Is it ‘proofy’ enough?

One thing I struggle with is getting my pupils to make the leap from creating examples of a proposition to giving a proof of it. I found the ‘Make 37‘ activity was great for getting them to start to explain ‘why it will never work’, instead of producing endless examples of it not working.

However, few of them were able to move from ‘an even number of odds can’t make an even’ to an algebraic proof (some version of 10(2n+1) = 37 -> 20n=27 -> no integer solution or “10(2n+1) divides by 2, so it is even, so it can’t be 37” would have sufficed).

In addition to appreciating any advice or anecdotes on how to take the step to formal proofs, I would also like to know people’s opinions on ‘picture proof’ (e.g. proving that angles in a triangle must have a sum of 180 by taking the exterior angles and ‘shrinking’ them inwards to show they make a full turn – there is an animated example of this on mymaths). Mathsisfun do what is, to my mind, a more rigorous approach, but it still relies on ‘looking right on this occasion’.

Is it sufficient to stop with ‘word’ and ‘picture’ proofs, or is it inherently valuable to push on to algebraic versions?

Extension task:)  I puzzled over this with a colleague from another school – we could only show it with a picture. Is there a better way? Can it be done other than by pointing at an illustration? To our embarrassment, it’s from the Junior section of the UKMT mentoring resources. I’m avoiding the advanced section for now.

Answers to this questions would be appreciated, as the solution won’t be released for a few more weeks!

5 Comments

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5 responses to “Is it ‘proofy’ enough?

  1. Scott

    Really admire that you’re stressing the importance of proofs – amazing! I found this which is quite fun to use with a higher ability group:

    http://uncyclopedia.wikia.com/wiki/Proof_that_the_angles_of_a_triangle_sum_to_360_degrees

    I used some of it in class and set some as a homework – they had to try and spot the mistakes in these proofs.

    Re picture proofs: non-normally sufficient to show any general proposition. However, vastly important to help you see the way through a problem! I think the picture proof used there is essentially a child friendly statement of the rigorous proofs which use the language of Eucliden geometry:

    http://www.cut-the-knot.org/triangle/pythpar/AnglesInTriangle.shtml

    Re parallelogram: I wasted a good 30 mins trying to show this algebraically and ended up in an absolute quagmire and much more confused than when I started. If you draw a picture and label all the points and lines then you can prove the proposition by using general Euclidean statements about intersection and collinearity of points.

  2. Probably way too late, but I solved the parallelogram problem.
    Not too tricky at all once you realise that two circles touch if the sum of their radii is equal to the distance between the centers.
    Coupled with the fact that opposite sides of a parallelogram are of equal length, and there we have it.

  3. I think (it’s rather early!) that I got there too, and had illustrated the point – I was just trying to decide if it was ‘enough’ or if it was possible to write it as a set of equations that prove it to be always the case (although, if it requires the equn of a circle it’s way beyond the scope of KS3/4!).

  4. Scott – “I wasted a good 30 mins” – yup…same here…and no better off for it!

  5. I have asked a lot of these questions myself. I understand the purist’s desire to see formal deductive proof done, but I question the value of it. Especially since when those who are going on to bigger and better math will be retrained by their professors anyway.

    This year I have tried pushing a more informal version of proof, still deductively-reasoned, but less formally-structured. Allowing the students to apply their own style seems to have made them more willing to try them. Disengagement was the biggest problem with the formal structures before.

    I describe my efforts and the results in a few posts. I also ask some questions…

    http://thegeometryteacher.wordpress.com/2012/12/07/when-measuring-is-okay/

    http://thegeometryteacher.wordpress.com/2012/11/16/proof-the-logical-next-step/

    Hopefully we can keep this conversation going and come up with some meaningful way to make proof meaningful, valuable, and accessible.

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