Addresses Keynote Address Problems Address Although the conference sponsors are responsible for some necessary condensing of the following three addresses, the conclusions and interpretations are those of the authors and do not necessarily represent the views of the conference sponsors. Needed Emphases in the Teaching of Mathematics and HIGHLIGHTS OF THE ADDRESS Dr. W. L. DUREN, JR. 1. Considering the Nation as a whole, high schools are not staffed with teachers who are adequately prepared to teach mathematics. 2. If qualified teachers were available, it would be desirable for high schools to teach advanced (college) algebra, analytic geometry, calculus, sets, matrix algebra, and statistical methods and to stress disciplined mathematical reasoning. 3. Suggested ways for increasing the number of qualified teachers: Raise salaries. Raise qualification requirements. Make summer and academic year institutes more generally available. Expand commuting-type inservice training facilities. Increase the capacity of undergraduate and graduate schools for teachers. Import teachers from Europe. Devise some new types of teacher-education programs. 4. Suggested ways for training teachers: Through foundation, State, and other aid, prepare student textbooks (and teacher editions) and workbooks that would be more theoretical, more rigorous, and more complete than current ones are. Provide local seminars in which teachers would work through the new textbook materials. Provide film and kinescope expositions of 11th- and 12th-year high school courses. Provide in each State a traveling supervisor who would consult with local teachers and organize small regional conferences. Provide home study courses for mastery of the new mathematics. Provide Skinner-type teaching machines to help teachers develop good techniques. Award degrees and certificates only on the basis of rigid examinations. 5. Any kind of inservice education program is likely to give most benefit to teachers in larger administrative units, and hence to teachers who usually have least need for improvement. 6. The lectures now being given by many institutes are at such an advanced level that they are not appropriate for high school teachers. THE ADDRESS Mathematics is recognized as that high school subject which contributes most to the education of a future engineer or scientist. In fact, if a student is not already good in mathematics before he leaves high school, college mathematics can do little to help him become either one or the other. True, it can add to the competence of a student who has already achieved some mathematical mastery, but it can retrieve for the engineering and science professions very few students who have failed to get a good mathematics education in high school. Thus it is that junior and senior high school teachers are perhaps the key persons in building the foundations for our national scientific and technological strength. In the United States a student takes up the calculus for the first time around age 19-if he takes it up at all. In Europe, by contrast, a student begins the subject at age 16 or 17, and it is not unusual to find a bright one beginning it at age 14. To be sure, we try to teach the calculus to a larger percentage of our school population, but the fact remains that our theoretical scientists are delayed at least 2 years in getting to this essential tool subject. Physicists who have been working in the Massachusetts Institute of Technology Physical Science Study Group have a feeling that university mathematicians in the United States do not as a whole favor the idea of having the calculus started in high school. The reason that these mathematicians don't want to see the calculus taught there, at this time, is that so few of the present high school mathematics teachers are competent to teach the subject. In many cases the latter would do more harm than good were they to attempt to teach it. On the other hand, the reason that the university physics teachers demand it is that they cannot advance the study of physics unless the students coming to them already have some knowledge of the calculus. Again, to keep our place in this world we must have a substantial number of our students beginning the serious study of physics in high school. Indeed, we probably have to expect them to do at least 2 years of high school physics. There is no reason to believe that the students lack intellectual capacity or lack interest. The main trouble is that they lack mathematical preparation. We have come to one of those times when the curriculum must be cleaned up. We must remove the deadwood, keep the substantial body of old mathematics whose value is ageless, but express it in a more precise and more powerful language and effect intellectual economies in the way it is learned and handled so that we can introduce those new ideas which are more valuable than the deadwood we remove. It has always been the history of intellectual progress that this sort of acceleration by economizing and simplifying things has enabled |