ground of entering freshmen vary a great deal, it is necessary to have a flexible program, especially for the first year courses. The following is a suggested outline of courses for a five-year program. A teacher would receive a conditional certificate at the end of the traditional four years. Four summer school sessions would then be required to become fully certified as a mathematics major. The course would result in a master's degree if desired. Courses which appear to be especially appropriate for junior high school teachers have been selected. I. Freshman year: 1 Unified Mathematics-Algebra, Trigonometry, and Analytic Topics in Teaching Junior High School Mathematics_. Advanced Calculus Since drafts of papers were not requested in advance of the Conference, reports on some of the papers presented are necessarily given in less detail. Eugene W. Hellmich, Northern Illinois State College Hellmich, speaking on the calculus in teacher education, emphasized the need of teachers for a course in the calculus and analyzed the contests and point of view desirable for teachers in such a course. It was recognized that the calculus course has traditionally been greatly influenced by what was assumed to be the needs of engineering students. Possibilities of special courses in the calculus for students entering various fields were explored, but the majority of those taking part in the discussion appeared to favor a single course developed around ideas and experiences which would be beneficial to engineers, teachers, social scientists and other students, alike. Attention was called to the work on the first two years of college mathematics being carried out under the sponsorship of the Social Science Research Council. (See Items, vol. 9, No. 2, June 1955: "Recommended Policies for the Mathematical Training of Social Scientists.") G. Baley Price, University of Kansas Price discussed inservice training for mathematics teachers. He reviewed past, current, and needed programs for teachers on the job, stressing the special importance of the availability mathematics courses for experienced teachers, carrying graduate credit when possible, at this time of "ferment" in secondary school and college mathematics curricula. It was pointed out that, in addition to their own Universal Mathematics, the MAA Committee on the Undergraduate Program has a strong interest in teacher education. He described important contributions which the National Science Foundation has made to the improvement in inservice teacher education, through their academic year and summer institutes, and will make through their new program to be inaugurated in the fall of 1957 providing for evening or Saturday courses for inservice teachers. 1 If a freshman enters with advanced standing due to work completed in high school, he may start his mathematics with the 2d year subjects. REPORTS OF THE RECORDERS I. Mathematics, freshman and sophomore years Leader: H. Glenn Ayre, Western Illinois State College. Recorder: Carroll E. Flanagan, Wisconsin State University, Whitewater. It was agreed by the group present that discussion should be centered around three main topics: I. Identifying the problems in teaching freshman and sophomore level mathematics. II. Current practices in meeting these problems. From these, several were Problem No. 1.-How do we induce small high schools to offer an adequate mathematics program? It was pointed out that some schools offer only one year of mathematics-no opportunity being given for pupils to study geometry. It is difficult to get legislation which would require the teaching of mathematics courses. Indications of the recognition of the problem of inadequate mathematics opportunities by various groups (educational and otherwise) are becoming more frequent. This is encouraging. A Wisconsin law provides that a high school student may take a correspondence course (from the university) in a subject not offered in his local high school and that the local board must pay the cost of the course. Improper counseling and guidance in many schools steers pupils away from the desire for mathematics courses. Colleges might help alleviate the problem by 1. Providing correspondence courses; 2. Television courses; 3. Filed services such as publications, counseling and guidance of high school supervisors and teachers; and 4. By exerting pressure through college entrance requirements. Problem No. 2.-Should there be a general requirement of mathematics for graduation from college? Most teachers' colleges require a mathematics course for graduation. Some allow a choice of mathematics or science or a substitution of foreign language for majors in some fields. There is a trend toward making the required course a mathematics appreciation course or a general mathematics course. College credit is usually given for these courses although they usually do not count toward mathematics major. There seems to be a general upward trend in the percentage of students electing to take mathematics courses. It was suggested that the required course for poorly prepared students be of a different nature from the traditional freshman level course. Perhaps a course based on the development of basic mathematical laws and their application to simple ideas would be appropriate. There was general agreement among the conferees that all college graduates should have some basic knowledge of mathematics. Problem No. 3.-Do colleges desire to have high school graduates with advanced standing courses in mathematics, or do they create too many problems for college mathematics departments? Most colleges are glad to get students with advanced standing in mathematics (i. e., high school courses in trigonometry, analytical geometry, and calculus). Colleges need to know the nature and calibre of the advanced high school courses or give an achievement test. What the colleges do with the advanced standing students depends upon the local situation-the types of courses the college offers and facilities for proper placement of the entering students. Problem No. 4 was an extension of the previous problem. How can the colleges fit the accelerated student into the modern mathematics program? It was suggested the entering accelerated student might be given background courses including topics from modern mathematics. Sometimes he is put into regular advanced courses in college mathematicshis placement depending upon a proficiency test. Usually college credit is not given for the advanced courses taken in high school. (Credit is usually not an important aspect for students.) It was suggested college credit be given if a certain degree of proficiency is evidenced on an examination. Some colleges give high school graduates an opportunity to try the advanced college level courses in summer school before enrolling for the regular academic year. Most (80 to 90 percent) of the schools represented by the discussion group sectioned entering mathematics students by means of a test, high school records, or both. The experts are not yet in agreement as to just what modern mathematics should be included or how it should be presented, but the consensus was that some topics from modern mathematics could be integrated with topics from the traditional courses. Certain phases of geometry, for example, might be omitted or condensed to provide room for the new topics. A few colleges still require solid geometry for entrance, but the trend is definitely away from this. Some aspects of this course might be integrated with other courses. Many groups, engineers, scientists, mathematicians, are pushing the introduction of modern mathematics topics. It is said students develop mathematical concepts faster. More work needs to be done on integrating modern and traditional mathematics and presenting it in a unified treatment. Proper revision of courses is important. Colleges need to provide in their mathematics departments for the training of prospective high school mathematics teachers so that they can go into high schools and teach modern mathematics. Critic teachers and supervisors of student mathematics teachers need to be well founded in mathematics as well as the principles of teaching. A mathematics supervisor from the college mathematics staff is highly desirable. Well written articles aimed at encouraging proper practices in the supervision of mathematics student teachers might be beneficial. One of the best ways to improve the mathematics program in high schools is to train students in the methods and materials of modern mathematics in our college mathematics courses. Colleges and high schools are encouraged to study the problem of introducing modern mathematics and to experiment, but progress should be made cautiously so that the program does not slip backward rather than forward. It was suggested some modern topics might be introduced in the advanced courses and then gradually pushed down into lower level courses. The Illinois project was recommended as being worthy of study and investigation by all interested parties. The problem of the shortage of mathematics teachers was discussed briefly. Suggestions which might be studied included (1) Large lecture sections with small discussion groups with laboratory sections for remedial work; (2) television courses and films of presentations supplemented by opportunities for private conferences. Other plans for teaching mathematics classes and saving mathematical manpower need to be studied, but any successful plan needs to provide ample opportunity for the student to have his questions answered by qualified mathematics teachers. II. Mathematics, junior and senior years Leader: T. C. Holyoke, Miami University, Oxford, Ohio. The problem.--How much mathematics and what kind of mathematics du we want our future teachers of secondary school mathematics to have during their junior and senior years in college? Assumption.-The mathematics teachers whose training is under discussion are to be the teachers of college-bound students. Discussion. There was common agreement evidenced in favor of a stronger mathematics program during these two years. The present offering of the colleges represented is varied, both in extent and content. Dissatisfaction with the course in college geometry was expressed by several participants. There was lively discussion on the merits of theory of equations, mathematics of finance, differential equations, slide rule, field work, and statistics. Mr. Landin, University of Illinois, proposed that the present varied offering for the third and fourth year be replaced with some courses that would answer the following three questions: 1. What constitutes a mathematical system? 2. What is a number? 3. What is the distinction between a number and a symbol for a number? He described the courses now in progress at the University of Illinois. Some topics included are set theory, logic, axiomatic systems, and classical problems of antiquity. Recommendation. The following recommendation was unanimously adopted: There should be three semester hours of work in each of these four basic areas: 1. Mathematical systems (a) Axiomatics (b) Logic 2. Real numbers 3. Modern algebra 4. Geometry (axiomatic treatment) In addition, there should be a total of six semester hours of work elected from these areas: 1. Advanced calculus (including metric topology) 2. Applied mathematics (e. g., statistics, applications to physics, numerical analysis, etc.) 3. Geometry (including topology) 4. Algebra Comments The above recommendation was agreed upon with the realization that it will not be practicable, at present, for most of the institutions represented. Some deterring factors are inadequate staff, lack of suitable published materials, and existing requirements on the secondary teaching curricula. Discussion of these problems led to the question of the offering during the freshman and sophomore years, although this question was treated in another discussion group. The following recommendation was adopted: "Full credit should not be given for courses offered in college that are highschool mathematics courses. This includes intermediate algebra." [Carnegie Corporation of New York Quarterly, October 1957, vol. V, No. 4] "He was President of the United States in the eighteenth century." Does that sentence read like mathematics to you? Well, it's the beginning of a lesson in painless algebra. There's nothing in it to memorize. No tables. No laws. No mystifying symbols. It's merely a statement that can be either true or false, depending on who is "he." If the name of Abraham Lincoln is put in the pronoun's place, the statement is patently false. If "he" is George Washington, the statement is obviously true. This is plain, simple English, the language every American high school freshman understands. Mathematics is a different language, but it can be just as easy to read. Equations are merely statements about numbers that are true if you put in the right names; and numerals are only the names you use for numbers. Numerals aren't "real" any more than names of people are real. You can see and touch the person Paul Jones, but the name Paul Jones itself is a mental invention. And although you can count the 10 fingers on your two hands, if you try to put one of these fingers on the number ten you find that the number exists only in your head. You can see a name for ten but you can't see the number ten. High school algebra, then, is nothing more nor less than the business of making statements about numbers that are true when you think up the right names. Take, for example, the equation: x plus 4 equals 9. The '4' and the '9' are names of numbers. If the language were English, the 'x' would be a pronoun. But the language is mathematics. So the 'x' is a "pronumeral." The only name you can put in place of that pronumeral to make a true statement is a name for the number five. Sould anything be more natural? Wouldn't mathematics be plain fun-even easy-to a youngster who approaches it this way? Yet many teachers will have trouble accepting this method because they have been taught that a child doesn't really understand anything that he can't explain in words. Prof. Max Beberman and a little band of mathematics experimentalists at the University of Illinois have set out to prove that this old rule of teaching is wrong. In an attempt to make math enjoyable to American children, they are turning the high-school math curriculum inside out to make it fit what adolescents actually feel and think as opposed to the reactions ascribed to them by grownups. “When you and I were thirteen, fourteen," Beberman reminds his fellow pedagogues, "we were enchanted by the mysteries of life. We weren't concerned with practicalities. We didn't ask ourselves how we would earn our living after we finished school. We wanted to know if there was a God, and if there was, what did He look like? Was He maybe like the light of day or the sound of the wind in the trees? Why was grass always green? Why were clouds always white or grey or sometimes pink-never blue like the sky behind them? The more abstract a question was the hotter we argued it. Angels absolutely delighted us; how many of them could dance on the head of a pin? "Numbers are as abstract as angels. Mathematics is as creative as music, painting, or culture. The high-school freshman will revel in it if we let him play with it as an abstraction. But insisting that he pin numbers down is like asking him to catch a butterfly to explain the sheen on its wings-the magical glint of the sun rubs off on his fingers and the fluttering thing in his hands can never lift into the air again to renew his wonder." Rather than asking a pupil to explain the fascinating tricks he sees numbers perform, Beberman and his colleagues discourage verbalization as long as they can without frustrating creative curiosity. "You don't ask a boy or girl to dissect his response to the first seascape he sees or the first string quartet he hears," the professor shrugs amiably. "Why expect a description of a mathematical operation?" Not only is nonverbal awareness enough in the beginning, it is preferable, because the language of mathematics is the most exacting tongue man has ever devised, and it can be mastered only gradually through an initial intuitive appreciation followed by long and patient practice. To grownups who instinctively reject this philosophy, Beberman, using a favorite practice of his colleague, Miss Gertrude Hendrix, will ask: "Tell me, what is the sum of the first 60 odd numbers?" When a startled hearer pleads for time and a computer, Beberman seizes a tablet of paper and a pencil. "Let's list the numbers here," he proposes, writing: He pauses momentarily and says, "Whenever you think you can tell me the sum of the first 60, stop me" and goes on Slowly it dawns on his watching companion that each number listed in the right-hand column is equal to the opposite number in the left-hand column multiplied by itself. Beberman stops writing and says, "Fine. Now how would you describe what you recognize to be true?" Try it yourself. You will probably start by saying, "The numbers are squared" or something like that. If you aren't a mathematician, you may spend some time before arriving at the succinct ultimate of mathematical precision, which reads: "The sum of the first 'n' odd numbers is n ('n' being the pronumeral for the name of any whole number you care to put in its place). Nevertheless, you will recall that you did see, merely by glancing at the paper, an almost magical relationship between the two columns of numbers. And you understood the relationship sufficiently to predict that the sum of the first 60 odd numbers would be 3,600. That tantalizing relationship between the two columns of numbers delights an adolescent mind. It tempts the question, "What makes it come out that way?" If the child is encouraged to guess the whyfor-as Beberman pupils are he soon will be formulating the underlying principles of mathematics for himself, in his own ingenious and sometimes in ingenuous language, instead of waiting to hear them pronounced by the teacher in words that inevitably bear the unwelcome sound of laws fixed by grownups. What's the problem? "Why?" is a question which too often atrophies and dies after we leave adolescence behind. Adults who were never freely exposed to math's fascination |