Cultivating original thinking

An interesting thing happened today.

A girl I was working with, who’s had virtually no schooling in geometry, made a brilliant “leap” in a geometry problem.

This student was starting a unit on fractions, and the curriculum was describing a standard way of conceptualizing fractions. The book was showing that if you divide a given area into a number of equal-sized pieces, and if you shade some of those pieces, the ratio of shaded pieces to the whole area offers one way of conceptualizing a fraction. To kick things off, the book asked students simply to split a rectangle into four equal pieces.

I was expecting my student, Karyn, to split the rectangle up like this:

Rectangle Split-Up

Rectangle Split #1

Instead she split it up like this:

Another way to split up a rectangle into 4 equal parts

Rectangle Split #2


I was immediately intrigued, for while I sensed that all four of these triangles had the same area, I didn’t immediately see how to prove this.  Of course I knew that triangles A and C were congruent (by ASA), as were triangles B and D. But that didn’t help me prove that A and B have equal areas, or the same for B and C, C and D, or D and A. I mentioned this question to Karyn.

As I was pondering this, Karyn took out her pencil and started making marks on her diagram. In just a few moments, she showed me her sheet of paper and said, “Does this work?”

When I looked at her diagram, I saw that Karyn had made a vertical line and a horizontal line, thereby dividing the entire rectangle into 8 little triangles, all of which were clearly congruent. Her diagram looked like this:

Intuitive proof that the two differently shaped triangles are congruent

Rectangle Split #3

Then she said, “Doesn’t this show that triangles A and B are the same? They both have two of the little triangles.”

Of course Karyn was right. All of the big triangles — A, B, C, and D — clearly contain two of the little triangles. And since all of the eight little triangles are congruent (by SAS), that means that A and B must have equal areas, even though they have different shapes. Of course I was impressed. In fact, Karyn’s approach reminded me of many proofs of the Pythagorean Theorem, proofs in which you see that various pieces that don’t appear to add up to another piece — actually do.

The reason I’m bringing this up is that it tells me something about human thought. This intuitive-style proof was thought up by a student who had had virtually no training in geometry. I, myself, with all of my training, did not see this proof as quickly as Karyn did. This makes me think that there’s a certain intelligence that comes from being able to see things fresh, without other thoughts and constructs limiting the mind’s ability to roam free. Karyn was able to tap into that freedom of thought, and it helped her see a beautiful little proof.

Of course, those of us who have been schooled cannot un-school ourselves; it’s just not possible. But perhaps we can keep in mind that sometimes, when we are stuck, it might be best to look at a problem with the same kind of freshness and originality that Karyn employed. I’m not entirely sure how one gets oneself to do this. But I would suppose that it’s a way of thinking that we can cultivate, if we wish to do so.

If any of you have any suggestions on how we can keep our minds youthful and agile, feel free to share your ideas in the comments to this post.


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One Response to “Cultivating original thinking”

  1. Karl Breiner says:

    My son, who is in first grade, is learning fractions by using shapes. One of his problems could be summed up as follows: “4 mice have a rectangular cookie. Share the cookie among the mice such that each gets the same amount.” On the rectangle, my son divided the cookie exactly the way your student did. The teacher marked it incorrect. When my son brought the test home, I explained to him why he was, in fact, correct. Although he has no concept of area or multiplication, I was able to demonstrate why the triangles are equal by cutting them out of a sheet of paper and re-orienting and re-arranging them (much like your #3).

    Now here is the thing.. I would classify split #1 (surely the answer the teacher was expecting)as the average solution. Split #2 is genius since it is simpler and achieves the same result. Split #1 really requires 2 tools- a straight edge and a ruler to bisect the lines while split #2 only requires a straight edge.

    This is one of the problems I see with math education. Everyone loves out-of-the-box thinking, but no one actually *encourages* it.