In my last “smart and thoughtful post” (the parlance that edu-pundits use when they refer to each other’s writings), I talked about “understanding vs procedure”. The quote at the end from a teacher in New Zealand seemed to ruffle the feathers of some who took to Twitter to state that they believed otherwise.
In all these discussions of “understanding”, those who believe it is not taught and that students are doing math without knowing math rarely if ever explain what they mean by understanding in terms of how it translates or transfers to problem variations or new areas in math. For example, a student who has learned the invert and multiply rule for fractional division may not be able to explain why the rule works, but may have an understanding of what fractional division represents. The student then uses the latter to solve problems requiring fractional division.
Anna Stokke, a math professor at University of Winnipeg has also addressed the issue of student understanding in math and echoes what the teacher in New Zealand said. She has kindly given me permission to quote her:
When we teach, most of us generally do teach students why things are true but I sure don’t want my students going through the understanding piece every time they solve a problem. What a waste of time! The point, I think, to get across to students is that there is a reason why everything in math works the way it does and you could figure this out if you need to (because you WILL almost certainly forget).
With the consultants I’ve met, who always push this stuff and insist that kids aren’t fluent unless they can explain everything to you, it seems that they themselves just figured out that there are reasons behind procedures in math as adults. Then they’re angry that their teachers (supposedly) didn’t explain all these things to them. They’re certain that they would have liked math more and done better if only their teachers would have focused on understanding. So, their mission is to make sure that all kids are forced to explain their thinking at every step. Pure torture, really.
Funny thing is, that the understanding piece is a lot more difficult for students. They generally don’t like it and it’s something that really comes with much experience and mathematical maturity. It won’t make students like math more if we spend more time on understanding…it will just confuse and frustrate them more. In my experience, I’ve found that students like step-by-step procedures and algorithms more than anything else.
What people in the “understanding uber alles” crowd likely mean when they talk about understanding probably has to do with words. They would probably be happy with words that didn’t ensure that the kids could actually DO the problems: a “rote understanding”.