All year long, the most common “proof” students offer has been to do an example with numbers. On “show that the distance between (a,b) and the origin is $\sqrt{a^2+b^2}$, I’ve been getting: “I chose random numbers for a and b, so I used a=4 and b=7. Then I plugged in and got $\sqrt{16+49}=\sqrt{55}$.” No mention of the Pythagorean Theorem we’d been playing with for the past two days.

To my students—most students?—mathematics is “Substitute, then simplify.” It’s all math has ever been. I have been aware of this, but my response has been inadequate.

• Item: I would put two numerical examples on the board, then a third using variables, crowing that “if you do the same thing with only letters, then you’re doing it for all possible numbers at once! Do it once or twice with numbers, then do the same thing with variables! This is called generalization!” Result: I had one student do a numerical example similar to the problem above, then go on to say “I wanted to solve it with variables, so I made 16=U and 49=V, so the answer is $\sqrt{U+V}$.”
• Item: The idea of a derivation is entirely absent. Later in the year I juxtaposed three student examples of finding a distance to show them that the distance formula is simply the Pythagorean theorem. Students were flabbergasted. I got compliments on my lecture. They had never seen anything like it.
• Item: After developing a meaning for the distance between a point and a line, I asked them, “What is the distance between $(x_1, y_1)$ and y=d?” and gave them some time to work on it. They were totally stumped. “Ok, let’s draw a picture.” Blank stares. “What’s up?” “We don’t know what d is—you have to know what number to use!” Ok. Let’s graph y=2. Fine. How ’bout y=-4. Great. What should we do for y=d? Someone, quiet, tentative: “Draw a flat line?” (indicating horizontal with his hand). Does it matter where I put it? No. What about here on the x-axis? No, don’t do that, that makes it look like zero, and it might not be zero.

In my mind, once you do something with specific numbers, it’s just one more small step to do it again with symbols. After all, it’s the same thing. But that’s the rub: to them, it’s not the same thing!. Torigoe (Thanks, Mylène) has got me thinking about what might be going on.

It never occurred to me that carrying symbols through a problem is more demanding than substituting numbers into as many variables as possible, as soon as possible. By stopping up all those leaky variables with numbers, I can stem the flood of confusion behind a nice high dam of something with just one variable to solve for. The structural relationships among whatever I’m modeling (say, distance between objects on a plane) are eliminated, so I don’t have to think about them anymore.

Anymore? At all! This explains everything! It’s why my precalculus students can’t articulate relationships between abstract quantities. They’ve never had to! So they don’t know how.

At least not yet. The technique of “give one specific problem, then a second, then a general one” and hope they make the leap has not worked. So how can I build the capacity to reason with symbols?