Which way pre-college mathematics?
The Mathematics Teacher 47 (5) (1954), 303-307.
It is encouraging that many mathematics teachers, in both high schools and colleges, are dissatisfied with mathematical education at all levels, and that they are .searching for ways to bring it in line with the increasing demands of the times and with the dramatic development of mathematics in this century. In spite of long hours, large classes, low salaries, distracting extracurricular activities, disciplinary problems, the legitimate needs of the majority who are not going to college, and the general anti-intellectual atmosphere, high-school mathematics teachers are doing a commendable job of preparing pre-college students. ... The present crisis in mathematics is due not to any deterioration in the work of mathematics teachers, but to an urgent national need for more and better mathematics at a time when administrators and the public have for years slighted mathematics and, indeed, discouraged all vigorous mental effort in the high schools. The high-school program seems better suited to meet an urgent shortage of athletes and club-joiners than of scientists and other professionals. In athletics and other "co curricular" activities, there are numerous awards and keen competition for them. In contrast, excellence in scholarship is permitted, but mediocrity is considered more "democratic." Our high schools, while claiming to prepare students for real life, have established a system of status and rewards that is just the reverse of that in the world. When school is over, income and status for most people depend on how they do on the job, whereas in school success is made to depend on how they do in recreational activities. The contrast is especially sharp for pre-college students, whose future is determined primarily by the soundness of their understanding of subject-matter and only secondarily by their extracurricular experiences. It is high time that scholarship be appropriately recognized, publicized, and rewarded as the main job of students. Athletic and other extracurricular activities should be given a subordinate place as essential parts of a rounded education.
Undergraduate Research in Mathematics.
The American Mathematical Monthly 65 (4) (1958), 241-246.
A creative mathematician is the intersection of several unlikely events. For the most part we are ignorant of the nature of these events and of their probabilities. Some of them are at present quite beyond our control. An example is the probability of a genetic composition necessary for intensive and highly abstract thinking. Others are clearly subject to our influence. For example, the probability of an early acquaintance with living mathematics and with the joy of mathematical achievement is determined by educational practices. ...
Examples of research activity.
After solving the familiar problem of the sailors on a desert island who divide up a pile of coconuts, one student conceived the idea of generalizing the problem to an arbitrary number of sailors and an arbitrary sequence of discards before each division of the pile. By using difference equations and some ideas from number theory he worked out an algorithm for solving any problem of this kind. His formulation of the problem and his method of attack were conceived without any faculty assistance. A faculty member helped him to polish up some details of the solution. He presented his results at a state meeting of the Mathematical Association. In a course in number theory, the professor wrote every day on the side board some difficult problems which the students might attack if they became bored. Students who did such problems were invited to present their results to the class. One such presentation inspired a freshman student to generalize the problem and discover new results. His work stimulated a sophomore to discover that the number of representations of an integer as a sum of consecutive integers is equal to the number of odd divisors not including one. The professor collaborated with the students in polishing up the proofs of these results. The later discovery that the result had been published about thirty years ago, far from discouraging the students, confirmed the significance of their original work.
Finding out about "modern mathematics".
The Mathematics Teacher 51 (2) (1958), 93-95.
In the last few years there has been considerable discussion about the possibility of introducing into the secondary school curriculum topics such as logic, theory of sets, Boolean algebra, and the set theoretic approaches to relations, functions, and other topics of elementary mathematics. For lack of a better term such material has come to be called "modern mathematics." It is only in very recent years that these topics have appeared in college courses; and, until very recently, there were virtually no books dealing with this material at the elementary level. Hence it is in evitable that few teachers are familiar with it. However, several books written recently for use in elementary college courses could also be used by high school teachers as sources of information on the ideas involved and even as sources of materials that could be used in the high school classroom.
Small Versus Large Classes.
The American Mathematical Monthly 69 (5) (1962), 433-434.
Since the teacher shortage is causing many departments to consider large classes in mathematics, our experience with a freshman lecture of 250 may be of interest. The entire group met together for a lecture three times a week. In addition each student had a weekly two-hour laboratory with about 15 or 20 students led by a regular staff member. Detailed assignments in two textbooks, along with sample quizzes and laboratory problems were contained in a syllabus supplied to all students. Daily papers were turned in at the lecture, corrected by a crew of about 10 readers, and returned to the students through campus mail the same day. A laboratory session began with a substantial quiz which was immediately discussed by the instructor. The rest of the session was devoted to student questions and the working of problems with help from the instructor and undergraduate student assistants. ... Our conclusion is that students may be taught as effectively in large sections (and our methods could have been used just as well with a class of 2000 as with one of 200), provided the work is very carefully organized and provision is made for easy feedback and consultation. The smaller size class group is more enjoyable, however, for both students and staff
Undergraduate Research: Some Conclusions.
The American Mathematical Monthly 75 (1) (1968), 70-74.
Original contributions to mathematics by juveniles have a long tradition. Indeed precocity and early involvement in creative work appear to be characteristic of practically all of the most productive mathematicians of the past and there seems no reason to think that mathematics has changed enough to make this any less likely in the future. The term "undergraduate research" has a somewhat different meaning. It refers to an educational activity designed to increase the output of creative mathematicians and scientists by giving young people the experience of original work in addition to the standard courses in which they "learn" what has been discovered by others. Undergraduate research activities may, and often do, lead to new publishable mathematical results, but this is not the goal. The activity may be considered successful if it enhances and accelerates the student's development as an original thinker. the important manifestations are an effort involving independent work and results that are original for the student, regardless of their newness to other mathematicians or significance for current mathematical research.
An ideal programme?
None exists but a very good one would involve the following:
- A student run colloquium.
- A student run publication.
- A student mathematics club that supports these activities and engages in others appropriate to the local situation.
- A faculty member who keeps a friendly eye on things and helps as needed.
- Some link of these activities with the curriculum.
Ideally, research-like activity should be part of every course (the quantity and quality varying with the context) and should be officially recognised by graduation honours, special courses, and the like. In short, undergraduate research should become a normal part of the curricular and extra-curricular educational process.
The American Mathematical Monthly 77 (10) (1970), 1120-1121.
The experience of examining virtually every English mathematical book published during a four year period has been most enjoyable and instructive. A goodly proportion were undistinguished pot boilers. Some were mathematically and pedagogically unsound, the first most common for the numerous remedial level books, the second too often true of advanced texts and treatises. Nevertheless the general impression is of improving quality. There is a healthy trend away from a show of rigor and toward genuine motivation, clear explanation, and closer links with applications. More books at the elementary level reflect genuine meditation about the problems involved, and more advanced books display scholarship as well as mere technical proficiency. It is to be hoped that these healthy tendencies will continue, since the unquestioning acceptance by the public of anything dished out by mathematicians, including publications and graduates, is likely soon to become a thing of the past. Sources of funds, employers, and students are likely to be increasingly insistent in their questions about the social utility of mathematics and mathematicians
History in the Mathematics Curriculum.
The American Mathematical Monthly 81 (8) (1974), 899-90.
The history of mathematics has always played a role in the mathematics curriculum insofar as teachers have found it useful to introduce historical information in courses organized along essentially deductive lines. Separate courses in the history of mathematics have also long occupied a modest position in the undergraduate curriculum. Less common, but beginning to gain popularity, are courses in which content is arranged historically. Such courses do not ignore the logical component in mathematics, but they present it and all other aspects of the subject in historical perspective. It is the purpose of this note to describe these roles of the history of mathematics in the undergraduate and graduate curriculum at the University of Toronto.
Four courses with significant historical content and orientation are offered to undergraduates. The first, entitled "Introduction to Mathematics," is designed for first-year students ... The course is described: "The nature and role of mathematics, illustrated primarily by the development of numerical and geometric ideas. Lectures, films, study of mathematical literature, and the writing of an essay. Tutorials will provide the opportunity for doing mathematics as well as talking about it." The course does not attempt to present a systematic chronological history of mathematics, but rather to give a survey of mathematics as a living, historically developing component of culture. ... The second course, entitled "Development of Analysis," is intended primarily for second-year students and may be elected by those having taken at least one university mathematics course. It focuses on the development of the basic themes and concepts of calculus, including a study of 18th and 19th century rigor, and the development of the concept of the integral in recent times, with detailed examination of selected examples ... The third course, intended primarily for third-year students, is entitled "20th Century Mathematics." Its purpose is to give the student "a survey of the different trends in the mathematics of this century, their interplay, social function, and effects in science and technology. ... At the senior level is offered the first course in the history of mathematics.
What is good history and who should do it?
Historia Mathematica 2 (1975), 449-455.
The answer to this question can be given in two words: it depends. Or in five: it depends on the purpose. And one might add that the infinity of possible motivations suggests caution in asserting dogmatic criteria. The alleged invariance of absolute standards may be mere inflexibility in wishing to ban personally uninteresting or offensive purposes. ...
Are there any universal criteria of quality in historical work, counterexamples to my initial claim that "it depends". Perhaps the following might be candidates:
- There should be a purpose, stated or implicitly clear, and this purpose should be achieved.
- Assertions should be supported by evidence and argument. This requirement corresponds to insistence on proof in mathematics. In learned articles it involves careful citation and close reasoning. In popularizations it is sufficient to indicate where the missing evidence can be found.
- There should be clarity and complete disclosure. This corresponds to the mathematical requirement of rigour. ...
When it comes to purpose, I can only state my preferences. I believe that history can and should be socially useful, to historians of science, to policy makers, to students and users of mathematics, to the educated layman, and above all to the mathematicians who are its most reliable consumers and the creators of its raw material. ...
The mathematician is quick to detect mathematical errors in historical exposition, and he may even blame the historian for an error that he is merely honestly reporting without comment because he is interested in describing and understanding the past rather than judging and correcting it. On the other hand, the historian finds hilarious such naive historical mistakes as assuming that words have fixed meanings or that a brilliant mathematician of past centuries must have understood a concept or had a proof because these would be evident to lesser lights today. Historians become bored with the mathematician's preoccupation with priorities, which are important to the persons concerned but become elusive when viewed historically and throw little light on historical issues. And of course the historian is as sensitive to technical errors in bibliography and to inept or uncritical use of sources as is the mathematician to clumsy manipulation of symbols, even though the one is no more the essence of history than is the other of mathematics. On the other hand, historians are tolerant of mathematical errors if they do not seem seriously to affect the historical purpose, just as mathematicians are tolerant of historical sloppiness that does not impinge on the mathematics.