(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 a 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. This report was published in THE MATHEMATICS TEACHER in May, 1959.
"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 aproblem. 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 "Report of the NCTM's Secondary Schools Curriculum Committee" referred to earlier.) 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. (This appeared in the February, 1994 issue of FOCUS.)
"From proof that's tough we will abstain."
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.
"Appeal to the hand instead of the brain."
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 unforeseeable. 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, sent by mail to those who requested it.
(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 the article "Dumbing Down of American Students" by nationally syndicated writer Eugene Kennedy, CHICAGO TRIBUNE, 2/21/93). 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.
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 authorative 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. -------------------------------------------------------------------------------- I will send copies of any reference mentioned here to anyone who sends me a self-addressed envelope.