Friday, April 6, 2012

Proof by Contradiction

This post at The De Morgan Journal introduces a paper by Oleksiy Yevdokimov, on teaching math as relationships between structures. One example the author uses reminded me of my Linear Algebra students' struggles with proof and Hao's advice (in the comments) to work with them a bit on formal logic.

Do you like this example as much as I do?

I am driving a car with an 8- year-old child inside. I deliberately start naming all green cars we see around as being “red”. Red cars continue to be red ones. The child identifies the problem immediately and has a growing concern about what is happening with me. Soon we approach a street intersection with traffic lights ahead. The red light is on. I slow down and eventually stop, waiting for the green light. When the green is on, I start driving and hear a huge sigh of relief. The conclusion follows: “You are joking!”

The child’s reasoning is based on proof by contradiction: “If you are in trouble with colours, you won’t drive across the intersection when the green light is on”. 

From the child’s point of view her reasoning does not have any relation to mathematics. From the teaching point it shows that many conceptual constructions in mathematics can be successfully introduced rather sooner than later. For example, the use of proof by contradiction presented here, as well as many other useful methods and structures, can be seen everywhere through elementary and higher mathematics. The teacher’s task is to keep focus on them all the time—while moving from topic to topic, extending the content knowledge and improving problem-solving skills.


  1. I like it, Sue! Reminds me of this old one: Ask someone to spell T-O-P-S three times. Person does so, then ask him/her, "What do you do when you come to a green light?"

    I referred back to your post on writing proofs and we were discussing this very topic on #mathchat yesterday, how important writing is in mathematics, the rigor of proving something driven by intuition and curiosity. Students across the curriculum struggle with writing explanation and proofs as if they were lacking academic language and known axioms/postulates, and we don't explicitly teach it with examples.
    Thanks, Sue!

  2. >we were discussing this very topic on #mathchat yesterday

    I must not understand twitter. I went there looking for the discussion and didn't find it. (??)

  3. Am so new to Twitter, so I don't know, but @ColinTGraham was not feeling well so we went ahead without a moderator. It was my first #mathchat, don't know how long one normally lasts, but yesterday's session did not seem long.

  4. This reminds me of a game I loved called Zendo.

    It was interesting to see the different ways people approached the problems. Many used a straight forward logical approach, but there are always a few who will do a gestalt leap of intuition and get it right.

  5. Zendo reminds me of Eleusis. I followed that page to a site full of mazes, and a cognition research site. Fascinating reading for my last day of break! Thanks.


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