Guest post by Ze'ev Wurman

Review of "Focus in High School Mathematics," National Council of Teachers of Mathematics, 2009

This document stands out more for what it omits rather than for what it includes. It is full of paeans to reasoning and sense-making that it--quite properly--hails as the hallmark of mathematics. Yet reasoning and sense-making are not exclusive to mathematics. They are the cornerstones of history, of physical sciences, of economics, and even of political science. The reason mathematics is unique, that it has been called the Queen and Servant of Science, is not because it is focused on reasoning and sense-making, but because it is focused on formal reasoning--on mathematical proofs.

Yet reading the new NCTM document one might well not notice the concept of proof. The word "proof" appears a scant 27 times in the body of this 145 page document, one third of them in the short opening chapter which ties reasoning and sense making to proof. Yet even there, the first appearance of "proof" is immediately followed by "however", as in "However, mathematical reasoning can take many forms, ranging from informal explanation ..." (p.4) Another half of the occurrences of "proof" is found in the single geometrical semi-proof example in the text, with the other five incidences sprinkled elsewhere. Contrast that with almost 500 occurrences of "reasoning" and over 200 occurrences of "sense making" in this document.

In fact, much of the document is spent trying to confound reasoning and sense-making (which are present in every discipline) with formal proofs that are unique to mathematics. This is presumably made in an effort to make mathematics more accessible, yet it amounts to pouring out the baby with the bathwater. Without exception, all of the discussion is centered around so-called "real world" examples, and abstraction is essentially unmentioned. Reasoning and sense making are indeed important, in mathematics as in history, yet they are not the defining characteristics of mathematics, as NCTM pretends they are. The formal aspects of mathematical proof, and its powerful abstractions, are what makes mathematics uniquely effective. Spending almost 150 pages with barely a mention of prerequisite mathematical knowledge and skills, and with nary a paragraph promoting proof or abstraction as the keys to mathematics, and to its enormous utility in all domains of life and science, is NCTM's disservice to all of us.

At this point it is almost pointless to mention the other deficiencies of this document: The exclusive focus on reasoning and sense-making comes with the deprecation of almost anything else in mathematics, and particularly skills and procedural fluency. Readers are warned that students may "capriciously invoke incorrect or baseless rules" without understanding (p.12) yet nowhere is a similar warning of careless or unsubstantiated reasoning given. Little time is spent on mentioning content, with the exception of promoting--without any research support--yet more probability, statistics, and discrete mathematics in the curriculum, topics that NCTM has been promoting for decades as its antidote to the "traditional math." Some things never change.

## Saturday, October 10, 2009

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Thanks for this review. It confirms my take on the NCTM document.

ReplyDeleteWhat is wrong with high schools focusing on mathematical concepts, content, and skills?

page 4 reasoning -- the process of drawing conclusions on the basis of evidence or stated assumptions.

What is it called when math education experts and decision makers draw conclusions in the absence of evidence, or the presence of fraudulent evidence, or false assumptions?

It is scary to think of the decisions that will be made based on the existence of this document. The majority of those decisions will not make any reasonable sense.

I am a college math professor. The NCTM types focus on reasoning, etc., but when these kids get to college they can't compute anything and even those who make it to Calculus are extremely weak. Besides emphazing proof (which any such document should) it should make it clear that students need to be strong at compuation. Thanks to the NCTM -like math de-education, we have had over 20 years of students who are lacking basic arithmetic, algebraic/trigometric skills and have absolutely no idea what a mathematical proof is. The Harvard Calculus text was the completion of the demolition of mathematics - make math open to all by taking out all theory and computations. What is left is math-lite, if that.

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ReplyDeleteThey are the cornerstones of history, of physical sciences, of economics, and even of political science. The reason mathematics is unique

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