I came across a good problem in John Sweller's Story of a Research Progam.
In a 1982 experiment to which cognitive load theory can be traced back, Sweller asked subjects to transform a given number into a goal number where the only two moves allowed were multiplying by 3 or subtracting 29. Each problem had only one possible solution and that solution required an alternation of multiplying by 3 and subtracting 29 a specific number of times. All problems had to be solved by this alternation sequence because the numbers were chosen to ensure that no other solution was possible. Sweller was surprised that very few students discovered his rule, that is, the solution sequence of alternating the two possible moves. It was, quite rightly, obvious to him that if he had simply told students to solve each problem by alternating the two moves until they got to the target number they would have immediately learned the rule. This was the source of his objections to problem-based learning.
Be that as it may, he's got a good problem here! Not the one of guessing his rule, which I agree, isn't particularly fit for problem-solving. But the mathematics - having a starting number and two operations and seeing where that leads. That seemed like something worth investigating. Sweller did his experiment with college students; I wondered how my class of 8 and 9 year olds might approach the mathematics.
The -29 makes the arithmetic a bit heavy. But what if we switched to -3 instead, and ×2 instead of ×3? That would make the arithmetic more manageable. We would have a common subject to talk about if we all began with the same starting number, and it made sense for this to be one.
I needed to make the whole thing intelligible to the students, so we began by looking at function machines first.
Emily Allman had shared pages from one of Mitsumasa Anno's books where there are a wonderful series of function machines. We looked at them closely.
This one features doubling:
Children worked in pairs, with whiteboards, some doubled and subtracted three, some just doubled. The ones that chose to double were getting to some pretty big numbers!
After a while I collected in some of the numbers that the pairs had found:
Some were still looking at doubles. One pair noticed that the last digit of the doubles was in a pattern after 1: 2,4,8,6,2,4,8,6.
I encourage the students to write down what they had found:
We were on the verge of finding out not only what was obtainable from the two machines starting at 1, but also why. We could see that a certain pattern of numbers was obtained from doubling. We could also see that there were three sets of numbers, the multiples of three, the numbers one under a multiple of three and the numbers one over the multiples of three. But when I asked the students what they noticed, no-one was particularly going in this direction. I often think this is a good indicator of whether to pursue an avenue, so I felt that pushing further would be overstretching things, and we should quit while we were ahead. It had been an interesting and profitable exploration.
We did return to the idea of repeated doubling later in the week with a reading of the wonderful book One Grain of Rice by Demi:
In a 1982 experiment to which cognitive load theory can be traced back, Sweller asked subjects to transform a given number into a goal number where the only two moves allowed were multiplying by 3 or subtracting 29. Each problem had only one possible solution and that solution required an alternation of multiplying by 3 and subtracting 29 a specific number of times. All problems had to be solved by this alternation sequence because the numbers were chosen to ensure that no other solution was possible. Sweller was surprised that very few students discovered his rule, that is, the solution sequence of alternating the two possible moves. It was, quite rightly, obvious to him that if he had simply told students to solve each problem by alternating the two moves until they got to the target number they would have immediately learned the rule. This was the source of his objections to problem-based learning.
Be that as it may, he's got a good problem here! Not the one of guessing his rule, which I agree, isn't particularly fit for problem-solving. But the mathematics - having a starting number and two operations and seeing where that leads. That seemed like something worth investigating. Sweller did his experiment with college students; I wondered how my class of 8 and 9 year olds might approach the mathematics.
The -29 makes the arithmetic a bit heavy. But what if we switched to -3 instead, and ×2 instead of ×3? That would make the arithmetic more manageable. We would have a common subject to talk about if we all began with the same starting number, and it made sense for this to be one.
I needed to make the whole thing intelligible to the students, so we began by looking at function machines first.
Emily Allman had shared pages from one of Mitsumasa Anno's books where there are a wonderful series of function machines. We looked at them closely.
We tried out a few other machines, where I asked for inputs and gave the outputs. We did the same in pairs creating our own machines and functions. We got inside a big cardboard box to be the function machine, receiving numbers through a hole on one side and posting them out modified through another hole.
The next day, we were ready for this:
After a while I collected in some of the numbers that the pairs had found:
We returned to our search for a while longer. I went and got a hundred square. Stopping the class, I asked which numbers they had managed to find now. We turned the ones we had found over to the red side, leaving the others blue. They were starting to tumble now.
Then AP noticed something. There were some blue diagonal lines left, the ones we hadn't managed to get to. LD noticed the red-red-blue pattern. MT noticed that the blues were the numbers in the three times table.
Time had run out, so we returned to it the next day, to record what we'd been exploring, and extend it too. The students were starting to feel confident with ways of reaching numbers:
We did return to the idea of repeated doubling later in the week with a reading of the wonderful book One Grain of Rice by Demi:
There's a real wow! in this book when the commonplace doubling 2,4,8... soon becomes... millions. Actual gasps from the students and the pull-out spread of elephants delivering hundreds of millions of grains of rice!
With older students, I might be inclined to push for the structure of the network of numbers that are obtainable:
Perhaps if I'd labeled the numbers in two colours rather than in just one it would have made it more within our reach. To see that there are two kinds of not-three-times-table number. Some people use "threven" for the multiples of three, "throver" for the numbers one over the threvens, and "thrunder" for the numbers one under the threvens. We might have been able to see that starting at one and doubling and doubling again alternates between the throvers (red) and the thrunders (orange), and subtracting three doesn't change it.
What other questions could we ask? How would varying the initial number affect this structure? How would changing the amounts by which we multiplied and the amount subtracted alter the pattern? It would be good to view using mod 3. What pattern of numbers in mod 3 do you get by doubling? Subtracting 3 doesn't alter these, so it becomes clear why some numbers are unobtainable.
But anyway, it was a pleasurable search, everyone got involved in a succession of individual, paired, group and whole-class thinking, going off to investigate, then gathering together to record ideas, crediting of those who took the thinking further, and then returning to paired work search further.
Knowing the structure of the numbers obtainable with ×2 and -3 is of course, as Sweller says, not an essential piece of information. But this kind of task brings with it all sorts of other benefits. Students starting from a simple question, and working together to decide which direction to go in, exploring, discovering new territory, noticing features and patterns, becoming familiar with the territory, proposing generalisations, investigating further. They were powerful learners.
So fascinating! There's so much to notice and so many ways to represent what's happening. And it implies that problem of what other machines can we build...
ReplyDeleteI was imagining a grid where to the right was x2 and down was -3. When I've had kids play with the Collatz conjecture, they get fascinated by how the network connects. "Oh, I got 17, and I know what happens from there." Of course, that's a specific rule, and here you have choices. What might a rule be for these machines?
Did any challenges seem natural? Like which numbers can't you get starting from 0 or 1, or given a number what's the best path to some target?
Also wondering how engaging this was for all the students with just the naked numbers. Of course, if anyone's classroom has kids interested in number for themselves, it's yours!
Thanks, John. They seemed engaged. It's hard for me to say whether they would be more so with say a real-world context (and within that whether aspects of real world could be a distraction as well as other aspects introducing important questions and caveats). As you know, I do a lot of lessons with 'stuff', so it's not all numbers numbers numbers anyway.
DeleteI had to resist the temptation to show my concentric grid (I think it was a temptation anyway!) although in hindsight a two-colour hundred square could have been good as the hundred square is a familiar representation.
I thought of Collatz. But there are some simpler chains we might look at first.
Fascinating, Simon and definitely one I'll try out. On the doubling 'wow!' I came across the pocket money question the other week: would you rather have 10 dollars a month for a year, or 10 cents in Jan, 20c in Feb, doubling each month until Dec? Haven't tried it in class yet (or on my kids!).
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