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Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural

Cool, i also had an "aha" moment while reading: “It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two equal rows, and square numbers have just the right number of stones to make a square! It’s really fun thinking about various operations in this way, and there are some beautiful proofs based on this technique. For example, why the sum of the odd numbers 1 + 3 + 5… Is always a square.” I should hang out with maths teachers more often!

roymurdock 11y ago
This is a really helpful way to think about natural numbers. I like to remind myself that we can gain efficiency through abstraction, but true, intuitive understanding comes through concretization/deconstruction. Interestingly enough, I use round stones to represent ideas when I meditate. As the ideas come to me, I pick them up, examine them, and weigh them. If the idea is pressing, I delve into it and think it through. If not, I put the stone down and wait until my mind picks up the next. Mental stones: some of modern life's most useful tools :)
j2kun 11y ago
I think the vast majority of math teachers don't go on to learn new things and perspectives on math after college.
thaumasiotes 11y ago
> and square numbers have just the right number of stones to make a square! This one bothers me, because I don't think it's right. You can define the triangular numbers easily: 1, 1+2, 1+2+3, 1+2+3+4, ... and it's easy to arrange that number of dots into a triangle. The difference between two consecutive triangular numbers is always increasing by one. The difference between consecutive square numbers always increases by two. The squares are 1, 1+3, 1+3+5, 1+3+5+7, ... and it's also easy to arrange square numbers of dots into squares. So the difference between pentagonal numbers always increases by three. The first few pentagonal numbers are 1, 5, 12, 22, .... But pentagons don't tile the plane. How do you arrange a pentagonal number of dots into a pentagon? (Wikipedia has a proposed solution on display at https://en.wikipedia.org/wiki/Pentagonal_number https://en.wikipedia.org/wiki/Pentagonal_number , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)

roymurdock 11y ago

This is a really helpful way to think about natural numbers. I like to remind myself that we can gain efficiency through abstraction, but true, intuitive understanding comes through concretization/deconstruction. Interestingly enough, I use round stones to represent ideas when I meditate. As the ideas come to me, I pick them up, examine them, and weigh them. If the idea is pressing, I delve into it and think it through. If not, I put the stone down and wait until my mind picks up the next. Mental stones: some of modern life's most useful tools :)

j2kun 11y ago

I think the vast majority of math teachers don't go on to learn new things and perspectives on math after college.

thaumasiotes 11y ago

> and square numbers have just the right number of stones to make a square! This one bothers me, because I don't think it's right. You can define the triangular numbers easily: 1, 1+2, 1+2+3, 1+2+3+4, ... and it's easy to arrange that number of dots into a triangle. The difference between two consecutive triangular numbers is always increasing by one. The difference between consecutive square numbers always increases by two. The squares are 1, 1+3, 1+3+5, 1+3+5+7, ... and it's also easy to arrange square numbers of dots into squares. So the difference between pentagonal numbers always increases by three. The first few pentagonal numbers are 1, 5, 12, 22, .... But pentagons don't tile the plane. How do you arrange a pentagonal number of dots into a pentagon? (Wikipedia has a proposed solution on display at https://en.wikipedia.org/wiki/Pentagonal_number https://en.wikipedia.org/wiki/Pentagonal_number , but the pentagons it constructs have no obvious internal structure. The square you construct from a square number of dots is symmetric wrt rotation.)