Math War in Morgan Hill

This page is a collection of the articles and letters I've written about the College Preparatory Mathematics program in the Morgan Hill (CA) Unified School District, most of which were published in the Morgan Hill Times. Others who want to express their opinions here, pro or con, please send to (include name).

The brouhaha over K-12 math cannot be said to have started with the 1989 standards of the National Council of Teachers of Mathematics, but they certainly intensified it. Now with its just-released new standards, the NCTM has backed off to some extent, but many of its supporters don't want to admit it, saying "we were misunderstood."

The 1989 NCTM standards, adopted in 49 states including California, gave a big boost to the idea of de-emphasizing basic computational skills and right answers, focusing instead on methodology. They spawned many math programs for schools, of which CPM, used in the Morgan Hill School District, is one example. These programs became the target of many parents and mathematicians, who viewed the NCTM standards as "dumbing down" math. The grade at which calculators were introduced, in some cases as early as kindergarten, became a barometer for the extent to which that had happened.

The reaction was strong enough to bring about new rigorous standards in California, but the momentum from the NCTM standards of eleven years ago is proving hard to stop. California's Superintendent of Public Instruction Delanie Eastin vehemently opposed the new standards, and many school districts are doggedly continuing programs built for the old NCTM standards.

The extent to which the new NCTM standards backtrack on their 1989 document is probably not much. Tom Loveless, quoted in the New York Times, said "This document does throw a bone to the basic skills crowd," suggesting that the argument is far from over.

The changes made in 1989 were motivated largely by a desire to alter the way math was taught so as to improve the grades of minorities and females, who tend to do less well in math. The result was an experiment in social engineering that went so far as to redefine the subject itself. One change, for example, was to introduce more exposition in the problems. This move was aimed at making math "friendlier" for girls, who tend to have better verbal skills than boys. The trouble with that particular change is that to the extent it succeeded, it did so by changing the subject from math to something else. If girls' grades did improve, in other words, it was probably due to the fact that the subject was no longer just math but an admixture of math and composition.

Diluting the subject for social reasons may achieve the object of improving grades, but it is a not-very-subtle form of fraud. Sooner or later, students who make their way in the university or the business world will have to confront the hard truths about math, one of which is that having the right idea but the wrong answer doesn't hack it.

Eighth-graders in Morgan Hill will continue to use math books that don't meet state standards for at least another year. That is the case because new textbooks approved by the Board for the Morgan Hill School District on March 13 did not include new books for eighth-grade math.

The non-adoption of new books occurred despite the fact that the State Curriculum Committee has reviewed and approved three programs that meet the new math content standards, and state money is available for those books. Consequently, eighth-graders in Morgan Hill will continue in the CPM program, which doesn't meet state standards, and the earliest time that they can benefit from the standards will be 2001, almost four years after adoption of the standards by the State Board in 1997.

Last month, the District furnished books it had selected for public review. For Algebra I, Grade 8, CPM books were displayed, with no alternate. That selection was later withdrawn, and no recommendation for new books for that course was presented to the Board.

To their credit, the Math Department at Live Oak High School selected a McDougal Littrell book for Algebra I that is aligned to the state standards. That selection, which was approved by the Board at the March 13 meeting, was made in the face of opposition by the administration.

Withdrawal of the CPM selection for 8th grade was probably due to the fact that CPM is not on the list of programs approved by the state. The significance of this is that the so-called Schiff-Bustamante funds, appropriated by the legislature specifically to encourage school districts to use the new standards, are available to buy books only for approved programs. CPM was not approved for the very good reason that the publisher did not even apply.

The delay in purchasing new algebra books for the eighth graders is consistent with the foot-dragging the District has displayed since adoption of the new standards by the state. It took them a year and a half to adopt the standards, and they did so then only with "District additions," which turned out to be mostly a useless hodge-podge of watered-down restatements of the state standards.

Adoption of the state content standards in social studies, English and math was by far the most significant development in California education in the last decade. The new math standards represent a turning away from the "fuzzy math" approach and toward mathematics more in line with the subject as mathematicians understand it.

The argument has always been more about what math is rather than how to teach it. The "how to" should be left largely to the teachers themselves, but the content, and only the content, should be spelled out in standards. The state standards were written on that basis, and all references to methodology were removed. Unfortunately, the additions inserted by the Morgan Hill School District violate this prinicple, and they are liberally sprinkled with references to calculators, manipulatives, and so forth.

District officials say they support the state standards, but their actions say otherwise.

The following letter appeared in the Morgan Hill Times on February 25, 2000. My reply is embedded in italics.

I can't take it anymore. I must respond to the "debate" that is taking place in your paper over the CPM math program.

I am currently a social studies teacher, but I have taught algebra in this district in the past using both the CPM materials and the more traditional textbooks that some are calling for. The traditional texts are easier to teach, but that's not supposed to be the measure of success in education.

Perhaps, but the subject here is choice of classroom materials, and ease of teaching, which means nothing if it doesn't mean more results are obtained for less effort, must be counted as a plus for traditional texts.

When I first started teaching math, I remember a feature in an area newspaper applauding the exemplary teaching of a San Jose junior high math teacher Jim Goth. What was so remarkable about this teacher was his use of manipulatives (hands-on materials) and activities that allowed his students to work together to explore the basic concepts of algebra. When his students learned a pattern, they didn't just memorize it, they understood it. And I thought I'd love to be a great teacher like that.

Ms. Wallace is speaking of pedagogy here, about which the teacher, I believe, should have a large degree of discretion. By insisting on group work and self-discovery, which have to do only with methodology, CPM excessively restricts teacher discretion. As to use of manipulatives, I'm dubious about their use for teaching algebra. They may indeed, as Ms. Wallace says, contribute to "learn[ing] a pattern," but is that algebra?

My style is to lean excessively towards lecture. (Just ask my students.) I can explain. I can demonstrate. I can answer questions. And more traditional math programs are designed for this style. But it doesn't work well for students who are not intuitively mathematical, and let's face it, that's a lot of students.

What Ms. Wallace has told us here is that the style she leans toward (lecture) "doesn't work well for students who are not intuitively mathematical." I have several problems with this: (1) While traditional math may be "designed for" lecturing, it is not mandatory. On the other hand, CPM is far more restrictive in regards to teaching method. (2) It is hard to see that the self-discovery method that Ms. Wallace so admires works better for students who are "not intuitively mathematical." In fact, self-discovery, it seems to me, depends quintessentially on mathematical intuition. (3) Ms. Wallace implies that she had to deal with a high proportion of students who were not mathematically intuitive, but the students who reach algebra in middle school, where she teaches, are among the more advanced by definition.

Not all students are fortunate enough to have a Mr. Goth. But with the CPM program, teachers like me can learn to provide some of what he provided. Yes, some students (and parents) protest. I can't tell you how many times students tried to cut off my explanations, even in the traditional program, with "I don't care why it works - just show me how to do it." The fact that they don't recognize the value of the challenge, that the true learning comes fiom the process of figuring out the problems through exploration and discussion, doesn't mean that the old system is better. It just means that they are kids, and it is unfair to exploit their immaturity in this debate.

The frustration that many students express should not be discounted so cavalierly. An integral part of the self-discovery method, after all, is to withhold information. Giving expression to the resulting frustration is not a sign of immaturity; I would call it the opposite. As to being "unfair" to students, is there not a tinge of arrogance in summarily dismissing their pleas because "[i]t just means that they are kids"?

Finally, let me say that I am offended by the implications that teachers who adhere to the CPM program are lazy or incompetent. It is far easier to tell students to open their textbooks and listen quietly while you demonstrate than it is to make sure you have purchased and, counted out enough beans, paper cups, or other lab supplies a day in advance - but it's not as effective and it's not as generous.

There is nothing in the "traditional textbooks" that prevents Ms. Wallace from teaching any way she wishes, something that cannot be said of CPM. But if I had a son or daughter who told me they were using beans and paper cups in algebra class, I'd be asking a lot of questions.

At the Morgan Hill school board meeting on Jan. 24, two Middle School math teachers defended their CPM program with some charts showing good test results from the Golden State Exam and SAT 9. I thought the numbers looked a little suspicious, so I wrote to Dr. McKennan, Superintendent, asking where they came from. I had not set out to trap her, but what I received turned out to be a good illustration of the un-math taught here.

On examining the charts she sent, I noticed something odd: some of the numbers are wrong. Not only are they wrong, but whoever wrote the accompanying narrative didn't notice, even though the information is obviously wrong and contradictory to boot.

The presentation chart for SAT 9 for Murphy Middle School shows two sets of numbers: percentiles for the average score for the school and "9th Grade, percent of students above average," each for 1998 and 1999. The percentiles for math are 70 for each year, a very good showing. The "percent above average" are 44 and 41, respectively, a rather poor showing. The narrative below the table of numbers says, proudly, "almost 1/2 are above average in mathematics."

The trouble with these numbers is that it would be almost impossible to have a class average score at the 70th percentile on the national scale, yet have only 41 or 44 percent score "above average." Furthermore, the reported data makes no reference to "average," in respect to the national scores, and it is probable that the writer was confusing average for median, a common mistake for those not versed in mathematical statistics.

A correct statement of what was intended would be "percent of students who scored above the 50th percentile," for which the numbers are 70 and 71, respectively, for 1998 and 1999. These numbers are practically the same as the percentiles for the average class scores, as they should be and as anyone familiar with statistics would expect.

Ironically, the incorrect numbers understate the results by a wide margin, arguing the opposite of what was intended.

All of this may seem a small point, but the mistake illustrates the very things I object to in CPM: carelessness with numbers and imprecision in the mathematical reasoning and presentation. Precision of thought is the heart of mathematics; a course without that may be something, but it's not math.

In the face of opposition by 450 students and many parents, the Morgan Hill School District is proposing to plow ahead with its CPM math program for Algebra I in the Middle Schools. Fortunately, the high school plans to drop it, at least for Algebra I.

Samples of books selected pending Board approval are available for review by parents and the public at the District office, so I went over to see them yesterday. Sure enough, the CPM books for Algebra I for Grade 8 are there. Next to them is the selection for Algebra I for Live Oak High School, which is a book and program by McDougal Littrell called Algebra I Exploration and Applications. It appears to be far more complete than CPM, with a generous supply of supplementary materials, including a practice workbook with much material specifically applicable for the 1997 California Math Content Standards.

The CPM offering consists only of books for students (two in Spanish), a thin parent guide, and a thick teacher guide in a binder. Although CPM was originated and is published in California, I could find no reference to the California standards in the teacher material.

The most important consideration for me is conformance with the California Math Content Standards, but it's hardly possible for an individual to make that evaluation. The State Board of Education does this for us, but unfortunately they haven't done it for grades 9-12. Algebra I for Grade 8 is one that the Curriculum Committee, which reports to the Board, did review, however, and several books for Algebra I, Grade 8, are on the list of adopted materials.

It is evident that the District is not particularly interested in public review of the proposed books. Display of the books was not announced in the most obvious place, the Winter issue of the District mailing to residents, and there was no other public announcement that I know of. Public review isn't very practical, anyway, and the best way for the public to judge the choices is through the evaluations made by the State Board.

Neither CPM nor the McDougal Littrell books are on the list called "AB 2519 Adopted Mathematics Instructional Materials," which can be seen at at http://www.cde.ca.gov/cilbranch/eltdiv/ab2519math.htm. The McDougal Littrell book is on a list of books offered but not approved, although there are three other ML algebra books, including Algebra: Structure and Method for Grade 8, that were approved. CPM did not apply. (Not surprising, of course, because CPM is precisely the kind of program the new standards were designed to get rid of.)

The Algebra I books the District plans to present to the Board for approval, then, can be summarized thus: one that was turned down by the state Curriculum Committee and one that didn't even apply.

A District official told me that they are not restricted to state-approved materials for algebra. This doesn't make sense to me for this case since the State Board has approved books intended for the course (Algebra I) and grade (8) in question.

I've been told that CPM has been upgraded to more closely conform to the new standards, and the new student book has a copyright date of 2000. A leopard can't change his spots, however, and it is still based on the self-discovery and group-think philosophy. The issues have been argued exhaustively, but my most serious objection is the attempt to apply the self-discovery method in large classes. The method strongly discourages lecturing for the obvious reason that lecturing would short-circuit the opportunity for the students to figure things out for themselves. It is a rather daring approach, in other words, where the teacher purposefully withholds information. I strongly suspect that it isn't unusual, when the students simply can't figure it out, the teacher can't either.

At the School Board meeting on Jan. 24, College Preparatory Mathematics, the math program at the middle schools and at Live Oak under fire by students and others, was defended by Sandy Snively, a middle school math teacher, and Jim Goth, retired math teacher. Both emphasized the advantages of self-discovery and group work, which together comprise the central ideas of CPM.

It is good for students to learn cooperation, but the purpose of math courses must be first to teach math. According to the 50 students who signed the letter objecting to CPM (subsequently increased to over 450, according to its authors), the emphasis on group work in CPM interferes with that main purpose.

Mr. Goth, in his guest column in the Times on Jan. 28, defended CPM because, as he put it: "Persistence and innovative use of strategies in difficult situations was truly a better measure of their mathematical abilities than how many correct answers." This statement contains two bad ideas that illustrate some of the fallacies in CPM and its companion programs, loosely known as fuzzy math.

The first bad idea is the excessive emphasis on "innovative strategies." It is a bad idea because:

It is a good technique to present students with problems that go a step or two beyond their understanding at a given point. It gives them a challenge and tends to sharpen interest and to remind them that they have farther to go. But mathematics is not an experimental science, and it is a rare student indeed who will "self-discover" the truths of mathematics bequeathed to us by mathematicians of the ages. To suppose that the "innovative strategies" that students conceive are of equal value bespeaks a breathtaking indifference, even disdain, for the great discoveries of Gauss, Descartes, Sir Isaac Newton, and the other great mathematicians.

What students should learn in high school math can be written down in concise form, and that's exactly what the writers of the state math content standards did. Excessive attention and reward for "innovative strategies" are diversions that can only detract from the main goal.

The second bad idea is the progressivists' breezy indifference to correct answers. I am reminded of the anecdote by the professor who denied appeals by students to give credit for "having the right idea" although their answers were wrong. His riposte: "The bridge will fall down." One wonders if the scientist who mixed English and metric units and destroyed a multimillion-dollar space mission learned math from Mr. Goth.

Ms. Snively justifies the group work in CPM by reference to the plea of employers for people who can "work with others." Whether that takes priority over technical skills, which I doubt, Ms. Snively fails logic here: social skills have nothing to do with mathematics per se, and the question which should be asked is whether the techniques dictated by CPM are the best way to teach math. If her students arrive in class so lacking in social skills that their ability to learn math is impaired, then teaching in lower grades is at fault and she may unavoidably have to fill the breach. However, to treat it as one of the main goals of math class is to redefine math as something it isn't.

In his column Mr. Goth seems to recommend that the District save money by keeping the CPM books and "adapting." I'm sure many teachers have adapted, through grit and hard work, bless them, but the new standards will make that more even more difficult. As to the economics, the monument that the District is preparing to build on Burnett Avenue will be a meaningless heap of brick and mortar if what goes on inside lacks intellectual discipline.

The Morgan Hill School Board approved new math standards for the MHUSD last June. They consist of the state math content standards approved by the California State Board of Education in 1997 with some requirements added by the District.

The new standards are important because they will guide the future of the math program throughout the District for all grades from K through 12 for several years. The District is currently in the process of selecting new books for some math courses, including Algebra I, and the most important requirement for that process is that they conform to the new standards.

The state standards are widely considered the best in the country, so it's a mystery why the District felt the need to add to them. Furthermore, a close examination of the additions shows that they are of very poor quality, and, contrary to the District's claim that they "exceed state standards," more often than not they are a step down.

1. Most of the additions amount to restatements of state requirements but are very poorly written, with far less precision and specificity.

3. Some additions pertain to methodology, not content, something the preparers of the state standards were careful to avoid.

4. Many additions have the coloration of fuzzy math (e.g., CPM), the discredited approach that was the main reason that the state wrote the new standards on an accelerated schedule.

Some parents who have been dissatisfied with the current programs may take comfort from the District's claim that its additions exceed state standards, concluding that any book selections must meet the state standards as a minimum. I'm not so sure; for one thing, the CPM faction, which includes the administration, is claiming that CPM meets the new standards. This is an astounding claim to me, in part because the promoters of "reform math" (of which CPM is one type) vehemently opposed the standards back in 1997 on the basis of their belief that the new document was a thorough-going rejection of their approach. They were quite right in that belief: dissatisfaction with CPM and other similar programs was the principal motivation for the new standards.

No reasons for the additions are given in the 56-page document that I obtained from the District office. They are listed beside the state requirements in general categories but unevenly in the grades. For example, Grade Five has 24 additions and Six has two. Mathematical Analysis, Probability and Statistics, and Calculus have none that are not duplicates of state requirements. The only discernable theme in the additions is the kind of fuzziness for which reform math is noted. One of the more distinct examples of that is:

which appears in K, 1, and 3 (apparently not needed at 2). The District is in serious trouble indeed if it has teachers who need to be told this.

So that parents of older students won't feel slighted, see this one for Algebra I:

For contrast, here is the state standard that appears opposite the previous example:

4. Students simplify expressions prior to solving linear equations and inequalities in one varible such as 3(2x-5) + 4(x-2) = 12.

This one is typical in precision and specificity of the state standards. While the examples of the District additions I gave above are among the worst, the contrast in quality is striking and pervasive.

The outstanding example of regression is the standard for addition and subtraction facts with sums to 20, which appears at Grade 1 in the state standards and Grade 3 in the District additions.

An unintended effect of the additions is to give the public a view of the quality of the mathematics staff in the District. That view is depressing indeed, especially considering that we should be able to assume that the teachers who prepared the document are among the most qualified and likely to wield decisive influence on the selection of books for some time to come.

I suppose I was the last to know, but the Morgan Hill School Board approved new math standards last June. I recommended some time ago (and repeated Jan. 16 on this site) that the District adopt the state standards passed by the California State Board of Education back in 1997. So did they do that? Well, yes and no: Yes, they did adopt the state standards, but no, they couldn't leave them alone. They added items where, they say, "District standards exceed state standards." The question is whether their additions are exceedances or adulteration.

More important is the question of why the District felt the need for additions to the world-class standards prepared largely by a group of mathematics professors at Stanford. There is no hint in the preamble, and, suspicious as I am, I'm wondering if the idea is to enable CPM (and Quest, the elementary program, if I remember correctly).

The MH standards are the product of the District Mathematics Task Force, which, according to the preamble to the standards, "reviewed, discussed and in some cases added to the State standards." The Task Force was composed of math teachers, the membership, I suspect, having more to do with a combination of aggressiveness and compliance with the District's "party line" than any other qualification. They labored for at least a year, the only substantive result, if it can be called that, being the things they added to the state standards.

The standards are organized in two columns in which the left column has the state standards by grade and subject and the right column shows the district's additions, rendered in italics. The purpose and rationale of the additions is not explained in the document. The only clue is the heading of the district column, which says: "Italics indicated district standards exceed state standards."

One of the "additions" is introduction of calculators in the third grade. It is a bulleted item buried at the end of a long column of District additions:

Once you introduce calculators in elementary, you can forget pencil and paper arithmetic.

Many probably feel that the "calculator issue" is blown out of proportion, but one of the strongest points in the rationale for the state standards is that calculators are not used in elementary school at all. If students are given the option of using calculators, very few will take the trouble to go through the laborious tasks of multiplication and division on paper, and they won't learn them.

A favorite pastime of reform math proponents is to sneer at long division, but look at what R. James Milgram, professor of mathematics at Stanford University had to say about that (1999 Conference on Standards-Based K-12 Education):

DYNAMIC SKILLS ASSOCIATED
TO LONG DIVISION

The process of long division is one of successive approximation, with the accuracy of the answer increasing by an order of magnitude at each step.
The skills associated with this process become more and more fundamental as students advance.
- They include all infinite convergence processes, hence all of calculus, as well as much of statistics and probability, to say nothing of differential equations.
Long division is the main application of the previously learned skills of approximation.

Other "additions" are steps backwards. Compare, for example, a district addition for Grade Three to an item in the state standard for Grade One:

Grade One, Number Sense (State Standard)	Grade Three, Number Sense (District addition)
2.1 know the addition facts (sums to 20) and the corresponding subtraction facts, and commit them to memory	- Know addition and subtraction facts (sums to 20)

The district addition appears under the heading: "Italics indicate district standards exceed state standards." Perhaps that should read: "...district standards subduce state standards."

Many of the additions strike me as trivial, e.g., "- Use concepts and skills they have acquired through previous activities" (K, p. 3). Well, yes, but do we really have teachers who need to be told this?

Students demonstrate facility with algebraic manipulations, perform algebraic computations easily and routinely, and rewrite expressions and equations to gain information and find solutions.

These cover more or less the same ground (again, begging the question, why bother?), but the state wording is a model of precision and the district wording is mush. Specifically, what is meant by "gain information" or "find solutions," hanging in mid air? And how do you measure "easily and routinely"? Well, nothing, nothing, and you can't. Another subduction.

Under Mathematical Analysis, the district additions say: "Our Math Analysis course includes standards under Linear Algebra and Probability and Statistics:" What follows then is verbatim copies of some, not all, items from state standards on Probability and Statistics, Advanced, and all items from Linear Algebra. If the point here is simply to convey the idea that the MHUSD curriculum is organized differently from what's implied in the state standard, it's trivial, completely unnecessary, and out of place. But what of the missing items? If they intend to describe the curriculum, then they need to identify where the other items are covered. In addition, the idea of copying items is ridiculous, forcing the reader to make word-for-word comparisons to be satisfied that they really are the same. If they were interested in conveying information, and not mere puffery, they would identify the items by number, since all items in the state standards are numbered.

These criticisms are not unimportant. If mathematics standards writers can't present their product with clarity and precision, how can it be any good?

So back to the question: What is the purpose of the additions, which not only add nothing of value but which are, well, fuzzy?

One answer is that they may provide an opening, just enough of an excuse, to plow ahead, doggedly, with fuzzy math. Is that it?

The debate about how to teach math in Morgan Hill schools has been seething for a while but largely confined to district staff and teachers. No more. Since appearance of the guest column in the Morgan Hill Times on January 7 written by three Live Oak High School students and endorsed by fifty or so others, it is in the open. Where it belongs, I might add.

The students' column was a plea to the school district to replace the "reform math" program, known as CPM, in use at the middle and high schools here. They provided what Karen P. Anderson, in a letter to the Times on Jan. 14, aptly called "compelling testimony" for replacing CPM. Indeed it was: As students in the higher math courses at Live Oak, they are in an ideal position to know the value of the CPM courses, and their conclusion was that it was miserable.

Math wars in the United States have been raging for decades between those who promote "fuzzy math" and those who argue for "back to basics." It all began in the 1960s, if not sooner, when the first incarnation of "new math" was introduced. Satirist Tom Lehrer from that era weighed in (with apologies to the editors of the Wall Street Journal, who dug this up recently) as follows:

The choice of overall standards and general direction is the responsibility of the local School Board, acting for parents and citizens in general. On this question, the California State Board of Education (CSBE) has pointed the way with the excellent math standard it adopted in December, 1997. That standard has been widely acclaimed to be among the best in the nation, and the Morgan Hill Board of Trustees should take the lead of the CSBE and adopt the state standard.

There is no excuse for not carrying out that action forthwith. The CSBE approved a new math framework consistent with the new standard in December, 1998, and approved new instructional materials aligned with the standards in July, 1999. Furthermore, the STAR test, implemented beginning in 1998, contains questions that reflect the 1997 standard, and the state is, according to the paper by Milgram and Norris cited below, funneling "massive state resources" to districts to assist them in implementing the new standards.

For background on the issue, an excellent source is www.mathematicallycorrect.com, which makes the case against reform math. For a summary of the history and status of the new California standards and tests, refer to the paper by R. James Milgram and Vernonica Norris (California Standards and Assessments, Oct. 21, 1999).

Response to letter in Morgan Hill (CA) Times signed by fifty math students at city's high school, protesting the math program.

A more authentic criticism could hardly be imagined. I'm speaking of the plea by fifty Live Oak High School students who signed the guest column on Jan. 7 to replace the CPM math program there. These students of calculus and the other higher math taught at Live Oak are in the best position to know, and they realize that they are victims of the sloppy form of math taught in the Morgan Hill School District.

There is no subject for which solid understanding at each stage is more important than mathematics, and the students' statement that CPM "does not prepare students for the challenges of higher level math" is a devastating indictment.

CPM, which stands for College Preparatory Mathematics (so ill named that one is tempted to wonder if it is a cruel joke), relies too much, as the students well described, on self-discovery and group effort. These pedagogical techniques have their place in teaching, but CPM focuses so much on methodology that the rigor and discipline that are the essence of mathematics are completed lost.

The experience of these students is not new. It is no less than a lack of intellectual discipline that is a perennial problem of schools, particularly public schools. It was well put by Jacques Barzun forty years ago when he said, speaking of many teachers at the time, that "their neglect of intellectual discipline amounts to denying the young the benefits of the long collective effort of Intellect which is their birthright." The figure-it-out-for-yourself philosophy of CPM is precisely the kind of denial Barzun was speaking of.

In the same book, Barzun spelled out the consequences of the anti-intellectualism that characterizes so many schools: "The outcome is that the educated young are self-educated, not only in the trite sense of the phrase, but in the sense that at some point in their career they confront their nurtured incapacity, and with gritted teeth set out to repair one or more of the deficiencies -- linguistic or mathematical -- which we have come to regard as normal and even praiseworthy." It is tragic that our best students find themselves in this position even before they finish high school.

Many programs for teaching math in the style represented by CPM came into being in the early 1990s and were "trendy" at that time. Their utter failure became clear in a short time and was thoroughly documented. One result was adoption in December of 1997 by the State Board of Education of an excellent set of rigorous standards for K-12 math that have been lauded across the country. Unfortunately, Morgan Hill public schools seem to be stuck in the fad of CPM.

The fault for the sad state of the math program in Morgan Hill lies with the teachers, administrators, and the Board of Trustees who promoted this program and have failed to replace it. Adoption of CPM as the great hope of past years might be excusable, but its continuation for years past the point where its deficiencies became blazingly obvious, with class after class becoming casualties, is inexcusable.

In my comments of June 25, I asserted that the Morgan Hill (California) school district's math program "runs counter" to all six philosophical points for a good standard extracted from a Fordham report that evaluated state standards. I gather from speaking to one official of the district that there may be some dispute about that; i.e., whether in fact Morgan Hill's program is consistent with the state standard, and, by inference, at least some of the six items I cited.

The District's math program uses Addison-Wesley's Quest 2000 for the elementary grades and CPM (College Preparatory Math) for secondary. Both are based on principles of "reform math," which was endorsed by the math framework adopted by the state in 1992. That framework, due to be revisited in 1999 (scheduled every seven years), was preempted by the standard adopted by the state Board last December. The unusual action of adopting a new standard out of order was prompted by widespread dissatisfaction with the 1992 framework.

Quest 2000 and CPM are full-strength reform math programs that feature the constructivist approach (coaxing students to figure things out for themselves), working in groups as opposed to lecturing, problem-based "real-world" learning, de-emphasis of drill in arithmetic, emphasis on use of calculators beginning in the very lowest grades, fuzzy answers, and emphasis on exposition of solutions.

The six items I cited are listed below (some shortened for brevity) with the reasons why I contend they are not subscribed to by the district.

New standards for mathematics for K through 12 were adopted by the California State Board of Education last December after an intense battle involving a special commission, several drafts, public comments, and uncountable meetings. Despite the messiness of the process, the result seems to be quite good. According to a recent Fordham Foundation report that evaluated K-12 math standards for 46 states and the District of Columbia, the new California standards are the best in the nation.

Unfortunately, that won't do Morgan Hill students any good because, speaking for the District, Superintendent Carolyn McKennan said, "we don't agree with the state standards." What she meant is that the Morgan Hill Union School District is wedded to the concepts of what District people call "reform math," known by others variously as "fuzzy math," or "new-new math." (The last is designed to distinguish between this latest version of "reform math" and the previous one, which was called "new math"; hence "new-new math.")

It was an unpleasant surprise to many of us who welcomed the new standards to discover that the battle at the state level was only the beginning, and it has to be fought again district by district.

A minor skirmish has broken out, sparked by a group of parents with children at Paradise Valley Elementary. Whether it will amount to anything remains to be seen. Two meetings were held with Superintendent McKennan, and she promised to provide an answer to their concerns by the end of June, later changing that to the end of July. (The meetings covered many other topics, but math was the main issue concerning curriculum.)

The Fordham report (see at www.edexcellence.net/standards/, also available by calling 1-888-TBF-7474, single copies free) makes interesting reading. Standards were evaluated for all but four states that declined to provide them. For comparison purposes, the standard used in Japan was evaluated by the same criteria.

The most striking result is the dismal showing on the whole. The average score was 6.5 of a possible 16.0. The median was 5.5, meaning that 23 states were below that. California was at the top with a perfect 16.0, besting even Japan at 15.0.

The report spells out the evaluation criteria in detail and indicates a thorough job. A key question, of course, is what the authors consider a good standard to be, especially content. Their viewpoints are spelled out clearly in four sections that explain their evaluation criteria. A few examples may suffice to convey the sense of their views:

The first item of President Clinton's education "program" (something it isn't, but that's another story), is national standards and tests for English and math for 4th and 8th grades, respectively. The president deserves due credit for promoting this idea, especially considering that he has been consistent on this issue for many years.

Like so many other issues of the day, however, "the devil's in the details," and it appears that defining the standards will be as contentious as ever.

Lynn Cheney, former chairwoman of the National Endowment for the Humanities, recently criticized the president's choices for the committee that will oversee his national eighth-grade math exam. Ms. Cheney contends that people dedicated to the "whole math" approach dominate the committee and that the project is therefore doomed before they begin.

There have been several attempts to implement standards. The current one in California is by the 21-member Academic Standards Commission, which recently released its initial drafts for math and reading standards for K-12.

The main contention in the math standards effort is between those who favor teaching children "how to think" as opposed to the traditional approach. The latter is criticized as "rote memorization," the prime example being drill in arithmetic, such as the multiplication table.

The "how to think" school of thought is variously called the new-new math, whole math, fuzzy math, or the constructivist approach. It is supported by the National Council of Teachers of Mathematics (NTCM) and was adopted in large part by the California Department of Education as a "reform" in 1985 and more determinedly in 1992.

The Morgan Hill School District adopted a form of the new-new math called CPM, which stands for College Preparatory Mathematics. At Live Oak, it replaces Algebra I, Geometry, and Advanced Algebra. It is full-strength constructivist math, with primary emphasis on self-discovery and group effort.

The constructivist approach favors the idea of coaxing students to discover mathematical realities for themselves. It is based on the theory that things which students figure out for themselves are learned better and are more useful than things simply told to them. This idea leads to less lecturing by teachers and more group effort and practical "real-world" examples.

CPM was created by a group of teachers in 1989 and has been in fairly wide use in California since around 1993. Lecturing is strongly discouraged, and most of the work, including even some exams, is carried out in groups of four students. The courses consist almost entirely of problem solving, from which the mathematical principles are supposed to be extracted by example. Formal structure and drill are essentially eliminated.

A number of parents' groups have formed to oppose new-new math, claiming that it is drastically watered down and ineffective. One of the more prominent of these is HOLD (Honest Open Logical Debate) in Palo Alto, one of whose members was appointed to the Academic Standards Commission's subcommittee on mathematics. The battles are incredibly intense, even within the Commission itself.

The main objection of these parents is the de-emphasis - they say complete absence - of drill and structure. The pedagogy is also criticized because of its group approach: thinking that you can put teenagers in lightly-supervised groups, encourage them to talk, and expect them to stay "on task" is, well, ridiculous.

Many teachers chafe under CPM because it dictates not only content but pedagogy. Traditionally, teachers had considerable discretion in how they taught. Under CPM, everything is specified, practically down to the teacher's script.

Paradoxically, CPM places greater dependence for its success on the teacher's ability in math. This is so because the mathematical principles are substantially extracted from the problem-examples and escape less qualified teachers. They get by, however, by simply following the script. This is no small matter, because 51 percent of math teachers in California have neither a major nor a minor degree in math. In fact, this may have more to do with our problems with math teaching than anything else.

It is well known that parental interest determines educational success more than any other factor, and that's no doubt the reason students in Palo Alto do the best in the county. The fact that parents in Palo Alto are also the most vocal in their dissatsifaction with the new-new math should give pause to parents elsewhere.

I attended the lecture by education consultant Dr. Ruth Parker Aug. 26 at Live Oak called "Mathematics for the Future." I'd seen a paper she'd written, so I knew she was a supporter of the new-new math, or, as she and her colleagues prefer to call it, reform math.

I had already formed an opinion about it - I don't like it - but I resolved to listen and try to understand what she had to say. She didn't convince me.

Dr. Parker acknowledged that there is considerable controversy about reform math and had nothing good to say about its opponents, targeting especially newspaper columnists "who turn out to never have taught math to children." She has the zeal of a missionary and asked her audience to spread the word for the good of the cause.

New new math is based on the idea that children should learn by figuring things out for themselves. Parker gave many examples of what she considers brilliant insights which children have come up with. These were indeed interesting, but one wonders how many children can't figure it out for themselves and just sink into a fog.

Much of the time was spent on alternate methods of doing arithmetic, mainly adding and multiplying. She spoke disparagingly of "the American algorithms" for these and showed how one can do just as well working left to right as the standard right to left. This method has the advantage that one gets a better feel for the magnitudes of numbers - what she calls "number sense" - and can avoid blunders by keeping perspective along the way.

I can see that point and might agree except that she is not saying that we should teach adding and multiplying this other way. The enemy in her mind is any set recipe: kids should be stimulated to come up with their own ideas.

Her favorite example is what she calls "the turkey problem," which she has been using for many years. It is a "word problem" involving proportion and fractions. It can be solved any number of ways, but the traditional approach is to equate two fractions, one containing an unknown, and to solve for the unknown by "cross-multiplying." Dr. Parker is proud of the many imaginative ways students have solved it, most using graphical techniques. She has nothing good to say about the traditional approach, and one gets the feeling that a student who did it that way would be sent to the corner.

Again, these are interesting demonstrations of resourcefulness and ingenuity, but is it mathematics or just parlor games? She spoke reverently of mathematics, but she and I don't have the same thing in mind. To her mathematics is a way of thinking. To me it is a logical structure. A discussion about how to teach it is pointless because we're not talking about the same subject.

The "way of thinking" philosophy discounts the importance of the basics. Learning rote algorithms is a waste of time because it contributes nothing to the ability of the students to reason for themselves; to what the National Council of Teachers of Mathematics calls mathematical power: "the ability to explore, conjecture, and reason logically."

Those skills are valuable, even indispensable, but mathematics is not just a way of thinking; it is the magnificent set of truths bequeathed to us by the great mathematicians of the ages, and no amount of coaxing will allow children to reconstruct those truths out of their own heads.

The trouble with the cute methods that students devised for solving the turkey problem is that they don't work for all cases. The cross-multiplying method which Dr. Parker so sarcastically maligns does. Mathematics is a science, and one of the central issues in science is: Does it apply in the general case? Cross-multiplying does; the others don't. QED.

Education is especially prone to fads that roll through every few years, each promising to solve all the problems. Mostly, they turn out to be only new brands of snake oil, and I predict that new new math will be submerged in a few years by yet another approach "based on the latest research," regaled by education consultants, and destined to meet the same fate a few years later. It is a tribute to the human mind that the country survives in spite of it.