About a year ago, Elizabeth Green published a piece in the New York Times magazine entitled Why Do Americans Stink at Math? Just a couple of months ago, the Notices of the American Math Society published some reactions to her piece. And I, in turn, have some reactions to what's in some of these reactions--in particle to Bill Jacob's discussion of how teachers should promote conceptual understanding:
Imagine a third-grade class being asked, "How many legs on there on three spiders?" Children who draw three spiders may first count the legs, but the context can elicit many strategies. Three groups of eight legs can e viewed as six groups of four when four legs are drawn on each side of a spider and viewed as a unit. A row of three spiders could be viewed as having two rows of twelve legs (top and bottom), or the legs could be counted as twelve pairs. A skilled teacher can pull from various groupings of the legs a spatial understanding of why the equivalence of 8 × 3 = 4 × 6 = 2 × 12 = 12 × 2 arises, beyond merely having the same value.Jacobs also writes about the virtues of emphasizing how subtraction can be thought of as "difference on a number line" so that later students will understand why slope can be expressed as (y1 - y0) / (x1 - x0):
Learners who only understand subtraction as removal and not as difference in a measurement context will miss the meaning of (y1 - y0) and (x1 - x0) in the slope expression.Both of these concepts--the different multiplicative groupings that produce highly divisible numbers like 24; and the fact that subtraction represents not just removal but measurement differences--strike me as relatively easy to grasp in isolation, even for third graders. In other words, they seem like the kinds of concepts that, like function and slope, are relatively easy to understand in and of themselves, especially when presented concretely, as Jacobs is advocating.
What's challenging about representing slope as (y1 - y0) / (x1 - x0), I'm pretty sure, isn't the concept of subtraction as measurement difference, but the symbolic abstraction involved in the algebraic expression.
And what's challenging about different multiplicative equivalences like 8 × 3 = 4 × 6 = 2 × 12 = 12 × 2 is remembering that you can use them to as tools to quickly simplify complex expressions.
These and other concepts are like some of the simple building blocks of computer programming--like div and mod and lists and arrays--or, for that matter, the simple building blocks of engineering--like levers and pulleys and valves and gears. All these are simple concepts in and of themselves, but very powerful tools for solving complex problems--especially when understood abstractly. The challenge is to figure out when and how to use them.