These pieces originally appeared as a weekly column entitled “Lessons” in *The New York Times* between 1999 and 2003.

[ THIS ARTICLE FIRST APPEARED IN *THE NEW YORK TI*MES ON JUNE 27, 2001 ]

## To peace on math’s battlefield

Few disputes in education are as bitter as those between back-to-basics and teach-for- understanding factions in math. Each blames the other for low scores. Each complains that it has been caricatured by the other.

The back-to-basics group wants more memorizing and practicing rules for multi-digit addition and subtraction, long division and multiplication. But most proponents of basics also say pupils should know how arithmetic applies to real situations.

The teach-for-understanding side wants more exploration of math problems so children can develop their own insights for solving them. But most proponents of understanding also say pupils should learn the conventional rules.

Yet sometimes when this debate filters down to classrooms, the caricatures become all too real. Many teachers simply drill students in procedures that are soon forgotten. Others expect children to invent techniques that civilization took eons to evolve. Unsophisticated followers of each camp share the blame for poor math performance.

Into this morass has come a little book, “Knowing and Teaching Elementary Mathematics” (Lawrence Erlbaum, 1999), by Liping Ma, a Carnegie Foundation researcher. Both sides in the math wars claim Dr. Ma as their own. Districts have distributed her book to teachers. Its broad appeal offers some hope for common ground in math education.

Dr. Ma says that, yes, children should learn to apply arithmetic rules fluently, as back-to-basics crusaders urge. But, she adds, Americans often teach the procedures poorly. Students lack proficiency not from inadequate drill but because too many teachers themselves do not understand the mathematical principles behind the rules.

This won’t be solved by hiring teachers with more advanced college math credits. Nor will it do simply to demand more pure pedagogical training. Rather, elementary-school teachers need deeper understanding of the superficially simple arithmetic they cover.

Consider a subtraction problem taught to second graders:

62 5612

-49 done like this: -49

13 13

Most teachers explain: Borrow a 1 from the tens’ place, leaving only 5 tens; then write the borrowed 1 next to the ones’ place to make a 12. Pupils can memorize this method, practice and become proficient in it. But with no proper theory underlying the gimmick, children do not learn why they should use it, and develop no foundation for higher mathematics.

Second-grade teachers may engage pupils’ interest with an ill-considered metaphor, for example saying that the digits on top in two-digit subtraction are like neighbors, one of whom goes next door to borrow some sugar. Dr. Ma notes that “this arbitrary explanation doesn’t contain any real mathematical meaning.” Worse, it misleads by suggesting that the 6 and the 2 in 62 are independent numbers, not two parts of one number.

Dr. Ma contrasts this with how teachers in her native Shanghai typically handle the problem. There, the metaphor of borrowing was abandoned in a 1970’s math reform. Instead, Chinese teachers speak of breaking down a higher number, explaining that the 6 in the tens’ place is actually made up of 60 ones. The number 62 can be regrouped in many ways: 60 and 2 is the same as 50 and 12, 40 and 22, etc.

After this explanation, children can learn the mechanics — putting a line through the 6 and writing 5, putting a small 1 before the 2 to create 12 — in a way that makes mathematical sense.

The difference between borrowing and regrouping may seem small. But regrouping numbers is a basis of higher math (like factoring in algebra). Second graders taught regrouping will understand arithmetic well enough to proceed to more advanced topics. Second graders drilled in borrowing may never make it to algebra.

Dr. Ma’s most shocking conclusion is that most American schools don’t teach mathematical foundations of arithmetic because teachers themselves weren’t taught those principles. Pupils are shown only what teachers know: to do operations by rote, using tricks (like borrowing sugar) to help remember rules.

The solution is not, as some think, to hire teachers who had more college math. The American teachers Dr. Ma observed took more math courses than the Chinese who began teacher preparation after ninth grade.

Rather, elementary schools and teacher colleges alike must offer deeper understandings of basic math. And, Dr. Ma says, teachers need more common planning time to discuss arithmetic lessons and how children comprehend them.

Without such reforms, we will continue fights over whether children should be taught arithmetic rules or theory. What Dr. Ma shows is that we need both.