I’ve been having lots of conversations with our director of computational sciences about computational thinking. Among many things, we have been talking about, “What are the beginnings of computational thinking and how do we foster those beginnings?” I’ve come to see that the beginnings of computational thinking involve thinking about arithmetic and algebra as strongly interconnected and thinking about computation as involving creativity and insight.
Here are few examples that we discussed this week.
How would you calculate 21 x 19? Of course, there are many ways to do it. You could add 19 twenty-one times, or add 21 nineteen times. You could do 21 * 20 and then subtract 21. One interesting way we can think about 21 x 19 is as (20 + 1) (20-1). The reasons this is interesting is because it takes form (x+1)(x-1) = x² -1, which is then just 20² – 1 = 399. Of course, we can generalize this to any distance from known squares, so that 18 x 22 = (20-2)(20+2) = 20²-2² = 396. With this method, even products like 63*57 and 112 * 88 are a cake walk.
Another question we talked about was 26 x 27, which is of the form (x+1)(x+2) = x² + 3x +2, which gives us 25² + 3(25) +2 = 702. Less compelling, but still an interesting and different way of thinking about it than the standard algorithm.
The point of all of this isn’t just to figure out how to multiply numbers quickly. Rather the point is to (i) begin thinking about how to break down complex calculations into collections of much simpler ones, (ii) to begin to recognize how classes of similar problems might all be solvable by a common algorithm, (iii) to come recognize that there are often many different algorithms that can be used to carryout the same calculation and (iv) to begin to make contact with the idea that efficiency of an algorithm can depend greatly on what kind of problem you have and its structure.
I know there are lots of physics teachers and physics education researchers concerned with computation thinking. What do you guys think?
“One interesting way we can think about 21 x 19 is as (20 + 1) (20-1). The reasons this is interesting is because it takes form (x+1)(x-1) = x² -1…”
I like that because it’s easy to picture a rectangular grid (on square graph paper) that is 21×19 – and if I take a row OFF of the long side and add it as a “column” to the short side, then it’s a 20×20 grid, except that I have one too few “squares” to do that… so 20×20 = 21×19+1. (Which is also interesting because it reminds me that the lowest perimeter-to-volume ratio for a rectangle is the square – I can start to see why that must be true.) And then it’s easy to see – without doing any algebra/calculation – why any size “missing square” will work.
I read your post from the perspective of a dorky parent, not a physics teacher. I was doing some of these tricks with the kids when I was small – not in multiplication, but in addition and subtraction. It’s the same class of tricks, doing 8+7 as 8+2=10 and 10+5=15. It’s a mindset game, of breaking things into the appropriate pieces and using those pieces to do your work for you.
At some level, isn’t that activity what we’re claiming physics is for? “It’s important because it teaches problem solving!” Well, yeah, but learning to identify the problem and reframe it into a solvable problem, that happens with a dorky dad doing math games with his 6 year old, you doing these math games around squares, and us solving physics problem. It’s learning to have a fluid perspective toward many different “algorithms” (as you call them), and learning to test and toss them out, on the fly, as you find the one that’s going to work.
I’m going to suggest that the problem of your process-oriented problem solving work in the course whose content you don’t control is that it so often only trusts in one algorithm, while your teaching as you’re describing it pushes toward exploring many different possible approaches, most of which haven’t been learned well enough to be an algorithm, yet. In other words, you are doubly hindered in your work – students may not have the algorithms yet and they may not recognize that there are other ones to explore. You’re trying to teach both topics in a setting where neither is valued, and the One True Algorithm is being promoted as the solution path for a given problem.
… In other dorky math games, learning to take square roots by knowing the squares of all the numbers up to 32 has always been a useless talent of mine. Quick: square root of 9.5! Well, it’s about pi, because I know pi squared is about the value of “g,” but more precisely, it’s less than the square root of 961/100, so it’s just under 3.1, and to find out how much I just have to…. (okay, I’ll shut up now).
By the way, 112*88 required a jolt of recognition. I’d only used your trick for numbers close together, but to do it with ANY number separated by an even count (24 here, so 24/12 centers me at 100), well, shit, I never thought of THAT. Here I am, using your trick all the time for closely spaced numbers, and never realizing I could generalize.
Funny how that works out.
The folks at the Cognitively Guided Instruction (CGI) research project at UW-Madison have worked hard on describing these connections between algebra and arithmetic for teachers. Worth a look-the book I have in mind is titled, Thinking Mathematically. It’s probably in your college library. Published by Heinemann.