Sunday, 17 March 2019

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.

Saturday, 2 March 2019


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.

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?
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.

Monday, 14 January 2019

High Context, Low Context

Thanks to Michael Pershan for writing a lovely piece about ancient Greek mathematics. He links to and comments on aspects of an article, The two cultures of mathematics in ancient Greece by Markus Asper. 

Asper says:

The words of, for example, Euclid or Archimedes appear to be of timeless brilliance, their assumptions, methods, and proofs, even after Hilbert, of almost eternal elegance…Recently, however, a consensus has emerged that Greek mathematics was heterogenous and that the famous mathematicians are only the tip of an iceberg that must have consisted of several coexisting and partly overlapping fields of mathematical practices.
The part of the iceberg that is underwater, so to speak, is practical mathematics, the kind that builders need for construction, that commercial calculatiors performed on a sand tray with pebbles and that surveyors used when stretching out ropes to measure land.
Some of the books of the theoretical, abstract mathematicians have come down to us; the everyday mathematics of craftsmen, engineers, accountants and surveyors is all but lost to us.
Euclid distilled the discoveries of the two centuries before him into a careful tower of propositions, each careful built on the one below, logically building upwards by detailing one step at a time:

The fragments of builder wisdom that have turned up in rubbish tips in Egypt couldn't be more different: recipes for how much stone you'll need for a particular sized house. There's not even a general formula.

High Context, Low Context

At this point I want to bring in the anthropologist Edward T Hall's description of high context and low context cultures. For Hall, “a high-context communication or message is one in which most of the information is either in the physical context or internalised in the person, while very little is in the coded, explicit, or transmitted part of the message”. In a low-context communication on the other hand, meanings are explicitly stated through language. Hall was interested in cross-cultural mis-understanding and understanding. I want to mention maths education in the light of his distinction.

Euclid is low-context: he tries to make every little step explicit for us.

The builder's recipes on the other hand were high-context. The builder works with his apprentices on site and such recipes as were written down would be adjusted and reworked within a social, oral, gestural, practical context.

Can you see where I'm going with this?

Euclid has a priveleged position in the history of mathematics and of learning mathematics, and it's style, impersonal, logical, abstract, has permeated mathematical writing. Moreover, the textbook and the test have, for additional reasons of their own, adopted the same low-context mode - everything you need to know must be contained in the writing

This kind of low-context writing, that has to contain all the information that would normally be carried by a social situation, the possiblity of clarificaton, questioning, gradually tuning in, is hard to read for the uninitiated. It's not our normal kind of communication.

Why do we funnel children through a low-context communication, the word problem, the odd-one-out verbal reasoning question, the test question, when we're in a high-context situation? We're like the builders, the students are like our apprentices, we have a social, oral, gestural context in the classroom, we have lots of materials. We can present fragments - an image, a numberless problem, an equation. There doesn't need to be a question. Through the unfolding of the lesson in the high-context culture of the classroom we can gradually, through dialogue and practical work, build the knowledge together.

Tuesday, 27 November 2018


Rather than triangles, which was on the Grade 3 (English Year 4) plan, I felt that hexagons would give us more scope for focusing on properties. It's worth looking at Christopher Danielson's lessons on this. I wasn't really expecting to get on to proof, but if a conjecture came up, I would be happy (it didn't this time!). We started with a Which One Doesn't Belong?
I mentioned that three of them were hexagons; they didn't have to be regular to be hexagons. 

We asked the classes to make hexagons from pattern blocks, to make one whose shape was not chosen by anyone else. This had advantages. The hexagon, kind of, 'belonged' to the student. We got a range of different hexagons that we could construct and measure easily, and would also, mostly, tile. But I'm still wondering about the gains and losses from this approach, and I'll come back to that.
Finally, after a number of repeats and non-hexagons that we didn't spot straight away, we had a set:
which I reproduced for the class without the colour, so that we'd be focusing on the component shapes less:
- still time for another pattern block hexagon WODB though:
Notice that 'looks most like a hexagon'! We were going to shake off that regular hexagon thing...

We looked at side length, counting the length of almost every side of the pattern blocks as one. I also talked about perimeter. (BTW: my opinion is that it's really good to introduce this separately to area. Although it goes together with area in our minds like salt-and-pepper, my feeling is that some students get a little which-is-which?) We documented this for our hexagon, later adding on parallel lines and symmetry:
Then, starting from the known fact that a square's corner is 90°, I asked the class to find the angles on the other pattern blocks, which, together, step-by-step, they did. The big breakthough was when a pair realised that three of the acute angles of the thin rhombus fit into the squares corner. Then that was a measure for all the others.

So, the next day, we applied this to our hexagons. I'm pleased about this part of the sequence, because it seemed to make it possible for everyone.

Time for another WODB then next day, with four of our hexagons with angles shown.
I hesitate to add this bit, because I'm unusually lucky to have a colleague in secondary who laser cuts me shapes out of acrylic, and that's not accessible to many. But the kids were really excited when these arrived...
 Immediately they began tessellating them:
 We recorded this later:
 This one AK couldn't get to tile:There's a couple we had difficulty tiling:
Martin helped out with one:

And Hana helped out with the other with a tiling that includes some other pattern block shapes:
Rod did the same:

All this led on nicely to our other tiling work with the pattern blocks.

To review what we'd done, I made little books about the 25 hexagons (taking too long on all the images, which was useful in class but I don't recommend to others!) and asked children to write about different aspects of them.

 The books showed that the students had a lot to say about the various properties we'd looked at.

We also finished off with a WODB that students annotated individually. This told a mixed story. I'd chosen some hexagons that weren't within the pattern block family.  The students did notice lots, and showed their knowledge through their annotations:
But some of them also treated these hexagons as if they had the same uniformity that the pattern block hexagons had. Some saw, for instance, the diagonal lines of the green shape as parallel. Some tried to see the hexagons as composed of pattern block shapes. And some tried to measure the perimeters - oh dear, I hadn't thought of that, and hadn't built in whole-number lengths! I kind of feel this was unfair of me, but it has made me more aware of something.

It seems we have some things to unlearn now. Another thing: for our brief estimation actiivity the following week I used Estimation 180's What degrees of the pie have been eaten? So many students put 90° or 60° - and suddenly I realised, now they think all angles are going to be in mutliples of 30°!

So we may need to unlearn:
  • all lines that look vaguely parallel are parallel;
  • all hexagons are made up of pattern block shapes;
  • all side lengths are whole numbers;
  • all angles are multiples of 30°.
Luckily, it's not hard to find counter-examples that undo most of these overgeneralisatons that I've led them into. The undoing could even be an impactful way of learning these important points. So, I'm not too disheartened.

But I'm still mulling this over. Would it have been better to have given a selection of more varied, non-pattern-block hexagons, like Christopher Danielson's
They would have lost the personal creation, the ease of measuring, the tiling perhaps. But they would have had a bigger, more representative example space.

Or I could have chosen a different playground for hexagons. A geoboard maybe, or as in Lana Pavlova's lovely hexagon work, the tangram. Or moved more quickly through differrent embodiments of hexagon.

All this connects with some of my recent thinking about examples and generalisation. Have a look at this, from Thinkers by Chris Bills, Liz Bills, John Mason and Anne Watson:
I'm now much more aware of how our choice of example, the variation we include, carries its own potential misinformation. Students at the start of this felt that a hexagon was a regular hexagon - that's what they'd been shown. We show a range of rectangles and don't include a square. We call the blue and light brown pattern blocks rhombuses, but (natually) call the square a square, even though it's a rhombus too. We look at fractions as circles. Students are going to see features of the examples we present as essential to the properties we want them to become familiar with. Some of those features will be, but others won't. 

So the 'problem' with my hexagon lesson is perhaps a more general one...

How would you do a set of lessons on hexagons? What do you think about this problem with examples? As usual, I'd value any comments. 

Saturday, 20 October 2018

A hundred pattern

Back in Grade 3 (English Year 4) now. We've been doing quick dot images once a week, looking at the different ways the students see the images without having to count every dot.
We've presented it to our parents and the school, from Grade 1 to 5, getting them to talk to each other about how they saw it.

It was time to try and count something bigger, to give the classes a bigger challenge. Could they use their ability to 'conceptually subitise' to help solve a more challenging problem? For a while I thought this would be a good one. The holes in the drain cover are conveniently - almost - in groups of ten
But then I remembered this pattern with Cuisenaire rods. It was one that had started off a wonderful discussion on Twitter (See Dan Finkel's vlog of it). But it had those triangles, a bit like the triangles of dots we'd just looked at. Would the students be able to use some of the pattern in this to make their counting easier? I was not sure whether this would be a bit too much of a leap...
 So we could write on it more easily, I photographed it again on white.
The question: if the white rod = 1, what is the whole thing?

Students worked in pairs to try and work it out.
Some pairs broke the image down to help to see it.
I didn't expect that some pairs would rebuild it...
... and some would rearrange it.

It was really interesting to get an insight into thinking. Tracy Zager has just tweeted a flow chart about mathematical thinking that includes the need for the teacher to see the students' thinking in some way.

This student had worried me a few days before when she'd asked me what half of 20 cm was. But now she was working out what three times 15 and three times 18 were. She didn't quite get to the correct answer, 100, because she didn't include the one in the middle. But she's done all but a tiny bit of the thinking! In fact, most of the students who got it wrong, were just getting one small thing not quite right. Her partner thought it was 102 because he counted the middle one three times!
Another approach was to add up all rods of each colour separately and then total them:

Interesting how they'd written the threes down as twos first of all, and then appear to have compensated for this mistake at the end by adding another nine on.

I pulled the class together and shared these two main strategies. I think in hindsight I would like to have dwelt on this more. I would have liked the class to have looked at what people were doing with the rods too, but it was all really busy! Perhaps we can return for a recap after the two week break.

Someone muttered something about failure. I'm not sure how convincingly I answered this. I said that I thought that even when people didn't get 100, they'd done a lot that was right on the way to 100. He seemed to accept this, but I'm sure there's more work to be done establishing a culture where we focus on showing our thinking rather than thinking about right/wrong.

What we did do was really interesting. I asked the class to make their own hundred pattern, so that it wouldn't need to be counted. (Not just ten orange rods, I said!)

There were all sorts of great approaches:
And some more:
We had a look at that double staircase afterwards; it bothered someone that there was only one orange rod!
The student with the yellow circle was making more hundreds, this time making his fives differently. I wish we'd looked at this too. I would have liked to give him credit for not just doing what I'd asked, but for taking it further and trying a variation. We talked about the person, and some students asked me if we could have a go at doing some like this. There wasn't time this time, but I'm so pleased how many different kinds of creations there were, and also that there was talk of coming back to it in a different form! (Time for some #hudnredface images too perhaps?!) I was pleased that there'd been a lot of choice in the two tasks - first in how they solved the question of what the picture represented, then of how to represent 100 differently. There had been so many different approaches to both!

One thing. There were mistakes. But a lot of them were what you might call silly mistakes, ones that the students realised were wrong as soon as it was mentioned. Counting the central white rod three times. Putting just eight tens in their design rather than ten. We all make mistakes like this, and maybe we can as a class get more forgiving about them. Also, having more practice with seeing numbers and counting flexibly should make the mistakes happen less often.