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Math: Not So Formal!

In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite – my new Three R’s. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms – three third grades, one combining the third and fourth grades, and one fifth grade.– Louis P. Benezet, describing his innovative program in 1929.

Allegedly, formal math teaches children how to think logically. In practice, for most children, it teaches how to grope for the “right formula” – and to then misapply it.

Benezet wrote three articles, combined here: Benezet: Informal Beats Formal Math.

These articles recount his experiences with a variety of classes, some of which use the traditional formal method of math instruction; others use the new Three R’s: Read, Reason, and Recite. Students in the informal classes – most of whom were learning English as a second language – had richer vocabularies and stronger mathematical reasoning skills They had incorporated math understanding into their language; it became second nature to them.

Benezet encountered much resistance. He took his critics around, and posed a problem to several fifth grade classes:

A pole is stuck in the mud at the bottom of a pond. There is some water above the mud and part of the pole sticks up into the air. One−half of the pole is in the mud; 2/3 of the rest is in the water; and one foot is sticking out into the air. Now, how long is the pole?”

Traditionally-taught students floundered – they tried various operations at random, without actually understanding the relationships – half plus one half equals one whole, for example.

The informally-taught children – who spent four years learning how to express themselves and to reason – homed more directly into the correct approach:

One−half of the pole is in the mud and 1/2 must be above the mud. If 2/3 is in the water, then 2/3 and one foot equals 3 feet, plus the 3 feet in the mud equals 6 feet.

Teach the facts, or the concepts? I’ll go with the concepts every time. Benezet, Correa, and many others have come to similar conclusions: students who understand what they are doing, and can articulate why it works, are much more confident and correct when reasoning about math problems, especially those much-dreaded “word problems.”

The difference is so marked that we should not even be having this debate, nearly 100 years after Benezet’s results – but we still do.