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A Mathematician Looks at the Standards

by Judy Roitman, University of Kansas

A report prepared for the National Institute for Science Education at the University of Wisconsin at Madison. Dr. Roitman is a Professor of Mathematics at the University of Kansas and a member of the EXTEND National Advisory Board.

I. Context

  1. Why do we have national standards?
  2. Politics
  3. Theory
  4. Doing mathematics

II. Specifics

  1. A problem
  2. Theory applied
  3. Technology
  4. Three good examples of bad things
  5. Stacking the deck
  6. Check Original
  7. Content
  8. Applications
  9. Pedagogy
  10. Assessment
  11. Teacher Preparation
  12. Equity
  13. Support

Summary and conclusion


I Context

This section briefly addresses the historical, political, philosophical, psychological, and personal issues that form the context of my views of standard-based mathematics reform.

1. Why do we have national standards?

In the early 1980's the popular perception arose that American education was in serious trouble. In international comparisons our students ranked low. In mathematics, the problem that captured the nation's imagination (quoted extensively in the national media) was the bus problem. Here is one version: 121 students are going on a trip, a school bus can hold 23 students, how many buses do the students need? A disturbingly large number of students would answer "5 and 6/23", or even (rounding down) "5." Our kids could calculate, but they couldn't make sense.

The documents proclaiming the nation's concern had quasi-official sponsorship: A Nation At Risk was sponsoredby the National Commission on Excellence in Education, and Educating Americans for the 21st Century was sponsored by the National Science Board.

Note that if everything had been going well, no- one would have proposed national standards. National standards exist because there is a perceived need to change, and I will slide among the notions of "standards," "standards- based reform" and "current reform" acknowledging this relationship.

2. Politics

In 1986, the National Council of Teachers of Mathematics began the process of writing national standards for mathematics education. These standards appeared in three volumes: Curriculum and Evaluation Standards (1989), Professional Standards (1991), and Assessment Standards (1995); the latter somewhat supersedes the evaluation standards of the first volume. At the same time many states were rewriting their state guidelines. The California Framework (which appeared just before the NCTM Standards) has generated the most noise, but is not untypical: many if not most states in their mathematics guidelines were heavily influenced by the NCTM Standards, and many if not most states formulated their guidelines under the strong influence of outcomes-based education.

Thus our national standards were born in an interesting political situation. They are not sponsored by a quasi-official body, but by the organization which represents teachers of K-12 mathematics and researchers in learning school mathematics. This has obvious advantages (accountability to teachers, sustaining long-term interest) and obvious disadvantages (difficulty in involving other interested communities, credibility). While national standards are not intrinsically connected to outcome-based education, there is a strong association in the public mind between the two. In reaction to standards and their various implementations, there are turf battles among mathematicians, teachers, mathematics educators, parents, and business leaders--who should have the strongest voice in mathematics education? There is the classical American conflict over jurisdiction--what authority should national standards have? state standards? Decisions on education are ultimately made by local school boards, with states holding financial and accreditation sticks to keep school boards in line. There are efforts towards national consequences to local actions, such as national teacher accreditation, but it is not clear whether these will be meaningful on the local level.

The increasingly ascendant role of business is worth noting. We were a nation at risk in part because businesses found that our high school graduates needed serious remedial education. In many discussions of standards-based reform, the needs of business are cited as justification for changes in emphasis, e.g., businesses give a higher value to communicating technical knowledge clearly than to doing arithmetic efficiently and correctly. The NRC launched state coalitions for mathematics and science education (which have recently become independent through their own national organization) to bring business, practitioners (scientists and mathematicians), teachers, and education researchers together. The agenda for my state coalition's July 1996 meeting has four reports. One is the treasurer's report, and the other three are titled "The roles of postsecondary education in workforce development," "The roles of business in workforce development," and "The roles of K-12 education in workforce development." These topics would not have appeared five years ago.

The political context of standards-based reform makes it important to distinguish among several notions that are often confused: what is actually written in the various standards documents, what people think is written in the various standards documents, and how what people think is written in the various standards documents is actually implemented. Underlying these notions is the definition of "standard," which I will not attempt.

3. Theory

Carefully reading through the various Standards volumes, I oscillated wildly among enthusiastic approval, confusion, and strong disagreement. For a long time I found this puzzling, until the chance e-mail receipt of a paper on cognitive psychology focused my attention on underlying assumptions. The choices for underlying assumptions in education are many, they are often contradictory, and they are often unstated, especially in documents not meant for specialists in education research.

Theoretical beliefs about education have both philosophical and psychological components, and it is not always possible to tease them apart. There are ontological and epistemological considerations. There are issues of political philosophy--is the goal of education training or learning? training or learning for what purpose? There are conceptual issues: for example, what are the units of learning-- facts? tasks? cognitive processes? are there any units at all? Most philosophical/psychological theories have consequences for education; even logical positivism had an influence in the over-logicization of the New Math. When William of Ockham defends Platonism against constructivism in the Journal of Research in Mathematics Education [9], we have a welcome glimpse of the theory wars raging beneath the surface. Even a benignly titled article in the JRME (picked semi-randomly) such as "Mental computation performance and strategy use of Japanese students in grades 2, 4, 6 and 8" (see [10]) necessarily has implicit theoretical underpinnings.

I see four basic questions in mathematics education. Two of them--what is mathematics? what does it mean to learn mathematics?--will have different answers in different theoretical contexts. Consider a simple answer to the first question (having, of course, its own theoretical ground), that mathematics is what mathematicians do. This, of course, the answer I prefer. However, it is not much help, since mathematicians do so many differents things, since much of the mathematics useful to those who use mathematics is essentially ignored by mathematicians, and since towards the boundaries it becomes problematic to decide who is a mathematician--what about theoretical physics, for example, operations research, or statistics?

Two other basic questions are: what mathematics should children learn? and how should they learn it? These questions cannot be answered without reference to the first two questions. But these are the questions that any set of standards needs to answer. The need to accomodate different underlying philosophical and psychological theories is, I believe, what gives the various Standards documents their confusing nature; this is unavoidable.

I should state here my own theoretical predilections. I tend to like constructivism, but also distrust rigid adherence to ideology--my constructivism is radical enough to lead me to distrust intellectual constructs, including those of constructivism itself. With Bishop Berkeley I believe there are times when kicking a stone is a good philosophical argument. With Wittgenstein, I am a great fan of the notion of "use," giving that word the broadest sense possible, and cannot understand the notion of "meaning" without it.

4. Doing mathematics

I came to mathematics somewhat late; the first real calculus course I took was after graduating from college. I came to mathematics late because, even in an honors track in an academically demanding high school, school mathematics did not seem interesting. This gives me a predilection to side with education reform. Because I came to mathematics late, I am perhaps more aware of what I went through in internalizing mathematics (or, if you prefer, becoming acculturated) than most mathematicians. Gratifyingly, what I think of as the necessary processes of and attitudes towards mathematics, not just for mathematicians but for anyone who can be said to have a basic mathematical education, are richly reflected in the Standards. Let me state them here, with two caveats. Caveat 1: the language is generally mine and not necessarily the language of the Standards. Caveat 2: this is my own personal list, and makes no claims to being exhaustive.

Making sense. The first thing that struck me about the current reform movement years ago was the emphasis on making sense. It is this emphasis that was lacking in my own school experience, and led to my perception of mathematics as boring and barren. The movement from mathematics as received knowledge to mathematics as perceived knowledge is a basic and necessary move, and can be made within most philosophical orientations. (It is not, however, compatible with certain fundamentalist notions of knowledge, with obvious political repercussions.)

Reification. The objects of mathematics are real objects, in a psychological, not necessarily ontological sense--they feel real, we act as though they are real. For example, "number sense" is based on reification. For another example, many young children have not reified the notion of fraction--for them, 1/2 implicitly carries with it the question "1/2 of what?" When the concept of "1/2" takes its place in the number system as just one of many rational numbers, to be thought about and used as we think about and use all rational numbers, it has been reified. To take a third example, algebra cannot really be understood unless variables are reified-- "x" is not a placeholder standing in for some unknown number, but an object in its own right. Reification cannot be forced, but its encouragement is a major part of the art of teaching mathematics. In many places we find reification in various guises in the Standards, but there are places where my emphasis on reification will lead me to disagree with the Standards.

Making pictures. Quasi-concrete mental imagery is a major intellectual resource available to mathematicians, and is used in even the most abstract mathematics. (The late great mathematicial Paul Erdos, in giving talks on infinite combinatorics, drew almost the same diagrams every time--a few dots, a few lines, a few circles--which encapsulate very different meanings in different contexts.) The importance of making pictures out of the most abstract situations was kept secret from school mathematics, and one of the great strengths of the current reform effort is not only its emphasis on imagery and metaphor--usually through the forms of physical models and diagrams--but its stress that different ways of picturing a situation should be encouraged. Connected to this is the encouragement of informal arguments from an early age; the early stages of mathematical justification are similar to the oral (but not written) practices of many research mathematicians in their reliance on pictures.

Justification. Learning how to write acceptable mathematical justifications was the hardest part of my becoming a mathematician--it is a social process, and different cultures have different standards of logical robustness. But this is not to say that mathematical justification is arbitrary--the rules, while more subtle than many of us choose to acknowledge, evolved for good reasons.

Insistence on mathematical justification at all levels is another great strength of current reform, as is the recognition that the practical definition of "sufficient justification" will change with a child's growing mathematical development. But scattered in the Standards are some notions of justification that seem counter to established mathematical practice.

Applications. This widely used word does not seem to fully capture what really happens when mathematics is succesfully used in another area. It is not that we simply apply a technique over here (from mathematics) to a situation over there (in real life, or in another field of study, or in another area of mathematics.) Rather, the process is dual to reification-- abstract situations permeate the situations to which they are being applied. So, for example, the geometric situation "areas of rectangles" is an instance of the arithmetic notion of multiplication, and in turn illuminates notions of probability; trees become instances of fractals; motion and distance become comprehended through the notions of differentiation and integration; and certain forms of turbulence are conflated with certain differential equations. It is a kind of double vision I am after here, in which students do more than move from one mode of thought to another freely--the different modes are, rather, different languages for the same phenomena.

(Indeed, one of the great strengths of the Standards docments is their stress on all senses of the word "applications.") That mathematics adds powerful systems to our intellectual resources is one of the main reasons I believe all children should learn serious mathematics.

Disposition. This is a very powerful set of notions, well articulated in the Standards, which has unaccountably been trivialized by opponents of reform to the parody of short-attention-span mathematics--all play and no work. Disposition, rather, is a cluster of intellectual character traits--thinking for yourself, being skeptical of others' claims, not believing something until you really understand it, knowing when you don't understand something, lack of wishful thinking, persevering, learning from mistakes. (Wishful thinking, the belief that something is true because it is convenient, is the source of most mathematical error.)

As a kid I was told that mathematics taught logical thinking (Euclidean geometry) and accuracy (arithmetic). I didn't believe this and still don't--those tightly defined compartments did not generalize easily. But standards-based reform seems to have a better chance at teaching the sort of intellectual integrity and clarity that I believe is inherent only in mathematics (other disciplines have their own forms of integrity and clarity, of course, and that is why children need a broad education). This is other major reason I believe all children should learn serious mathematics.

II. Specifics

The context being set, we are ready to discuss specific issues.

1. A problem

Although I am generally pleased by the major directions of the Standards, it is undeniable that the Standards documents are peppered with statements which are mathematically questionable. Generally these are not anything as simple as a straightforward mathematical mistake. Their best description is as something no-one who really knew the mathematics would say--extremely difficult (even unsolved) mathematical problems may be suggested for exploration in a way that indicates that students should be able to solve them, or complex mathematical situations may be presented as if they were simple.

This sort of carelessness is perhaps to be expected in such an accumulation of pages, but it has two important consequences. Some mathematicians have devoted enormous amounts of time to finding these glitches, and cannot take the Standards seriously because of them. Other people, knowing less mathematics, can be misled. I would encourage the authors of future standards documents to be more careful.

2. Theory applied

Let me pick one series of the oscillations referred to in section I.3 to demonstrate the theoretical tensions I see in the Standards, and how I respond to them. Here are my reactions to parts of standard 12, geometry, grades 5-8.

For the bulleted standards: "explore transformations of geometric figures; represent and solve problems using geometric models" I wrote "Good." These standards represent a significant increase in mathematical content over the traditional curriculum.

Besides the quote "Geometry is grasping space... that space in which the child lives, breathes, and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it. (Freudenthal 1973, p. 403)" I wrote "Bull." Does anyone really live, breathe, or move better because they have learned geometry? This is contextualism at its most strained.

For the sentence "Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements" I wrote "?" What is a hypotheses if not a formal summary statement?

Three lines later, for the phrase "develop informal arguments," I wrote "Yes." It is the absence of experience with informal arguments that made formal arguments so meaningless for so many students (including myself).

For the sentence "Students should learn to use correct vocabulary, including such common terms as and, or, all, some, always,never, and if... then" I wrote "Great." The ability to use these words (= understand these concepts) precisely is one of the best gifts we can give to children, and essential to any logical clarity of thought.

But two lines later, when we are advised that words like "dodecahedron" are important, I wrote "Nah." This seems a hold-over from the "math is complicated vocabulary" school, in which vocabulary is emphasized and ideas are not. What is important is that a student at the appropriate level can describe these solids geometrically.

When it is suggested that the Pythagorean theorem can be discovered "through explorations, such as the one suggested in figure 12.2" I wrote "Overly optimistic;" figure 12.2 encapsulates a proof, and a somewhat hard one.

The sentence "Students can make conjectures and explore other figures to verify their reasoning," gets another "Nah." How can any finite collection of figures verify reasoning? This is getting the mathematical process backwards.

One line later, a paragraph recommending dynamic geometry software gets a "Good," as do the suggested explorations of the relations between perimeter and area, surface area and volume.

For "Which polygons will cover the plane and which ones will not? Why?" I wrote "Careful." We are walking a tightrope here between the highly nontrivial (e.g., for convex polygons) and the trivial (e.g., rectangles and triangles). Where are we supposed to go?

For the claim that "students can also consider why the square is used as a unit of area and the cube as a unit of volume," I wrote, "Huh?" What sort of response is of value here?

For the paragraph on symmetry, I wrote "Do more."

As for the last two sentences, "Experience with geometry at the 5-8 level should sensitize students to looking at the world around them in a more meaningful way," this is again contextualism at an extreme. There are better reasons for studying geometry seriously in middle school.

Several partial conflicts become apparent. On the one hand, there is a serious strengthening of the level of the subject matter. On the other, the issues of justification and verification appear in a strange fashion. Constructivist, contextualist, and traditionalist attitudes all jostle for space. No matter what the reader's orientation, her reactions will oscillate as mine did, as things she approves of appear, to be followed by things she disapproves of.

3. Technology

What should be done with technology? This is a serious question that has not been sufficiently addressed in the Standards, perhaps because when the Curriculum and Evaluation Standards went to press, so little had been done with or was known about technology. But technology has been seized upon by opponents of standards-based reform, who are trumpeting in the popular media what they believe to be the failures of its use as evidence of the failure of reform. So any serious discussion of technology in the classroom will need to be reflected in the popular media as well--no easy task.

Technology is neither benign nor malignant, but it is powerful. Furthermore, like the HIV retrovirus, it is protean; by the time you thoroughly understand how to use one version of it, you are several technological generations out of date.

While the Standards contains some good instances of using technology (along with some bad ones) it does not provide a unified discussion of sufficient depth of the issues raised by technology, and in places seems to assume that technology should be used in the classroom simply because it exists.

The most egregious instance of this is in the Professional Standards (pp. 82-83) where Pete Wilder "has read that calculators should be emphasized at the middle school level [and] has been reluctant to use them. His supervisor, Tim Jackson, has been urging him to use calculators whenever possible." The boldface note in the margins reads "The teacher is aware of the need to incorporate calculators into his teaching but is reluctant to do so."

This is troubling. Is Pete Wilder really aware of a need, or is he aware that he is supposed to do something without understanding why? I think it is the latter, and the rest of the vignette (which focuses on specific activities) does not address this fundamental issue.

There are four basic questions to answer about any use of technology in the classroom. (With slight changes, these are the four basic questions about using anything in the classroom --I have modified them from the first four questions about assessment from p. 4 of the Assessment Standards.)

These questions are benchmarks for using technology. They should be widely known, and examples of both good and bad uses of technology, explicitly referring to these questions, should be widely distributed.

Meanwhile, the bad press that technology has been given makes it urgent to communicate what is really known about its effects in the classroom and to continue such studies. There are subtle methodological issues in any serious studies in this area, and conclusions can never be as clear-cut as local school boards would wish, but a serious conversation with the public must be attempted. In particular, the public deserves to know what is known about how use of calculators affects children's facility with arithmetic, and how graphing calculators affect students' abilities to translate algebraic functions into pictures.

Vanguard attempts to use technology to substantially transform mathematics education--I am thinking specifically of the work of James Fey's and Kathy Heid's group in high school algebra, and Ed Dubinsky's group in abstract algebra, both using computer technology--need to be discussed widely (and dispassionately) in the mathematics, mathematics education, and school communities, apart from discussions focused on curriculum adoption.

Finally, we need to understand what support systems are needed to use technology thoughtfully, and this knowledge needs to be disseminated widely.

4. Three good examples of bad things.

In the name of tradition a lot of bad things happen in our schools, so it should be no surprise that in the name of reform some bad things have happened. I will present in this section three situations in the name of reform mathematics education which I find both problematic and typical of the mistakes that can happen under reform. One situation was observed. The other two are from reform documents, hence possibly hypothetical.

I will also propose benchmark questions for preventing such mistakes.

A classroom visit. The first situation was observed during a classroom visit I made to a fifth-grade teacher in a small rural community. She had drawn a complex pattern, reflected it across a line, duplicated it, and asked the kids to "color the inside."

There was no inside. Towards one edge of the paper the pattern curved tightly, so it looked like there was an inside, but as the kids moved along they began to realize that the pattern was opening up and didn't have a closed boundary. They had no idea what to do, and neither did she--"oh well, just finish it the way you want," she said.

What was her content goal for this activity? She had none. Her knowledge of reform was that you did less arithmetic and more... well, more stuff. Her pattern had symmetry, and symmetry was mathematics, and that was enough for her.

She had, of course, missed an important opportunity: when does a figure have an inside? She didn't know enough mathematics to have thought of this, nor did she understand this was important mathematics when I suggested it to her as a possible extension.

When I think of the need to communicate clearly what mathematics is under reform, I think of this well-meaning, dedicated, good-hearted woman (who, mea culpa, worked with me for three summers--so I cannot claim any easy answers to the problem she represents.)

When is the use of number in applications an instance of mathematics? The second example is from an early draft of the Assessment Standards. I know it is bad form to quote from non-final versions of documents--and I hasten to add that this example did not make it inot the final version. But this example is telling because not only one person but a committee of people thought it was good mathematics.

In this activity, a group of African-Amercian middle-school girls are reporting on the statistics about minority men, higher education, and prison. They engage in some very clever guerrilla theater with the class, they make a cogent sociological point, but mathematically this is as empty an activity as those fifth-graders trying to color an interior that didn't exist--data presented without mathematical analysis is not statistics, just as coloring a bar graph does not make it art. Where were issues of variance? of sampling? of categories (e.g., a federal prison is not a county jail, and Harvard is not ITT)? Why weren't the kids pointed towards these questions? If this really happened, it is another missed opportunity.

Another instance of this sort of problem occurs in the middle grades communication standard, where kids are asked how many hours they think teenagers watch TV a day and to compare their answers with the results from a national magazine. While the discussion goes on to say that "This exercise encourages students to... discuss appropriate survey techniques" I don't believe it does. The kids don't have access to information about what the magazine did, nor do they have the resources to conduct a comparable survey. Without such access and resources, this too becomes a exercise in social science, not mathematics, and a superficial one at that.

Too much mathematics; deviation from mathematical practice. The third activity is from the Assessment Standards, pp. 36-39. In it, students are asked to explore, using dynamic geometry software, the following (where P is a variable interior point of an acute triangle): (a) the sum of the distances from P to the sides of the triangle; (b) the sum of the distances from P to the vertices of the triangle, (c) the area of the pedal triangle, (d) the perimeter of the pedal triangle. They are supposed to make conjectures, make convincing arguments, support their arguments with data, and "explain a situation where someone would want to know this information." They do this after having been led through a similar exploration when the triangle is equilateral.

This is a very troublesome example. Let me briefly summarize its problems.

The first problem is that, paradoxically, the situation is too mathematically rich. With no suggestions of what's worth looking for, how is a student to find anything? A good student can spend hours looking in the wrong direction (why not? mathematicians spend decades, even centuries, doing this) and end up with a collection of aimless observations. Furthermore, the previous exploration of an equilateral triangle is misleading--the situation there is not like the general case.

The sequence of steps described--conjecture, make a convincing argument, support by data--isn't how mathematics works. First look at the data, then conjecture, then convincingly argue. Only when the problem is intrinsically finite can data really support a conjecture; this problem is intrinsically continuous and very far from finite. This notion that data can be used to justify a conjecture is one place where the Standards greatly deviates from mathematical practice, and reappears throughout the Standards.

I am, of course, aware that, psychologically, data can be a more compelling justification than abstract reasoning. But I claim that (1) this is either because the person is easily convinced or because he or she is at a developmental stage in which the abstract needs to be encapsulated in the concrete; and (2) one purpose of teaching mathematics is to get beyond this stage. The level is wrong. Even if the student comes up with true conjectures, what would constitute a convincing argument? I assume the student is expected to concentrate on minimizing (for (a) and (b)) and maximizing (for (c)); I don't know what (d) is about. But this is hard mathematics. These are unexpected results. Their proofs are non-trivial. Helping students learn this stuff, whether constructively or in straight lecture, takes a lot of thought from the teacher. Expecting students to do it on their own as part of assessment is inappropriate.

Expecting teachers to know what this is about is also inappropriate. Few teachers--few mathematicians, for that matter--will have had a chance to be familiar with this material. If something like this is suggested for either curriculum or assessment, the mathematics needs to be clearly explained.

How to avoid similar examples. These examples essentially fail because they don't answer at least one of four basic questions--what is the mathematical point? what is an acceptable mathematical justification? can we expect kids to do this? have we provided enough mathematical explanation for teachers? While these questions are implicit in many standards-based documents, we need to pay more careful attention to them.

5. Stacking the deck.

In the Curriculum and Evaluation Standards, before the individual standards are explicated for the different levels (K-4, 5-8, 9-12) there is a summary chart consisting of facing pages, one labelled "Increased attention," and one labelled "Decreased attention."

I have no quarrel with anything that is supposed to receive increased attention. The suggested curriculum is good, authentic mathematics, and the instructional practices are clearly pointed towards making mathematical sense of things. Despite claims that standards-based reform means a lowering of standards, if everything that is supposed to receive increased attention really does, our current students will in many ways know much more mathematics by the time they graduate from high school than my generation did.

My quarrel is, instead, with the pages labelled "Decreased attention." The deck is rhetorically stacked, so that "decreased" can easily become "no." Bad words appear, such as "rote," "isolated," "routine," "by type,"-- everyone knows these are bad words--and by association everything on these pages becomes suspect. But in fact this material is a mixed bag.

Let me deal with each level separately. To make this section easier to follow, I will put in italics the notions that are slated to receive reduced attention.

K-4. Personally, I never want to see anyone use key words ever again; this practice is indefensible. Estimation should have context; rounding is seldom useful. Division facts are really multiplication facts and should not be treated in an isolated fashion. But I do think there are times when worksheets and written practice are helpful, and when kids need to focus on paper-and-pencil computations. There are times when you do have to tell the class something (e.g., about pi). (There are good activities for motivating the hypothesis that, over all circles, perimeter divided by diameter is constant. But is this hypothesis true? And exactly which number is this constant? That is that has to be told.) Often in mathematics--almost always in arithmetic--there really is only one answer, although there may be many ways of getting there, and sometimes there really is a best method. I like long division for two reasons: it is an early and well-motivated example of a complicated algorithm, and it lays important groundwork for algebra, both in the obvious sense of division of polynomials, and in the more subtle sense that understanding it contributes to a general mathematical sophistication. For similar reasons, I want kids to do paper-and-pencil computation with fractions. If early attention to reading, writing, and ordering numbers symbolically is done in context, as in whole language, then what could be wrong with it?

5-8. Manipulation of symbols is terribly important, as a skill in its own right, in order to do other interesting work, and as a step in the reification of symbols. Algorithms, formulas, vocabulary, facts and relationships need to be remembered, and for most of us that means consciously memorizing them. Some questions really do have only yes, no, or a number as responses. Here is a very important one at a more advanced level: what's e^i(pi)? The answer (-1) is a profound piece of mathematics.

9-12. About a quarter of what is listed here to be de-emphasized strikes me as very important. Under algebra, simplification of radical (and other) expressions, factoring, and operations with rational expressions are instances of algebraic manipulations which are themselves necessary steps in the reification necessary to understand algebra--being able to freely manipulate algebraic expressions is cognitively similar to number sense, and I am disturbed that it seems to be absent on the "Increased attention" side. Geometry from a synthetic viewpoint is important, and can be done by the increased attention given to the development of short sequences of theorems and to deductive arguments. Two-column proofs should not only get decreased attention but be eliminated. I agree that analytic geometry and functions should not be isolated, but should be integrated with the rest of the curriculum.

As for Euclidean geometry as a complete axiomatic system, yes, it should appear only as a piece of history, but my reason for this is somewhat maverick--if it is presented in a way that can be absorbed by 9th or 10th graders, then some things have to be fudged (which astute students will notice), and you end up with so many axioms that enquiring minds will wonder why you bothered in the first place. Applications of trigonometric sum, difference, double-angle, and half-angle identities to specific examples is important: the mere fact that these identities exist is remarkable, and students should have some immersion in them. There is nothing wrong with using formulas to model real-world problems--that is the essence of mathematical modelling. And expressing function equations in standardized forms is an important conceptual step in turning algebra into geometry (it even shows up on page 101 of the Professional Standards).

What's going on. The motivation of these lists is clear and even commendable. In general the thrust is to get away from rote exercises--I am not the only adult who has no fond memories of page after page of trigonometric identities, and the cartoon "Hell's library" (in which every book is labelled "Word problems") has been widely distributed; someone must find it funny. But just because something can be taught, and often was taught, by rote methods does not mean it is bad in itself. Much that is essentially good, even fundamental (such as algebraic manipulation) is being tarred with the brush of the bad. As long as there is no distinction between what should really be thrown out and what needs to be taught differently, important school mathematics will be in danger of disappearing from school curricula, either at the district or at the individual classroom level. Many mathematicians and parents, even teachers, are convinced that this has already happened. I am not so sure, but I am worried.

6. 0.31 x 0.588

Let me focus on a particular problem to which "instructional time should not be devoted" (Standards, p. 96) as an example of the importance of mathematics which the Standards either de- emphasizes or throws out, and how such material can and should be incorporated in standards-based reform. This is the paper- and-pencil computation of 0.31 x 0.588

Why is such a problem important? After all, anyone not a calculation prodigy, unlucky enough to face such a problem in real life, would use a calculator.

But this is irrelevant. To compute 0.31 x 0.588 by hand requires either a deep understanding of place value or sophisticated skill in symbol manipulation or both, and that is what this problem is really about. Would I have children work sheet after sheet of such problems? No, not even without time pressure. But would I have them work a few problems like this in small groups, reporting to the class how they solved them, and then work a few on their own to make sure they understand what's going on? Absolutely. As part of the standards relating to fractions, decimals, and arithmetic, I would expect all children in the class be able to do problems like this--not necessarily quickly, but correctly--throughout their lives.

Nearly everything that I would rescue from the "Decreased attention" charts has similar justifications, and can be handled in similar ways.

7. Content.

If the summary charts in the Curriculum and Evaluation Standards are radical and might raise fears of a diluted curriculum, the actual boldface lists of topics defining each standards are both conservative and ambitious-- there are even two tracks in 9-12, for college-intending and others. Topics slated for decreased attention in the summaries indeed appear (e.g., synthetic geometry), so we know in some cases that "decreased attention" does not mean "no attention". Reasoning ranges from informal to very formal indeed (including axiomatic systems and mathematical induction). Even infinite series is in there.

I have only three quarrels with this material, all at the 9-12 level. Two quarrels are that symbolic manipulation isn't appreciated sufficiently, and that perhaps more mathematics is proposed than can realistically be achieved. Should kids planning on college really "prove elementary theorems within various mathematical structures, such as groups and fields," "represent finite graphs using matrices," "solve problems using linear programming and difference equations," and "interpret probability distribusions including binomial, uniform, normal, and chi square?" Almost none of this is beyond the capabilities of motivated high school students (I'm not so sure about the groups and fields), but all of it? Along with everything else?

The third quarrel is with the words "verify" and "validation" which appear many times in the 9-12 standards. I'm not sure what they mean, and what is expected of students when they are used.

8. Applications.

While the Curriculum and Evaluation Standards reminds us that "not all problems require a real-world setting," (p. 77) there is a strong impetus in current reform (based in contextualist theory) to try to root classroom mathematics in real-world problems, especially in middle schools which attempt integrated curricula.

The Standards documents are themselves fairly balanced on applications--real-world applications are no more (and, I hasten to add, no less) standards-based than theoretical mathematics. That the momentum towards applications- based curricula is done in the name of standards-based reform is unfortunate.

There is one crucial place in the Curriculum and Evaluation Standards that can give rise to this misapprehension, the discussion of why "the educational system of the industrial age does not meet the economic needs of today" (pp. 3-4). Three of the four new social goals serve business needs: the need for mathematically literate workers; lifelong learning (which is connected with "changes in technology and employment patterns" and not learning for its own sake), and equity (which "has become an economic necessity"; maybe that is what it takes to finally gain what should be a right). The next section goes on to establish "learning to value mathematics" as the first new goal for students. Perhaps we have something very close to a political contradiction here--can we simultaneously serve the needs of Boeing and create a society of, say, Thomas and Thomasina Jeffersons?

For a beautiful example of an applied problem that involves very deep mathematics, see "Lightning Strikes Again!" from Measuring Up in which 4th graders have an opportunity to move from simple arithmetic calculations to working out the intersection of two circles.

For a beautiful example of serious and difficult mathematics motivated by a simple-sounding application, see the airport problem in Connected Geometry.

9. Pedagogy.

Here, as with applications, the Standards do not say what they are charged--by both supporters and detractors--with saying. They do not say that all mathematics learning should take place through activities in small heterogeneous groups in which students develop all of the ideas, with the teacher acting only as a moderator. Yes, there is a constructivist orientation in the Standards, but nowhere is it exclusive, and the Curriculum and Evaluation Standards reminds us continually that all forms of instruction are useful--although this is contradicted somewhat by the bias on the "Decreased Attention" pages.

But occasionally a more dogmatic attitude creeps in which is disturbing. For example, on page 142 of the Professional Standards, Rich says that he was "really reluctant to use that activity because it didn't seem like exploration. It made me feel that I would be directing the students toward a single result..." But there are many times when directing students towards a single result is exactly what is called for. Furthermore, just because students are going to inevitably find the same result doesn't mean it isn't exploration. And, finally, sometimes exploration isn't called for.

Where the constructivist bent is seen most clearly is in the Professional Standards, where most of the vignettes are about teachers becoming more constructivist in their methodology. This is understandable. Even now, many teachers have few sources of information on constructivist methodology, and there was a clear need for such information in 1991.

It should be noted that the key issue in many of these vignettes is how to guide exploration and discussion. Contrary to parodies of constructivism, children are not left to their own devices, nor do they work exclusively in small groups.

10. Assessment.

As a mathematician, I am not used to thinking comprehensively about assessment, and I learned a lot from reading the Assessment Standards. The basic notions in this document seem unassailable, and I was especially pleased to see the emphasis on performance assessment, and citations of assessments from other countries.

But assessment is another place where what the Standards say is not what they are perceived as saying. Somehow there is a perception that standards-based assessment is inherently trivial; does not allow for arithmetic calculation or algebraic manipulation by hand; invites subjective judgement; and is designed to make all children look good.

I believe that these misperceptions have several roots. One is a key-word approach, in which certain terms (e.g., "open-ended," "equity") are given different meanings than they have in context. The other is a not unreasonable concern that something which seems difficult (e.g., creating a robust rubric for a problem with complex or multiple solutions) may not be possible. The third is a philosophical position (which neither I nor the writers of the Standards share) that there is something called objective assessment which can be used to categorize students and place them in appropriate educational programs. (One sign of this philosophical difference is that the Standards say very little about assigning grades, while several of the critics of reform do not speak about assessment but about grading systems.)

This last desire--to put kids in the appropriate classes--has roots in real, even poignant, situations. (As a formerly precocious child, and as the mother of a child with learning disabilities, I have too much familiarity with both ends of this particular spectrum.) Correct placement is indeed very difficult, as is teaching outliers. Perhaps this is one issue that is not sufficiently addressed.

There is one very important issue in the Assessment Standards which is handled somewhat cavalierly, and that is the issue of time. Having begun to use some alternative assessments myself, sparingly, and with only two classess of about 25 students each a semester, I can tell you how time- consuming this is. I can't imagine my son's junior high school teachers--7 classes a day , about 30 students in each--doing it on a regular basis. As with the plethora of interesting ideas for curriculum at the 9-12 level, this seems too much to expect.

11. Teacher Preparation

I could pick some nits, but basically the Professional Standards outlines a solid mathematical background for mathematics teachers, which is most welcome.

I am puzzled, however, by the comment on p. 139 that "Since the spirit and content of the coursework described above can be very different from traditional courses, every effort should be made to develop new courses that reflect these differences."

Except for the call for manipulatives in probability and statistics (which would be good for all students) I don't really see much if any difference between what is recommended for teachers and what we teach in our regular courses. There is a danger that an entirely different track for future high school teachers would be perceived as lower level than the regular mathematics track, and I know from experience that in courses created for teachers there is often pressure from the students to be relevant to exactly what they will teach This can get pretty strange--our pre-service students regularly complain about having to learn transformational geometry, even when we assure them that they will be teaching it themselves. They didn't learn it themselves in high school, why should they believe us? I suggest that a mix of courses within the mathematics department, some with a pre-service emphasis, others for all mathematics and mathematics education majors, may be the best solution.

12. Equity.

Racial equity is a serious issue for this society, which faces the great contradiction of a national rhetoric steeped in equity and historical roots steeped beyond inequity in genocide and enslavement. As for women, in no society have we had an easy time of it.

So I naturally welcome the emphasis in the Standards on equity (even with the corporate sponsorship on page 4 of the Curriculum and Evaluation Standards).

I have, however, a major concern about equity.

This is concern about the essentialist view, which seems to have its attractions in education - women think like this, African American men think like that--and is closely connected to cultural stereotyping. We need to guard very carefully against essentialism, even as we recognize that, yes, our society is made up of different cultures, these cultures have different rules, and when rules collide there are problems. The desire for easy answers here is what makes essentialism attractive, but there are no easy answers.

Concern for equity has given rise to one of the most emotional critiques of standards-based reform, the claim that it hurts disadvantaged, especially minority, kids. The charge is that trivial curricula and overly easy assessments give the impression that these kids are learning, when in fact they are not. In this view, the various standards about equity are viewed as hypocritically creating demands for false entitlements ("I have a right to pass algebra," and not "I have a right to learn algebra"). These charges have, as far as I know, not been directed at the national standards, but at the California Framework. Those making them are quite sincere, and are armed with stories of parents and teachers begging the schools to deviate from the Framework and teach their children substantive mathematics.

13. Support

The final topic I wish to discuss is the last section of the Professional Standards, the section entitled "Responbilities." This sets forth the responsibilities of policymakers in government, business, and industry, the responsibilities of schools, the responsibilities of colleges and universities, and the responsibilities of professional organizations.

My only comment here is that most of these groups were not seriously consulted, so no matter how laudable the recommendations, they are necessarily moot.

III. Summary and Conclusion

The ambiguity of the notion of "standards," coupled with the (never quite explicit) clash of theoretical positions would make any standards document impossible to agree with completely. Within these constraints, the NCTM Standards generally stresses what is mathematically important and is to be applauded for seriously attempting to create a culture of doing mathematics in the classroom. I have some disagreement with content emphasis (symbolic manipulation, difficult arithmetic, algorithms) and method (some of the notions related to proof and justification); other mathematicians will have other complaints. There are places where the documents could be written better--more carefully, less ambiguously, or with less bias. But the over-all framework is a good one, especially if it continues to be revised, and especially if those in charge of the revision process listen seriously to mathematicians, educators, and teachers with diverse viewpoints.

The debate has been muddied, however, by confusing Standards documents with other reform documents, with various interpretations of reform, and with classroom practices justified in the name of reform. The extremism of much of the rhetoric that attacks or justifies reform is a serious problem. What we have learned from studies in mathematics education needs to be communicated to the general public as clearly as possible, especially on such contentious issues as constructivist pedagogy, technology, and assessment.

As a mathematician, I have focused on the Standards documents, knowing that they are only a part of the picture. And as a mathematician I like to end papers with questions. I will end this one with two, whose answers need a very different expertise than I can bring to the table: how are standards actually implemented? and what over-all systemic changes have been/should be made so that the standards movement can succeed?

Acknowledgements. I would like thank the Systemic Reform Team of the National Institute on Science Education for commissioning this paper. I would also like to thank a few of the many people whose conversations with me over the years have helped clarify my thinking on these issues: Susan Addington, Dick Askey, Hy Bass, Becky Corwin, Al Cuoco, Jan Dicker, Joan Ferrini-Mundy, Charlotte Keith, Billie Manderick, and Linda Ware.


  1. John R. Anderson, Lynne, M. Reder, Herbert A. Simon, Applications and Misapplications of Cognitive Psychology to Mathematics Education [1], preprint.

  2. Michael W. Apple, Do the Standards Go Far Enough? Power, Policy, and Practice in Mathematics Education, Journal for Research in Mathematics Education 23 (5), November 1992.

  3. Educational Development Corporation, Connected Geometry, Janson Publications, Dedham MA, 1996.

  4. Mathematical Sciences Education Board, Measuring Up: Prototypes for Mathematics Assessment, National Academy Press, Washington, D.C., 1993.

  5. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, Va., National Council of Teachers of Mathematics, 1989.

  6. National Council of Teachers of Mathematics. Professional Standards for Teaching Mathematics. Reston, Va., National Council of Teachers of Mathematics, 1991.

  7. National Council of Teachers of Mathematics. Assessment Standards for School Mathematics. Reston, Va., National Council of Teachers of Mathematics, 1995.

  8. National Commission on Excellence in Education, A Nation at Risk: The Imperative for Educational Reform, Washington, D.C., U.S. Government Printing Office, 1983.

  9. National Science Board Commission on Precollege Education in Mathematics, Science, and Technology, Educating Americans for the Twenty-first Century: A Plan of Action for Improving the Mathematics, Science and Technology Education for All American Elementary and Secondary Students So That Their Achievement Is the Best in the World by 1995, Washington, D.C., NSF, 1983.

  10. Robert E. Orton, "Ockham's Razor and Plato's Beard," Journal for Research in Mathematics Education 26 (3), May 1995, 204-229.

  11. Robert E. Reys, Barbara J. Reys, Nobuhiko Nohda, and Hideyo Emori, "Mental computation performance and strategy use of Japanese students in grades 2, 4, 6 and 8," Journal for Research in Mathematics Education 26 (4), July 1995, 304-326.

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