Trying to be Charitable, Dept.

I came across this article from over a year ago which appeared in the N.Y. Times about how we are not teaching students to solve math problems the right way.
In it, the author features a quote from an email she received from Tracy Zager, an author of math ed books:

“I really feel I have no way to have an impact on this teacher’s blind spot since it is shared by all math teachers and so many other teachers: If you don’t understand, it’s your fault. It was never a sensible idea to try to have students memorize first and understand later; this approach to mathematics instruction is structurally flawed. I really feel for these parents and this kid, but the frustration they face is inevitable. If we teach kids math without understanding, we build on a house of cards.”

I try to be charitable on Christmas and ask myself what would Jesus do if he read something like this. I have to say, my first reaction is He would say “I don’t have time for this.”
I looked at the comments from a year ago and was pleased to see that I addressed it already. So I’ll repeat what I wrote in comment here so you can all rest easy and have a nice Christmas:

The email quoted at the beginning of the article states: ‘I really feel I have no way to have an impact on this teacher’s blind spot since it is shared by all math teachers and so many other teachers: If you don’t understand, it’s your fault.’

I do not agree with the generalization that ALL math teachers blame poor learning outcomes on students. I certainly do not, nor do many math teachers I know.
I also find that Tracy Zager [the person who wrote the email quoted in the article] gives a typical mischaracterization as to how math is taught; i.e., that students memorize first and understand later. A glance at math texts from previous eras shows that there were explicit explanations of what is happening with specific procedures and algorithms. Also sometimes understanding comes before the procedure, sometimes after. And for some students, depending on the procedure, it might occur many years later.

Traditional math taught poorly continues to be the definition for traditionally taught math–the mischaracterization of traditionally taught math prevails. For the record, I teach the conceptual context of particular procedures, as well as what’s going on in various types of word problems and how to analyze them mathematically. But  despite “teaching the concepts” students still gravitate to the procedural because they want–and need–to know “how to do the problems”.  Who can blame them? After time, some understanding of why certain procedures does occur. And for some students, it may or may not. Case in point: I will demonstrate what a negative exponent means; i.e., x²/x³ can be represented by x*x/x*x*x, which simplifies to 1/x. We can also apply the rule of quotients of powers and subtract exponents: we then get x-¹. But after a day or so, students have forgotten this and have to be reminded of the procedure. After about a week or so, it begins to sink in. But the “understanding” part while helping some students certainly doesn’t help all.

In the words of the teachers at the no excuses school Michaela in London which takes a traditional approach to teaching: “Just tell them”. And sometimes that means telling them a few times. It doesn’t mean skimp on the understanding. It means do both. I have practiced this philosophy for years as well as other teachers I know. If the only way students can do a problem is to practice the procedure, then go with it. Stop whining that “They don’t truly understand”.
I majored in math, but didn’t understand why the invert and multiply rule for fractional division worked until 10 years ago. So shoot me.


3 thoughts on “Trying to be Charitable, Dept.

  1. Many things i learned in high school, then university, only truly began to sink in a few years ago, and I’m 50. Did my teachers fail in teaching me understanding along with ensuring I memorize facts? No. I was blessed to have truly exceptional teachers, ensuring that they instil knowledge on me, which they knew would help kickstart the understanding along the way. Both are intrinsically linked; we cannot separate the 2 if the first step is done right, i.e. present the information and just do the work.

    For all this discussion surrounding empowering kids, and having them compete on a global stage, it boggles the senses then how so many believe they must talk down at kids, and treat them as if they cannot comprehend what’s going on. EVERYTHING these days MUST BE explained to kids: think about that for a moment. Did our parents, or our grandparents have a lengthy explanation every time we messed up? Or showered us with endless praise on the rare occasion we did something right? Maybe they were on to something. Sometimes less is more; learning is the same way.

    Just do the work. The understanding comes through that. Too bad so many grown ups continue to ignore this and continue to foolishly believe that “they” are the experts and continue to dumb down our kids by their own foolish behaviour.


    • Thanks for your comment. I think there is a large degree of mischaracterization of traditionally taught math, so that it is thought of as not teaching any understanding at all. As I’ve discussed–and demonstrated with text books from previous eras–this was certainly not the case. Traditionally taught math done poorly should not be the proxy or definition of traditionally taught math. On the other hand, the prevalent belief of math reformers that if students understand, then there is less need for memorization is only partly right. Sometimes understanding is part and parcel to the procedure (for example multidigit addition and subtraction, or multidigit multiplication) and therefore is an aid in executing the procedure. In other cases, the underlying concept is more abstract. As I discussed above, many teachers do in fact teach the concept/understanding. Some students get it immediately, some later, some never. The last two are generally taken as evidence that traditionally taught math is a failure.

      I often hear those who embrace the philosophy that we are teaching math wrong that “I wish I had been taught this way”. They are referring to the diagrams and alternative methods that are taught in lieu of standard algorithms in the belief that the standard algorithms are “rote learning” and eclipse “deeper understanding”. What they fail to see is that they are viewing such methods through the lens of an adult who has achieved some degree of mathematical maturity, which has not yet developed in many students in the lower grades. And for some of the people exhorting the philosophy of “understanding”, the fact that they now can see these concepts is due in part to the underlying traditional approach to math that they may have received.

      The proponents of the understanding philosophy may in fact be inflicting harm on many students. “Math taught poorly” may therefore be inherent to the alternatives to traditionally taught math that place such over-emphasis on understanding. And as I’ve said many times before, the successes that such proponents claim as a result of their methods may be due to other influences, such as outside tutoring, or help from parents.


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