Wednesday, August 5, 2015

Stop belaboring the concepts: the limits of "conceptual understanding"

In an earlier post, I discussed two mathematical concepts that are easy to grasp in isolation and that therefore shouldn't be belabored ad nauseam: the different multiplicative groupings that produce highly divisible numbers like 24; and the fact that subtraction represents not just removal but measurement differences. The more I think about this, the more similar mathematical concepts I come up with: concepts that are easy to grasp via concrete examples, but often excessively belabored by teachers, delaying the more challenging abstractions and applications of these concepts to more mathematically complex situations.

Such concepts include:

1. The number line
2. fractions
3. negative numbers
4. place value
5. the axioms of arithmetic
6. sets and subsets
7. functions; domain and range
8. slope

What's challenging and interesting about negative numbers, for example, isn't that they represent numbers less than 0 or numbers on a particular side of 0 on the number line, or that they have such concrete instantiations as distances below sea level or temperatures below freezing. What's challenging about negative numbers is grasping that a negative times a negative is a positive and correctly distributing and multiplying out the negative numbers in a complex expression. A class that spends two weeks on ways on which negative numbers correspond to distances relative to sea level is wasting precious time and making students think that negative numbers are boring.

What's challenging and interesting about place value isn't the concept of groups of 10, 100, 1000, etc., or how 123 is 1 hundred plus 2 tens plus 3 ones, but the use of place value by the standard algorithms.

And what's challenging and interesting about sets and functions and slope aren't the concrete examples that teachers, rightly, use to introduce them, but their more mathematically abstract instantiations: for example, the connection between if A then B and A ⊆ B, or the slope of a slope in a non-linear function.

In general, when students struggle to do problems involving these various concepts, the answer is to spend more time, not on the concepts themselves, but on worked examples and practice problems. The best way to get better at math problems, in other words, isn't to spend hours depicting and discussing what a fraction is, what a function is, or multiple ways to multiply numbers, but to do lots of math problems that involve these concepts in mathematically challenging and interesting ways.


lgm said...

I had to explain why a negative times a negative is a positive to my child. His teacher left the definition out. Working more problem sets wouldnt have helped as it would have just resulted in slightly more rote memorization of cases. He needed good teaching to know it and use it down the road. The answer for me is thus a decent textbook, to make up for being stuck in a class of uninterested students who just want a pass; or grouping by instructional need, which would be a politically incorrect honors program. Having neither from the public school, we bought a used Dolciani and the child learned. Khan takes too much time...reading and discussion is faster.

Auntie Ann said...

First thing our kid's 5th grade teacher said when they started the fractions unit was: When I was your age, I thought fractions were really hard.

Anonymous said...

Having a teacher who had to struggle with learning something in school is not necessarily a bad thing as long as he/she has worked through the problem to the point of being an expert.

As a homeschooler, I know that I have the most trouble teaching the subjects I found most intuitively obvious in school.

Auntie Ann said...

But starting a discussion of a topic by telling kids it's really hard, sets their minds against the lesson.

Anonymous said...

Did she happen to finish her thought by saying "but now I see that they are easy, and when we're finished, you're going to think they're easy too"?

Sometimes kids don't report everything the teacher said.

Barry Garelick said...

I had this same conversation (re: teachers saying that they found something difficult) with Sybilla Beckmann, math professor at U of GA and one of the writers of the CC math standards. She felt that it helped motivate students by seeing that they weren't alone, and that even the teacher struggled with it. Really depends on how the message is framed, I suppose. Students may focus on the negative part of the statement to the extent that they fail to hear the happy ending part.

Anonymous said...

When I was a kid, my sister struggled with math. I didn't. (I'm a female.)

When my sister struggled with math, my mother tried to empathize. My mom said, "I really had a hard time with math when I was a kid." My sister wishes my Mom hadn't said that. She internalized it, and my sister still has a negative perception of herself and math.

I never cared whether or not my Mom admitted that she had been bad at math. My reaction to my Mom? "Well, you struggled with math. But I didn't."

We were the same family. Same upbringing. Same gender. And my sister and I had two totally different reactions to the same statement.

In a class of 30 kids with totally different backgrounds, I can imagine how many different reactions there would be to a female teacher admitting that she struggled with math.

Anonymous said...

P.S. I'm a different anonymous than the other anonymous. I posted once on August 6th at 6:12 and also now.

lgm said...

Yes, I can imagine many different responses. As a parent, if I went to open house and the teacher told me they struggled with fractions as a child, I would be expecting to afterschool math the entire year, as that message is a signal that the struggle was over multiple years. That tells me they dont have the ability to understand my child who 'gets' arithmetic intuitively and doesnt need dumbed down nomenclature or a year targeted at getting strugglers and special needs a pass, using such things as the 'nines finger trick' instead of actually leading the children to conceptual understanding.