Math-du-jour theories simply don’t add up

Hey there, time traveller!
This article was published 25/06/2013 (4656 days ago), so information in it may no longer be current.

Math support teacher Neil Dempsey (Right idea on math, wrong way to get there, Free Press, June 21) says memorization of math facts and bringing back standard algorithms (vertical addition with a carry; vertical subtraction with a borrow; vertical multiplication and long division) are a step in the wrong direction. This trope is all too common in the educational-consultant class. It arises from wrong-headed educational theories that defy both mathematical and common sense.

Not to mention logic: Dempsey writes “They do need to know their facts and be able to use a method of computation that is efficient and reliable.” And how shall they “know their (math) facts” without actually — you know — memorizing them?

What he’s talking about is teaching kids to perform fast intermediate steps instead of knowing, for example, that 6 + 8 is 14. This ensures, so the theory goes, that students “understand” what they are doing. But what it actually does is establish a lifelong mental habit of cluttering one’s limited working memory with trivia, impairing later processes that involve single-digit arithmetic.

A team of researchers in cognitive science including the renowned Daniel Ansari of the University of Western Ontario, published some relevant findings on this very question in the Journal of Neuroscience this past January. An MRI was used to study brain activity of students performing very elementary arithmetic, enabling them to sort students neatly into two groups. Brains of students in one group showed activity in a region associated with fact retrieval — they were recalling memorized facts. The other group showed activity in a region associated with numerical processing — showing that they were “working out” things such as 4 + 7 or 6 x 8.

The same students were later tested in the much more advanced material in the math portion of the Grade 10 PSAT test. The group employing automatic recall of elementary facts significantly outperformed those carrying out intermediate steps.

Other studies in cognitive science show similar improvements in efficiency as early as Grade 2. There is no mystery here: It’s all about whether students are steered toward economy of thought and given the right foundation for later, more demanding tasks.

Obsessing over the details of single-digit arithmetic for several years without closure or mastery is exactly the wrong way to teach. Most teachers and parents grasp this instinctively, and one wonders why expensive research is needed to prove such an obvious point. But Dempsey and much of the consultancy class, whose math-du-jour theories fester within their echo chamber, presume to have transcended common wisdom.

Similarly, standard algorithms support a conceptual framework for arithmetic and later algebra. They are building blocks for later success, particularly in technical studies for certification in many professions. Avoiding these procedures is analogous to forbidding an entire food group. Behold the Atkins Diet of math education.

Members of the education establishment have explained the algorithms had to go because they were being taught “without understanding.” Talk about throwing out the baby with the bathwater. Instead of disparaging canonical, elegant and efficient procedures, why not help teachers teach them with understanding, as they ought to be taught, and as the best teachers have always done? This is a matter of helping teachers do their job. Why make life so difficult for everyone?

Dempsey says “Finland and Japan move away from memorization of math facts.” Really? I would be interested in seeing evidence to that effect. Even if it were true, this is not the case in other, even higher-performing countries. Singapore, Shanghai and Korea are the best in the world. Singapore’s model curriculum requires memorization of the times tables by the end of Grade 3, mastery of the standard algorithms for whole-number arithmetic by the end of Grade 4, and of fractional and decimal arithmetic by the end of Grade 6. Quebec, which outperforms all other Canadian provinces, requires memorization of multiplication tables (0 to10) by the end of Grade 4 and mastery of associated division facts in Grade 5.

Manitoba parents can now breathe a sigh of relief. Times-table memorization is now required. Expect no more stories of parents being told they are harming their children by teaching the algorithms at home. In the past, we’ve been told things such as “If the students learn the algorithms too early they will prefer them over the ‘strategies’ we’re trying to teach.” Well, obviously. This is, indeed, the point: Who would not prefer well-defined, elegant and efficient methods over ad-hoc tricks that only work in special cases, or only with small numbers? And they are not, as some insist, “hard to learn,” though apparently they can be poorly taught. Our children are not stupid; let’s teach them the best approaches, and provide teachers with resources to do so effectively.

The U.K. has already made this move after deciding their experiment with so-called reform math was a failure. Most U.S. states have embraced the algorithms again after a decade of this madness. It’s high time we got with the program, and it’s good to see our Department of Education paying attention to the counsel of common sense and the best available research.

Many things remain to be improved, but in our view, this is a step in the right direction for which the provincial government should be commended.

 

Robert Craigen is an associate professor of mathematics at the University of Manitoba and a co-founder of WISE Math, online at wisemath.org.