Count the Tropes, Dept.

Yet another in an unending series of articles about math education and what parents should (and should not) be doing to help.  This particular one is from Chicago Parent and contains the usual tropes/mischaracterization about how math used to be taught and why the new ways are so much better.  We start with the usual ever-popular one:

STOP TEACHING THE TRICKS: A large amount of research has gone into the progression of teaching students mathematical concepts. The shift has moved away from teaching students to blindly follow rules and toward making sure they understand the larger mathematical ideas and reasoning behind the processes.

I have written extensively about how math was taught in the past, citing and providing examples from textbooks from various eras. It might be interesting to look at some of the books used in previous eras that have been described as teaching students to “blindly follow rules”.  Many, if not most, of the math books from the 30’s through part of the 60’s were written by the math reformers of those times.  It makes the most sense to start with the series I had in elementary school: Arithmetic We Need.  The reason is because not only is it from the 50’s, but also one of the authors was William A. Brownell, considered a leader of the math reform movement from the 30’s through the early 60’s. Today’s reformers also hold Brownell in high regard, including the prolific education critic Alfie Kohn, who talks about him in his book The Schools Our Children Deserve .

In arguing why traditional math is ineffective, Kohn states “students may memorize the fact that 0.4 = 4/10, or successfully follow a recipe to solve for x, but the traditional approach leaves them clueless about the significance of what they’re doing.  Without any feel for the bigger picture, they tend to plug in numbers mechanically as they follow the technique they’ve learned.”  He then turns to Brownell to bolster his argument that students under traditional math were not successful in quantitative thinking: “[For that] one needs a fund of meanings, not a myriad of ‘automatic responses’. . . . Drill does not develop meanings.  Repetition does not lead to understandings.”

Brownell, however, requires students to do the practice and exercises held in disdain by those who believe traditionally taught math did not work.  The series contained many exercises and drills including mental math exercises. Such drills might appear to run counter to Brownell’s arguments for math being more than computation and “meaningless drills,” but their inclusion ensured that mastery of math facts and basic procedures was not lost. Also, the books contained many word problems that demonstrated how the various math concepts and procedures are used to solve a variety of problem types. Other books from previous eras were also similarly written—most authors were the math reformers of their day—and provide many counter-examples to the mischaracterization that traditional math consisted only of disconnected ideas, rote memorization, and no understanding.

But let’s move on. It gets worse:

…By teaching the trick before a child has this foundation, you may be inadvertently doing more harm than good. Students become reliant on tricks and fail to master the conceptual understanding needed to use the tricks appropriately.  Remember, kids will be more successful in the future as a problem-solver than as a memorizer.

What this article and many look-alikes caricature as “tricks” are actually mathematically sound algorithms. The idea that teaching standard algorithms “too early” eclipses the underlying conceptual understanding most likely stems from Constance Kamii’s infamous study “The Harmful Effect of Teaching Algorithms to Young Children” which was published by the National Council of Teachers of Mathematics (NCTM) in an annual review.  It has become the rallying cry that has garnered more believers than the idea that the substance called “laetrile”, extracted from apricot pits, is a cure for cancer.

With respect to the math books of earlier eras, they started with teaching of the standard algorithm first.  Alternatives to the standards using drawings or other techniques were given afterwards to provide further information on how and why the algorithm worked.  This is opposite of how reformers are advising it be done now.  What happens, typically, is the first way a child is taught to do something becomes their anchor, with everything else being supplemental.  By teaching the supplements first, there is a mix-up of main course versus side dish, with many students unable to tell the difference.  The popular theory is that students now have a choice and can pick the method that works for them.  I have tutored students showing profound confusion, asking me what method they should use for particular problems, feeling that various problems demand different versions of the same algorithm.

Note also the ever-popular warning against memorization: Memorizers are not problem solvers apparently. Well, sure, there may have been teachers who taught math poorly and had students memorize day in and day out with no conceptual context. To listen to the people who write these articles, it seems that the nation was plagued with such teaching, as if poor teaching was/is an inherent quality of traditionally taught math.  I would argue otherwise and go so far as to say that many of the “understanding-based”, student-centered, collaborative techniques that dominate many of today’s classrooms are inherently ineffective and damaging.

Memorization is the seat of knowledge. Eventually students just have to know certain facts and procedures and do them automatically. The idea that memorizing eclipses the understanding of what, say, multiplication is presumes that students are taught the times tables with no connection to what multiplication is, and what types of problems are solved using it.

Other than that, the article is pretty good, I suppose.

 

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