What If Those Math Standards Are Wrong?
Even in the faddy world of K-12 education, the "standards'' issued by the National Council of Teachers of Mathematics have met with rare acceptance. Seldom has so profound a change in conventional wisdom and standard practice had such homage paid to it, so little resistance shown to its onrush, so few doubts raised about its underpinnings. Republican and Democrat, textbook publisher and test maker, governor and businessman, federal official and local school board member--just about everyone is rushing to implement "the N.C.T.M. standards.'' What's more, they're ceaselessly cited as the example par excellence of what national education standards should look like. Gov. Roy Romer of Colorado, for example, has said so hundreds of times, as has outgoing U.S. Secretary of Education Lamar Alexander. We've seen national, state, and local groups, struggling with standards in their own domains, cite the N.C.T.M., embrace the N.C.T.M., and yearn to emulate the N.C.T.M.
We'd better hope the N.C.T.M. has got it right. If not, American education's lemming-like rush to follow its lead could find us hurtling off a precipice.
Worries about the N.C.T.M. approach to math began to stir in me several years back. They took two forms then; I've lately added a third.
The first had to do with what I've come to realize is vast misunderstanding of what the N.C.T.M. has actually wrought. It is centered in confusion between "content standards'' and "student-performance standards''--to borrow the terminology of the National Council on Education Standards and Testing, co-chaired by Mr. Romer.
Oversimplifying only a bit, content standards describe what schools should teach and--presumably--their pupils should learn. Retro examples might be "state capitals'' in grade 4 and diagramming complex sentences by grade 7. Content standards are about curriculum: its goals, frameworks, scope and sequence, etc. They are intended mostly for educators.
Student-performance standards are something else. They involve how well youngsters must do in order to be said to have met the expectations of the content standards. How many state capitals must that 4th grader actually know? Must she have them memorized or is it okay to match up states and capitals from two lists? If she only gets 42 of them right has she fulfilled the standard? Must all those names be spelled properly? As for diagramming, just how intricate a complex sentence do we have in mind? How many must the student diagram?
How many errors are acceptable? Such are the issues we must resolve when we set student-performance standards. Note, though, that only when these standards are in place can students (and parents) see how well they--and their schools--are doing vis-a-vis what's expected of them. Student-performance standards are truly about results and outcomes. They're meant mostly for laymen.
To date, what the National Council of Teachers of Mathematics has provided are content standards only. Educators are properly grateful, as are many policymakers. But what most nonprofessionals have in mind when they talk about education standards are student-performance standards and those--regrettably--the N.C.T.M. has not yet given us.
So how satisfactory are the content standards? We now come to my second worry. It flickered to life when I began to hear of elementary school classrooms where teachers were passionate about "problem solving'' but where students were counting on their fingers as late as 3rd or 4th grade because they hadn't learned rudimentary "math facts'' to the point where these came automatically. I contrasted this with what I knew of Japan's "Kumon'' math program and with the teaching strategies developed by John Saxon, an eccentric textbook author who is shunned by the math establishment but whose pupils seem adept both in basic arithmetic and in solving complex math problems. (The math establishment despises Kumon, too, of course.)
Was it possible, I asked, that children taught according to N.C.T.M. standards might have all sorts of imaginative ideas about tackling a problem yet seldom get the right answer to it because five times 11 was beyond their ken?
Math sophisticates pooh-pooh my concern, arguing that of course the N.C.T.M. intends both results, that arithmetic and accuracy are not being sacrificed on the problem-solving altar, indeed that these will develop hand-in-hand.
That, of course, is what everyone yearns for: deft skills and reliable "math facts'' combined with imagination and deep understanding. And in the hands of terrific math teachers, that's pretty much what happened long before the N.C.T.M. was heard from. But U.S. schools don't boast a surfeit of such teachers, especially in the early grades, and in trying to compensate for this kind of shortage (not just in math) we've tended to lurch from one extreme to another, grabbing for the latest miracle cure, forcing tradeoffs, opposing "a'' to "b'' rather than melding them. For a vivid example drawn from another field, look at the endless war between the "whole language'' crowd and the partisans of "phonics,'' notwithstanding ample research showing that both are needed for young students to read effectively and enjoyably.
Great teachers do use both in language-arts lessons, just as they attend both to problem-solving strategies and rapid calculation of the right answer in math class. They know these fierce arguments involve spurious choices and phony tradeoffs. But what happens when their professional association slips--or is perceived as slipping--over to one side? In particular, what happens to millions of children whose less-than-gifted instructors rely on prepackaged programs, the latest nostrums, and what others tell them is the approved way to proceed?
So long as many teachers are dependent in this way, it's vital to ask of any new approach being thrust upon the education world whether it has been fully tested with students to insure that it yields the desired results--and is not just being promoted because it appeals to grown-ups caught up in ideological battles.
Which leads to my third and newest anxiety about N.C.T.M. math, seeded by a flawed but compelling (and widely ignored) book by Siegfried Engelmann called War Against the Schools' Academic Child Abuse (Halcyon House, 1992).
Professor Engelmann, of course, is the father of dozens of instructional programs, especially for the primary grades, the best known of which are äéóôáò reading and math. He is also one of the world authorities on "direct instruction,'' a highly structured approach that relies on clear expectations for teachers and students, tight performance requirements, "behavioral'' (rather than "developmental'') instructional practices, and strong emphasis on accountability for results.
Here's what Mr. Engelmann has to say about math a la N.C.T.M.:
The Standards de-emphasize anything teachers have failed to teach. ... The most serious problem with the Standards, however, is its arrogance. In the tradition of the sorting-machine, it assumes that it can derive a curricular reform through metaphysical masturbation of words, not through experimental evidence about what works and what doesn't. The writers of the Standards did not first verify these activities, suggestions, and standards by first demonstrating that they worked and that they created kids who performed well in math. Instead, they made it up and then presented it as an authoritative document.
If Professor Engelmann is right, we may be buying a pig in a poke, a radical yet unproven overhaul of math curriculum, instruction, and assessment that massages the nerve centers of the "math community'' but won't necessarily produce more numerate young Americans.
Mr. Engelmann asserts that the N.C.T.M. approach has a lot in common with the debacle known as "new'' math. "The manipulatives, the exposures, the acting-out, and the moral insistence on problem-solving,'' he writes, "has been a theme of math educators since the mid-60's. The approach is actually one of the reasons kids currently don't know long division and are not proficient at paper-and-pencil work in math.''
The kind of instruction that Mr. Engelmann favors--direct instruction--isn't popular with today's educators. It smacks of rote learning, drill and practice, even memorization, thus of a "canon'' of skills and knowledge that every teacher should impart and every pupil acquire. This is unfashionable. It's not what we find in the N.C.T.M. standards. But it's performance-oriented, hence amenable to assessment--including the kinds that emphasize right answers and thereby lend themselves to accountability, high stakes, and other such scorned practices.
Unfashionable to be sure. But we err when we slight the acquisition of facts, specific knowledge, and simple skills, both as building blocks of more complex intellectual structures and as potent motivators. Many teachers and parents can attest to the satisfaction that kids get from knowing things: precise, definite things that they know they know, can tell they're good at, and from the accumulation of which they can gain a sense of steady progress--in contrast to the subtleties and ambiguities that experts favor. As an example, I recently observed the ardor, pride, and feeling of accomplishment palpable in an elementary school in the much-afflicted South Bronx, a school that is using E.D. Hirsch's "core knowledge'' program.
E.D. Hirsch isn't the main point, though, nor is Siegfried Engelmann, nor even the N.C.T.M. What's important is whether U.S. youngsters actually reach higher levels of skill and knowledge. As yet--a full decade after the National Commission on Excellence in Education labeled us a "nation at risk''--there's scant evidence that our reform strategies are working. The cures we've tried have done little to boost outcomes. To that glum news some people respond by seeking (as often before) to ease the press for results and go back to indices of input, effort, and intention. Others want to replace the measuring sticks, hopeful that different assessments will reveal--and perhaps stimulate--better results from today's voguish curricular and pedagogical strategies. A few, however, are turning away from those strategies themselves, returning to what Marilee C. Rist, in a useful article in The Executive Educator, terms "learning by heart.'' Memorization. Direct instruction. Recitation. Plenty of practice. And gobs of core knowledge.
We oughtn't dump all our eggs into that basket, either. Or any other. No single container is capacious enough. Diverse classroom strategies should be welcome--so long as solid learning occurs. The reason for standards isn't to impose a regimen of what Diane Ravitch terms "pedagogical imperialism.'' Rather, it's to be clear and prescriptive about ends--and then laid-back and versatile about means.
I doubt this was intended, but the N.C.T.M. may have given a boost to such imperialist tendencies in math. By focusing on content rather than performance standards, the organization has probably led its members and followers to dwell overmuch on what happens in the classroom instead of the results attained there. "Problem solving'' works in some situations, to be sure, but "learning by heart'' may accomplish more in others. Usually both are vital. Teachers must feel free to adapt their strategies to specific situations, not harnessed to a single pedagogical approach.
Says Thaddeus Lott, the maverick principal of the Wesley School in Houston, an institution attended by hundreds of "at risk'' youngsters, a place where DISTAR is used in both math and reading--and where test scores are soaring: "You don't send a guy to dig gold without the proper tools; and you don't build a house without a saw and hammer.'' By giving his children the tools they need, he is empowering them to build all sorts of structures. But it takes courage to stand up to conventional wisdom. And today that wisdom insists that the N.C.T.M. and its ilk have things figured out just right and that everyone had better do things their way. What if they turn out to be wrong?
Vol. 12, Issue 17, Pages 26, 36