“How can we lose when we are so sincere? ”
-Charlie Brown
The errant mathematical texts and the mathematical anomalies alluded to would not by themselves a reform make, had not the NCTM Standards [N1] given the reform a voice and an identity. The impact of [N1] has
been spectacular. Ever since 1989, when [N1] appeared, one would be hard pressed to find a research paper in mathematics education, an education document, or a school mathematics textbook that does not pay at least lip service to [N1]; and this includes the so-called “traditional” texts such as [RH]. The NCTM Standards are the reform.
The goal of [N1] (and the companion volumes [N2] and [N3]) was to redress all the ills of the “traditional” curriculum. In order to overturn all this in one fell swoop, NCTM came up with a new curriculum, a new pedagogy and new assessment techniques ([N1]–[N3]). Here we shall concentrate mainly on the NCTM Standards [N1]. By and large, it is probably not controversial to assert that the curriculum of [N1], if carefully executed, would remove the mindless drills, the dry memorization and much of learning-by-rote. It is more successful than the traditional curriculum in reaching out to the lower 50% of the (mathematics) students. Those who used to be turned off by math now find the Standards-inspired materials much more “user-friendly”. The Standards have also brought renewed enthusiasm to some teachers and made others think hard about education again. Yet, even in the midst of such gains, one senses trouble.
The most common criticisms of NCTM Standards [N1] are its inflated prose and its carelessness in the formulation of various recommendations. This carelessness has inspired mischief and lent legitimacy to mischiefs already committed. For example, [N1]’s advocacy of “open-ended” problems “with no right answer”; its unfounded faith in the propriety of technology in mathematics education beginning with kindergarten (“There is no evidence to suggest that the availability of calculators makes students dependent on them for simple calculations”); its recommendation to give “decreased attention” to “two-column proofs”; its repeated admonition to downplay “memorization of facts” and “memorization of formulas”; etc. These have all been cited, rightly or wrongly, to justify some or all of the practices. However, the most substantive defects of the Standards are global in nature, in the sense that they are not tied to a particular chapter or verse, but concern the sum of all the parts of [N1].
A first major defect lies in its insistence that (real world) problem solving must be the focus of school mathematics ([N1], p.6) and the particular way this decision is implemented. A second major defect is that the floor of mathematics education has been set far too low (cf. [N1], p.9). The ceiling, which is described as “the NCTM Standards for College-Intending Students”, is consequently also dragged down. A third major defect is [N1]’s failure to confront the pressing issue of how one single mandated curriculum can be used for all students, no matter their mathematical capabilities. All these defects are inter-related.
The eleven-page Introduction of [N1] sets the tone: this is to be a curriculum with an uncompromising emphasis on producing a “technologically competent work force”. Subsequent discussions and examples are rooted in this leitmotiv. Readers are reminded at every turn that mathematics is a powerful tool to solve real world problems. In contrast, one encounters little discussion of the need for mathematical developments of a mathematical idea. Bluntly put, the Standards read like a vocational-training manual that casts occasional side-glances at students’ intellectual development. (The discussion in the classic treatise [BE] on this topic is very relevant here.) When the New York Times carried news of the proof of Fermat’s Last Theorem (FLT) on the front page, it was acknowledging the fact that there exists an intellectual component to mathematics that even laymen (including high school graduates) must reckon with. Sadly, students coming out of the Standards curriculum would have little idea why a quaint statement such as FLT should be of interest to anyone (especially since it obviously holds up when a few integers are plugged into the Fermat equation on a computer). The preceding paragraph is not an indictment of application-oriented curricula per se, only of the particular way [N1] wants such a curriculum implemented: it allows the utilitarian impulse to overwhelm the basic educational mission, with the result that basic ideas and skills not directly related to the so-called real world problems often get left out. By yielding to the temptation of “doing just enough to get the problems solved”, the curriculum of [N1] ends up presenting a fragmented and amorphous version of mathematics.
What the Standards should have done is to bring the idea of mathematical closure to the forefront. In other words, if certain tools are developed for the purpose of solving a particular problem, then the solution needs to be rounded off with a discussion of the tools themselves in the context of the overall mathematical fabric. How are they related to other mathematical techniques and concepts? Are they part of a general structure? Is the idea behind them applicable to completely different situations as well? And so on. Unfortunately, the idea of mathematical closure is never broached in [N1].
For the sake of argument, let us take an approach to teaching classical music in school analogous to that used by [N1] to teaching mathematics. Then classical music would be presented only as it serves commercial purposes. The greatness of Beethoven may have to be authenticated by facts such as his 9th Symphony having been used in the Huntley-Brinkley Show and the Beatles’ movie Help!. Rossini’s worth would be shown via The Lone Ranger, while Richard Strauss would be immortalized by 2001 and Mozart by Elvira Madigan. And so on, ad nauseum. No doubt this would make classical music accessible to more students than ever before, and we may even talk ourselves into believing that we have achieved the goal of “Class 4
musical music for all !” But is this really all we want to get out of a classical music education? Why then should we allow the same thing to happen to mathematics?
When an educational document consistently presents mathematics as a toolshed instead of the edifice that it is, its wholesale revision is overdue. A simple example would perhaps clarify the meaning of mathematical closure. On p.152 of [N1], there is a discussion of the problem of finding the roots of the cubic 5x3 − 12x2 − 16x + 8 = 0 in the context of Grades 9-12. In the view of NCTM, all that the best of the non-college-bound students need to know about this problem is summarized in the following paragraph (my paraphrase of Level 4 on pp.152-3):
Assign students to a group project of constructing an algorithm for approximating the real roots, such as the bisection algorithm. Pay special attention to the proper expressions used to record this algorithm. Once this is done, test the procedure by computer implementation.
That is all: just a computer procedure to approximate a real root. From the narrow perspective of treating mathematics as a tool to solve real life problems, this is of course sufficient. However, from the point of view of mathematics, shouldn’t a student be interested in roots of polynomials in general? Fourth degree? Odd degree? Other roots, once one is found? Rational roots? Total number of roots? Not every detail need be explained, but even the average student will have his life improved by the mere knowledge that there are such questions, often with answers, e.g., that the factor theorem and the quadratic formula predict that the above cubic will be completely solved (approximately) once a single root is found. Students would benefit from the exposure to this bread-and-butter kind of mathematical thinking along with the basic technical skills that are developed in the process. This is the floor appropriate for all students in this particular instance. Why is the acquisition of this kind of “higher order thinking skill” not the compelling message of the Standards? Now part of the preceding mathematical discussion is indeed in [N1] (Level 5 on p.153), but (one infers) is reserved in [N1] for what it calls the “college-intending students”. It hardly seems reasonable that such simpleminded mathematical deductions and questions should not be made available to all students. Furthermore, the ceiling in this case is set far too low. Except for the Fundamental Theorem of Algebra, proofs of most properties of polynomials should be given and even the Cardan formulas and other intellectual
triumphs such as the works of Abel and Galois should be discussed to indicate that mankind does not always think of mathematics exclusively as a tool for solving real world problems.
Let me give one more short example of the inappropriateness of either the floor or the ceiling set by [N1]. Consider the comment on p.165 of [N1]: “College-intending students also should have opportunities to verify basic trigonometric identities, such as sec2 A = 1+tan2 A, since this activity improves their understanding of trigonometric properties and provides a setting for deductive proof.” If the proof of such a trivial consequence of sin2 A + cos2 A = 1 is now reserved only for college-intending students, how does NCTM expect the average high school graduate to understand sec2 A = 1+tan2 A ? To graph (sec2 A) and (1+tan2 A) separately and observe that the two graphs coincide, as suggested in [PEL]? With this in mind, we find it hard to believe that “the mathematics of [the Standards’] core program is sufficiently broad and deep so that students’ options for further study would not be limited” ([N1], p.9).
In education circles, a great merit of the Standards is seen to be its success in producing a curriculum for all students, including all those “who aren’t getting it in math”. In brief, the coded message behind these words is the elimination of tracking, the separation of school students into different classes according to ability. True enough, [N1] sets the floor and describes the ceiling, but it stops short of describing how to implement this abstract idea of addressing both the floor and the ceiling of a curriculum in the same classroom except to offer the disclaimer that “it does not imply that students of all performance levels must be taught in the same classroom” (p.130 of [N1]). What it should have done is to make a clear cut recommendation for a program-with-choices for the last two or three years of high school. Mathematics courses should bifurcate in those years into the general track and the scientific track, and students should be allowed to choose between
the two, the same way college students are in choosing among different kinds of calculus classes. It is a matter of record (according to Zalman Usiskin) that no other developed country practices one-curriculum-for-all in the last years of high school.
What has been happening to reform curricula in the absence of a clear directive for program-with-choices is that the floor takes precedence, the ceiling gets ignored, and the serious and gifted students end up being shortchanged. This problem is becoming so serious that it has alarmed not only educators and psychologists (cf. [RE], [GA], and also [PEN]), but also the U.S. Department of Education. Partly in response to the reform, the latter addresses the “quiet crisis” of our neglect of the top students in the refreshingly straightforward document [NA]. I will simply quote a passage from [NA] to serve as a critique of the Standards:
Ultimately, the drive to strengthen the education of students with outstanding talents is a drive toward excellence for all students. Education reform will be slowed if it is restricted to boosting standards for students at the bottom and middle rungs of the academic ladder. At the same time we raise the “floor” (the minimum levels of accomplishment we consider to be acceptable), we also must raise the “ceiling” (the highest academic level for which we strive).
One final comment on [N1]–[N3] may not be out of place here. Any attempt to improve mathematics education must address at least three main issues: to insure that the teachers can do justice to mathematics, to induce students to work hard, and to improve the curriculum and assessment methods. The documents [N1]–[N3] deal with the last of the three, while the first two have fallen by the roadside. However, the critical importance of students’ willingness to learn in any meaningful discussion of education has not been overlooked by people outside the reform. The articles [AN1], [AN2] and [BA] (among others) are powerful reminders of the folly of ignoring the student factor in the present reform. One would like to respectfully suggest that the NCTM Standards on assessment and teaching ([N2], [N3]) should waste no time in confronting this topic head-on in their forthcoming revisions.
In addition, teachers’ inadequate knowledge of their subject was in fact a main concern of A Nation At Risk ([NAR]) and is at the root of many serious problems in mathematics education. Yet, in its 195-page volume on teaching ([N2]), NCTM — the National Council of Teachers of Mathematics — saw fit to devote only 8 pages (pp.132-139) to this most pressing of all instructional issues. More damaging is the fact that the vignettes in [N2] all seem to point to the failure of pedagogical methods as the root cause of poor performance in mathematics instruction, whereas even a casual inspection of the typical classroom would convince an observer that the lack of a firm grasp of mathematics is most often the culprit. The negligence is the more surprising because the curriculum of the Standards in fact makes a greater demand on the teacher’s command of the subject matter than the traditional curriculum. Teachers who are already having difficulty with the old curriculum would be even less prepared for the new tasks NCTM sets for them.
“We already have consensus from the major mathematics organizations . . . ” Price was referring to the statement on p.vi of [N1]: “This document is significant because it expresses the consensus of professionals in the mathematical sciences for the direction of school mathematics in the next decade.” Further down the page, one finds that AMS, MAA and SIAM are listed as Endorsers. This is how we mathematicians enter the picture in education.
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