The balance between “answer getting” and “mathematical thinking”

So – two similar posts on twitter this morning just a few minutes apart:

David Coffey here:

@mathtans The sooner folks (kids) understand that there isn't always a "right" or "best" answer the less we have to undo, IMO.

— David Coffey (@delta_dc) October 21, 2014

and

Keith Devlin here:

Students in my Math Thinking #MOOC now at the stage of recognizing that there is more to math than getting a right answer. Lot to unlearn.

— Keith Devlin (@profkeithdevlin) October 21, 2014

I struggled with the balance between answer getting and mathematical thinking just this morning with my older son.  I had him working through an example problem in the “angle-angle similarity” section from Art of Problem Solving’s Geometry book.   This is a topic that he’d not seen before, but he seemed to be able to work through the first couple of sample problems reasonably well.  The problem we recorded gave him a little more difficulty, though – particularly when it came to identifying how two similar triangles would match up.

I’m not satisfied with how I interacted with him in this problem and that dissatisfaction has been nagging at me all morning.  I’m not sure what specific thing I should have done better, but I definitely wish I had found a better balance between the answer getting part of this problem and the math ideas.