Experimental mathematics programs and their consequences

New York City

June 6, 2001

Investigations In Number, Data, and Space (TERC)

Dwight Parker Robinson Professor of Mathematics

Harvard University

I shall devote my opening remarks to the TERC Investigations curriculum. Many of you know that TERC does not include textbooks in the usual sense. It consists mainly of a collection of teachers' manuals, roughly a dozen for each grade level. The curriculum is highly scripted, as they say in education circles, meaning that the teachers are told in great detail what they are supposed to do.

One of the most striking features is the emphasis on "group learning" and "discovery learning" - the students are supposed to arrive at mathematical facts and procedures through a process of communal discovery - and this emphasis goes so far as to discourage almost completely direct instruction by the teacher. In the TERC manuals, there are elaborate descriptions of the role of the TERC teacher, using verbs like "listen", "facilitate", "observe", but never the verbs "explain" or "teach". A TERC teacher doesn't explain, and a TERC teacher doesn't teach! I don't want to be misunderstood: group learning and discovery learning are parts of the tool chest of every accomplished teacher, but it is folly to turn these techniques into an ideology. If we mathematicians had to re-discover mathematics on our own, we would not get very far! And indeed, TERC does not get very far. By the end of fifth grade, TERC students have fallen roughly two years behind where they should be.

"Discovery learning" is in, but any kind of memorization is out. Addition facts and multiplication facts can be rediscovered on the spot, the TERC manuals seem to say, so there is no need to commit them to memory. The TERC authors are also opposed to the teaching of the traditional algorithms of arithmetic, such as long addition, subtraction with borrowing, and the usual pencil-and-paper methods of multiplication and division. Not only do they refuse to teach the algorithms, they make clear their preference not to have the students learn them outside of the classroom, either.

Why do mathematicians believe in memorization and algorithms? The ability to compute is a basic skill, absolutely necessary to succeed in high-school algebra, just as mastery of high-school algebra is crucial for success in college mathematics. When I say basic skill, I mean a skill similar to speaking and reading, something one does without first reflecting about it. The TERC authors maintain that they are very interested in developing the students' computational skills. But when one looks at the TERC manuals, one finds a highly self-conscious approach to computation, a far-too-great reliance on mental crutches, such as number blocks, clock faces, paper strips. Most disturbingly, the practice problems are ridiculously easy, and there are not enough of them, not enough by a long shot!

The TERC authors proudly boast that they aim to develop the students' mathematical thinking. TERC's defenders accuse critics (like me and my colleagues) of advocating mindless drill. How can that be? We, the mathematicians, unconcerned about mathematical thinking? Of course not! The TERC manuals, on the other hand, reveal a complete misconception of what constitutes mathematical thinking.

The pedagogical practices advocated by TERC are truly extreme. Many teachers have enough sense to mitigate the worst excesses, sometimes consciously, sometimes subconsciously. But there are also true believers among the TERC teachers, and then you see real problems. It was a problem of that sort that got me involved in math education!

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