NYC HOLD Honest Open Logical Debate on math reform
Are our school’s math programs adequate?
Experimental mathematics programs and their consequences
Do NCTM Standards-Based programs prepare students for calculus?
Professor of Mathematics
In the current debate on mathematics curriculum reform there are many points of agreement, the most important of which is that traditional methods and materials have proven inadequate for conveying mathematics to students and must be modified in order to achieve that goal. Furthermore, most participants in the discussion would agree that students should succeed at and enjoy mathematics and should understand the mathematical methods they are using. Unfortunately, the definition of success in mathematics, and even what is meant by ‘understanding’ of mathematics, is itself a point of contention. The question to be addressed herein is much more focused. What is the effect of proposed modifications of K-8 curricula on children’s preparation for the mathematics they will confront in college should they pursue a program of study with a calculus requirement.
I will address
this question by drawing on thirty years’ experience teaching all levels of the
calculus sequence at the
There is an odd perception is some of discussions of mathematics reform that concern with calculus preparation is somehow elitist. In fact, a large percentage of students are involved, because many majors and programs, including science, engineering, medicine, architecture, computer science, business, and secondary math education require one or more semesters of calculus. Indeed, in Fall, 2000, over 462,000 students in four year colleges enrolled in first semester calculus, compared to a total of 1,156,000 students who began freshman year at those institutions.
between NCTM Standards-based curricula
and college mathematics was addressed in the draft document of the report
issued in Spring, 2001 by a commission
of experts convened by Schools Chancellor Harold Levy and chaired by CUNY
Chancellor Matthew Goldstein, as follows:
“ Whenever an emphasis is placed on ensuring that applications are made to ‘real world’ situations …less emphasis is placed on arithmetical or mathematical ideas and the formal, abstract contextual settings sought particularly by students who will go on to become scientists, engineers, mathematicians, computer scientists, physicians, and educators of mathematics.”
“Despite their many strengths, the NCTM standard do not contain the rigor, algorithmic approach, formal methods, and logical reasoning which are required of this small but critically important portion of the population.” (emphasis added)
The above passage, omitted from the final commission report, expresses the profound concern of experts that the NCTM standards offer a vision of mathematics instructions that doesn’t include appropriate preparation for college mathematics. What is the basis for such a disturbing suggestion?
Let’s start by examining a typical page from the TERC teachers’ manual, a page that purports to demonstrate examples of children’s invented solutions to multiplication and division problems. Note the ubiquity of children’s English language description of their mathematical activities, in accordance with TERC doctrine advocating that children tell a story about what they are doing. What is not obvious at first glance is in fact quite disturbing: the problems have been carefully chosen to be amenable to nonstandard solution methods. Indeed, the showcased children’s solution techniques would not work if applied to randomly chosen problems. The TERC solutions may be useful as pedagogy, but they don’t provide a consistent procedure for actually solving math problems. For real world math, in college or anywhere else, the TERC methods work only sometimes, and that’s simply not good enough.
At this point I want you to contrast the pages from the TERC manual with the mathematics that students will experience in college. The page of mathematics on the screen is taken from a nicely written first semester calculus final exam paper. What you see contains the solution to one and a half questions out of a total of ten that appeared on a recent final exam lasting two hours and twenty minutes. I mention these details to give a sense of the quantity of mathematics that students must set to paper. What about the quality?
The most blatant contrast between the college exam paper and the students’ TERC solutions is that there is no English language explanation on the calculus final paper. At first glance, one might think that the calculus final, unlike the TERC example, doesn’t tell a story. Quite the opposite is true: the calculus problem solution tells a complex and elegant story, but condenses the narration into a universal language of mathematics that is written in symbols rather than in words. English language commentary, were it included, would be irrelevant and confusing, to say nothing of tripling the amount of writing. The language of mathematics is symbols; the language of TERC problem solving is English. It is this dichotomy that lies at the heart of mathematics professors’ concerns that TERC and similar programs provide an inadequate base for moving from the study of arithmetic to the study of algebra and calculus.
Does the absence of ordinary language on the calculus final suggest that the student was performing by “blind rote” or without proper understanding? Of course not! Her solution is beautifully organized. She understood exactly what needed to be done and when to do it. She
analyzed a real life problem;
represented that problem by a diagram;
selected variables and wrote algebra expressions to convert the problem to symbolic representation;
selected the appropriate methods for transforming the symbolic representation to a solution; and
carried out the transformation correctly to solve the problem.
That’s quite a bit of work in ten minutes, but that’s what’s required in calculus. Symbolic representation and complex symbol manipulation are an absolute prerequisite to success in college mathematics.
Let’s consider a number of specific ideas of curricular reform that are discussed in the mathematics education literature and compare their implementation in TERC with what actually goes on in college mathematics.
1. Mathematics should be taught using stories, pictures, and symbols. In TERC, nearly the entire emphasis is on stories and pictures. That’s useful for pedagogy, but at some point there has to be a transition from concrete to abstract symbolic representations of problems. In calculus, stories and pictures play a small part, whereas about 99 percent of typical student work involves symbol manipulation.
2. In TERC, children use blocks, pictures, and fraction strips to model the operations of arithmetic. That’s fine for motivation; most teachers use such methods to one degree or another. However, it’s much harder to represent five sevenths by cutting up a fraction strip than it is to represent one third or one half, the typical easy fractions of the TERC curriculum. That curriculum shows pervasive bias toward choosing only the simplest examples, and seems to ignore the fact that “unfriendly” fractions and numbers are the norm rather than the exception.
The benefits of both modeling and visualization are limited. Indeed, excessive reliance on these techniques can be dangerously counterproductive. Any effective curriculum must employ a clear strategy for helping students graduate from reliance on mathematical crutches. After all, when children study algebra in high school and calculus in college, they need to work with x, y, and z. If they have been trained to rely on models and pictures, children who encounter variables will be in serious trouble: there’s no way to draw pictures of x, y, and z.
3. In TERC, children invent their own procedures for solving problems. As noted above, these procedures work only for limited classes of problems. In contrast, real mathematics requires problem solution methods that work both efficiently and with perfect consistency. That’s why the standard algorithms were developed. They organize the basic operations of arithmetic in the most efficient manner possible for solving complex problems.
4. TERC students are not taught the standard algorithms for either multiplication or division. Why, in fact, are these “rote procedures” necessary? Aren’t they a waste of time if they can be carried out easily by pushing buttons on a calculator? Or so goes the claim of a significant portion of the mathematics education community.
This argument is both shallow and specious. As Alan Siegel has said, mathematics is layered and deep. There is a continuum of thought and logic that runs from elementary school arithmetic to high school algebra to college calculus. The algorithms of arithmetic accomplish much more than obtaining the right answer. They establish patterns of algebraic thought and execution that reappear repeatedly throughout the precalculus curriculum and as such are critical to students’ mathematical development. For example, if students don’t learn to multiply 37 times 22 using the standard algorithm division algorithm, they will encounter great difficulty when a minor variant of that algorithm is used to solve the related problem 3x + 7 times 2 x + 2 in high school algebra. There’s a direct connection. Similarly, long division matters, not because children need the answer, but rather because they need hands-on experience with the logic included in the method. A child who hasn’t become proficient with the standard long division algorithm will be totally lost when the problem transforms to the polynomial division problem in high school. In turn, that polynomial long division algorithm is needed at three distinct places in the college calculus curriculum. The instructor will assume that students know it cold: those who don’t will be lost.
The reputations of standard American algorithms, especially the one for long division, are at times impugned by math educators’ assertions that different algorithms are used in different countries. This statement arises from a misperception that the term “algorithm” refers to the format, or tableau, that is used to present the solution. In fact, the algorithm is the sequence of logical steps involved in the solution. All countries use the same division algorithm. The only variations are minor differences in the number of intermediate steps that are recorded and in their placement on the division tableau. Students who learn to do long division in Russia and the United States, where the division tableaux are in fact rather different, can easily make the transition from one tableau to the other because both utilize precisely the same division algorithm, whose use has been universal (or at least planet-wide) for centuries.
5. TERC profoundly de-emphasizes computation. Overall, the total number of computational examples provided with K-5 TERC student sheets is modest. Of course, teachers can supplement these sheets with classroom activities and games. Of critical concern, however, is the student sheets’ bias toward the selection of easy problems. Indeed, in the entire set of published materials, the number of examples that require students to perform any of the following single-digit multiplications:
6 x 6, 6 x 7, 6 x 8, 6 x 9, 7 x 7, 7 x 8, 7 x 9, 8 x 8, 8 x 9, and 9 x 9
either as standalone problems or as part of a multi-digit problem, is less than twenty. By a huge margin, that’s much too little practice to prepare children for the increasing demands of symbol manipulation they will face as they progress through algebra on their path toward calculus.
6. TERC emphasizes fluency; mathematics requires proficiency. What’s the difference?
When I use the term proficiency, I mean that a child masters basic arithmetic facts and responds to questions such as “6 times 7” reflexively, within a second or two at most. Communication with TERC curriculum developers reveals that they care relatively little about this type of performance. Indeed, they are on record as being concerned more with “fluency,” understood as follow. When a child is asked “6 times 7” he tries to work out a clever method of approaching the problem, e.g. “3 times 7 equals 21, then double the answer to get 42.” The amount of time spent is of minor concern: what’s important is the thought process.
This is a perspective that I characterize as interesting, at best. In fact, speed is essential. It’s fine for children to use such strategies, but only as a step along the way to mastery. Pre-algebra mathematics requires mastering and working with a foundation of basic arithmetic facts that can be summoned automatically for the solution of more complex problems. As the philosopher and mathematician Alfred North Whitehead shrewdly observed in his Introduction to Mathematics:
It is a profoundly erroneous truism repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations that we can perform without thinking about them. Operations of thought are like cavalry charges in a battle--they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
Whitehead is saying, among other things, that mathematics problems must be solved by omitting unnecessary details. In our setting, the mathematics that leads up to calculus is structured and hierarchical. Students must master and absorb each stage to the point of automaticity before proceeding to the next stage: proficiency is essential.
“Fluency” as advocated by the TERC developers can indeed be desirable. To the extent, however, that it encourages a consistent preference for a derived solution over a memorized one, advocacy of fluency at the expense of proficiency is downright dangerous for students who must gradually develop the patterns of thought and practice that are essential to the efficiently and accurate solution of complex multi-step problems.
In closing, I would like to add a personal note. Frequently discussed in discussions of curricular reform are the pain and boredom of memorizing multiplication tables. Indeed, this sort of memorization is more difficult for some children than for others. But there is no royal road to proficiency in mathematics. My concern is with the later part of a child’s journey, when he or she arrives in my calculus class. At that stage, unfortunately, I encounter a lot of suffering: not the pain of students struggling on a test, but rather the shock of flunking calculus and being forced to change majors. All too many students encounter a sudden and rude awakening, one that amounts to “You can’t be a doctor [an engineer, a chemist, an architect]; sorry, you don’t have proficiency.”
When I all too frequently am forced to be the agent of such a message, I share my students’ pain. The sad reality is that their pain could have been avoided had they studied a balanced K-12 mathematics curriculum that suitably emphasizes algebra and formal skills.
Curricular reform is needed. Sadly, at least from the perspective of children who want to pursue mathematics based careers, current efforts at reform are moving in precisely the wrong direction, by de-emphasizing the critical role played by symbol manipulation and algebraic fluency. Instead, curricular reform should acknowledge the importance of these skills and seek innovative and effective methods of helping students develop their proficiency in arithmetic and algebra. Educators and theorists must awaken to the reality that every calculus course requires its students to enter with a substantial reservoir of formal skills.
Perhaps the most misused term in the mathematics education literature is “(conceptual) understanding.” In fact, there is no such thing as mathematical understanding without formal skills. A student or teacher who talks about a problem does not understand the problem until he or she reduces the statement of the problem to symbols and then finds an economical way to transform those symbols to a solution. Talking math is not doing math. College level mathematics without symbols is cocktail party mathematics, no more, no less.
Perhaps the most puzzling feature of mathematics curriculum reform has been the absence of meaningful participation by the community of academic mathematicians. It is imperative that children who are preparing for college mathematics study curricula that have been devised in meaningful co-operation with college mathematics and science teachers. This has not been the case. Indeed, the de-emphasis on symbol manipulation skills in most if not all NCTM Standards-based curricula poses a clear and present danger, both to American children and to the future American technology. To help children succeed in mathematics, we need to have curriculum development and revision that focus on improving the study and development of algebraic skills rather than throwing them away.
quoted statement’s characterization of the referenced portion of the student
population as “small” is misleading. In fact, 462,000 senior college students enrolled
in first semester calculus courses in Fall 2000, as compared with a total of