Last month, I was helping some 6th-graders prepare for their final exam when it became clear that their teacher had utterly failed them. “You poor souls!” I said. “Orphaned by intellectual negligence!”
They blinked hopefully, as children do.
“Who was it?” I demanded. “Who taught you – or should I say, failed to teach you?”
“Um…” they hesitated. “You?”
“Well, whoever it was,” I said, “he denied you the chance to experience the beauty and centrality of the distributive property. For this, he shall have my undying scorn.”
The philosopher Alfred North Whitehead once described European philosophy as “a series of footnotes to Plato.” Twist his arm enough, and perhaps he’d have consented to describe algebra as “a series of footnotes to the distributive property.”
We use it frequently in arithmetic, but rarely name it as such. For example, try calculating 17 x 6 mentally.
One common strategy is this:
Another approach is this:
Both are just applications of the distributive property. You’re exploiting the fact that “a+b” groups is the same as “a groups” + “b groups.”
In fact, to avoid using the distributive property, you’d have to do something a bit unusual, perhaps like this:
Now, my 6th graders are pretty slick at the numerical version of the distributive property. They gobble up problems like this:
But they falter at the same step that haunts students worldwide: converting numerical instincts into algebraic generalities.
The distributive property is like most algebraic facts: just the crystallization of a familiar thought pattern from arithmetic. But that’s not how most students learn it. The easier short-term path is to see it as a new, disconnected rule for manipulating symbols: a(b+c) = ab + ac.
I fear this purely symbolic understanding is a broken futon, liable to collapse under enough strain. In particular, it seems to breed the “everything is linear” mistake. (One of my recurring nightmares.)
It’s crucial to get this right, because (to mangle the words of William Carlos Williams) “so much depends upon the distributive property.” Just look:
Now, most of my colleagues aren’t as worried about this as I am. They see a natural order for learning math: First, become adept at symbolic manipulations. Then, later, come to understand the deeper meaning. Trying to do it all at once – manipulation and meaning from day one – leaves students befuddled, befogged, and belligerent. Better to reach for the big ideas only once you’re comfortable with the mechanics.
They’re not wrong. Lots of effective mathematicians learned their craft like this.
But I can’t figure out how to teach that way. “No black boxes” has long been my motto. Intuition before formalism. Never charge forward with symbolic manipulations until you understand what those symbols actually symbolize.
This approach generally works for me. It means puzzling out notions of area before introducing compact formulas like “A=bh/2.” It means playing around with prime factorization before developing an algorithm for finding a highest common factor. It means baking understanding directly into a student’s thinking, rather than waiting until the cake is settled and then trying to sprinkle a little understanding over the top.
But when it comes to the distributive property, I still don’t have this figured out. Area models don’t seem to do much for my students. My wordy explanations (“it’s seven bags, each containing x + 7!”) are little better. I’m still seeking tasks that can sharpen and hone their thinking about distribution.
But I guess that’s the perpetual state of the teacher: still seeking.
On a different note:
A few weeks ago, the folks at Brilliant.org asked me to help announce their “100 Day Summer Challenge,” a free collection of math puzzles targeted at high schoolers to keep them sharp and math-enthused over the summer. I was about to say “no thanks” (as I do to all such publicity requests) when I realized that the problems are actually quite slick and fun. And what is this blog for, if not peddling addictive stuff to high schoolers?
So here’s the cartoon I drew for them. I encourage you to check it out!