Comments on: Take what you don’t know…

By: Inquiry standards for math « Work in Pencil

Inquiry standards for math « Work in Pencil — Mon, 19 Jul 2010 02:36:34 +0000

[...] (or, as Sam says, “take what you don’t know and turn it in to what you do know.” Spans connecting [...]

By: Math Teachers at Play #5 « Let’s Play Math!

Math Teachers at Play #5 « Let’s Play Math! — Fri, 17 Apr 2009 11:17:14 +0000

[...] Sam Shah offers a valuable problem-solving mantra in Take what you don’t know…. [...]

By: samjshah

samjshah — Sat, 11 Apr 2009 05:18:30 +0000

Thanks y’all. It seems like my feeling about how I could frame what we do resonated with you — which means I’m probably doing something right. I’d love to take a few days this summer and see if I could map out the Alg II or Calculus curriculum based on this one premise. Can we see a direct line, for example, between solving and what we did earlier in the year? I think I’d argue yes. A basic lineage would look like:

complicated integral –> simpler integral via u-substitution –> integration as antidifferentiation –> differentiation –> limits.

Do they see how much they’ve done — taking what they don’t know and turning it into something they do know? I have to highlight this more – make it more fundamental to how I frame what I’m teaching.

@RobertTalbert: point (“Knowing what methods are true and which ones are false, and which ones work better in various situations, is a precondition for understanding process”) definitely taken.

By: David Cox

David Cox — Thu, 09 Apr 2009 16:43:52 +0000

I love this. If we do a good enough job of scaffolding the material students would naturally see this. Man, if it were just that easy!

By: Kate

Kate — Thu, 09 Apr 2009 14:27:37 +0000

You are channeling the master! Your post reminded me so much of this passage that I had to go look it up:

The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems. Thus, it is often appropriate to start the work with the question: Do you know a related problem? If we succeed in recalling a formerly solved problem which is closely related to our present problem, we are lucky. We should try to desrve such luck; we may deserve it by exploiting it. Here is a problem related to yours and solved before. Could you use it? - George Polya, How to Solve It

By: Robert Talbert

Robert Talbert — Thu, 09 Apr 2009 13:23:59 +0000

I totally agree with you — the essence of problem solving is fully understanding the unknown in the problem and exploiting all connections to the known.

The one possible (probable?) pedagogical issue here is that students do tend to lock in on methods rather than processes at any available opportunity, and framing math problem solving as turning one thing into another will sound temptingly methodical to students. When you ask them to solve log(2x+1) + log(x-1) = 2 and say “turn it in to what you do know”, what may students will start doing is trying anything that looks like a method that could possibly bring a quick end to the problem. You know what will happen: The student will say something like:

log(2x+1) + log(x-1) = 2 -> log((2x+1) + (x-1)) = 2 [factoring out log]
-> log(3x) = 2
-> 3x = 2/log [Has the pain started yet?]
Therefore x = 2/(3 log).

Then the student will say, “Why are you throwing a fit at me? I’m doing just what you said — turning it into to something I do know!”

So I think you’re right, mathematical problem solving has to be thought of more in terms of process than method — but method is and will always be really central to understanding the process. Knowing what methods are true and which ones are false, and which ones work better in various situations, is a precondition for understanding process.

By: Adam Glesser

Adam Glesser — Thu, 09 Apr 2009 13:19:58 +0000

Brilliant post. You should send a condensed version into the I Want To Teach Forever blog’s “52 Teachers, 52 Lessons” contest. It is awfully inspiring.

By: doug

doug — Thu, 09 Apr 2009 12:16:50 +0000

I also find myself saying similar things to my classes. I think you are dead on when you say “It’s how we solve math problems”. During a calculus review the other day, I told the students that the main difference between people who struggle with a topic and those who don’t is not usually a matter of the latter group knowing more rules or formulas or what not. Instead, it’s that the people in the second group are much more adept at recognizing how they can turn a seemingly difficult problem into one they know how to solve.

By: Matt E

Matt E — Thu, 09 Apr 2009 11:09:44 +0000

Yes, yes, and hell yes.

I haven’t gotten it boiled down to mantra-size yet, but I do say this often to my classes. Come to think of it, I remember when I interviewed for my current position, I taught a Geometry class the Law of Sines, and I did exactly this. “You’ve only been using trig functions with right triangles. THIS is not a right triangle, so we can’t use trig on it as-is. But perhaps we could CREATE some right triangles…” I think I even may have said something like, “This is what we do in math, we take a problem we don’t know how to do, and turn it into one we DO know how to do.”

So you can imagine how this post resonated with me!