A brief note on the existence and the importance of standard algorithms for mental arithmetic. This was a contribution to a discussion on the Kto15 list in August, 2001.

See the Education Page of Bas Braams - Links, Articles, Essays, and Opinions on K-12 Education - for related matter.

Dear Colleagues,

One should be careful in these discussions about standard algorithms. There are well-established standard algorithms for paper-and-pencil arithmetic. There are different standard algorithms for mental arithmetic, and in mental arithmetic there is also much scope for nonstandard shortcuts. Pupils should learn both paper-and-pencil and mental arithmetic.

The standard algorithms for paper-and-pencil addition, subtraction and multiplication operate right to left; one column at a time in the case of addition of many numbers. The standard algorithms for mental arithmetic operate left to right. The standard algorithm for division operates the same way for paper-and-pencil and for mental arithmetic; the two cases diverge only when it comes to the multiplication step that is a component of the division algorithm.

The example in the text quoted by Elizabeth is a small problem, 3*36. Pupils should be able to apply the standard algorithm of mental arithmetic and think to themselves "90, plus 18, makes 108". They should know the multiplication tables by heart, so as not to forget the 90 while they are figuring out 3*6=18. In the early stages of their training they should also do examples such as 3*36 by standard paper-and-pencil arithmetic, working right to left, but they should develop to the stage where 3*36 is within their facilities for mental arithmetic.

If the example were instead, 3*39, then the pupil might use the standard algorithm for mental arithmetic and think "90, 27, makes 117", but a superior thought process is for the student to think "120-3=117".

So there are multiple algorithms for arithmetic, and students should be acquainted with these multiple algorithms and should have facility in both paper-and-pencil and mental computation. The algorithms can and should be taught and practiced.

There is nothing fuzzy about this. The fuzzy confusion sets in only when it is said that students should discover their own algorithms, or when it is said or implied that the multiple algorithms are equally good, or that students can only understand well what they have discovered for themselves.

Let's be clear also that there are, in principle, many algorithms for paper-and-pencil arithmetic. One can add a column of numbers working left to right, or working on one pair of numbers at a time. But the right to left column method is clearly superior, and should be taught as the standard algorithm.

Bas

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Bastiaan J. Braams
- braams@math.nyu.edu

Courant Institute, New York University

251 Mercer Street, New York, NY 10012-1185