What are the actual advantages to making the student represent this symbolically as x — 20 = x — .2x?

For students comfortable with that level of abstraction, the equation is faster than guessing and checking for the break-even point.

And I think someone like Bret Victor is pretty satisfied with this student’s response. Are you?

I’m not sure about that.

As I read Victor’s treatise on abstraction, my sense is he would create a 2D visualization of the break-even point for every value n (where in this case n = 20). Then he’d use that visual as a club for whacking teachers. “Why are you teaching abstraction?” Imaginary Bret Victor would say. “Look at the power of the visual!”

Obscured by his technical artistry, though, is the fact that the visualization wouldn’t have been possible if he hadn’t already known how to use the abstraction.

So I’m thinking of a kid who gets it, and says something like:

“At $100, they’re the same. Anything less that that, go with $20 off, anything more go with 20% off.”

And I think someone like Bret Victor is pretty satisfied with this student’s response. Are you?

What are the actual advantages to making the student represent this symbolically as x – 20 = x – .2x?

You’ve written before (and quoted Einstein I think?) that the formulating of a problem can more important than the solution to it. I wonder, though, about the kid who first solves this problem, next generalizes (“$100 is always the break-even point”), but gets stuck at modeling it algebraically and even more stuck trying to manipulate whatever it is he’s modeled.

You need to be careful in your use of “small” and “large.” An iPod is small (yet expensive) compared to a large bouncy ball (inexpensive).