Professor Jack Price, President of NCTM
California State Polytechnic University
Pomona, CA

Dear Jack:

I read your President's Message in the December News Bulletin. It confirms my long-held opinion that you are a man of sound judgment. After gently indicating that there is no such thing as a core curriculum for all students you endorse tech-prep education. So do I.

Now let's look at other situations which should be among the Council's major concerns, but apparently are not.

1. The duplication of content from year to year in mathematics texts for grades six, seven, and eight is so great that, lacking labels, it is actually difficult to arrange them in the intended order.

Some of the time in these precious learning years should be spent developing the gradually formalized natural language so essential for mathematical discourse. This includes set language, extensive two way language-symbol translation, a vocabulary which includes such key words as "and", "or", "not", "implies", "if", "then", "converse", "inverse", "contrapositive" and which confers the ability to formulate the contradictions of conjunctive, disjunctive and implicative statements (see p.1 of Enclosure #1 The Language of Mathematical Exposition). This is the language of rational thought. While it applies to all fields it is probably best learned in the setting provided by early secondary level mathematics. In fact, this may be the crucial time to learn it. Presented here, it greatly facilitates the student's thinking, not only in secondary mathematics (see p. 2-4 of enclosure #1) but in any situation where rational thinking is invoked. It is our mission to present this language.

We are not doing this. Instead, much precious learning time is squandered by the repetitive, largely unstructured, manipulation-oriented presentation of mathematical facts, including many later encountered in tenth grade geometry. This outrageous waste of time which, could be spent in vocabulary and language development, should be a matter of grave concern for the NCTM which, according to the Standards*,is dedicated to the improvement of writing skills.

(*Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, 1989.)

Any attempt to build writing skills without this essential language is bound to fail. One dire consequence of this failure is that the student has no basis for understanding the lexical reasoning and proof which should begin in ninth grade algebra and be the dominant theme in tenth grade geometry and beyond.

2. The gradual elision of lexical reasoning, logic and proof from secondary school mathematics is an unmitigated disaster. Due to the current over- emphasis on problem solving, manipulative techniques and applications, the student of school mathematics does not get nearly enough experience with the various aspects of proof.

There also seems to be a widespread belief that even the most elementary fundamentals of logic needed for mathematical discussion are too difficult for secondary school students. (Actually its their omission that renders mathematics unintelligible.) This belief may stem from the fact that in the past the abrupt introduction of proof in tenth grade geometry, without the language needed to render it understandable, has led to frustrating and even traumatic experiences for both students and teachers.

Whatever the reason, proof in the applications, problem-centered domain of secondary school mathematics is postponed -- suppressed -- downgraded. (We won't have any proof questions on the exam, will we?) There is even strong support for the idea that we should not presume to do much with proof at the secondary level. And the progressive emasculation of American texts in school mathematics, which began in the late sixties, will continue apace! (Algebra without structure -- geometry with little or no proof!)

I ask you this question: Can we suppress proof without distorting school mathematics and seriously impairing the students' ability to understand it? Mathematics is essentially a structured hierarchy of propositions forged by logic on a postulational base. For how long do we protect our students from the pedagogical consequences of this fact? And indeed, whom are we protecting? Those who advocate this language of logical discourse are seeking clarity of exposition rather than pretentious rigor in proof.

Also, due to our preoccupation with applications, there is not nearly enough time spent in deriving key propositions and theorems. When you and I say that we understand the theorem "The determinant of a square matrix is zero if and only if its rows are linearly dependent", we are saying that we understand how this theorem fits into a hierarchy of propositions -- and could, given time, derive it from first principles.

Why should our students be any different'? This nagging question remains: What basis do they have for understanding anything without seeing how it fits into a structure based on something?

Yet when a high school student says that he understands the formula for the cosine of a difference, he generally means that he has memorized the formula and can apply it. He generally does not mean that he can derive it from first principles. Very often he does not realize that this essential dimension of understanding even exists; i.e., his understanding is deficient. So he gets by on memory and facility until the cumulative effects of these deficiencies ultimately overwhelm him, and he leaves mathematics in frustration and despair.

Consider now what is happening in what was formerly known as "demonstrative geometry" (Grade 10). There are still a few geometry texts such as Moise & Downs (a derivative of the first SMSG geometry text Circa 1959) and Rhoad, Milanskas & Whipple that try to give proof its due. But the trend is definitely to reduce emphasis on proof.

The older texts such as those mentioned in Enclosure 2 (Constructions in Plane Geometry Revisited) are so much more demanding than those currently in use that any attempt to use them now would lead to mass defenestrations from our geometry classrooms--probably led by the teacher.

Many recently trained high school geometry teachers have not studied Euclidian geometry since they were high school students in the same course they are now teaching! (Whose fault is this?) As a result they lack the background they so urgently need for effective exposition. And they are desperately afraid of proof which is the culmination of exposition. (Yes, Virginia, there was a time when math teachers were expositors.) Publishers, sensing this situation, have virtually eliminated requirements that the students provide proofs for demanding originals and solve challenging constructions. I have just discovered that modern geometry texts do not provide the student with the where-with-all to solve construction problems of the kind shown in enclosure #2 (Constructions in Plane Geometry Revisited)

This is sad. Of course, some might say that these constructions and originals derived from the old texts (1930-50) are obsolete. They are not obsolete--unless thinking is obsolete. Maybe we are being told that, in the world of manipulatives and electronics, we do not need to think for ourselves anymore. If so, how thwarting for the developing mind! How numbing for the human spirit!

In place of the old, beautifully structured books laden with demanding and deeply instructive problems we now have 600-800 page "teacher-friendly" books whose wide margins or extensive appendices, in the teacher edition, contain answers to the trivial exercises in the student's text. I wonder how C.M. Austin, founder of NCTM, would react to these newer, massive, expensive, typographically spectacular, multi-colored coffee-table books that masquerade as geometry texts in the United States.

Along with this degeneration of geometry texts we sense a strong desire, in some quarters, to replace demonstrative geometry with a laboratory course in which we use manipulatives and electronic devices to "rediscover" many of the geometric facts previously encountered in junior high school. What a stultifying operation!

"Indeed, methods and gimmicks are a popular cop-out in teachers education programs. Universities seem to produce teachers who cannot understand the theory, research or principles underlying their subject, but rather want methods and techniques to satisfy and pacify their charges." Gerald L. Peterson, Saginaw Valley State University in National Forum, Summer 1992, p. 48.

This whole deplorable situation can be aptly described as "The Elision of Proof" and "The Scandal in School Geometry." The Council's failure to deal with it belies its oft-expressed concern about developing "mathematical power".

3. The Need for Reform

Reform! Don't we wish that our students could again attain the higher performance levels they reached in the 40s, 50s and early 60s? Instead of talking about "reform" we should be talking about regaining lost ground.

Keep in mind that when the Standards writers talk about reform they are talking about curriculum and teaching reform not about social reform that would send kids to school ready to learn (Goal 1). Has the council considered the implications of this situation? (See Enclosure 3 Crisis in school mathematics due to social conditions for which standards provide no remedy)

4. What would you, as President of the Council, say to a school superintendent who asks for a basis in research or experience that supports the recommendations contained in the Standards? (See Enclosure 4 The NCTM's own Research Advisory Committee says that Standards recommendations lack research support).

5. In their zeal to cope with the" drill-master's" approach to teaching mathematics which relies largely on developing manipulative skills, the writers of the Standards may be dangerously over simplifying a very important issue: the "trade off between proficiency and comprehension." (See Enclosure 5 Kilpatrick's statement)

6. Many, perhaps most, of our speakers at local, regional and national meetings of the NCTM have had little or no experience applying, UNDER NORMAL TEACHING CONDITIONS, the procedures they advocate.

This is especially true in the field of "assessment." These speakers should be required to take the oath described in Enclosure 6 (Comments on NCTM’s Standards on Assessment). (I took the oath on 10/26/94 before I spoke on "Constructions Revisited".)

The assessment procedures required in the Standards are not only highly subjective, they make such heavy demands to the teacher’s time that it would be utterly impossible to use them under normal teaching conditions.

Judging by the program, the Boston meeting is largely a convocation of math educators who have convened to talk to each other about topics they know very little about. Most of them having taught at the college level for several years could not possibly have had the classroom experience that would enable them to take my oath. Having appeared on our program they can say, "I am in the van of the reform movement in school mathematics."

7. The NCTM ignores the 20th century history of school mathematics. Now in my 86th year, I remember most of this history--and I made some of it at both State and National levels (See Enclosure 7 My Vita with Addendum) The NCTM Board should study this history. If they did they would be less likely to classify as "innovative" ideas that have been hardy perennials throughout this century. Example: Integrated Math.

I first heard of this idea from Professor E.R. Breslich of the University of Chicago in the late 30s at a meeting of the Men's Mathematics Club of Chicago*. Since then it has surfaced repeatedly. The names Swenson, Howard Fehr, Phil Jones, Philip Peak and Vernon Price come to mind. The last three wrote a series of texts based on this principle. Moreover, this list includes the names of three former Presidents of the Council.
*Now called "Metropolitan Mathematics Club of Chicago".

Did the Standards writers give credit to their distinguished predecessors in the field as ethical practice requires? Did they say, "We agree with these proponents of integration in the field of school mathematics and we think it should be tried again?" No. To do so would have been an admission that something worthwhile happened in school mathematics before 1987. An idea which is anathema to them. Instead they presented it as another innovation; another one of the "visions" with which they have been blessed, thus taking advantage of the public's ignorance on the subject. This is shoddy and reprehensible.

Another instance where old ideas are hailed as new is provided by the subject area called "Discrete Mathematics". Set language, counting theory involving permutations and combinations, Pascal’s triangle, proof of the binomial theorem for positive integer index, mathematical induction and probability have been taught in our secondary schools for many years in courses such as College Algebra. Texts such as Rosenbach and Whitman, Fine, Reitz & Crawthone come to mind. It is quite misleading therefore to suggest that the introduction of topics in discrete mathematics is a major change. I am reminded of Molier's seventeenth century play in which one of the characters reacts to the information that any language which is not poetry is called prose with the exclamation "Good heavens, for more than 40 years I have been speaking prose without knowing it." (Jean Batiste Moliere, "Le Bourqeuis Gentle-Homme", Act III, Scene 4, Circa 1670). Yes, and the ill-defined term "discrete mathematics" denotes a very wide range of traditional topics that we have been teaching in high schools for well over forty years without paying much attention to the label. {By the way, what do we call mathematics which is not discrete?}

8. What's in a name? A word by word analysis of the "National Council of Teachers of Mathematics."

(a) "National". The organization of which you are President is truly national in scope. In fact, it could be considered as international in some respects. Much of its growth springs from the work of the Committee on Affiliated Groups which was a high priority during my term as President 1962-64 (perhaps it was launched then, I can't remember). Anyway, I found a wonderful leader for this important committee. It was Isabelle Rucker, a southern belle from the Commonwealth of Virginia, who could charm the birds out of the trees. She got the CAG off to a roaring start and look at the result!

Also during my time, we appointed Jim Gates as Executive Secretary, thus putting an end to a sticky problem which had plagued three previous administrations (I have the letterheads to prove this). Jim has proved that he could function effectively in this responsible position in a national organization. (As Jim leaves it would be appropriate to publish a comprehensive audit of NCTM finances, showing where our funds come from and how they are spent. I'm sure Jim would want that.)

Your organization may keep the word "National" in its title.

(b) Council. "An assembly convened for consultations, advice or agreement." Your organization was a council once. It's meetings provided teachers with a place to convene as peers, in a collegial atmosphere to trade experiences gained in the classroom and to discuss perennial controversies about effective procedures. No more. The "council's" steam-roller promotion of the Standards sweeps all else aside. There are no controversies anymore. Everything was settled when the Standards Report was published. If you do not regard the Standards as holy writ, if you even question its sweeping and largely baseless recommendations you are a non-person so far as the "council" is concerned.

(c) Teachers. The Standards recommend variations of the teaching process which, although used occasionally and under appropriate conditions in the past, are now over-emphasized. An example is the extensive utilization of cooperative learning groups. This is a tremendously time-consuming procedure. This recommendation emphasizes once again that the Standards' writer's attitude toward that precious commodity, learning time, is about the same as that of our Federal Government toward our tax money. Spend it lavishly and often not too wisely. Cooperative group learning is an "unfunded mandate" insofar as teacher time is concerned. Let us keep in mind that parents do not send their kids to school to learn from other kids. Moreover, cooperative group learning has a totally unacceptable corollary: group examinations where group performance affects or even determines the individual student's grade.

Indeed, many of the instructional procedures advocated by the Standards cannot be characterized as "teaching" in the accepted sense of the word. The dictionary definition depicts the teacher as a director, one who imparts knowledge, an instructor. Teaching suggests the personal relation of master and pupil. (One might proudly say, "I studied under Benoit Mandelbrot.") (Or at an earlier time: "Mr. Fount Warren, my geometry teacher in high school was great. He was a tough grader but he was fair. He really knew his subject and his enthusiasm for it was contagious. Like a great coach, he pushed us to ever higher levels of understanding. One day toward the end of the year a student asked respectfully, 'Mr. Warren, don't you get tired of teaching the same geometry year after year?' He thought for a moment and then replied somewhat as follows. "There are great classic theorems. Every time I present them I see something new. No, I never tire of them any more than a musician would tire of the great symphonies by Brahms, Beethoven or Bach." "I shall always be grateful to Mr. Warren for the indispensable help he gave me in organizing and focusing my ability to think and to communicate my thoughts to others."

On page 245 of the Standards we bid good-bye to the teacher as a "Directive Authority. "The procedures recommended in the Standards (without research justification) require such a drastic modification of the teacher's role that they can no longer be regarded as "teachers" in the accepted sense of the word, "facilitators of learning" perhaps but not "teachers". Much nearer "A guide on the side" than "A sage on the stage." One wonders if these bumper-sticker slogans really convey the "spirit" or is it the "vision" of the Standards. Or is someone suffering from a terminal case of the "cutesies". Well, two can play at that game. Consider the following deadly doggerel.

Set to music, this would make a highly appropriate theme song for the Standards-dominated "Council."

Well, it appears that we can no longer use "teachers" in our official designation.

(d) Mathematics. When we speak of Standards in school mathematics one expects a set of performance goals to be met by the student.

Examples from physical education:

• The candidate will run a mile in x minutes.
• The candidate will do x push-ups in one minute.

Examples from school mathematics:

• The candidate will demonstrate his understanding of the relation between the coefficients and the roots of the quadratic equation.
• The candidate will use matrix methods to solve systems of linear equations.
• The candidate will use a graphing calculator to find the real roots of a polynomial equation.

Consider also one from the Report of NCTM’s Secondary Mathematics Curriculum Committee (1959), a portion of which is enclosed (Enclosure 8 Excerpt from the Report of the NCTM’s Secondary School Curriculum Committee).

"The student should be able to recognize and formulate mathematical problems."

(Note that this report is much better written than the "Standards." For one thing it is not so mercilessly repetitive.)

It is altogether fitting and proper for the NCTM to set such performance goals -- provided it is done in consultation with the mathematical community, and is periodically revised.

But the NCTM has not set such performance standards for students of school mathematics to meet and be judged by. Instead, it has attempted to set standards from something which should not be standardized; the teaching of mathematics.

The teaching of mathematics, or of any subject for that matter, is an art not to be done "by the numbers" as I am sure you have heard Professor Smith say many times. We should set performance goals for students and rely on the inexhaustible ingenuity of mathematically competent classroom teachers throughout the land to find many ways to achieve them. (If they are not mathematically competent the NCTM should be concerned about this.) In former years, their efforts were enhanced by our NCTM meetings which provided an open forum for the exchange of teaching ideas. This multiplicity of effective teaching strategies should be well recognized by the Standards writers who have loudly proclaimed their "newly discovered" notion that students should be encouraged to find several ways to solve a problem. It renders the idea of setting standards for teaching rather vacuous.

I spoke earlier of "consultations with the mathematical community." Where is your liaison with mathematics? On my Board of Directors there were three mathematicians. On your Board there are none. On my Secondary School Curriculum Committee there were seven mathematicians, and on the sub-committees there were four more. (See Enclosure 8) On the NCTM Commission on Standards, there are possibly two, one of whom recently left the chairmanship of perhaps the best college math department in the nation to accept a job inside the Beltway.

Clearly our liaison with the mathematical community needs rebuilding. In rebuilding it, turn to mature, well-established mathematicians with unassailable research credentials such as Householder, B.W. Jones, Magnus Hestenes and R.H. Bing, who served on our Secondary School Curriculum Committee. Consider Deborah Tepper Haimo, a mathematician of the first rank and Past President of the Mathematics Association of America. In her retiring presidential address she gave a powerful argument for retaining an emphasis in proof in mathematics. (Enclosure 9)

Another question: Why are strong math majors who were evidently attracted to mathematics by the logically structured proof courses which follow calculus, so reluctant to introduce students to formalized language, logic and proof at the high school level? This is a mystery. Of course, there is one scary explanation. Maybe strong undergraduate majors in mathematics are not as prevalent as one might expect among the 15-20 thousand people who will attend the Boston meeting of mathematics teachers. Who are these people anyway?

Clearly you have lost your focus on the subject we are supposed to be teaching. You have misrepresented its essential nature and virtually abandoned the idea that mathematics is a structured subject whose central and defining idea is proof. This great idea can be developed in ways that are effective in varying degrees for students of every ability level. It is only when this is done that school mathematics makes its unique and indispensable contribution to the education of all youth.

This loss of focus is manifested in many other ways.

(a) Failure to promote the defining and intrinsic values of 9-12 mathematics. If structure, proof and enhancement of the thinking process do not justify teaching 9-12 mathematics as a separate subject, to be taught by teachers who can exploit and develop these great ideas, what does? Having abandoned this position in favor of an inductive, MANIPULATIVE*, problem solving approach based on MANIPULATIVE techniques, the NCTM has no answer.

(*In the descriptions of the last 203 sessions of the Boston meeting, the word "manipulative" and its derivatives appear 70 times. The word "proof" and its derivatives not at all! By the way, didn't our Third Yearbook stress "The Folly of Emphasizing Applications"? Maybe we should reissue this.)

As a consequence, we are confronted by an increasingly successful and commercially profitable effort "to convert 9-12 school mathematics into a laboratory subject" which it should not be. High school math departments are being absorbed into math-science divisions at an alarming rate. The mathematics department at Lyons Township High School, La Grange, Illinois, which I chaired from 1956-68, went down the division tube last September. If you view this trend with indifference you are saying, in effect, that 9-12 school mathematics has no intrinsic values that justify its being taught as a separate subject. This would be a bizarre position for a group of math teachers to take.

Mathematics Department Chairmen who promote this inductive, manipulative, problem solving approach are mindlessly sawing off the limb that presently supports them.

The following sequence is developing now.

Math instruction emphasizes applications for justification and motivation ---> school mathematics (grades 9-12) valued only as a tool for other fields ---> Absorption of math departments into "Divisions" ---> De-emphasis of mathematical structure, theory and proof through the use of calculators and computers which get results quickly and largely obviate the need for knowing "Why." (Teachers of other subjects think that they can punch these little keys and elicit stored programs as well as we can, and they are probably right.)

At the 9-12 level, it is really not practical to seek out problems and then try to develop the mathematics needed to solve them as required by the "Agenda for Action", which advocates organizing school mathematics around problem solving. Such problems are valuable only insofar as the student encounters similar ones in the future. Instead, our purpose should be to exploit the structured nature of our subject and its amenability to logical analysis to enhance the students ability to think. This so he will be able solve problems which are today unforeseen and unforseeable. Listen again to the great Dieudonne who was, as you know, one of the leading spirits in the Bourbaki movement.

"For indeed what good do we seek--Certainly it is not to introduce them (students) to a collection of more or less ingenious theorems about the bisectors of the angles of a triangle or the sequence of prime numbers, but rather to teach them to order and link their thoughts according to the methods mathematicians use because we recognize in this exercise a way to develop a clear mind and excellent judgment. It is the essence of the mathematical method that ought to be the object of our teaching, the subject matter being only well-chosen illustrations of it." *

*See also Enclosure 2.

(b) Still another indication of loss of focus is our preoccupation with peripheral issues such as "equity", anxiety", "diversity", and "multiculturalism". These are social issues. We should try to teach our students how to think, not what to think. Let us try to remember this even though many college and universities seem to have forgotten it.

The elision of proof and the prevailing aversion to abstract thinking serve to destroy the aesthetic values of school mathematics. This is sad indeed. Listen to Davis and Hersch:

"Blindness to the aesthetic element in mathematics is widespread and can account for a feeling that mathematics is dry as dust, as exciting as a telephone book. Contrariwise, appreciation of this element makes the subject live in a wonderful manner and burn as no other creation of the human mind seems to do."

(c) Proposing methods and procedures which are no more applicable to mathematics than to many other subjects. These include; cooperative learning, assessment of disposition, portfolios, rubrics, etc. These are general methods, equally applicable to other subjects and probably even more appropriate for content subjects such as those in the social studies field.

(d) Neglect of the gifted student. When the press interviewed you about the outstanding success of our mathematics team in world competition (five perfect papers!) I hope you had the good grace to explain that the NCTM had nothing to do with this. While most NCTM leaders are elitists in their daily lives, they are acutely uncomfortable with the idea of an elite group of mathematics students -- and even more uncomfortable with the intensive training methods that were used to bring them up to speed. Your indifferent and even hostile attitude toward able mathematics students is part of a larger malaise of our educational establishment which is gradually coming to the attention of the general public (see Enclosure 10). It is also manifested in your neglect of the brilliant mathematics students who belong to Mu Alpha Theta, of which, I believe, you are cosponsor along with the Mathematical Association of America. These are your "mathletes". One would expect that an organization focused on mathematics would encourage them and would showcase them in the regional and national meetings. I am proud of my contribution to Mu Alpha Theta and I resent the way you have neglected and ignored it.

In view of this almost total loss of focus, we had best drop "Mathematics" from our official title.

Recognizing the value of the "NCTM" acronym, I suggest it be retained to represent "National Congress for Technical Manipulation."

Some of your advisors may say, "This is an angry old man who cannot accept innovative ideas." Where did we get the idea that innovations are necessarily good? They probably said the same thing about the elderly Germans who opposed the rise of the Third Reich under Hitler.

Other advisors may say, "Let's ignore this indictment as we have others from the same source in previous years. We have nothing to fear.. He has no way to bring it to the attention of the general public." This may work for a short time, but in the long run it is a losing strategy.

In the meantime, I respectfully request a point-by-point response from you with the assurance that it represents the consensus of your Board of Directors.

Sincerely,

Frank B. Allen

President of the National Council of Teachers of Mathematics, 1962-64

P.S. I have developed suggestions for recovery from our present theoretician-induced crisis. They involve goals for students, teachers who are skilled expositors and students who learn to listen so that they can first follow authoritative instructions and perhaps later give them. The kind of disciplined learning situations which, if the present trend continues, will soon be found only in private schools, where 22 percent of the NEA members (twice the national average) now send their own children.

P.P.S. I am sending a copy of this letter and its enclosures to Jim Gates in order to facilitate his sending a copy to each member of the Board of Directors at your signal.

Any impartial investigation of the NCTM, its policies, publications (particularly the Standards) and programs for regional and national meetings will abundantly verify the allegations made in this document. I think you realize this.

As I have said before, I seek no adversarial relationship with the Board. But I have no choice, I must oppose policies which I believe to be deleterious to the learning of secondary mathematics in the United States.