Note on Mr Ellwood's Remarks on Division.
Amer. Math. Monthly 1 (3) (1894), 74-75.
Reluctantly, feeling that it is almost unnecessary, a note is offered on Mr Ellwood's article. Such articles float through the primary journals of education now and then, and possibly do no harm that can be undone by replying to them. But in a mathematical journal such pedagogy should not go unchallenged. It is a trite idea that all advance in mathematics, and in all science generally, has had to contend with just such arguments as those of Mr Ellwood. The whole history of the development of mathematics has been a history of the destruction of old definitions old hobbies, old idols. ... But Mr Ellwood ... says, "Multiplication is a mere process of adding." Here is the idol, the sacred definition. He might as will have said, "One is not a number," because nobody ever thought of it as such until modern times. So the enlargement of the notion of multiplication is modern. But the enlargement is made, and it 'will never be unmade. Multiplication was, originally, what Mr. Ellwood says, but the world of to-day says, "nous avons change tout cela." And the world of to-day says $10 ÷ 2 = $5, and a good part of the world expressly distinguishes between the two notions of division in the primary grades. To try to establish the limitation proposed by Mr Ellwood is to let the tail try to wag the dog.
The International Commission on the Teaching of Mathematics.
Amer. Math. Monthly 17 (1) (1910), 1-8.
An address delivered before Section A of the American Association for the Advancement of Science, and The American Mathematical Society, in joint session, at Boston, December 29, 1909.
So much has been said and written of late concerning the work of the International Commission on the Teaching of Mathematics, and so closely connected are several members of this Association and this Society with the movement, that I find myself quite at a loss in attempting to impart any new information as to the inception of the work and the general purposes in view. Nevertheless a brief resume of the organization of the Commission may not be out of place, to be followed by a statement of the problems that particularly confront us here in America. At the Fourth International Congress of Mathematicians, held at Rome in 1908, the following resolution was adopted: "The Congress recognizing the importance of a comparative examination of the methods and plans of study of the instruction in mathematics in the secondary schools of the different nations, empowers Messrs. Klein, Greenhill, and Fehr to form an International Commission, to study these questions and present a general report to the next Congress."
The American Work of the International Commission on the Teaching of Mathematics.
The Mathematics Teacher 2 (2) (1909), 56-71.
The work in America has been organized on a different basis from that in certain of the European countries, and necessarily so. One reason for this difference is that we have nearly fifty state governments, each with its own system of education, a condition paralleled only by the German Empire, and even there to not the same extent. Another reason is found in the fact that educational matters are less settled with us than is generally the case abroad, so that we have no such bodies of accumulated material to which we can at once turn for information. Still a third reason for our different plan of attack is seen in the fact that we have not so large a body of trained university investigators in our teaching force, with time for preparing exhaustive discussions of special educational topics, as is found for example in Germany.
Certain Problems in the Teaching of Secondary Mathematics.
The Mathematics Teacher 5 (3) ( 1913), 161-179.
An address given before the New England Association.
It is not without considerable hesitancy that I come before an association like yours to speak upon some of the great problems that confront us in the teaching of secondary mathematics. This hesitancy arises from several causes, prominent among them being the feeling that I shall only be "carrying coals to Newcastle." For surely these problems are already in your minds, and many of you have pondered over their significance and their solution quite as seriously as I have and no doubt with a more satisfactory issue. I hesitate, also, because I can merely state them with no attempt at solution, mindful all the time of the ancient adage referring to questions which a wise man can not answer. But after all, there is a value in clearly stating from time to time the large questions that confront our guild, for if problems were never formulated they would never be solved, and it is upon associations like this that we must largely depend for the solution of the ones that I shall venture to lay before you.
What is to be the Outcome?
The Mathematics Teacher 9 (2) (1916), 77-79.
Recent experience in certain parts of the country shows that it is not unlikely that, within a few years, students may rather generally, at any rate for a time, be admitted to college without mathematics. This movement is intended to be democratic in its purpose, but it may well be doubted whether it is not more likely to turn out to be a phase of the return to the old aristocracy of education. By depriving students of equal privileges in the way of serious mental work, the danger is the same as that of placing the extremely difficult subject of vocational guidance in the hands of any except a highly trained, highly gifted, and carefully selected group of experts - that of depriving all of equal chance in the race of life. To reduce this danger to a minimum, it is essential that we carefully consider, in meetings of this nature, our attitude in the face of the attacks now being made upon mathematics.
Mathematical Problems in Relation to the History of Economics and Commerce.
Amer. Math. Monthly 24 (5) (1917), 221-223.
If students of the history of economics and commerce wish to find a new and interesting field for exploration, and one which is certain to yield results that are worth the labour of cultivation, they will do well to consider the history of problems in arithmetic and algebra as set forth in the manuscripts and early printed books that have come down to us. No doubt some of this field has already been explored, but it is quite certain that only a small portion has thus far come under cultivation. The manuscripts on arithmetic from the thirteenth century to the beginning of printing, the large number of books printed before problems began to represent past as well as contemporary conditions, and the more original text-books of later periods contain a considerable amount of material on the history of commerce and economics that no one seems yet to have studied with any thoroughness.
Introductory Course in Mathematics.
The Mathematics Teacher 11 (3) (1919), 105-114.
In spite of all that has been said in this country in opposition to mathematics in the past few years, the feeling of certainty still exists in the intellectual world that the science is not dead, is not dying, and is not stagnant; that it touches more lines of human interest today than ever before; and that its values have only been accentuated by the efforts made to relegate it to the position of formal grammar, formal rhetoric, and formal logic. Indeed, it may safely be said that mathematics stands more firmly today than ever before, not only in the minds of what is commonly called the intellectual class but in the opinions of the man in the shop and of the man who has so recently been in the trenches on the battlefields of France.
Certain Mathematical Ideals of the Junior High School.
The Mathematics Teacher 14 (3) (1921), 124-127.
The possibility of a new and stimulating course in mathematics in the Junior High School is the most encouraging feature of our recent advance in the teaching of the subject. Such a course enables the schools to break away from the present initial course and prepare the pupil for the best that mathematics has to offer in the high-school period. It enables them to take the best that other countries have to offer and to introduce it by the natural steps that psychology shows that the youth is ready to take. It allows mathematics to relate itself to the interests and apparent needs of young people instead of being presented as a purely abstract science, since the early stages of a rational course can easily be made much more apparently practical than is the case with the present course.
The Mathematics Teacher 14 (8) (1921), 413-426.
... what is the duty of teachers of our science? To preach? -- that should be the last thought. The greatest sermons are preached in silence. The most ancient religions that we have, if there be more than one fundamental religion, have always recognized this fact. And so it must be with us, - that we should teach "the science venerable" not merely for its technique; not solely for this little group of laws or that; not only for a body of unrelated propositions or for some examination set by the schools; but that we should teach it primarily for the beauty of the discipline, for "the music of the spheres," and for the faith that it gives in truth, in eternal law, in the infinite, and in the reality of the imaginary; and for the feeling of humility that results from our comparison of the laws within our reach and those which obtain in the transfinite domain. With such a spirit to guide us, what teachers we would be!
The Next Step in Content in Junior High School Mathematics.
The Mathematics Teacher 15 (1) (1922), 26-27.
The problem resolves itself into recognizing that mathematics permeates every science that we have, every department of business activity, and, indeed, most of the other fields of human interest, and that the first step is to ascertain precisely what phases of mathematics are needed by the average, well-informed citizen in order that he may understand any one of the manifold lines of human activity in which the science plays an important role. We have made a fair beginning in the solution of this problem, but that which constitutes a solution today ceases to be a complete solution tomorrow because of the rapid changes in the applications of mathematics to the various lines of human activity.
Suggestions on the Arithmetic Question.
The Mathematics Teacher 18 (6) (1925), 333-340.
As to arithmetic, the question has not been discussed with anything like the care given to the mathematics of the secondary school. The psychologists have begun to give it some attention, but they approach the problem from only a single angle, and their conclusions as to what should be taught have generally been open to grave criticism. Even their conclusions as to how the special branches should be presented have, in general, shown an unfortunate ignorance of the needs of practical life, these needs not infrequently tending to render invalid the reasoning which seems to characterize our educational laboratories.
The Call of Mathematics.
The Mathematics Teacher 19 (5) (1926), 282-290.
Why is Mathematics Studied? Ever since man came to think in the abstract, to think of the number two as distinct from two objects, to create for himself units of measure that possess some approach to uniformity, to think of time, and to be aware of such concepts as lines and angles, mathematics has been an object of his study and the basis of most of the natural sciences of antiquity, becoming the very essence of all science of the present day. In every generation men have arisen to question its value; when these men have died, their protests have died with them; and in our time other men have arisen to ask the same question, and with them the protests will die as with their ancestors. The question is, however, a fair one; it would be asked by the mathematician if it were not asked by those to whom the science was so poorly taught, or so unsuccessfully, as to warp their judgments and to encourage their element of destructiveness. In the one case the query would be raised in a sympathetic frame of mind; in the other case, in a hostile spirit; but in any case it is legitimate and it demands a frank answer.
Esthetics and Mathematics.
The Mathematics Teacher 20 (8) (1927), 419-428.
In considering mathematics in relation to the beautiful, the range of possibility is so vast that a brief article like this can hardly be expected even to list the salient points of contact. The field might properly include all that we designate as the fine arts or, to use the more expressive phrase of the French, the beaux arts. Painting, for example, might he considered with reference to the works of that great genius in science, in mathematics, and in art - Leonardo da Vinci. Sculpture might equally well be included because of the mathematical principles employed by that majestic user of ponderous masses, Michelangelo. Architecture might have place with reference to the works of that Oxford professor of mathematics, Sir Christopher Wren, who rebuilt ecclesiastical London; engraving, with reference to that gifted artist of Nürnberg, Albrecht Dürer, who published the first modern work on curves; music, with reference to the fact that it always ranked as a branch of mathematics until the sixteenth century; decoration, with reference to the geometric designs found in all ages and reaching their highest degree of perfection in the works of the Moslems; and literature, with reference to the mathematics of poetry, and the poetry of mathematics. Indeed, we might properly include the beauties of nature, where mathematics plays a part of which we are usually quite unconscious.
Introduction to the Infinite.
The Mathematics Teacher 21 (1) (1928), 1-9.
In elementary mathematics we loosely speak of infinity as a number without limit, or as a line which has no end, and we tell our pupils that it is represented by a symbol, . We thus give a definition without meaning to those who hear it, and a sign without significance to those who see it. Our duty should be something far different. Just so far as we can adapt our work to the minds of our pupils we should make the effort to get much nearer the soul of mathematics than is the case in most of our classes under present conditions.
The Lesson of Dependence.
The Mathematics Teacher 21 (4) (1928), 214-218.
A child in the seventh school year learns how to find the circumference of a circle. He sees no particular reason for learning it; he does not use it, nor is it used in his home. If he enters certain types of industry, he will need it, but probably not otherwise. If his teacher connects it with the speedometer on his father's car, it immediately has some interest; and if it is similarly related to a bicycle, the interest is increased and the way is open to other applications that give reality to the abstract.
Time in Relation to Mathematics.
The Mathematics Teacher 21 (5) (1928), 253-258.
Because time is constantly referred to by the world's great poets and because it has for thousands of years been the subject of speculation by the world's great philosophers is no sufficient reason for considering it with relation to mathematics. Since, however, time is infinite, and infinity plays a major role in all branches of our science, it is not without value to the teacher to spend a little of time itself in considering its relation to mathematics, for it is a mathematical certainty that the time we spend will not lessen in the least that which is to follow through the ages, and beyond.
The International Congress of Mathematicians, Zurich, September 4-12, 1932.
Science, New Series 76 (1977) (1932), 468-471.
If any evidence were needed of the rapid development of mathematics in the last third of a century it might well be found in a study of the programs of the several sections of this Congress and of the number and nature of the papers read. In the early years of these quadrennial meetings four sections were considered sufficient; at present there are twice as many. At first the countries represented were chiefly European, the number of members from other continents being relatively small; but in this congress between 40 and 50 nations were represented and the attendance approximated 650 men and 200 women.
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