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 
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
.”
- 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
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?
[...] to spread through the educator blogosphere. Dan Goldner, in his blog Work in Pencil, wrote a post Substitute, then simplify, which discusses the insights he has gotten into his own teaching: It never occurred to me that [...]
Dan
I wrote this in a comment on Dan Meyer’s blog and it is even more appropriate here in light of your remarks. I teach an Honors Calculus class and one of my students – a high achieving junior – was struggling to do what I was asking her to do. She told me, in a fit of frustration, that she thought that math was just knowing what formula to use, remembering the formula and accurately plugging numbers into it. I have been fighting this fight for most of my 25 years of teaching and expect to continue fighting it for the next 20 or so. We HAVE to be able to convince our students that the process is more powerful than isolated conclusions that they might be able to accurately draw in specific situations. One of my favorite examples – along the lines of your y = d example – is to ask precalculus students for solutions to the equation px^2 + qx + r = 0. The answers I get make me want to give up.
Jim – I know what you mean by “fit of frustration”. I had a student (a senior!) move from smiles to near tears almost instantly when I started calling a point (a,b) on her paper when she could plainly see it was (3,4). “Well, what if we didn’t know that?” “But we *do* know that!” she said. I backed off because her point of view was legitimate and I didn’t want to turn math into nonsense for her. So I had set it up wrong – the question I was asking wasn’t motivated by the information she had.
One baby step I’ve had them take is to repeat a proof I’ve shown the class, but with the names of the variable / parameters changed.
In beginning algebra and intermediate algebra, I love showing my students how the quadratic formula is derived. I tell them they won’t be tested on it, to get their anxiety levels down, go through it very slowly, asking them for lots of help, during one class. Then, during the next class, I get them to tell me all the steps. The whole thing has worked much better since I switched to James Tanton’s lovely method for completing the square.
I saw the same thing you’re talking about in my Calc II course, when I asked them how we’d prove the Pythagorean formula. I saw it early in the semester just past, and determined to work on that with them, but felt a bit stumped.
I’ll be teaching Linear Algebra this coming semester (for the first time in 10 years). I’m looking forward to it, and thinking a lot about logic, and how I might want to teach it. I hope I can do it justice.
Thanks, Sue – What your comment brings up for me is that I think a lot of this hinges on how to model the inquiry process for them without taking away the chance for them to do it themselves. I was at some Math Circle demos at the AMS/MAA Joint Meeting and saw the Kaplans and James (speaking of James!) both do the facilitated-group-discussion-math-circle-thing I know you’re familiar with, and they all left open questions. Then last week in one class at school we were discussing whether the interior angles of a triangle add to 180 degrees. “Yes, because our teachers said so” was the best proof I was getting. So we started talking about it and I started scribing Math-Circle style, and we got two ideas on the board that didn’t work. I then let the class go to independent work, and everyone dropped it … except for two students who went on to come up with some cool ideas on their own.
What were their cool ideas? It’s exciting to me when math circle approaches work in the classroom. I’d love to hear more.
One or two drew some lines that, after some conversation, ended up being parallel to the base of the triangle so they could use alternate interior angles. Another is working on a strategy starting with a 45-45-90 triangle and extending all lines to show that if he changes one of the 45s to 44 degrees, the other one must become 46.
[...] Cf. Substitute, then simplify. [...]