At some point during the process of teaching multiplication to our children, we really need to come to terms with this question:
What IS multiplication?
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“What’s my answer? It’s not one that society’s going to like. Because society expects — demands, even — that mathematics be concrete, real-world, absolute, having definitive answers.
I can’t give a definitive answer.
Multiplication manifests itself in different ways. So maybe the word ‘is’ there is just too absolute. And it’s actually at odds with what mathematicians do.
Mathematicians do attend to real-world, practical scenarios — by stepping away from them, looking at a bigger picture.”
Continuing on my theme of times table facts, here’s the inimitable James Tanton:
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“If our task is to memorize this table, please make it about mathematics — about thinking your way through a challenge, and what can I do to make my life easier.”
You may also enjoy my blog post series about working through the times tables, paying attention to mathematical relationships (and a bit of prealgebra) along the way.
With the pandemic still raging, most of us will have to adapt our normal holiday traditions to fit the new reality. We may not be able to have a big family gathering (except over Zoom), but we can still enjoy great food.
So for those of you who are planning ahead, here is a mathematician’s menu for next week’s Thanksgiving dinner.
Michael and Nash have been creating and posting new math games with astonishing regularity throughout the pandemic. Their YouTube channel is a great resource for parents who want to play math with elementary-age children.
Today’s entry: Closest to Ten, a quick game for addition and subtraction fluency with a tiny bit of multiplication potential.
And here’s one of my favorites for older players: Factor Triangles, a card game for 2-digit multiplication.
Check out their channel, and have fun playing math with your kids!
Do you have trouble believing that math can be beautiful?
In “Inspirations,” artist Cristóbal Vila creates a wonderful, imaginary work studio for the amazing M.C. Escher. You’ll want to view it in full-screen mode.
Read about the inspirations, and then try making some math of your own.
“I looked into that enormous and inexhaustible source of inspiration that is Escher and tried to imagine how it could be his workplace, what things would surround an artist like him, so deeply interested in science in general and mathematics in particular. I imagined that these things could be his travel souvenirs, gifts from friends, sources of inspiration…”
I had forgotten this video, and then rediscovered it yesterday and loved it just as much as ever. Perhaps you’ll enjoy it, too — especially if you think of yourself as “not a math person.”
Annie Fetter is talking to classroom teachers, but her message is just as important for homeschoolers. Math is all about making sense. Let’s help our kids see it that way.
“Sense-making is the first mathematical practice for a reason. If we don’t do this one, the rest of them don’t matter. If we’re not doing this, our children are not going to learn mathematics.”
“When I began my college education, I still had many doubts about whether I was good enough for mathematics. Then a colleague said the decisive words to me: it is not that I am worthy to occupy myself with mathematics, but rather that mathematics is worthy for one to occupy oneself with.”
I would like to win over those who consider mathematics useful, but colourless and dry — a necessary evil…
No other field can offer, to such an extent as mathematics, the joy of discovery, which is perhaps the greatest human joy.
The schoolchildren that I have taught in the past were always attuned to this, and so I have also learned much from them.
It never would have occurred to me, for instance, to talk about the Euclidean Algorithm in a class with twelve-year-old girls, but my students led me to do it.
I would like to recount this lesson.
What we were busy with was that I would name two numbers, and the students would figure out their greatest common divisor. For small numbers this went quickly. Gradually, I named larger and larger numbers so that the students would experience difficulty and would want to have a procedure.
I thought that the procedure would be factorization into primes.
They had still easily figured out the greatest common divisor of 60 and 48: “Twelve!”
But a girl remarked: “Well, that’s just the same as the difference of 60 and 48.”
“That’s a coincidence,” I said and wanted to go on.
But they would not let me go on: “Please name us numbers where it isn’t like that.”
“Fine. 60 and 36 also have 12 as their greatest common divisor, and their difference is 24.”
Another interruption: “Here the difference is twice as big as the greatest common divisor.”
“All right, if this will satisfy all of you, it is in fact no coincidence: the difference of two numbers is always divisible by all their common divisors. And so is their sum.”
Certainly that needed to be stated in full, but having done so, I really did want to move on.
However, I still could not do that.
A girl asked: “Couldn’t they discover a procedure to find the greatest common divisor just from that?”
They certainly could! But that is precisely the basic idea behind the Euclidean Algorithm!
So I abandoned my plan and went the way that my students led me.
