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

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

David Coffey here:


Keith Devlin here:

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.

2 thoughts on “The balance between “answer getting” and “mathematical thinking”

  1. Hi Mike,

    I suppose you could suggest a rotation of the small triangle 90 degrees on Point D and then determine the scalar required to expand that triangle to match the second largest triangle. A more complex transformation of the small triangle would put it “in line” with the largest triangle; an appropriate scalar for expansion could then be determined for that as well.
    I think that would have some value for transformations he sees later on.


  2. Hello Mike,

    I liked the way, your son was making a sincere attempt to think/persevere till the end. Of course, credit goes to the teacher for sowing the right seeds, and properly 🙂

    Yes, I too feel something went missing from your side — twice….But I am very happy that you reflect upon this and do sense/feel this 🙂 First time – when he erred on figuring out the correspondence (though he had written it correctly — but, after looking at his explanation later, I even doubt if the statement he made in the beginning reflected his clear understanding.. I would not leave any ambiguity and try to verify/ensure this…) And secondly when he had seen some interesting relations with the BIG triangle… This could have been nicely taken forward, though this problem did not demanded it.

    Geometry directly on the board is really confusing at times. So I would in such cases when students struggle and are really unable to visualize/get through, would switch to concrete, at least for some time. How about giving him the paper cutout of these triangles and then encouraging him to figure out/ write/ explain the expansion/ correspondence? I am sure this experience would go deeper, help him understand the nitty-gritties of smaller and longer legs of right angled triangles in case of similarities.

    And I loved the way he pointed out the factor 4/3 without writing the ratios of corresponding sides…

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