My long-time math crony Barry Garelick has recently alerted me to claims about how students don’t understand the “equals” sign and therefore require lessons on the underlying concept of mathematical equality.
(These claims are based on studies that potentially confuse conceptual understanding of the equals sign with the ability to do basic arithmetic or involve additional confounds like the effects of tutoring.)
The notion that teachers should be devoting class time to conceptual understanding of the equals sign reminds me of this old post from Out in Left Field.
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:
- The number line
- Negative numbers
- Place value
- The axioms of arithmetic
- Sets and subsets
- Functions; domain and range
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 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.