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.