Some observations on the 1997 battle of the two
Standards in the California Math War

H. Wu

1.

In October of 1997, the Standards Commission of California submitted to the State Board of Education a set of Mathematics Content Standards [1] which took the Commission more than a year to complete. But within ten weeks, the State Board released a revised version of its own [2], first the portion on grades K--7 and then that on 8-12. The reaction to the revision was swift and violent:

The interest engendered by these two sets of Standards has remained unabated in the intervening months. For example, in the February issue of its News Bulletin, NCTM has weighed in with unflattering comments about the Board's revised version [3]. Because education is a very political issue, there is no need to bemoan the fact that opinions are often delivered without relation to facts. However, a set of mathematics standards for schools also deserves a critical inspection from the mathematical and educational perspectives, one that is based on facts and not on hype. With this in mind, this article takes a close look at both sets of standards from a scholar's perspective. Section 2 details some of the mathematical flaws in the Commission's Standards, and Section 3 contrasts these flaws with the clarity and the overall mathematical soundness of the Board's revision. In Section 4 there is a discussion of possible additions to the California Mathematics Framework Draft [4] that would enhance and provide balance for the Board's Standards.

The Commission's Standards is a thoughtful document. In both the Interim Report from the Commission Chair to the State Board and the Introduction to Mathematics Standards, one sees clearly the care that went into the enunciation of the goals, the work that had been done to achieve them, and the work that is still needed in the days ahead for their implementation. Even if one disagrees with some of the details, one can applaud the overall soundness of purpose and the conscientious effort that went into the writing. Yet there are also grievous defects in the document that made its revision inevitable. This is a classic example of how good intentions are shipwrecked by questionable execution. Parts of the document are extremely controversial, such as the omission of the division algorithm in the lower grades1, the omission of the Fundamental Theorem of Algebra in the upper grades, or the mixing of pedagogical statements with statements on content. There is also a pervasive ambiguity of language that makes the document less than readable in many places, e.g., were the authors aware that the word "classify" has a precise meaning in a mathematical context? Or, what is a 7th grader to make of "identify, describe, represent, extend and create linear and nonlinear number patterns? " However, this article chooses to focus attention on the numerous mathematical defects because they are more susceptible to a dispassionate discussion.

But first, are mathematical defects really that important in a set of mathematics standards? Such a question, were it posed thirty years ago, would have been met with howls of derision. Times have changed, however, and there are those who claim that it is not what is taught, but how it is taught that matters (cf. e.g., [10], especially pp. 203-8). Risking some sneers from my colleagues, let me therefore affirm that indeed I believe getting the mathematics right is very important in any mathematics standards, in the same way that correct pronunciation is critical to being a good teacher of a foreign language. Are there people who feel comfortable about sending their children to a French class taught by a teacher who mispronounces a word every other sentence? While the general public cannot conceive of a set of mathematics standards not being mathematically correct, the fact remains that mathematically correct public documents on mathematics education are more rare than people realize. For example, the mathematics standards of most of the states from around the nation exhibit mathematical ignorance (cf. [11]). Even the NCTM Standards [5] is no exception to this rule: there is an outright mathematical error at the top of p. 136, and many discussions show a lack of understanding of the underlying mathematics (e.g., p. 149 and bottom of p. 165; cf. also [12]). This is why when this article comes around to affirming the mathematical correctness of the Board's Standards later on, such an affirmation must be taken as strong endorsement.



2.

The mathematical flaws in the Commission's Standards [1] are of two kinds. First there are the local ones, i.e., those which contain obvious errors which can be corrected without causing damage elsewhere. A colleague has estimated that there are over a hundred of these, and that is a conservative estimate. Since it is impossible to be exhaustive, we will only exhibit a few that are easily understood even when taken out of context. Starting with the Glossary at the end, we find, for example:

Since this definition of an asymptote does not specify that the distance between the curve and the straight line has to decrease to zero, it would make the line y=-1 an asymptote of y=1/x for x>0. By dictating that the "axioms" of an axiomatic system must be "self-evident truths", this definition excludes the axioms for non-Euclidean geometry from being an axiomatic system. After all, the statement that given a line and a point not on the line there are infinitely many lines from the point not intersecting the given line is certainly not a `self-evident truth".2 This term has a precise technical meaning in symbolic logic, and its definition is nothing this simple. Perhaps the authors had in mind "recurrence relations" instead. Assuming this to be the case, then the correct definition would change "the previous value" to "previous values". Otherwise, even the Fibonacci numbers would not fit this description.

Next, we turn to the Standards proper and look at some representative local flaws. It may be noted that the following examples do not include any that might have been the result of carelessness, such as that about the asymptotes of a polynomial (Clarification and Examples:3 for 1.1 and 1.2 in Algebra and Functions of grades 11/12).

It is not common to equate "predict outcomes" with mathematical reasoning. One would gladly overlook this as an inadvertent error but for the fact that the same sentence appears nine times all through grades K through 8

This passage is supposed to clarify the content of the Standards, but it has achieved the opposite effect of obfuscating it. It would take many pages to write an analysis that does this passage justice, so here is a very abbreviated account (but see [13]). First of all, mathematics deals with precise statements, and if we are going to educate our children at all, we would do well to teach them the necessity of eliminating the inherent fuzziness in many everyday utterances before transcribing them into mathematical terms. "Her classroom is bigger" is clearly a case in point. Faced with such a statement, a set of mathematics standards has the responsibility to instruct children of grade 3 to make sense of the word "bigger" before proceeding any further. If they interpret "bigger" to mean "more area", then they should measure the respective areas. If they interpret "bigger" to mean "longer perimeter", then measure the perimeters. The basic message is therefore that each answer would be correct according to whichever interpretation is used. Furnishing such an explanation would seem to be the minimum requirement of a mathematics education for the young. Now look at the passage above: it tells teachers and students alike to accept an instruction that has no precise meaning ("bigger") and immediately proceed to "find the answer", and worse, "prove that your solution is correct using mathematics". If a teacher in an English class shows students a black box without telling them what is inside other than that it is an expensive piece of jewelry, and asks them to write an essay to describe the latter and justify why their description fits the object, there would be an uproar. Yet when the same thing happens in a set of mathematics standards, we have people leaping to its defense and calling it "world class". Why is that? The trouble with both standards is that there is no relationship whatever between perimeter and area, or between volume and surface area, unless it be the isoperimetric inequality. However, the latter would be quite inappropriate for students at this level. What could the authors have in mind? What does "fractional value" refer to in this case? Does each piece count as one unit, or is the area of each piece being sought in proportion to the whole area? What kind of "equivalence" between the pieces is intended here and why has it not been clearly defined? Since a fraction is a ratio of integers, how can there be any difference between them with respect to their mathematical operations? Some educators, it is said, have begun to advocate that fractions are not ratios. If so, then we must redouble our efforts not to allow such ideas to creep into any mathematics standards. There is no explanation of how a 6th grade student could "determine estimates of pi" with this kind of accuracy, 3.14 or 22/7, especially the latter value. Is such a precise estimate even remotely conceivable? The order of operations to evaluate algebraic expressions is a matter of definition, and is not a technique. Moreover, to say in a mathematics standards document that knowing the simple definition of the notation is more helpful in the situation of 3(2x+5) than in 3(2x+5)2 is to undercut its own credibility. One would guess (although that is asking a lot of the general reader of the Standards) that the "relationship between the time and the height of water" is that the height is a well defined function of time. This function happens to be quadratic, but what could it mean to find an equation that would generalize the situation? If the intended message is that the zeros of a given polynomial can be approximated by examining the intersection of its graph and the x-axis, then this statement is very poorly phrased. If the intended message is something else, then obviously this statement needs to be completely re-written.

Next we examine a different kind of mathematical flaw: the global ones. Their corrections would involve changes in several related parts. The first such example occurs in grade 7:

Now Grade 7 is not the usual place to find references to simple transformations in the plane and their images. What is meant by a "simple transformations"? Has it been defined? Has the image of a transformation been discussed? It turns out that "simple transformations" are defined nowhere in the Standards, but one could guess from related comments that the authors had in mind reflections and translations. It is difficult to decide whether the authors were unaware of the need to fulfill the minimum mathematical requirement of clarity, or simply considered such matters unimportant. Could such negligence be nothing more than a momentary lapse? Not likely, because one also finds in grades 9/10 another reference to "transformations" in the plane with no explanation:

A pertinent related issue in connection with the above standard in grade 7 is how much coordinate geometry has been developed up to that point so that students may appreciate such a discussion. The answer appears to be "not enough". The first introduction of coordinates in the plane takes place in the 4th grade under, of all places, Algebra and Functions:

Special attention should be called to the fact that the important idea of "algebraicizing" the geometric plane occurs here almost as an after-thought in a discussion on variables and mathematical symbols. Since it is now fashionable to talk about "conceptual understanding", one can say unequivocally that such a set of mathematical standards displays a lack of conceptual understanding of mathematics. But to continue with the present discussion, in the standards of grade 5, one finds "write the [linear] equation and graph the resulting ordered pairs of whole numbers on a grid" (Algebra and Functions, 2.2) and in grade 6, more graphing of linear functions and "single variable data" is called for in Algebra and Functions and Statistics, Data Analysis and Probability. This would seem to be the extent to which students have been exposed to coordinate geometry before they are asked to contemplate the image of a transformation in the plane.

Consider now a second example, which is the way the Commission's Standards approaches the Pythagorean theorems, a fundamental result in school mathematics. The first mention of this theorem is in grade 7:

(One's first reaction to the last sentence---"Experience with ..."---is: "How very Californian!") This standard certainly makes it sound as though the Pythagorean theorem is a tool already familiar to the students, and it took some time to find out that in fact this is the first time the theorem is discussed. One could bend over backward to give a benign interpretation of this standard as: "State the Pythagorean theorem and verify it empirically by direct measurements". Few readers, however, would recognize that this is the intended message. Because this theorem is so surprising to a beginner, one would expect a demonstration of its truth early on. For example, the so-called "tangram proof" using four congruent right triangles nestled in a square is so elementary that it could be presented to 4th or 5th graders. One finds instead that when the theorem is mentioned again in grade 8 and for the first time in grades 9/10, no proof is mentioned: It should be obvious that this standard in grade 8 merely repeats what is already in grade 7. What purpose does this serve? In addition, is it good education to ask students to believe in the converse of this theorem in grades 9/10, as indicated above, without first giving them a proof of the theorem itself? It remains to point out that only later in sub-Standard 4.4 of Measurement and Geometry, grades 9/10, do we find: "prove the Pythagorean Theorem using algebraic and geometric arguments".

It was mentioned earlier that the Commission's Standards omits the long division algorithm in the early grades except for the case of a single digit divisor (grade 4). With that in mind, let us look at what happens in grade 7.

Yet the mere fact that a fraction yields a repeating decimal depends on the understanding of the sequence of remainders in the division algorithm. How are students going to understand that terminating and repeating decimals represent fractions without first knowing this algorithm by heart? Furthermore, in grades 11/12, we have: Perhaps not enough thought was given to the fact that, without learning the division algorithm for integers, it may not be possible to teach synthetic division for polynomials.

Incidentally, the preceding two examples from the Commission's Standards show an all-too-common sloppiness of language: "equivalent relationship among ...", "relationship between terminating and repeating decimals ...", and "attributes of the equation and graph" are too vague for a set of mathematics standards.

As a final example, let us look at how the Commission's Standards handles the concept of a function. Although the term "functional relationship" is used already in grades 4 and 5 ("the functional relationships within linear patterns" in grade 4, and "solve problems involving functional relationships" in grade 5), a knowledgeable reader could conceivably deal with such missteps by ignoring them. (The Board's Standards in fact simply deletes all such references.) However, in grade 6 of Commission's Standards, one finds:

Since it calls for a direct confrontation with the concept of a function itself, this standard is less likely to be ignored and the potential damage is consequently greater than before. Are students to learn about the definition of a function, or are they not? That is the question. The hazy conception of mathematics itself as exemplified in this instance (and elsewhere too, of course) is unnerving to the mathematically informed. If one cannot resolve this issue here, what about the next one in grade 8? Because this explicitly asks students to distinguish between a relation and a function, nothing short of a full-scale investigation of the functional concept would suffice. But should one do this in grade 8, and is this really what the authors had in mind? The answer seems to be supplied, however indirectly, by the following standard in grade 9: It would appear that here is the first time that students learn what a function is. If this is to be believed, then what is one to make of all the rumblings on this topics in grades 4 through 8? But if not, i.e., if a function is supposed to have been defined earlier, then what is such a standard doing in grades 9/10?

I hope the foregoing gives some idea of the magnitude of the problems besetting the Commission's Standards. At the same time, it should be pointed out that these problems are probably not detectable by someone who is not mathematically knowledgeable. The criticisms of the Board's Standards coming from educators and politicians are therefore understandable to a certain degree. By the same token, this gap in mathematical knowledge then imposes on those of us in mathematics the obligation to serve as intermediaries between the Standards and the public. Regardless of our philosophical orientations in matters pertaining to education, we should have spoken as a single voice in detailing the glaring mathematical failings of the Commission's Standards in order to furnish a valid platform for the ensuing debate. We should have been the voice of reason to inform and to mediate. The mathematical community in California may well ask itself at this point if it has indeed met its obligation, and met it well.4


3.

Now a brief look at the Board's Standards [2]. The overriding fact is that no document of this nature can be expected to be without blemish, and it would be foolhardy to look for perfection or to argue that this set of Standards is close to perfection. What is important is to ask whether it has fatal flaws, and whether in the main it points in the right direction of a sound mathematics education. The answers to both are easily no and yes, respectively, and their justifications will emerge in the succeeding discussion.

An important point regarding the Board's Standards is that, in reading this document, one does not wince in embarrassment over mathematical errors. Let us first start with the portion on grades K--7. This portion is very close to the Commission's Standards, and the only difference between the two is that the Board's version eliminates the ambiguous and superfluous, corrects the erroneous, and deletes the Clarifications and Examples in the right column of the original. I will have more to say about the latter presently, but let us sample some of the differences. It was mentioned above that in grade 4, the Commission's Standards incorrectly asks for "the relationship between the concepts of perimeter and area". By comparison, the Board's version now reads:

It is clear, and it is correct. More than that, 1.2 and 1.3 anticipate students' possible confusion, and 1.4 emphasizes the importance of applications and the general principle of progressing from the simple to the complex.

Another example is the Board's correction of the error committed in the Commission's version regarding the introduction of coordinates in the plane in grade 4. Now it is accorded a standard all its own and is placed correctly in the strand on Measurement and Geometry.

Note that sub-standard 2.1 pays special attention to the tactile aspect of learning mathematics: use graph papers and draw the first ten points (by hand). We should be grateful that it does not say: enter these data in a graphing calculator and watch the graph emerge on the screen. Moreover, sub-standards 2.2 and 2.3 again anticipate students' confusion by singling out two key points for discussion. There is no question that this is an education document that truly tries to educate.

As a final example, let us look at how the Board's version discusses in one instance the issue of mathematical reasoning:

In plain English---readable English---this standard lays out a step-by-step method of doing mathematics. Educational writing can be no better than this.

It is improvements of this nature that make the Board's Standards [2] a superior document over the Commission's Standards [1] in grades K--7. Yet, intense criticisms were already pouring in as soon as the K--7 portion of the Board's Standards appeared. Looking at the facts, how does one presume to claim that this set of standards is "basics only", or that it "almost cuts out almost everything that is not related to computation and the memorization of formulas" ? Obviously not on account of the standards themselves, but one explanation is that some people reacted strongly to the deletion of the Clarifications and Examples that are in the Commission's Standards.

It is time to point out that whereas in other states the Mathematics Standards must stand alone as the sole guide-post for mathematics education, we in California have two documents: the Standards and the Framework [4]. In this arrangement, the curricular comments on the Standards, including examples, properly belong to the Framework, which is yet to be approved by the Board. It serves no purpose to criticize the absence of examples in the Board's Standards when they have merely been moved to a companion document. If one's goal is to improve the Board's Standards rather than stir up controversy, the natural thing to do would be to make concrete suggestions for changes in the existing Framework Draft. This article will make several such suggestions.

Let us complete our brief survey of the Board's Standards by looking at grades 8--12. There is a basic change of format here, in that the grade-by-grade account in the Commission's version is replaced by a listing of topics in the traditional strands across the grades: Algebra I, Geometry, Algebra II, etc. The justification is that since at present an overwhelming majority of the schools teach mathematics in the traditional manner while others do so in an "integrated"7 manner, listing only the content of each subject would provide maximum flexibility. Instead of prescribing one particular approach to the curriculum, it throws the door open to many approaches. Such a change is a defensible one, and is in any case not one to make a lot of fuss about. With this understood, one can immediately appreciate the clear and uncompromising demand that the Board's Standards places on students' all-around mathematical competence---not the formula-laden, rote-learning variety, but the genuine one. Students must be technically proficient, and they must also know what they are doing. For example, consider the discussion of the quadratic formula in Algebra I (which contains twenty-five standards):

It does not say: derive the quadratic formula and use it to solve all quadratic equations. Instead, it makes students learn the important technique of completing the square first. Five standards later---which presumably indicates that one gives students time to digest it before proceeding---it asks for a derivation of the formula. Then, and only then, does it mention using the formula to solve equations. Does a document that handles the learning of a formula in this sensitive manner strike anyone as a "back-to-basics" document that emphasizes memorization and computation? Next, a similar example in a different subject: The unequivocal demand on students' ability to write down proofs and counterexamples is important in this day and age of diminished standards when proofs produce allergic reactions in many education circles. One can quibble with the precise meaning of standard 4---and more of this later---but that is not the same as insinuating that these Standards ax the development of mathematical understanding in the students. My personal opinion is that these are thoughtful standards, but their virtues are by no means apparent to the general public. Perhaps for this reason, the torrent of abuse heaped on these Standards took over the front pages of many newspapers for several weeks. Here are some reminders: It may be noted that the NCTM editorial [3] endorsed the preceding statement by Luther Williams.

One may ask, in light of all the flaws in the Commission's Standards and the obvious emphasis on mathematical understanding in the Board's version, how people could bring themselves to make indefensible statements about the high quality of the former and the unworthiness of the latter. There are probably political and psychological reasons that are beyond my power to probe, but as an educator, I would like to offer a speculation on how this has happened. I believe there is a fundamental misconception about mathematics education that has sprung up more or less in the past decade, which is that there are conceptual understanding and problem solving ability on the one hand and basic skills on the other. Furthermore, this misconception postulates that it is possible to acquire the former without the latter. It is likely that the explicit requirement of fluency in basic skills in the Board's Standards was seen by some as an artificial obstacle intentionally set up by elitists to thwart students' "mathematical empowerment". Hence the resulting furor. One can acquire some appreciation of mathematics without mastering technical skills, in much the same way that one can instantly recognize an opera in recordings of "operas without the human voice"8 and even enjoy it to some extent. But if we wish to educate students properly about the art of the opera, using such recordings "without the human voice" is not recommended. In the same way, a correctly written set of mathematical standards has to be more like the Board's version rather than the Commission's. In mathematics, understanding goes through technique, and technique is built on understanding. That is the way it is.


4.

It is time to take a critical look at the Board's Standards [2] and make explicit some of the concerns adumbrated earlier. There is no pretension to being comprehensive in this critique, however. Almost all the recommendations below are concerned with what to add to the Framework Draft [4] in order to round off the Board's Standards. Since this Draft is still in a state of flux, it is quite possible that these recommendations have already been anticipated by those in charge of [4]. If so, then nobody would be happier than I to have been rendered irrelevant in this undertaking.

First, a minor concern. By the end of the 5th grade, the Board's Standards mandates that "students increase their facility with the four basic arithmetic operations applied to positive and negative numbers, fractions and decimals." In principle there should be no problem with this goal. In practice, students in other countries usually achieve this level of competence in the 7th grade. For this reason, we may wish to watch carefully how this stipulation would play out in the classroom.

It is gratifying to know that examples which would clarify the terse statements of the Board's Standards will be incorporated into the final version of the Framework [4], but let me explicitly lobby for more clarifications along this line. There is no doubt that in order to help educators across the state understand the Standards, especially in grades 8--12, more guidance in the Framework is needed both in the details and in the overall planning. Regarding the former, a statement such as standard 4 in Geometry (Grades 8--12), "Students prove basic theorems involving congruence and similarity", means many things to many people. Should the AAA theorem for similar triangles be proved, for example? Depending on how this statement is approached, it can be a difficult theorem. Or take standard 2 of Algebra II: "Students solve systems of linear equations and inequalities (in two or three variables) simultaneously, by substitution, graphically, or with matrices". This may or may not be calling for some discussion of linear programming, and since matrices appear here for the first time, the question naturally arises as to how best to handle it. We must remember that these Standards are pioneering something new in California, and pioneers have to be transcendentally clear at each step or they run the risk of having no followers on their trail. I wish to drive home this point by comparing with what I consider a very admirable set of mathematics standards, the 1990 Mathematics Standards of Japan [6]. There the statement about similarity (in grade 8!) is equally terse:

There is a big difference, however. The Japanese change their standards every ten years and, because they already have a well established tradition, the changes are gradual and minor by comparison with the kind of sea change we have over here. Moreover, they have excellent textbooks (cf. [7]--[9]9) already in place, so there is no great need to spell out everything. By contrast, we are almost starting anew in California, especially in these turbulent times in education. There is therefore very great need for the Board's Standards to be absolutely clear.

On the matter of overall planning, the Standards intentionally eschew any prescription on how to teach students in grades 8--12, whether in the traditional way or the "integrated" way.10 The intention for greater flexibility is admirable, except that in the absence of a tradition, the added flexibility may turn out to be a curse. For example, the Standards specify that each discipline (Algebra I, Geometry, etc.) need not "be initiated and completed in a single grade". It would appear that this specification makes it possible to describe the desirable content of each discipline without undue regard to the time limitation of fitting everything into exactly one year. Perhaps for this reason, there are more topics in Algebra II than can be reasonably completed in a single year. How to teach this material in more than two semesters then becomes a challenge which few schools could meet. Also Algebra I asks that "Students [be] able to find the equation of a line perpendicular to a given line that passes through a given point." No matter how this is done, it would involve theorems about similar triangles. Does it then imply---contrary to the traditional curriculum---that Geometry may be taught simultaneously with Algebra I? The Framework would have to give more explicit instructions on how to bring this off. Finally, it appears that the forthcoming 10th or 11th grade statewide mathematics test would include some statistics. Is the Framework going to suggest ways of teaching statistics in the early part of secondary school if the traditional curriculum is followed?

Considerations of this nature bring out the fact that the traditional method of offering year long sequences on algebra and geometry is too rigid to be educationally optimal. While none of the current "integrated" models in this country seems to be entirely successful, the argument cannot be ignored that we should pursue the kind of integrated mathematics education that has been in use in Japan or Hong Kong for a long time. The Framework would be fulfilling its basic function if it could nudge California in this direction in a forceful manner.

An idea that has undoubtedly occurred to many people is how much the standards of grades 8--12 in the Board's Standards read like a "Manual for Pure Mathematics". One almost gets the feeling that this document could not bring itself to face the relationship between school mathematics and practical problems. It is now incumbent on the Framework to restore the balance between the pure and applied sides of school mathematics. While it is true that the reform exaggerates the role of "real-world" problems in mathematics, ignoring them altogether is for sure not a cure either. We would do well to remember that the overwhelming majority of school students will be users of mathematics, and that as future citizens they need to be shown the power of mathematics in the context of daily affairs. But all through grades 8--12, I seem to see only three explicit references to applications:

I hope I am not over-using the Japanese model if I look at the corresponding situation in [6]. The description of the Content of the Standards in [6] is every bit as abstract and "pure" as the Board's Standards, but The Construction of Teaching Plans and Remarks Concerning Content after each of grades K, 1--6, 7--9, and 10--12 pays careful attention to the bearing of "daily affairs" on the curriculum. For example, here is what is said after grades 7--9: The tone makes it abundantly clear that this is no mere lip service to applications, but that the applied component is central to the whole curriculum.

I hope that the Framework will be equally emphatic on this point in order to make clear that the relation of mathematics to daily affairs is also central to the Mathematics Standards of California.

A final comment is on the contentious subject of technology. From K to 12 in the Board's Standards, I could detect only the following two references to technology:

I fully expect to be shown to be wrong, that there is another place where technology is mentioned. Nevertheless, I hope my point is clear. If this reticence is not complemented by a strong message in the Framework on how to confront technology, then we would be conceding that we, as educators, do not know how to deal with the technology around us. But the computer and the graphing calculators are here to stay, and the younger generation is besieged on all sides by them. It would not be an effective education policy to retreat and abdicate responsibility exactly when we are supposed to come forward to provide guidance. We do not want any kind of technological debauchery in the mathematics classroom, but neither do we want to make technological prudes out of our students. What we want are students who are technologically informed, especially about the role of technology in mathematics, but we won't get them if we continue to pretend that technology does not exist. I am being intentionally suggestive in my use of language in order to force the comparison with sex education. In both situations, it is better to keep our students informed than to let them pick up the wrong information in a state of prevailing ignorance.

Allow me to cite for the last time the Japanese Standards [6]. Part of The Construction of Teaching Plans and Remarks Concerning Content also deals with the technological issue after each of grades K, 1--6, 7--9, and 10--12. Here is what is said after grades 1--6 and 10--12, respectively.

The Board has already wisely decided that no state test in grades K--6 would use calculators. This general policy on technology, sensible as it is, needs to be supplemented by a more comprehensive one which gives guidance not only on when not to use it but also on when to use it. For example, encouraging teachers in K--6 to use problems with more natural---and therefore more unwieldy---numerical data by enlisting the help of calculators is a beginning. In the presence of the no-calculator-in-tests rule, students would get a clear perspective on what they need to know regardless of technology, and on how they can use technology to their benefit when the need arises. Encouraging students in calculus to use calculator to estimate the limits of sequences while also holding them responsible for proofs of convergence is another example. Doubtlessly, thoughtful educators will be able to formulate similar specific recommendations in other situations. As the preceding passages from [6] indicate, we must make active use of calculators and computers to improve the effectiveness of teaching and learning, and what better place to launch this idea than in the Framework?


5.

It is very likely that another person who is willing to read the Board's Standards carefully would come to slightly different conclusions about its strengths and weaknesses. It is even more likely that, in that case, the differences can be calmly discussed and the resulting discussions would benefit the next generation in the long run. One can either avail oneself of this opportunity to improve education in California, or one can turn one's back to the welfare of the young and act irresponsibly.

This then brings me to the news release about U.S. 12th-grade performance on TIMSS on February 24. Gail Burrill, the President of NCTM, made the following comment on the TIMSS result: "What's important is that we are working together toward a common goal of excellence in mathematics. The recent math wars have done nothing to improve mathematics education." These are sobering statements. On the one hand, Ms. Burrill's optimistic view that we are already working together toward a common goal in mathematics education could not have been based on the reckless public condemnations of the Board's Standards that have just transpired. NCTM's editorial [3] has not exactly contributed to producing harmony either. On the other hand, the math war in California did manage to reverse the disastrous trend initiated by the 1992 Mathematics Framework for California Public Schools. While much work remains to be done to achieve a balanced mathematics education in California, this achievement of the math war alone would give the lie to the assertion that math wars have done nothing to improve mathematics education. Nevertheless, educational reconstruction should be our common goal at this juncture, and the battle over the Standards is in this light nothing but a distraction. In his address before the Annual Meeting of AMS-MAA on January 8, 1998, Secretary Richard W. Riley had sounded the same theme of reconciliation: "This leads me back to the need to bring an end to the shortsighted, politicized, and harmful bickering over the teaching and learning of mathematics. I will tell you that if we continue down this road of infighting, we will only negate the gains we have already made -- and the real losers will be the students of America." In all our education activities we should think of our children first. No, we must. If there is any lesson to be learned from the battle of the Standards, it is that it serves very well as an object lesson on how not to behave in the future.

Acknowledgment: I could not have written this article without the support of Henry Alder, Dick Askey, Wayne Bishop, and especially David Klein. Subsequent corrections by Roger Howe also contributed significantly towards an improved presentation. I would like to express my heartfelt gratitude to all of them.


Bibliography

  1. California Academic Standards Commission, Mathematics Content Standards, October 1, 1997. Available at
    http://www.ca.gov/goldstandards
  2. The California Mathematics Academic Content Standards as adopted by the California State Board of Education, February 5, 1998. Available at
    http://www.cde.ca.gov/board/board.html
  3. New California standards disappoint many, NCTM News Bulletin, Issue 7, 34 (1998), 1 and 5.
  4. Mathematics Framework for California Public Schools, Kindergarten Through Grade Twelve, Field Review Draft: October 15--December 15, 1997, California Department of Education.
  5. Curriculum and Evaluation Standards for School Mathematics, National Council of Teachers of Mathematics, Reston, 1989. Available at
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  7. K. Kodaira, ed., Japan Grade 7 Mathematics, Japan Grade 8 Mathematics, Japan Grade 9 Mathematics, The University of Chicago Mathematics Project, Chicago 1992.
  8. K. Kodaira, ed., Mathematics 1, Mathematics 2, (Japan Grade 11 Mathematics), American Mathematical Society, Providence 1997.
  9. K. Kodaira, ed., Algebra and Geometry, Basic Analysis, (Japan Grade 11 Mathematics), American Mathematical Society, Providence 1997.
  10. Rita Kramer, Ed School Follies, The Free Press, 1991.
  11. R. Raimi and L. S. Braden, State Mathematics Standards, Fordham Report Volume 2, No. 3, Thomas B. Fordham Foundation, Washington D.C., 1998.
  12. H. Wu, Invited comments on the NCTM Standards, available at
    http://math.berkeley.edu/~wu
  13. H. Wu, The role of open-ended problems in mathematics education, J. Math. Behavior 13 (1994), 115-128.
  14. California Mathematicians Respond, available at
    http://ourworld.compuserve.com/homepages/mathman



Department of Mathematics #3840,
University of California,
Berkeley, CA 94720-3840
wu@math.berkeley.edu
April 11, 1998

An expanded version of a colloquium lecture at the California State University at Sacramento, February 12, 1998


Footnotes

1According to Commissioner Williamson Evers, "the omission of long division with two or more digit divisors was a conscious decision" by the Commission. See [14].
2Dictionaries usually define "axiom" as "self-evident truths", but since dictionaries aim merely to inform the laymen, such lapses are marginally excusable. However, in a set of mathematics standards which must address the professionals---mathematics teachers and mathematics educators,---there is no place for this kind of error.
3The Commission's Standards is published in a two column format which displays the mathematics standards on the left and the "Clarifications and Examples' on the right.
4On February 2, an open letter to CSU Chancellor Reed signed by over 100 mathematicians was released to the public; it expresses sentiments in support of the Board's Standards. See [14].
5There is an unfortunate linguistic slip here: "draw ten points" is undoubtedly what is meant.
6As of February, 1998.
7The meaning of this word has to be carefully qualified because there are several "integrated" approaches to mathematics in secondary schools.
8A popular undertaking by conductors such as Carmen Dragon and Andre Kostelanetz in the 50's and 60's.
9Their original publication date is 1984.
10See footnote 7.