The Government Flunks Math
Earlier this year, James Milgram of Stanford University got curious
about something called the Connected Mathematics Project (CMP), an intermediate-school
math curriculum lately developed at Michigan State. So he carefully
analyzed CMP's sequence of 24 student booklets. And in one of the seventh-grade
units, he found the following review exercise, a problem of basic algebra
and arithmetic that depends for its solutions on an equally basic understanding
In 1980, the town of Rio Rancho, located on a mesa outside Santa Fe, New Mexico, was destined for obscurity. But as a result of hard work by its city officials, it began adding manufacturing jobs at a fast rate. As a result, the city's population grew 239 percent from 1980 to 1990, making Rio Rancho the fastest-growing "small city" in the United States. The population of Rio Rancho in 1990 was 37,000.
A. What was the population of Rio Rancho in 1980?
B. If the same rate of population increase continues, what will the population be in the year 2000?
Reasoning that Rio Rancho's population was 2.39 times larger in 1990 than in 1980, and would be 2.39 times larger again in 2000, the CMP booklet goes on to recommend dividing 37,000 by 2.39 to arrive at the answer it lists for question A (15,481) and multiplying it by the same amount to get answer B (88,430).
Except that both answers are wrong, by a wide mark. Deeply, essentially wrong, in fact: since, as every schoolchild was once drilled to know, increasing a number by 239 percent produces another number not 2.39 but 3.39 times its size. I guess we should be glad, James Milgram mordantly notes, that the population did not increase zero percent. In which case, by the logic of CMP's instructional materials, Rio Rancho's residents would all have died.
This gasp-inducing error turns out to be merely a surface symptom of
the CMP curriculum's paramount, underlying flaw: a nonchalance about -
no, outright hostility towards - the precision, coherence, and content
of mathematics as an academic discipline worthy of study in its own right.
Throughout the booklets, CMP students are asked to do a great lot of group
"investigation" into otherwise classic math topics. But those topics are
never explicitly defined as such, and the standard algorithms they involve
are never introduced. Is whole-number factorization into primes - the fundamental
theorem of arithmetic, which CMP only implicitly establishes with "experiments"
proposed for sixth grade - an inviolable principle? The booklets do not
say. And they are silent by design. CMP's teacher manuals advise a passive
approach to pupils because "showing them how to do something" only produces
an "impression" of success. Parents are then warned not to fill in the
gaps: "It is important that you do not show your child rules or formulas
for working with fractions," for example. Better that kids just figure
it out. Or fail to.
Having read this far, you have no doubt reached the not unreasonable conclusion that Connected Mathematics is a pedagogical disaster waiting to happen. You will therefore be distressed to learn that it has already happened; CMP is widely used in public schools across America. And you will be appalled to learn that CMP is likely soon to be still more commonly employed in our classrooms. Especially when we tell you why: On October 6, the U.S. Department of Education officially endorsed Connected Mathematics - as "exemplary" - along with nine other, philosophically indistinguishable kindergarten-through-twelfth-grade math curricula. Local school districts, if they haven't yet done so, should seriously consider adopting such a program, the department announced. "These are the prevailing standards in the country,"education secretary Dick Riley observed.
And, alas, he is right about that, which is the whole crux of the problem. In 1989, the National Council of Teachers of Mathematics (NCTM), reflecting the rigid utilitarianism and learn-by-discovery "constructivism" that has long dominated the nation's colleges of education, promulgated new and much-ballyhooed guidelines for K-12 math instruction. Numbers must be made "useful" and "engaging" to the many American students previously mystified or bored by them, the council proclaimed. Direct guidance by teachers at blackboards should now be minimized in favor of less stultifying "cooperative learning" driven by pupils themselves. NCTM derided math education's past "preoccupation with computation and other traditional skills"; children must be allowed to use calculators in place of paper-and-pencil procedures for all but the most rudimentary numerical operations. And so on.
The NCTM standards were initially welcomed as revolutionary by unthinking politicians of both major parties - and largely ignored by the general public. What those standards might mean in practice did not begin to penetrate popular consciousness until 1992, and then only in California, the first major state to conform its instructional objectives with the council's recommendations. California, to take just one of many bizarre particulars, decided that mastery of multi-digit long division would no longer be a goal for its high-school graduates.
And textbook publishers immediately got with the program. "Mathland," an elementary-school curriculum specifically created in response to California's 1992 math "framework," does not teach calculator-unassisted long division. To help students grasp the notion of really big numbers, Mathland has them count out a million birdseeds, one by one. To help students maintain a disappointment-free, exploratory feeling about these and other number problems, Mathland advises teachers never to "indicate in any way the rightness or wrongness of different answers." Mathland, in short, is an educational abomination.
In 1997, after an intense, grass-roots parent reaction against such stuff, California was forced to reinstitute more rigorous and traditional mathematics requirements and ban future purchases of Mathland-like curricula by its school districts. But by then 42 other states, operating in 42 separate vacuums of ignorance, had adopted their own NCTM-derived math benchmarks. And the cruddy textbooks to which those benchmarks correspond. Mathland may be expiring in California. But it is alive and well almost everywhere else.
And Mathland, too, amazing but true, has now been endorsed ("promising") by the U.S. Department of Education. These are the "prevailing standards in the country," after all. And they seem likely to continue prevailing from above until populist state-by-state rebellions defeat them from below. The NCTM has recently circulated a draft revision of its 1989 standards. The new document makes only the barest, cosmetic concessions to the council's critics.
Most notable among those critics have been research mathematicians like Stanford's Professor Milgram. Three weeks ago, Milgram helped write, and 200 of his professional colleagues around the country signed, an open letter to Secretary Riley urging the Education Department to withdraw its endorsement of the 10 "new new math" curricula. These people are the cream of American professional mathematics: the department chairmen of Caltech and Stanford; four Nobel laureates in physics; and two winners of the International Mathematical Union's quadrennial Fields medal. But they have complained to no avail. Riley's spokesman insists that "we stand firm behind the process that was used."
Which brings us to the only practical suggestion we can think to make to Congress. The "process" in question, a system by which the Education Department commissions panels of outside experts to recommend "exemplary" and "promising" school programs, is one directly authorized by law. The departmentís math curricula endorsements are the first, ill fruit of this system. Still pending are reports on science, safe schools, technology, and gender equity. It is too late to undo the damage to mathematics; that cat is unfortunately out of the bag. But before the damage spreads to other disciplines, Congress can do something simple and overdue. The expert panel system is self-evidently untrustworthy and dangerous. Congress should abolish it.
--David Tell, for the Editors