A two-machine problem

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:

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:

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:

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:

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:

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.

Magnitude

It came as a shock to me that there are people without numbers and counting. I first came across this in Daniel Everett's wonderful book about his time with an Amazonian people called the Pirahãs, Don't Sleep, There are Snakes:

But bigger surprises were in store. 

One of the first was the apparent lack of counting and numbers. At first I thought that the Pirahãs had the numbers one, two, and “many,” a common enough system around the world. But I realized that what I and previous workers thought were numbers were only relative quantities. I began to notice this when the Pirahãs asked me when the plane was coming again, a question they enjoy asking, I eventually realised, because they find it nearly magical that I seem to know the day that the plane is arriving. 

I would hold up two fingers and say, “Hoi days,” using what I thought was their term for two. They would look puzzled. As I observed more carefully, I saw that they never used their fingers or any other body parts or external objects to count or tally with. And I also noticed that they could use what I thought meant “two” for two small fish or one relatively larger fish, contradicting my understanding that it meant “two” and supporting my new idea of the “numbers” as references to relative volume – two small fish and one medium-size fish are roughly equal in volume, but both would be less than, and thus trigger a different “number” than a large fish. Eventually numerous published experiments were conducted by me and a series of psychologists that demonstrated conclusively that the Pirahãs have no numbers at all and no counting in any form.

Because they're sometimes cheated by traders, the Pirahãs asked Everett for regular lessons in counting to ten. But despite a year's worth of lessons they don't really learn to do it! These are people that are in many ways a lot smarter than us, people who can walk with nothing into the jungle and come back with all sorts of food, some of it carried in baskets that they've woven on the spot from wet palm leaves. These are happy self-sufficient people. But they don't do numbers.

The ATM has as its first guiding principle that, "The ability to operate mathematically is an aspect of human functioning that is as universal as language itself. Attention needs constantly to be drawn to this fact."

How does this square with the Pirahãs?

At first it bothered me. Could it be that in some way mathematics is not a universal? But then I realised that I have a kind of blindness with numbers too. Because numbers aren't there, I wasn't seeing the mathematics. The two go together for us, numbers and mathematics.

Talking about two, what about the Pirahãs' two, hoi? Everett says it's really about relative volume. And this is key: our lives, all human lives (and the lives of animals too) are full of thinking about magnitudes - volumes, areas, distances, times, weights - usually continuous magnitudes, ones where an exact whole number doesn't come into the thinking.

That basket that the Pirahãs made on the spot to carry what they'd gathered, the Pirahãs had to select the frond of the right width and break off the appropriate length. They had to lay it alongside other fronds and weave in and out until the beginnings of roughly the right size of basket appeared. All sorts of mathematical thinking here, even if numbers don't come into it. And of course, every step, every reach, every move in fact has all sorts of magnitudes involved. Our experience of our bodies in the world is full of them - so full that they're kind of invisible!

I've been thinking about this more after some tweets with Tali Leibovich-Raveh. She shared some articles that she has co-written: Magnitude processing in non-symbolic stimuli and From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition (pdf).

Both of them discuss how early number sense is studied. Often dot images are used:

OK - there's four dots and there's three dots, and four dots is more. But there's more going on here. The one on the left also has a bigger pink area, and covers more space (normally this 'convex hull' isn't coloured, but that doesn't mean it's not perceived). And though the amount of black is the same in the two images, the total length of the circumferences in the left hand image is greater. So when a young child indicates that there's more in the left image, we don't know whether they're solely responding to the number of dots. There's all sorts of magnitude 'mores' that they could be influenced by.

Tali Leibovich-Raveh goes on to argue that there's evidence that in fact it's the sense of continuous magnitude that is primitive, and that number sense is built on this.

When I read this, I started to think again about all sorts of things in this light.

Seeing a boy piling blocks up in the sandpit in these few seconds here:


He's not counting them. He's interested in height, specifically I think in how high up he can make it go. He knows (and here is one of those so-common it's invisible bits of mathematics) that if he adds to his height, he'll be able to add to the tower's height.

You start thinking about it, and magnitudes are everywhere. Taking a common list of play schemas:

• Transporting
• Enveloping
• Enclosing
• Trajectory
• Rotation
• Connecting
• Positioning
• Transforming

every one of them involves magnitudes of some kind or other (as well as arrangements and geometry and topology and patterns - but counting not so much). If we have a number-skewed idea of mathematics, we don't credit these play types for all the mathematics they contain.

As Tony Cotton writes:

I have described previously the enjoyment in observing my grandchildren creating patterns, experimenting with filling and emptying containers or loading toy trucks with rods. Interestingly, what they don’t do is count. They only count when asked by teachers or other adults. Counting is their lowest mathematical priority.

Watching funny cat videos afterwards, all sorts of jumps, cats squeezing through gaps and into boxes. They're funny when they go wrong, but in the wild a sense of magnitudes, a sense of timing, distance, volume is essential for survival. Will this branch be strong enough to hold me? Can I get through this hole? Can I jump this gap?

source

And what about maths teaching beyond the early years? This takes whatever innate mathematical abilities we do have and goes beyond. It's a cultural activity that co-opts mental processes that haven't evolved for school mathematics and uses them to build mathematical understanding.

I've posted before about how Cuisenaire rods in some ways bypass counting. In the light of magnitude thinking, I see their use as a kind of extension of the play we do so much of: judging lengths, filling containers, loading and unloading, putting things end-to-end and side-by-side, making arrangements based on size and shape and pattern.

Getting young children to play with the rods is always fascinating. I recently showed my Grade 3 class some pictures of when I visited them with Cuisenaire rods when they were in K3 and got them playing.

Building on the understandings that young children have is so important, and here there's a kind of natural transition between the world of playing with objects with continuous magnitudes to playing with wooden rods with discrete magnitudes. We're still in the realm of length and area and volume, still using our knowledge of placing things, of lining up, or building, of balancing.

Children are in familiar territory when they lay rods side-by-side. They see that the length of the red rod + the length of the red rod again is equal to the length of the pink rod.

Or they might say that two of the red rods are equal to the pink rod. Or that the red rod is half the length of the pink. Or they might get to know the numerical equivalents and see that four is double two. All this with very little counting.

I think those of us who emphasise physical and spatial resources in our mathematics teaching for other things than simply counting should take courage from these ideas about magnitude. Even experientially, apart from any research results, once we decide to see it, we can see how full our behaviour is of magnitudes. How ready we are to think in this way.