Repairing school mathematics in the US

The three-day meeting of the National Council of Teachers of Mathematics (NCTM) in Washington DC (April 2-4) should serve to focus the nation's attention on the crisis that exists in school mathematics today and on the NCTM's role in that crisis.

That a crisis exists cannot be denied. The meeting comes just a few weeks after the nation was stunned by reports of the abysmal performance of our best high school seniors on the mathematics examination given to them and their peers from twenty other countries, by the Third International Mathematics and Science Study (TIMSS). On general math skills these US seniors, some of whom are taking calculus, came in 19th in the 21-nation field, surpassing only Cyprus and South Africa. Our performance was really worse than it sounds because, as John Leo notes in the 3/9/98 issue of the US News and World Report, "If the Asian countries who ordinarily score very high on TIMSS exams, had participated, we might have been fighting for 39th place in a field of 41."

Now NCTM policies, as expressed in their three "Standards" reports, have dominated American mathematics classrooms since well before these seniors entered high school. The first of these reports "Curriculum and Evaluation Standards for School Mathematics" appeared in draft form in 1987 and the theories involved were taught in our Teacher's Colleges long before that. When our fourth graders scored a little above the average nation on the TIMSS math exams, the NCTM was quick to claim credit. When we came in 28th out of 41 on the TIMSS eighth grade math exams, there were many rationalizations. Some of these were embarrassing. Now is the time to ask the leaders of NCTM, "Is the disgraceful performance of our best high school seniors attributable in part to the policies you have imposed on mathematics classrooms throughout our nation?" [Hint -- "Our policies have not had time to operate" and "We have been misinterpreted" are not acceptable answers.]

NCTM leaders must admit that they have urged the application, on a national scale, of highly controversial methods of teaching before they have been adequately debated or even understood and before researchers have verified them by well-controlled and replicated research studies. Should the FDA allow a new drug to go on the market under such circumstances? Under NCTM domination, our entire school system has become a laboratory for the testing of untried methods. Consider just a few examples:

• The highly retentive memory of youth which has been used for centuries for learning the number facts and the fundamental operations (algorithms) of arithmetic, is being supplanted by the use of calculators in the early grades. This destroys the foundation on which the understanding of mathematical concepts can be built. Countries that score highest on the TIMSS tests do not allow such early use of calculators.

• There has been widespread application of the doctrine of "constructivism" which asserts that students understand and remember only those concepts that they "construct" or discover for themselves. No longer should teachers (or even books) disseminate information. Instead, students are often placed in "cooperative learning" groups without direct instruction from the teacher who is relegated to the role of "facilitator." This requires students to reconstruct the great ideas of the past, starting at ground zero and aided only by their equally uninformed peers. Many well-informed people view this new classroom situation with alarm. They say that it destroys the cumulative nature of knowledge, strikes at the very heart of the education process and, according to the TIMSS test scores, is not working very well. Yet, group learning is spreading like wildfire in our nations mathematics classrooms and teachers are becoming facilitators on a wholesale basis.

• The new math "facilitators" are now using "authentic" assessment (grading) systems which minimize the importance of correct answers and often include deliberately ambiguous questions (prompting the use of the term "fuzzy math"). They use subjective and inaccurage grading techniques that lack the reliability of objective tests. Worse yet are group tests, which often follow cooperative learning. They destroy the validity of course grades, mask individual performance levels, and make the assignment of individually prescribed remedial work impossible.

• There has been a downgrading of proof in plane geometry almost to the point of elision. The older geometry texts contained many challenging, highly instructive "originals" and construction problems requiring proof. During recent years the NCTM has watched, without protest, while these have virtually disappeared from the glossy 800-page "coffee table" books that pass for geometry texts in the US. At the same time they were speaking frequently about "mathematics as reasoning" and "higher thinking skills." This is an example of the vast discrepancy that exists between what NCTM says and the reality of NCTM-based programs.

• Social engineering has crept into the math curriculum, including irrelevant material about student attitudes and social issues and misguided efforts to build self-esteem. Under the guise of opening the door to higher level math for all students, we now have algebra and geometry students who are not prepared. This sends students the wrong message and hampers instruction. Recall that Jaime Escalante, of "Stand and Deliver" fame, showed us how students can earn their self-esteem and triumph when properly taught and motivated.

Lacking support in either research or experience, these NCTM-based programs are worse than just fads -- they are mistakes that have been systematized. They impair the quality and content of the mathematics our students are expected to learn. We must heed the warning provided by the timely publication of the TIMSS twelfth-grade test results and replace the fad-laden, NCTM-based programs now found in the mathematics classrooms of America.

California, perhaps having suffered the worst from these fads, is now moving in the correct direction. That state has, with the very substantial help of College and University mathematicians, adopted Standards that clearly describe the mathematics the student is expected to learn in each grade in elementary school and in each subject in high school (despite the name, the NCTM "Standards" do not do this). Recently adopted by the California State Board of Education, these new standards will soon lead to new programs to replace the weak NCTM-based programs. That this is a vast improvement, there can be no doubt. In a recent review of state standards, the Fordham Foundation reports that the new California Standards are number one, surpassing even the equivalent document used in Japan.

Links to the new California Standards and to two papers -- A Program for Raising the Level of Student Achievement in Secondary School Mathematics and A Plan for Improving Exposition in High School Mathematics -- are available from the Mathematically Correct web site. The papers deal with a national effort to place mathematically well-prepared teachers in secure classrooms, where they are free to use any methods of presentation that yield good results on objective tests.

Using these materials, concerned parents can, with the advice of local mathematicians, duplicate California's new directions in other states. Proceeding thus, on a state-by-state basis, we can replace fuzzy math programs with programs of instruction in school mathematics that will enable our students to compete successfully with their peers in other industrialized countries.

Frank B Allen is Emeritus Professor of Mathematics at Elmhurst College,
Past President of NCTM, and National Advisor for Mathematically Correct

See also Toward a Cease-Fire in the Math Wars

A condensed version of this report appeared as
'Standards' To Blame For Rise Of 'Fuzzy Math'
in Investors Business Daily, April 3, 1998