By Stanley Ocken
Department of Mathematics
The City College of the City University of New York
On September 12, 2006, the National Council of Teachers of Mathematics (NCTM) released its Curriculum Focal Points as a supplement to its K-12 curriculum standards written in 1989 and 2000. The new document is an important statement, but addresses with insufficient clarity the issues that made the prior publications controversial and divisive. Clarification is needed, and it is needed now.
The NCTM's stated purpose for the 1989 Curriculum and Evaluation Standards (the "1989 Standards") was to ensure that "computational algorithms, the manipulation of expressions, and paper-and-pencil drill no longer dominate school mathematics." That statement is shocking to mathematics professors, not least because reading, writing, and manipulating symbolic expressions are skills critical to the success of every calculus student.
Unfortunately, few if any math professors had heard of the NCTM in 1989, much less reviewed the 1989 Standards. Had meaningful review taken place, that document's language and perspective might have emerged in much more palatable form. But its implementation in accordance with the anti-algorithm, anti-algebra statement quoted above led directly to the development of content-barren K-12 math programs.
The 1989 Standards, as well as statements of NCTM officials, indicated clearly that algorithms and expression manipulation were to lose their dominance because they are associated with memorization, working by rote, subjecting students to drill, and, worst of all, doing mathematics without understanding. In fact, each alleged sin has its virtues when integrated into a well-designed curriculum.
More than a century ago, the eminent mathematician and philosopher Alfred North Whitehead debunked a myth that remains with us to this day. In response to the claim that people must always think about what they are doing, Whitehead wrote: "The precise opposite is the case. Civilization advances according to the number of operations that we can perform without thinking about them. Operations of thought... must only be made at decisive moments."
Whitehead's statement has real resonance in the present discussion. Certainly mathematicians and their students spend time thinking about what they are doing. But there are many kinds of thinking. Formal algebraic thinking, which to an outsider looks like manipulating numbers or expressions according to arbitrary rules, is hugely important in mathematics. Furthermore, it's crucial to perform most procedures automatically, in order to free the mind for the study of higher level questions.
Students should spend time memorizing and practicing algorithms, which are the systematic, efficient, and foolproof procedures for executing standard tasks. Facts are important: students need to know the multiplication tables cold. Facts and algorithms together are an indispensable part of the toolkit for solving both abstract and applied problems that appear, or should appear, on math exams beginning in fourth grade. Practice and repetition are essential to success in mathematics, just as they are essential to success in music, sports, and the study of foreign languages. Those who call this a "back to basics" recommendation may do so, but a better description is "back to mathematics with real mathematical content."
The NCTM Focal points are issued at a time when many parents are appalled by the lack of quality and content in their children's math programs. The Focal Points in grades 6 to 8 seem reasonable. But in earlier grades, they fail to disqualify very weak programs, still in use, that derive inspiration from and are consistent with the 1989 Standards.
In Grades 1 to 5, the Focal Points fail to explain that sustained experience with nontrivial multi-digit arithmetic problems is critical to success in algebra and higher mathematics. They omit all reference to memorization. They do not admit the interpretation that skills can be developed first and understanding filled in later. Some content material is vague or weak. Indeed, the description of whole number division activities with multi-digit dividends suggests that the hardest problem that students need to handle is 99 divided by 9, while the description of place value fails to name numbers above 1000. That is unacceptable. The NCTM authors should have incorporated existing documents, most notably the State of California's Mathematics Content Standards, which provide a crystal clear grade by grade delineation of computational and symbolic skills appropriate to a content-rich elementary math program.
Nevertheless, the Focal Points are a move in the right direction, if only because they seem to restore computational skills to a major place in the Grades 1-5 curriculum. A much clearer statement is needed, however, in order to undo the damage that has been done to parents, students and teaches by content-poor NCTM-inspired curricula.
The NCTM cannot move forward unless it explicitly repudiates the philosophy that motivated the 1989 Standards document. Real pedagogical leadership requires restoring computation, algorithms, and algebra to their critical place in a content-rich mathematics curriculum.
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