My Experience With the CEEB Commission Report

 

          On July 1, 1959 I became Acting Chairman of the department of mathematics at the University of Rochester, for one year.  My major task that year was to find and hire a real Chairman, to be brought into our department from outside, and I succeeded: Leonard Gillman was appointed in 1960 and in the fall of that year I reverted to the ranks.

 

          In about January of 1960, while I was still the Chairman,  I was visited by John Montean, a professor of science education in our College of Education, asking me to find someone in my department, preferably myself, who would accept NSF money for a “Math Institute”, something I had only dimly heard of through reading the Report of the CEEB Commission.  I had recently read the 1959 Report of the CEEB Commission, which, along with its larger, companion, volume called Appendices had been mailed to all members of the MAA.  It had impressed me greatly, and favorably.  In particular, it includes a long recommendation on the education of secondary school mathematics teacher beginning on page 50, describing fairly well how such Institutes as Montean had in mind should operate, and also urging courses, given by mathematicians, to provide such teachers with actual teaching-material that they had not received in their earlier training, mathematics that could be found in the Report’s Appendices.  To me, as I read it again, the Appendices, or major parts of it, seemed to fit marvelously as a syllabus for a course such as Montean was suggesting.

 

          The resulting NSF grant was to Montean, not me; he did all the administrative labor – almost.  The grant paid the tuition for 30 working high school teachers to come to a classroom each Saturday morning during the school year 1960-1961 for a course in “modern mathematics”.  The participants received a small stipend, too, and I received some payment  ----I don’t remember how much, but it was useful. 

 

There was one thing more:  Montean (a professor of Education) insisted that the course be given six hours of graduate-level credit; that’s what the teachers would want.  In New York, as in many other states, a Master’s degree is required for permanent certification as a high school teacher, and young teachers, temporarily certified, often take courses part-time towards that end, even if they finish up (or begin) with a semester in residence somewhere. The law intends that high school teachers be required to be academically more competent than one can expect of elementary school teachers, but in the case of mathematics and the sciences it is really impossible to ask a Master’s degree in the subject matter.  Persons of that competence in math or physics generally continue for PhDs or go into technical jobs paying much more than teaching in school.  Future teachers therefore mostly get their Masters degrees in colleges of education, where the mathematical content of their “methods” courses is not demanding.  At least, this was the case in 1960, and as for teachers already in service in Rochester-area high schools, virtually none of them could qualify for any graduate courses at all in our own program. (If they could, the entire Institutes Program would have been unnecessary.)  On the other hand, an in-service course in what is essentially high-school mathematics, however necessary and valuable, could not honestly be labeled a graduate course in our math department. 

 

Yet the course I had in mind was what the Appendices was written for. Except for its clipped exposition and its lack of enough exercises for the student, it is an admirable text, at least for the professor, who naturally finds it easy to expand on the exposition and supply more examples.  Much of the Appendices markedly resembles, in fact, a  textbook that became exemplary in the following decade or two:  Fundamentals of Freshman Mathematics , by Carl Allendoerfer and Cletus Oakley (McGraw-Hill, 1959).  This was no accident, since Allendoerfer was one of the more prominent members of the CEEB Commission.  But Freshman mathematics – for graduate students?

         

Well, I had to solicit an exception.  Technically speaking, the decision to award graduate credit in mathematics is a function of the Graduate Committee of our college (not the college of education).   They understood the delicacy of the problem as well as I and Montean; but this was a special case.  Since the course was of limited duration (it would never be given again, in fact), there would be no danger of Arts College graduate students discovering it as an easy way to pile up “elective” graduate credits.  Furthermore, I told the Graduate Committee, the NSF, and myself, I would be teaching a rare and necessary skill to these teachers, something the country would not have benefit of without this Grant.  Nobody in the College of Education was as qualified as me, I said modestly, and besides, the NSF requirement for the grant needed a mathematician. Being Acting Chairman helped, too, and so  my recommendation to the College’s Graduate Committee for a six-hour donation of credit for completion of my course was granted.  Two courses, actually, Math 212 and 213, fall term and spring term, three hours attendance each Saturday (9 to 12 a.m.) throughout the school year, award of three hours graduate credit for each course. 

 

The Arts College Graduate Committee did understand the scholarly hypocrisy invariably attached to having a College of Education graduate degree awarded equal status with one in math, history or engineering.  That  hypocrisy, which seldom has to be confronted as directly as in the time of the NSF Institutes of the newmath era, is something every university has to endure.  Its history is a long one whose telling is beyond the scope of this story.[1]

 

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          Looking back on this episode I see that I made one serious mistake.  I suffered from a democratic virtue in those days, the days of my youth, the 1950s, long before such democratic thinking became obligatory in the university turmoils of the 1960s and 70s. Thinking that the best educated high school math teachers had less need of my course than the average ones, and that the least educated would probably not be able to follow the course at all if its content was to be non-trivial, I selected an intellectually “middle-level” thirty persons from my very large group of applicants.

 

Montean had seen to it that this Institute was advertised in all the high schools of the region, and I must have received sixty or more applications.  Each one came with a Curriculum Vitae and all the college transcripts and course credit listings the applicant could amass for me.  Strange transcripts are of course hard to read, and I had to take account of the quality of the colleges my applicants had attended, the grades received, their SAT scores, if any; but all in all I probably did get the middle group.  What I didn’t get was what I had innocently expected to get: a group that would learn what I thought I was teaching.

 

          I should have selected only the very best.  I began the first Saturday session with exactly the thirty students I had accepted; 28 finished the first semester with a passing grade (quite minimal, and indeed fraudulently passing for some), and of these, only 20 finished the second course with a passing grade.  That was democracy at work, I supposed; in my usual university courses I had never had to face the problem of grading a group almost all of whom hadn’t learned the subject.  Fundamentally, the only listed failures in the present case were those who were too often absent, or hadn’t handed in at least half the written exercises (no matter how well done).  Half the rest got the lowest grade, “p” for “pass”, and the rest got A’s and B’s, sometimes decorated with a plus or a minus.

 

          I would estimate that about five of my students really gained something from this course, but I cannot be sure it was not more.  Spending a year hearing the words and looking at the diagrams, and attempting to solve problems – even without success -- might itself have been helpful to some of my near-failing students in later years, when called on to teach “trigonometry”.  For example, almost none of these high school teachers had previously known that sine and cosine could be defined for “angles” larger than 180 degrees; and, needless to say, they had earlier known nothing of the graphs of the trigonometric functions (radians were also a mystery), or of the exponential and logarithms, either (base e was another mystery).

 

          Those teachers were my age, after all, some of them older.  Probably all of them had learned, as I had, that “a logarithm is an exponent”, an abbreviation of the ungainly, though complete, definition, “The logarithm is the exponent to which the base must be raised to produce the, er…, the number you began with.” This definition, which I remember we stood by our desks and recited when I was in high school, suffered from the start by not saying just what the logarithm was a logarithm of, since the idea of “function” did not appear at all in high school math of the 1940s.  And since there was no sense, in such a recited definition, of the idea of function, no picture of a graph in our minds, the exercise of interpolation when it came time to use the tables, was to us a very long story.  How much easier to know the inverse relationships, log(exp(x)) = x and exp(log(x)) = x; how transparent the interpolation process when we had a picture of the two functions!

 

I tried to teach such things.  My goal was simplification, not filling the air with newmath jargon.  But I did not succeed. The main trouble was that they were overworked.  During the school year they taught a full load, and then on Saturday had to drive to my class besides.  They had little time for homework, the exercises for which I wrote out for them every week, duplicating them on a “Ditto” machine, something that no longer exists these days but serves well to make copies of handwritten text.  As the year went on I received fewer and fewer responses.  I expected the work I assigned to take three or four hours, plus what time was needed to review the class notes.  They didn’t have it, or didn’t have the strength for that much.

 

My class was not entirely lecture and homework; I included student problem-solving at desk or blackboard, but the results of in-class problem-solving were so dismal that I did that less and less as the year went on, hoping that more leisurely work at home, consulting familiar books of their own (They each had a copy of the CEEB Appendices, too, courtesy of NSF) or talking things over with each other where possible, would give them some ideas and some confidence.  By the end of the year I was receiving next to no handed-in solutions to my exercises. 

 

The typical class began with a discussion of the preceding week’s homework.  I would solve crucial problems at the blackboard, asking questions of the class as I went along, sometimes digressing, sometimes a Socratic procedure tending to the solution of the problem at hand.  Then I would explain the next bit of text, exhibit some problems with solutions, and have the students do something simple at their desks with the new definitions, in preparation for doing their homework.

 

          I’m afraid most of them were lost.  One day after class, two of my students, women older than me, came up to me with a question, something they had not understood the preceding week.  I was appalled; what they didn’t understand was at the heart of the matter.  They were in no condition even to begin that homework they were supposed to (but didn’t) hand in to me that day.  So I took a few minutes – this was Saturday and no school bells were ringing to push us anywhere – to begin at what I thought was the beginning, to gather up what they needed, illustrating it on the blackboard, too.  They stood there dumbfounded – and I was being ever so plain!  One of them said, “My, you’re so young, to know so much.”  It made me so sad, to know that there was no way to reach them with the basis of this course, at least in the time available.  They knew that, too.  There was no animosity in that exchange.

 

  There were, however, some simple exams and a final exam.  I needed something on which to base a grade, and I entered many little grades in my classbook (which I still have), some for exercises and some for short, mainly factual, quizzes.  By accident I have found a copy of what was probably my last effort, a ditto sheet of particularly ambitious exercises I handed out on April 15,1961.   It has been folded into my copy of the Appendices, which like some other of my old textbooks I have kept all these years. (I reproduce it below.)  Looking in my classbook today, I see that I have listed no grades for that last assignment, which should have been handed back to me the following week.  Many of the better students, who ordinarily handed something in, were absent altogether that following week.  Embarrassed, perhaps, at having nothing to hand in.  I think practically the whole class took a bye on this one. 

 

I have no later “homework” grades, either.  By April I had given up.  Except that I recall my discouragement with the course, I no longer remember exactly what happened in response to this or other assignments concerning elementary functions, the capstone of the course.  Nor do I remember how much of the Appendices I actually covered during the course, except that it was far from all of it.  Whether or not, or how much, they understood was something I no longer dared investigate by Socratic dialogue at the end of the course, as I was wont to do earlier.  My Socratic dialogue grew more and more like a monologue, my questions becoming more rhetorical than real as time went on.  But while there were no more than four who were handing  in the homework by the end of the term, I continued to assign it.

 

While the CEEB Commission Report lists exponential and logarithmic functions as part of their recommended 12th grade syllabus, the Appendices don’t have them.  Evidently I tried to include them, somehow, for here is that last assignment, verbatim, copied here from my ditto page:

 

Math 213       Homework     April 15, 1961

 

1.       Use a table of powers and roots to obtain a table of values for 3x when x = -2, -1, 0, 1/3, 2/3, 1, 4/3, 5/3, 2; plot the points on a good piece of graph paper and connect them smoothly to get a graph for y = 3x over the interval

–2 ≤ x ≤ 2.

 

2.       Draw a tangent line at (0,1) and estimate the slope of the graph there.

 

3.       Calling the answer to Question 2 by the name c, show that the tangent line at (1, 3) has slope 3c, and that the tangent line at (-1, 1/3) has slope (1/3)c.

 

4.       Use the graph to find an x such that 3x = 2, and to find a number u such that 3u =1/2.  Do you see anything that can be generalized?  If not, try again:  Find x such that 3x = 4 and u such that 3u = ¼.  State the general rule.  Can you prove it?

 

5.       Take another piece of graph paper and plot the log3 function, i.e., the graph y = log3x over the interval 1/9  ≤ x  ≤ 9.

 

Raimi

 

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          Yet the prescriptions of the CEEB Report remain today a good model for a pre-calculus program, though to cover it all, whether in a part-time Institute or as a full scale MWF college course, seems to me to be more than one year’s worth.  On the other hand, the sections on complex numbers are unnecessary for both the geometry and the analysis involved in a calculus course, and could well be replaced by a chapter on the exponential and logarithm functions; and I would certainly also include some graphing of rational functions with factorable or factored numerators and denominators.

 

My  Institute course didn’t try to include all the topics listed in the Contents of the Appendices, some of which isn’t exactly “pre-calculus”; but I did add the logarithms and exponentials.  I was anxious to give them some idea of the notion of function, with as many accessible examples as possible, so I omitted most of the suggested geometry

 

Here is the table of contents of the Appendices:

 

Contents of Commission on Mathematics Report: Appendices

 

Introduction, page vii                               

 

Algebra, page 1

 

An introduction to algebra

Sets, relations, and functions

A classroom approach to irrational numbers

The linear function and the quadratic function

Introduction to complex numbers

Limits

Permutations, selections, and the Binomial Theorem

Mathematical induction

The mathematics of collections of objects

 

Geometry, Page 108

 

Some reasons for modifying the traditional treatment of geometry

A note on deductive reasoning

The first sequence of theorems

Indirect proofs

Introduction to coordinate geometry

Theorems having easy analytic proofs

Outline of a unit in solid and spherical geometry

Geometrical transformations

Order relations in plane geometry

 

Trigonometry, page 178

 

Introduction to vectors

Coordinate trigonometry and vectors

Trigonometric formulas

Circular functions

 

Index, page 224

 

          I have italicized all those sections – there are only two of them – that sound at all like what was, in the 1960s,  being objected to by opponents of the newmath. Neither the CEEB Report nor its Appendices appear to me today to contain an overemphasis on structure à la Bourbaki, as the outcry that greeted its publication, especially in view of what came several years later, would make one think.  Morris Kline would make one think it was pure Russell and Whitehead.

 

Yes, the notion of “set” was essential to almost everything in the book; but those sections had not been included for abstraction’s sake at all, or in ignorance of the fact that mathematics has something to do with the real world.  In writing of a solution of an equation, for example – and this I saw myself while teaching my teachers – a student can be confused if he is not told what the sentence “Solve  x2 + 5x + 6 = 0” means.  It does not mean, “Take 5 and divide it by two, …”; it simply means, “Find all the x such that x2 + 5x + 6 = 0.

 

Yet, to my audience questions remained. “All the x”?  Is x a number, is it two numbers?  (Try telling them “x is a literal number” as was common in 1950 and see where that gets you!)  The phrasing, “Solve the equation …” is unclear unless there is the understanding of the intent of the symbol “x” in the problem.  With a minimum of explanation, the matter is clarified by the set-theoretic translation, “Find {x | x2 + 5x + 6 = 0}.”  Corresponding symbolism was becoming common in calculus books during that period, and if the teachers teaching it hadn’t made such a long story of the notion of “set” itself they would have made a better story out of such things as comparing {(x.y) | x2 + y2 = 4} with {(x.y,z) | x2 + y2 = 4}.  When I went to college I found it quite unpleasant that “ x2 + y2 = 4” was entitled sometimes to be a circle and sometimes a cylinder. Here in 1960 it became clear.  Should not the high school teachers know this?

 

          Even more elementary things sometimes are cleared up with set notation.  A common “definition” given for the idea of “circle” is “A circle is made up of all the points equidistant from the center.”  At least, you will sometimes hear this from students.  What is wrong with this phrasing is that it doesn’t define a circle.  A circle is a set of points, and to know a circle one must know a rule by which a tested point is known to be on or not on the circle.  A defined circle must have a defined center and radius, and a proper definition, in the plane, is meaningless if it does not invoke both data and give us the criterion, viz., “A circle with radius r and center C is {p in the plane | dist [p, C] = R}.”  From this the Cartesian equation may follow, of course.

 

          The judicious use of the word “set” can be a great help in stating almost everything in analysis; but it doesn’t need a chapter, or Venn Diagrams, for such a purpose.  The CEEB Commission could have gone a bit easier on the subject, and probably would have done so if it could have foreseen that its work would soon become something the authors of textbooks for middle schools and even elementary schools would be trying to imitate; but as writtenand it must be remembered that the Commission recommendations were intended for the teachers of high school college-preparatory students only – the Report and its Appendices were not the call for abstraction for abstraction’s sake that their critics imagined.

 

          Even so, when I tried to get these ideas across in my teaching about algebra and the elementary functions to the high school teachers in my Math 212, 213 of the year 1960, I failed.  I believe I would have done better if I had repeated the course with a group taken from my most skilled or best-prepared candidates.  I see now that the seeding of the high schools with teachers who do understand these things would have been of benefit to their college-preparatory students, and maybe to other teachers.  Too late to be wise;  I never taught in another Institute, summer or winter, and in 1975 the whole Institute program was closed down.   Federal money for Education, albeit under other headings, never was diminished, even in the Nixon administrations which closed the Institutes, but the invitation to mathematicians to participate in projects for the improvement of school teaching, and the improvement of textbooks, was replaced by 1975 by a corresponding invitation to professors of Education instead.  And there it has remained.

 

                   Ralph A. Raimi

                   Revised 21 December 2006



[1] See, for example, Cremin, Lawrence A., The Transformation of the School: Pro­gressivism in American Education 1876-1957.  NY, Knopf 1961; and Duren, Wm.L., the chapter, Mathemat­ics in Ameri­can ­Society 1888-1988, in Duren, Peter, et al (Editors), A century of Mathe­matics in America (Amer. Math. Soc. 1988), Part 2, pp 399-447, for some orientation concerning the intellectual decline of the profession of teaching in America.  Other, more recent works to similar purpose are by Diane Ravitch:   The Troubled Crusade, NY, Basic Books 1983, and Left Back: A Century of Failed School Reforms, Simon & Schuster, 2000. Finally, a polemic which damns the education schools from top to bottom is Albert Lynd’s Quackery in the Public Schools, Little, Brown, 1953.