the future.

  2. The Logical Eye and the Mathematical Eye. Their Outlook on Euclid's Theory of Proportion - Presidential Address 1928.

    1. My address last year was devoted to the consideration of a few rules or statements made in teaching, with, as it seemed to me, insufficient explanation, in the hope that I might suggest some line of action, in a few particular cases which would be helpful to members of our Association engaged in elementary teaching.

      The greater part of that address was concerned with the first of Euclid's two great pronouncements, viz., the Postulate of Parallels, and to the suggestion of an alternative treatment, based on Wallis's Postulate of Similarity, and it was shown that Wallis's Postulate, invented 264 years ago to explain Euclid's Postulate of Parallels, sufficed also to explain his Fourth Postulate, viz. that all right angles are equal.

    2. Today, with a similar object in view, in the hope of assisting those members of the Association who are preparing mathematical pupils intending to enter a University, and those of you who are teaching students of mathematics during the first year of a University course, I take up the second of Euclid's two great pronouncements, viz. the famous Definition of Proportion, the Fifth Definition of the Fifth Book of the Elements, and the argument based thereon.

      This definition, which is attributed by tradition to Eudoxus, the master of Plato, is, when it makes its first appearance in Euclid's text, no less startling than the Postulate of Parallels.

      The Two Views.

    3. The title I have given to this Address was suggested to me by a saying of De Morgan, which forecasts the treatment of the subject here adopted.

      It is on record that he said once:
      The two eyes of exact science are mathematics and logic; the mathematical sect puts out the logical eye, the logical sect puts out the mathematical eye, each believing it can see better with one eye than with two.
      The distinction between the views obtainable from the two eyes may be stated thus:
      From the time of Aristotle downwards logic has always meant the art of making correct deductions from the principles employed. It looks at the verbal treatment of the terms of an hypothesis and the development of all assertions which are necessarily included in the terms of a proposition without drawing on any other axioms or theorems for evidence. It is concerned only with the purely logical process, by which we make two assertions put together show their joint meaning, and express what, without deduction, they only imply.
      The mathematical eye looks at the matter reasoned upon, it examines the principles employed, it enquires in what way they are related to one another and to fundamental ideas; it seeks to determine whether the course of the argument is the most direct that can be taken.

      "In addition to the purely logical process of extracting implied meanings out of the expressions of the hypothesis, it appeals to propositions which are not in the hypothesis, and which, for anything the hypotheses tell us to the contrary, may or may not be true. Of course - not logic - but - reason requires that these propositions should have been previously proved, or assumed, on their own evidence expressly."

      In De Morgan's mind there was no conflict between the two studies, for he was not only a great mathematician, but a master of logic as well.

      My master, the late Professor Henrici, to whom I owe a great debt, told me that one day, when the conversation in the Common Room at University College had turned upon De Morgan, he expressed the view that De Morgan's writings impressed him as those of one whose primary interest was in logic whilst his interest in mathematics was due to the fact that it gave him such abundant opportunities for applying and testing his logical theories. Whereupon one of those present, who had known De Morgan in the flesh (I had not that honour) said, "He told me so himself."

    4. It was the logical nature of Euclid's argument that first attracted De Morgan's attention. In early manhood he wrote a little book entitled Connexion of Number and Magnitude, with the significant sub-title, "An attempt to explain the Fifth Book of Euclid's Elements" - significant because it implies the difficulty of the argument, to which he returns again and again in his work, and also because he describes it as an "attempt," pointing to the fact that he was not altogether satisfied.

      Five years later we find him writing a remarkable article in the Penny Cyclopaedia. This is written from the mathematical standpoint, that is, he tries not merely to draw conclusions from the definitions laid down, but to explain them.

      In one passage he constructs a certain object, namely, a row of equidistant columns, and a parallel row of equidistant railings, and a model thereof. From a consideration of the relation of the model to the object, he obtains, without drawing upon any definition of proportion, or anything more than the conception we have of that term prior to any definition (and with which we must show the agreement of any definition that we adopt), the conditions which Euclid lays down in the Fifth Definition, thus bringing the Definition into relation with our fundamental ideas.

      A complete description of the object and model, to which I can add nothing, will be found in Heath's great edition of the Elements of Euclid, monumentum aere perennius, therefore I will leave you to read this at your leisure.

Last Updated June 2021