Sunday, November 30, 2014

Formal proofs vs. intuitive proofs plus jargon

Formal geometry proofs are to some extent the opposite of intuitive. To prove that two figures are similar, for example, you have to throw out any intuitions you have about why they must be similar, and rely exclusively on the axioms and theorems of Euclidian geometry.

Consider, for example, two of the triangles in the problem below.


Intuitively, it’s clear that if you flip triangle BEC over on its bottom vertex, i.e., keeping vertex E stationary, then two of its sides (BE and CE) would lie along two sides of triangle AED (AE and DE).

 

From the given information, it’s also clear that BEC’s third side, BC, is parallel to the EAD’s third side (AD). That means that in the flipped configuration of AED, if you stretch AED downwards until BD reaches AD, the resulting triangle would completely overlap with triangle AED.

However, a formal proof based on established theorems bypasses these intuitions:



While some people complain that these proofs are overly formal and mindless because they ignore intuition, others, myself included, found these proofs compelling and mentally engaging for the very same reason: i.e., because they force you to abandon all preconceived notions and work things by logic.

Under Reform Math and the Common Core, unfortunately, these formal Euclidian proofs are waning. Replacing them are informal proofs that codify intuition via formal-sounding jargon: “transformations,” “reflect,” “translations” and “dilations.”

And so (as far as I can tell from what I'm seeing in today's Common Core-inspired rubrics) an acceptable proof for the above problem is a formalization of the intuitions I described above. BEC and AED are similar because you can reflect BEC on an axis of symmetry that goes through vertex E and is parallel to BE, and then dilate the triangle (by a scale factor of EA/EB) so that it completely overlaps with AED.

It seems to me that something key is lost when kids no longer get regular practice in deducing what does, and does not, follow from those basic assumptions that we all agree on.

1 comment:

Auntie Ann said...

I was looking at an article discussing the wordiness of CC exams and the requirements to explain in words, and thought that teaching higher math (algebra and up) in terms of formal proofs was a way out for traditional math fans.

Instead of vague wordiness, a student could work through a problem in the style of a formal proof, thus putting their steps in words to satisfy the exam requirements, but still teaching the techniques and standard solutions STEM kids actually need. Also, teaching to think through problems in reasonable, logical, and systematic ways benefits everyone.

(I don't recognize the A=~A reason (geometry was a long time ago!)