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For those of us who love maths, it's sad to see how many people
hate it, or have a phobia of it, or proudly announce "I was never
very good at maths at school." The question we often counter with
is "Would you be so proud of being unable to read?"
But what's true is that despite the dedication and best efforts
of maths teachers in schools all over the world, many people are
left with no real appreciation of what the subject is, or of the
beauty that's there to be experienced.
So what would be great would be to have a collection of activities
that have, at their heart, proofs. Proofs to make you go "Wow!"
Wouldn't it be great if just for one lesson a year a teacher had
a resource that let them present an age-appropriate, curriculum
appropriate proof that was accessible, engaging, and genuinely
beautiful.
Maybe that's not possible, but we can try. Let's gather together
a collection of candidate proofs, then bundle them to be accessible
to teachers so they can let the students see beyond the sometimes
motiveless manipulations and tedious tinkerings. We know that
teachers want to instill a sense of awe, wonder, and enthusiasm in
their students, so let's try to provide the resources they need to
do exactly that.
The proofs
So we need a collection of proofs to be bundled and presented, to
be made accessible and useable for time-pressed teachers. At this
stage we don't know exactly what will fly, or what will best engage
the students, but we can start by collecting candidates, and then
working to attach them to the curriculum and thereby determine the
best age group for them.
We need candidates.
Some Suggestions ...
So here are some suggestions:
- The proof that a 10x10 board cannot be covered by 1x4 strips
- Euclid's Proof of the Infinitude of Primes
- Pick's Theorem
- A proof without words to Nicomachus theorem
Others, currently ungraded:
- A chessboard with opposite corners removed cannot be covered
exactly by dominoes of size 2x1
- But any chessboard with one black and one white removed
can always be so covered
- For every number N, (N-1)!+1 is a multiple of N
if and only if N is prime. (Wilson's Theorem).
- If you take two copies of any map, scrumple one,
and place it on top of (and inside the boundary of)
the other, then there is at least one point that
"matches". (Brouwer's Fixed Point Theorem).
- Any "doodle" that can be drawn in a single pen-stroke
can be drawn without crossing over a line already drawn.
More, if you return to where you started, the resulting
division of the plane can be coloured with two colours.
- Any prime of the form 4k+1 can always be written as
the sum of two squares, but any prime of the form 4k+3
cannot.
- Given any six people, either there are (at least) three
who have all shaken hands, or there are "at least) three
none of whom have shaken hands.
- The number e is irrational
- There exist transcendental numbers
- Any traversable network has all nodes of even degree
(except perhaps 2), and any connected network with all
nodes of even degree is traversable (has an Euler Cycle)
- Construction of a regular pentagon.
A challenge for you ...
So, what mathematical proof could you present to a six-year-old?
Or an eight year-old? What is there in the school curriculum that
you think has a lovely proof that the students really ought to get
to experience?
Let me know.
Please.
Send us a comment ...
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