Teaching Arithmetic

I think it’s generally bad form to link to items that don’t deserve additional publicity, but this video opened my eyes to some real dangers for mathematics education.

In the video, a “meteorologist” for Seattle’s fourth-place local TV station makes the case that she knows much better than the faceless mass of educators who write math books. In particular, she decries the lack of emphasis on the traditional paper-and-pencil algorithms for long division and multiplication of large numbers.

There’s a reasonable argument that there was a time (at least 30 years back) when such paper-and-pencil calculations were important, but anyone who thinks that they are still relevant is completely out of touch with the modern world. Today there are two relevant types of arithmetic: mental arithmetic and automated calculation (using something with a microchip in it). Beyond issues of relevance, the teaching of arithmetic can serve as an opportunity to illustrate basic mathematical principles of symbolic representation, algorithms, and problem solving.

If anything, I think the curricula denounced in the video do not go far enough in discarding irrelevant paper-and-pencil algorithms. I’d much rather have students spend their time learning advanced mental arithmetic, displayed in surprisingly entertaining form in this TED talk by Arthur Benjamin. The issue is that what is efficient on paper is not necessarily efficient without it—mental arithmetic is limited primarily by the size of intermediate results which must be remembered. In complexity jargon, this means that good mental algorithms must have very low “space complexity”. Most mental arithmetic algorithms also have the advantage of working from left to right, so errors introduced along the way affect only the least significant digits. Students are much better off knowing that multiplying together two three-digit numbers will get you a result between 10,000 and 1,000,000 (and the first digit or two of the exact value) than needing to grope for pencil and paper before they have any estimate whatsoever of the answer.

As for the basic mathematical principles illustrated through arithmetic, I would also much rather have students tinker with a bit of light algebra to break down a division problem than teach them to apply a rigid algorithm. Expertise with this type of tinkering (dependent upon an understanding of what types of tinkering are permissible and why) is by far the most import skill I gained from my entire formal mathematics education.

As a final dig, anyone who announces (and circles) “22 remainder 1” as some kind of free-standing mathematical entity should stay well away from mathematics education. I realize this includes quite a few elementary school teachers, and I stand by my statement.