A Mathematician Looks at National Standards
by Judith Roitman - 1998
The purpose of this article is to present one mathematicianís view of the NCTM standards documents. There is a brief discussion of the history of the standards and a brief presentation of philosophical and personal context. The bulk of the article discusses specific notions, examples, concepts, phrases, and uses/misuses of the standards. An attempt is made throughout to diff erentiate among what the standards say, what people think they say, and what teachers do in the classroom under the influence of what they think the standards say.
The purpose of this article is to present one mathematicians view of the NCTM standards documents. There is a brief discussion of the history of the standards and a brief presentation of philosophical and personal context. The bulk of the article discusses specific notions, examples, concepts, phrases, and uses/misuses of the standards. An attempt is made throughout to differentiate among what the standards say, what people think they say, and what teachers do in the classroom under the influence of what they think the standards say.
This section briefly addresses the historical, political, philosophical, psychological, and personal issues that form the context of my views of standards-based mathematics reform.
WHY NATIONAL STANDARDS?
In the early 1980s 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 nations 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 6/23, or even (rounding down) 5 Our students could calculate, but they could not make sense.
The documents proclaiming the nations concern had quasi-official sponsorship: A Nation At Risk was sponsored by the National Commission on Excellence in Education (1983) and Educating Americans for the Twenty-first Century was sponsored by the National Science Board (1983).
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.
In 1986, the National Council of Teachers of Mathematics (NCTM) began the process of writing national standards for mathematics education. These standards appeared in three volumes: Curriculum and Evaluation Standards for School Mathematics (1989) Professional Standards for Teaching Mathematics (1991), and Assessment Standards for School Mathematics (1995); the third volume somewhat supersedes the evaluation standards of the first volume. At the same time, many states were rewriting their state guidelines. The California Framework has generated the most noise, but is not untypical: Many, if not most, states in their mathematics guidelines were heavily influenced by either the NCTM standards or by the viewpoints that gave rise to 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.1 They are sponsored not by a quasi-official body, but by the organization that 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 outcomes-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 leaderswho should have the strongest voice in mathematics education? There is the classical American conflict over jurisdictionwhat 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 toward national consequences to local actions, such as national teacher accreditation, but it is not clear whether these will be meaningful at 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; for example, businesses give a higher value to communicating technical knowledge clearly than to doing arithmetic efficiently and correctly.2 The National Research Council (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 coalitions July 1996 meeting has four reports. One is the treasurers report; the other three are entitled 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. While business is no stranger to American education, these topics would not have appeared with such prominence 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 standards, which I will not attempt.
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 philosophyis 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 (JRME) (Orton, 1995) 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 (Reys, Reys, Hohda, & Emori, 1995), necessarily has implicit theoretical underpinnings.
I see four basic questions in mathematics education. Two of themwhat 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.3 This is not much help, since mathematicians do so many different things; since much of the mathematics useful to those who use mathematics is essentially ignored by mathematicians; and since, toward the boundaries, it becomes problematic to decide who is a mathematicianwhat about theoretical physics, for example, or 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 must answer. The need to accommodate 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 ideologymy constructivism is radical enough to lead me to distrust intellectual constructs, including those of constructivism itself. With Samuel Johnson (Boswell, 1836), I believe there are times when kicking a stone is a good philosophical argument. With Wittgenstein (1967), I am a great fan of the notion of use, giving that word the broadest sense possible, and I cannot understand the notion of meaning without it.
I came to mathematics somewhat late; the first real calculus course I took was after graduating from college. I came to mathematics late because school mathematics (in an honors track in an academically demanding high school) 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 toward 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.
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.4)
The objects of mathematics are real objects, in a psychological, not necessarily ontological sensethey feel real; we act as though they are real. For example, number sense is based on reification: We can compare numbers, operate on them, and look at their properties because they are real. For another example, many young children have not reified the notion of fractionfor 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. (That the rational numbers are themselves created through reification of particular rational numbers such as 1/2, which is itself not fully reified until the whole system is, shows how complex this process is.) To take a third example, algebra cannot really be understood unless variables are reifiedx 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, most notably on symbolic manipulation.
Quasi-concrete mental imagery is a major intellectual resource available to mathematicians.5 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 metaphorusually through the forms of physical models and diagramsbut 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.
Learning how to write acceptable mathematical justifications was the hardest part of my becoming a mathematicianit is a social process, and different cultures have different standards of logical robustness. But this is not to say that mathematical justification is arbitrarythe 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 childs growing mathematical development. But scattered in the standards are some notions of justification that seem counter to established mathematical practice.
This widely used word does not seem to capture fully what really happens when mathematics is successfully used in another area. It is not that we simply apply a technique (from mathematics) over here to a situation over there (in real life, or in another field of study, or in another area of mathematicsone of the great strengths of the standards documents is their stress on all three senses of the word applications). Rather, the process is dual to reificationabstract 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; treebecome 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 freelythe different modes are, rather, different languages for the same phenomena.
That mathematics adds powerful systems to our intellectual resources is one of the main reasons I believe all children should learn serious mathematics.
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 mathematicsall play and no work. Disposition, rather, is a cluster of intellectual character traitsthinking for yourself, being skeptical of others claims, not believing something until you really understand it, knowing when you dont understand something, lack of wishful thinking,6 persevering, learning from mistakes.
As a child I was told that mathematics taught logical thinking (Euclidean geometry) and accuracy (arithmetic). I did not believe this then and still do notthose 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 the other major reason I believe all children should learn serious mathematics.
The context being set, we are ready to discuss specific issues.
Although I am generally pleased by the major directions of the standards, it is undeniable that the standards documents are peppered with statements that 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 sayextremely 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.
Let me pick one series of the oscillations referred to in section 1.3 of the Curriculum and Evaluation Standards (NCTM, 1989, pp., 112-115) to demonstrate the theoretical tensions I see in these standards and how I respond to them. Here are my reactions to parts of standard 12, geometry, grades five to eight.
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.
For 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, breath 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 contextualism7 at its most strained.
For the sentence, Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements, I wrote ? Summary statements are part of the standard academic format in many fieldsthis article has a summary statement, for example (and it felt quite strange to write one)but they do not exist in mathematics. In mathematics there is very little redundancy; statements generally summarize only themselves.
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, neuer, 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 holdover 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 difficult one. (Figure 12.2 contains three standard pictures often used in explicating and proving the Pythagorean theorem: a right triangle with squares attached to each side; and two rearrangements of the associated squares and triangle.)
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 backward.
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 readers orientation, his or her reactions will oscillate as mine did, as things one approves of appear, to be followed by things she or he disapproves of.
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 (NCTM, 1989) went to press, so little had been done with or was known about technology. But technology has been seized on 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. Any serious discussion of technology in the classroom needs careful explication if it is to avoid major distortion in the popular media.
Technology is neither benign nor malignant (the referee rightfully reminds me, and I warn the reader, that this is not an uncontroversial statement), 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 contain some good instances of using technology (along with some bad ones), they do not provide a unified discussion of sufficient depth of the issues raised by technology, and in places, seem 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 (NCTM, 1991), 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 margin reads, The teacher is aware of the need to incorporate calculators into his teaching but is reluctant to do so (pp. 82-83).
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 classroomI have modified them from the first four questions about assessment; see NCTM, 1995, p. 4). What mathematics is reflected in the use of technology? What efforts are made to ensure that the mathematics is significant and correct? How does the use of technology engage students in realistic and worthwhile mathematical activities? How does the use of technology elicit the use or enable deeper understanding of mathematics that it is important to know and be able to do? 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 given8 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 study 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 childrens facility with arithmetic and how graphing calculators affect students abilities to translate algebraic functions into pictures.
Vanguard attempts to use technology to transform mathematics education substantiallyI am thinking specifically of the work of James Feys and Kathy Heids (1995) group in high school algebra and Ed Dubinskys (1994) group in abstract algebra, both using computer technologyshould 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 must be disseminated widely.
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 that 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 children to color the inside.
There was no inside. Toward one edge of the paper the pattern curved tightly, so it looked like there was an inside, but as the children moved along they began to realize that the pattern was opening up and did not 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 morewell, 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 did not 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).
Number ≠ Mathematics
A set of numbers does in itself make it a math problem. For example, in an example in the discussion of the middle-grades communication standard, students are asked how many hours they think teenagers watch TV a day and to compare their answers with the results from a national magazine (NCTM, 1989). While the discussion goes on to say that this exercise encourages students to . . . discuss appropriate survey techniques (p. 79), I do not believe it does. The students do not 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 becomes an 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 (NCTM, 1995, 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 (p. 38). 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 is 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 misleadingthe situation there is not like the general case.
The sequence of steps describedconjecture, make a convincing argument, support with datais not 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 it reappears throughout the standards.9
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 do not know what [d] is about. But this is hard mathematics. These are unexpected results. Their proofs are nontrivial. Helping students learn this stuff, whether constructively or in straight lecture, takes a lot of thought from the teacher.10 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 teachersfew mathematicians, for that matterwill have had a chance to become 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 dont 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.
STACKING THE DECK
In the Curriculum and Evaluation Standards (NCTM, 1989), 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 labeled Increased attention, and one labeled 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 toward 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 labeled Decreased attention. The deck is rhetorically stacked, so that decreased can easily become no. Bad words appear, such as rote, isolated, routine, by typeeveryone knows these are bad wordsand 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 boldface the notions that are slated to receive reduced attention.
K-4 (NCTM, 1989), p. 21). 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 students need to focus on paper-and-pencil computations. There are times when you do have to tell the class something (e.g., π).11 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 subtler sense that understanding it contributes to a general mathematical sophistication; for example, ease with complex algorithms. For similar reasons, I want children to do paper-and-pencil computation with fractions. If early attention to reading, writing, and ordering numbers symbolically is done in context, then what could be wrong with it?
5-8 (NCTM, 1989, pp. 71, 73). 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 have 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: whats exi? The answer (-1) is a profound piece of mathematics.
9-12 (NCTM, 1989, p. 127). About a quarter of what is listed here to be deemphasized strikes me as very important. Under algebra, simplification of radical (and other) expressions, factoring, and operations with rational expressions are instances of algebraic manipulations that are themselves necessary steps in the reification necessary to understand algebrabeing able to manipulate algebraic expressions freely 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 maverickif it is presented in a way that can be absorbed by ninth or tenth 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 problemsthat is the essence of mathematical modeling. 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 [NCTM, 1991]).
What Is Going On
The motivation of these lists is clear and even commendable. In general, the thrust is to get away from rote exercisesI am not the only adult who has no fond memories of page after page of trigonometric identities, and the cartoon Hells library (in which every book is labeled 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. So as long as there is no distinction between what should really be thrown out and what has 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 (and will tell you all about through various Web sites and e-mail lists; see, e.g., http://ourworld.compuserve.com/homepages/mathman, the Web site of the group, Mathematically Correct). 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 (NCTM, 1989, p. 96), as an example of the importance of mathematics that the Standards either deemphasizes 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 is 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 thisnot necessarily quickly, but correctlythroughout their lives.
Nearly everything that I would rescue from the decreased attention charts has similar justifications, and can be handled in similar ways.
If the summary charts in the Curriculum and Evaluation Standards (NCTM, 1989) are radical and might raise fears of a diluted curriculum, the actual boldface lists of topics defining each standard are both conservative and ambitiousthere 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 in there.
I have only three quarrels with this material, all at the 9-12 level. Two quarrels are that symbolic manipulation is not appreciated sufficiently (already noted above), and that perhaps more mathematics is proposed than can realistically be achieved. Should students planning on college really (NCTM, 1989) prove elementary theorems within various mathematical structures, such as groups and fields (p. 184), represent finite graphs using matrices (p. 176), solve problems using linear programming and difference equations (p. 176) and interpret probability distributions including binomial, uniform, normal, and chi square (p.171)? Almost none of this is beyond the capabilities of motivated high school students (Im 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 am not sure what they mean and what is expected of students when they are used. They seem more appropriate to science than to mathematics.
While the Curriculum and Evaluation Standards (NCTM, 1989) 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 realworld problems, especially in middle schools which attempt integrated curricula.
The standards documents are themselves fairly balanced on applicationsreal-world applications are no more (and, I hasten to add, no less) standards-based than theoretical mathematics. That the momentum toward 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 (NCTM, 1989) 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; (p. 4) maybe that is what it takes, finally to 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 herecan 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 (Mathematical Science Education Board, 1993), in which fourth graders have an opportunity to move from simple arithmetic calculations to considering 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 (EDC, 1996).
Here, as with applications, the standards do not say what they are chargedby both supporters and detractorswith 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 (NCTM, 1989), reminds us continually that all forms of instruction are useful (although this is contradicted somewhat by the bias in the decreased attention pages).
But occasionally a more dogmatic attitude creeps in which is disturbing. For example, Rich states in the Professional Standards (NCTM, 1991) that he was really reluctant to use that activity because it didnt seem like exploration. It made me feel that I would be directing the students toward a single result (p. 142). But there are many times when directing students toward a single result is exactly what is called for. Furthermore, just because students are inevitably going to find the same result does not mean it is not exploration. Finally, sometimes exploration is not called for.
Where the constructivist bent is seen most clearly is in the Professional Standards (NCTM, 1991), where most of the vignettes are about teachers becoming more constructivist in their methodology. This is understandable. Even now, many teachers have few sources 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 I 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.
As a mathematician, I am not used to thinking comprehensively about assessment, and I learned a lot from reading the Assessment Standards (NCTM, 1995). 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 judgment, 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 that 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 that 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 desireto put students into the appropriate classeshas roots in real, even poignant, situations.12 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 (NCTM, 1995) that is handled somewhat cavalierly, and that is the issue of time. Having begun to use some alternative assessments myself, sparingly, and with only two classes of about twenty-five students each a semester, I can tell you how time consuming this is. I cannot imagine my sons junior high school teachersseven classes a day, about thirty students in eachdoing it on a regular basis. As with the plethora of interesting ideas for curriculum at the 9-2 level, this seems too much to expect.
I could pick some nits, but basically the Professional Standards (NCTM, 1991) outlines a solid mathematical background for mathematics teachers, which is most welcome. I am puzzled, however, by the comment, 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 (p. 139).
Except for the call for manipulatives in probability and statistics (which would be good for all students), I do not 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 (we may wish that danger away, but wishing will not make it so), 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 strangeour pre-service students regularly complain about having to learn transformational geometry, even when we assure them that they will be teaching it themselves. They did not learn it themselves in high school, so 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.
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 [NCTM, 1989]).
I have, however, a major concern about equity.
This is concern about the essentialist view, which seems to have its attractions in educationwomen think like this, African American men think like thatand 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, children. The charge is that trivial curricula and overly easy assessments give the impression that these children 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.
The final topic I wish to discuss is the last section of the Professional Standards (NCTM, 1991), the section entitled Responsibilities. 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.
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 stress what is mathematically important and are 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 better writtenmore carefully, less ambiguously, or with less bias. But the overall 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 overall systemic changes have been/should be made so that the standards movement can succeed?
This article was commissioned by the National Institute for Science Education in the spring of 1996, as a mathematicians reactionqua mathematicianto the National Council of Teachers of Mathematics Standards. Like state frameworks, the standards are designed to be regularly revised, and in 1997, the author became a member of the writing team revising the NCTM standards. Members of such a project implicitly agree to become less public with their own opinions, since it would be easy for their opinions to be confused with the opinions of the group. So it is important to declare that the opinions expressed in this paper are solely the opinions of the author in spring and summer 1996, are not necessarily shared by any other human being, and do not reflect the work of any of the various NCTM standard revision committees. All mention of the NCTM Standards in this article refers to the published documents from 1989 through 1995, and not to any drafts or discussions of a revised set of standards. This has set obvious limits on revision of this paper: only minor changes could be made to the original version. For this I apologize to the referees, many of whose good suggestions for substantive changes could not be used. 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. Finally, I would like to thank the referees for helpful comments.
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