**1. An Objective in Education**

C C MacDuffee, An Objective in Education, *The Mathematics Teacher* **38** (8) (1945), 339-344.

The philosophy of an education fitted to the capacity of the student has, up to the present, been largely developed in the interests of the weaker students. This is of course an essential step in the unique experiment in universal education for which the schools of the United States are now a vast laboratory. It is an experiment which would be attempted only by a great democracy, and it is an experiment which must be successfully completed if democracies are to survive and flourish. But in solving the problem of the poorer student, difficulties have been placed in the path of the superior student which in some instances make it difficult for him to obtain the education of which he is capable. Because the superior students are in the minority, and because they do not be come problem children, they are often woefully neglected. Because advanced subjects in a small high school have small registration, they are often abandoned and the time of the teacher is "more economically employed" with large classes of "future citizens." ... It appears that more and more educators are abandoning the thesis that all students should be forced to go as far as they can through the college preparatory high school courses. The principle of the course tailored to fit the capacity of the student seems to be attracting more and more adherents, not only among high school teachers, but also among college teachers. This plan of a double-track curriculum promises excellent results provided the express track is not neglected in favour of the freight track, and provided that students of ability are persuaded to purchase Pullman tickets and not bills of lading.An address delivered before a joint meeting of the Wisconsin Section of the Mathematical Association of America, the Mathematics Section of the Wisconsin Education Association and the Mathematics Club of Milwaukee at the Milwaukee State Teachers College on5May1945.

**2. Objectives in Calculus**

C C MacDuffee, Objectives in Calculus, *Amer. Math. Monthly* **54** (6) (1947), 335-337.

What should be the objectives in a beginning course in the calculus? That is a question which many college teachers ask themselves, and to which it is difficult to frame an answer. Calculus is the course for which the student has long been preparing through college algebra, trigonometry and analytics, and for many a student it is the last mathematics course which he will ever take. The amount of interesting and valuable material which is at that point open to him is large and beyond the capacity of the time available for its complete presentation. What gems shall be presented and which omitted is a problem which we all have to face. ... In these days of educational experimentation, various combinations of courses are being tried which have never been tried before. What could be more natural than a combination course of basic physics and calculus? This course would probably have to be spread over two years if it were to contain a complete course in both physics and calculus. It would have to be given by a man who is competent in and sympathetic toward both courses. He could not be a physicist who teaches a little mathematics as a "tool," nor a mathematician who "runs in a few illustrations from physics." But he would be able to develop physicists to whom mathematics is a mother tongue. Can you think of a better background for scientists of the present age? Regardless of the framework in which it is taught, the first course in calculus must be handled with a fine sense of balance. It should be rigorous up to the capacity of the student to appreciate rigor, and this rigorous treatment must be extended to the problems, not merely confined to the proof of the existence of the definite integral. But the fundamental and basic problem is to develop the student's intuitions so that mathematics is to him a spoken language. Then and only then is he in a position to appreciate the meaning of rigor. For is rigor anything else than clarity?The American Mathematical Monthly(1947).

**3. The Scholar in a Scientific World**

C C MacDuffee, The Scholar in a Scientific World, *Amer. Math. Monthly* **55** (3) (1948), 129-140.

A nation is just about as great as its universities. They are the ganglia in the central nervous system of the nation, whence come the nerves that stimulate its intellectual, industrial, and political life. Without the leadership of the universities, national life would probably continue for a while by inertia but would gradually slow down and succumb to the competition of rival nations. The function of the university as a teaching institution is purely incidental. Its principal function is to keep alive the great wealth of knowledge and culture that past ages have collected, and to add to that wealth through the encouragement of scientific research and creative art. Knowledge and art can be kept alive only by the creation of scholars, and the scientific horizon can be extended only by the creation of scientists. To create scientists and scholars is the primary function of the university. ... Everything that I have said so far is trite and generally admitted, but we in the United States do not make it part of our practical thinking. We do not believe that the production of scholars is in itself a worthy objective, nor that the scholar is worthy of his hire. Our colleges and universities are supported by state and private funds, but the stated objective is the instruction of vast numbers of young men and women in ways and means of earning a better living than their fellows. They are taught citizenship and the American way of life. I am not always clear just what this means. Occasionally it seems to be football and the cult of the gods and goddesses of Hollywood.Retiring address as President of the Mathematical Association of America, Athens, Georgia,1January1948.

**4. What Mathematics Shall We Teach in the Fourth Year of High School?**

C C MacDuffee, What Mathematics Shall We Teach in the Fourth Year of High School?, *The Mathematics Teacher* **45** (1) (1952), 1-5; 9.

I think I might as well start right out with a statement of what I think should be the content of the fourth year of mathematics in high school. I think it should be a course in which advanced algebra and the rudiments of analytic geometry are integrated, and which contains a few of the essential ideas of the calculus in the second semester. You are probably saying to yourselves that this is not a new suggestion, and I know that it is not. It has been and is being offered in many of the forward-looking schools of the nation. It has been offered in several of the schools of New York City. It is now being offered in several of the schools in Wisconsin, including the high school administered by the Department of Education of the University of Wisconsin. Hence it is not an untried experiment. But it is far from being standard procedure in our schools. ... may I ask you to believe in the gifted student, to believe in the importance of scholarship and the subject which you teach, to believe in the future of your country, and to stand fast in your beliefs though the rabble storm without the high school walls.An address delivered at the Twenty-Ninth Annual Meeting of the National Council of Teachers of Mathematics at Pittsburgh, Pennsylvania,31March1951.

**5. Teacher Education in Algebra**

C C MacDuffee, Teacher Education in Algebra, *Amer. Math. Monthly* **60** (6) (1953), 367-375.

The question which we have come here to discuss is the proper content of a course or courses in algebra for a student who has just completed elementary calculus. There are many possibilities, of which I list three, because they seem to represent most common procedures: 1. A problem-solving course in permutations and combinations, probability etc. with advanced skills as the prime objective. 2. A development of some of the more elegant topics in classical algebra, a course in which skills and theory are kept in balance. 3. An introduction to abstract algebra. Before deciding which of these procedures to follow, one must examine care- fully the objectives of the course. Some students at this level are aiming to take graduate work in mathematics, but more of them are not. In fact, we have graduate and undergraduate students from almost all scientific departments in the university and we cannot afford to ignore their demands for algebraic skills. It is this diversity of objectives which makes the problem difficult and calls for a compromise answer. ... The course which I have been giving at Wisconsin for the last couple of years is still entitled the Theory of Equations, but might more properly be called the Theory of Polynomials. This approach seems to unify the somewhat scattered topics in the theory of equations, and to give a deeper insight into the subject which is particularly valuable to those who go on in algebra and to those who contemplate teaching algebra.A lecture delivered at the Symposium on Teacher Education in Mathematics,27August1952, in Madison, Wisconsin.

**6. Mathematics Curriculum in Perspective**

C C MacDuffee, Mathematics curriculum in perspective, *The Mathematics Teacher* **52** (4) (1959), 265-267.

As soon as curriculum revision is suggested, many persons spring up with all the answers and with novel methods of approach which they have been waiting for years to try out. I know of no profession in which so many people are eager to try something new and drastic as in the teaching profession, particularly in mathematics. I am guilty myself, but I am not alone. My colleagues agree on almost nothing regarding methods of teaching. The remarkable fact is that so many different methods will work. ... It would seem, then, to be an almost impossible task to devise a curriculum suitable for all teachers and all students. In the first place, teachers differ sharply in their abilities, knowledge of mathematics, and interests. I wish it could be said that all teachers are fundamentally scholars. It is not true in all American schools at the present time, and a teacher with little genuine respect for scholarship cannot be expected to instil such respect in the minds of his charges. Then, too, the mathematical preparation of teachers varies much more than it should. Standards of accreditation mean little when half the mathematics teachers are not accredited in their subject. And, finally, good mathematics teachers do not always agree among themselves on the relative interest and importance of the various topics in the curriculum. ... genuine reform in mathematical teaching is intimately tied up with reform in all kinds of teaching. This is a long-range problem that will not be solved by any magic formula. It will not be solved until the citizens of the United States want it to be solved. As long as merely keeping the child happy is a more important objective than his education, we cannot make progress. When the nation is ready to pay teachers an appropriate wage and to render them the honour and respect that is their due, we can begin to go forward. When scholarship is honoured above athletic ability, and extracurricular activities are trimmed to size, the millennium will be near. Then, with the teachers uniformly good and the pupils eager and able, the sky will soon be so full of American sputniks that even the sun will hide its head.An address delivered27December1957, at Indianapolis at a symposium on mathematics instruction, a program of Section A of the American Association for the Advancement of Science.