Stanley Ocken, Professor of Mathematics, The City College of CUNY
To: Members of the Board of Education and other Interested Parties
Subject: Implications of CSD 2 mathematics programs for the new
Upper East Side High School
December 12, 2001
At its December 12th meeting, the Board will consider the Chancellor’s Resolution to establish in Community School District 2 (CSD 2) a new high school that is supposed to offer a “challenging college preparatory program,” according to the description in Item 16. I am writing to address the question of whether mandated K-11 mathematics programs in CSD 2 offer appropriate preparation for Advanced Placement and college courses in calculus, statistics, and physics. In my view, the answer to this question is an unequivocal NO. In support of this conclusion I shall draw on my experience as Professor of Mathematics at the City College of New York since 1971, close examination of the CSD 2 TERC Investigations K-5 curriculum, and careful study of the mathematics education literature that has supported National Council of Teachers of Mathematics (NCTM) Standards-based curricular reform in recent years.
My analysis is based on the role of pre-algebra and algebra in the mathematics programs that are being implemented in CSD 2. Algebra, the manipulation and solution of expressions and equations involving numbers, equations, and variables, is the fundamental tool of calculus. In turn, calculus is a prerequisite for careers in engineering, science, architecture, medicine, and computer science. The cohort of students pursuing these careers is a significant part of the college population. Unfortunately, American high schools have a poor track record of ensuring that their students graduate with algebra skills adequate for success in college mathematics. Indeed, during the thirty years I have been teaching at
In recent years, my students’ skills, as well as those of my colleagues at other branches of CUNY, have shown even further deterioration. As a result, it has become extremely difficult to teach and to test my calculus students in a meaningful way without failing a large portion of the class. The grim fact is that students who enter college with inadequate basic algebra skills seldom make up for the deficiencies imposed on them by their earlier mathematics education. Their ladder of opportunity to pursue rewarding careers is cut off at the base. Among the groups most severely impacted are the children of non-English speaking immigrants, children who traditionally have entered the mainstream of American society by pursuing careers that emphasize mathematical, as opposed to linguistic, competency.
New mathematics programs being tested in CSD 2 are of course not to blame for current college students’ poor algebra skills. Unfortunately, indiscriminate implementation of these curricula will significantly exacerbate students’ algebra deficiencies. Since these curricula are based on the 1989 NCTM Standards document, they focus principally on enhancing students’ quantitative literacy skills rather than on facilitating development of the formal mathematical skills crucial to success in calculus. Although the anti-formal bias of the 1989 NCTM standards was attenuated somewhat in their Year 2000 Revision, it is unclear to what extent, if at all, CSD 2 programs such as TERC, CMP, and ARISE will be modified to incorporate a greater emphasis on formal skills.
The three curricula just mentioned are among those alluded to as follows in the draft document of the report issued recently by a commission of experts convened by Schools Chancellor Harold Levy and chaired by CUNY Chancellor Matthew Goldstein:
“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 standards 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 Commission Report further asserts that quantitative literacy skills should be developed as a supplement to, rather than as a substitute for, pre-calculus skills. In my view, this warning will ipso facto be ignored by any school district that forces students into programs based on the 1989 NCTM-Standards.
Among the new CSD 2 programs, the TERC K-5 curriculum exhibits the most egregious deficiencies. Although this curriculum sometimes achieves its goals of engaging students and providing interesting in-class activities, it fails utterly to provide the firm bedrock of hands-on computation and symbol manipulation that serve as a foundation for students’ pre-algebra and algebra skills. One chilling fact: fewer than twenty computations in the entire set of TERC K-5 student materials require the use of the multiplication facts 6 x 6, 6 x 7, 6 x 8, 6 x 9, 7 x 7, 7 x 8, 7 x 9, 8 x 8, 8x 9, 9 x 9, either as standalone problems or as part of multi-digit multiplication problems. The bias toward easy rather than hard multiplication facts is pervasive and cannot be due to chance.
The TERC teacher manuals offer scenarios in which children develop their own strategies for multi-digit multiplication and division problems. Unfortunately, nearly all of the demonstration problems are carefully chosen to be amenable to trial-and error or other non-systematic solution methods. The showcased methods are in fact unsuitable for solving truly representative problems.
Furthermore, the TERC curriculum omits completely the standard algorithms for multiplication and division, apparently heeding the frequent assertion in the mathematics education literature that traditional algorithms are obsolete because they produce answers more easily obtained with a calculator. This uninformed and specious argument fails to recognize the standard algorithms’ critical importance to students’ mathematical development, for these algorithms encapsulate fundamental computational and algebraic ideas and experiences necessary for children’s future success in algebra and calculus.
Having referenced one misguided argument in the mathematics education literature, I feel obligated to note others as well:
There is an expanding body of misleading data and obfuscation in much of the “research” that is marshaled to support NCTM Standards-based curriculum initiatives. Among these is the questionable relevance of scores on standardized testing instruments, including those used in
NCTM Standards-based curricula consistently claim to enhance students’ conceptual understanding, a goal typically touted as a revolutionary advance over traditional adherence to “blind rote manipulation.” This is nonsense. When NCTM curricula such as TERC’s Investigations use the term “understanding,” they often refer merely to the obvious and pedagogically useful technique of furnishing concrete models for simple arithmetical examples, e.g. by using fraction strips to picture fractions such as 3/4 and 2/3. Every competent parent or educator knows that this is a good way to start. Unfortunately, a principal failing of Standards-based curricula is that students never move beyond, and so are forced to rely on, simple models and representations. As a result, when students confront purely symbolic representations that are not attached to physical models, they simply freeze. Their reaction, perhaps best characterized as “symbol shock,” is, in my experience, a primary cause of students’ failure to succeed in college mathematics.
It cannot be overemphasized that the essential methodology of college mathematics and science is to analyze a real-life problem, then to represent it by symbolic expressions, and finally to simplify and transform those expressions into a solution to the original problem. Successfully implementing this agenda requires a tremendous repertoire of purely technical skills that must become second nature and that can become so only with intensive practice. For example, students must be trained to recognize legitimate patterns and processes of symbol manipulation and to explain why specious manipulations, such as canceling the x's in the fraction (x + 5) / (x +1), make no sense algebraically and must be instinctively avoided. This and similar algebra rules must be appreciated on their own terms, as referring to symbols rather than to things. You can’t draw a picture of x!
As a professor of long standing in the CUNY system, I have never ceased to wonder why so many students enter college seemingly endowed with the ability to succeed at mathematics, but are subsequently prevented from doing so by their lack of mastery of basic algebra skills. The mathematical training of our city’s most precious resource, its children, has for too long been driven by the faddish and demonstrably ineffective policies and perspectives of some mathematics educators. I for one, together with many of my university colleagues, am prepared to devote considerable effort to the true reform of mathematics education in this city.
I applaud the Board’s intent to establish a new high school on the
It may be desirable to guide some students with special needs toward a curriculum that de-emphasizes preparation for college-level mathematics, although, in my view, such tracking should be undertaken only as a last resort. However, it is unconscionable for the Board of Education to consider establishing a new high school in, or to continue promoting the mathematics programs of, any school district that does not enthusiastically support rigorous mathematics curricula offering all students the opportunity to pursue mathematics-based careers.
Professor of Mathematics
The City College of the City University of New York