Saturday, 24 October 2020

Seeing the mathematical: Filling

I'm teaching in PK this year. It's my first year teaching the 3 and 4 year olds! Luckily, I've got a wonderful team to work alongside, who I'm learning so much from, and who do a lot to compensate for gaps in my knowledge.

One of the things that intrigues me is what mathematics looks like at this age.

I don't know the full answer to this - does anyone?? - but I do know it doesn't look like it does later on. Here are some of its characteristics, from my point of view.


  • woven into all sorts of other activities - art, building, role play, small world play, block play;
  • not mainly about numbers or counting;
  • mostly expressed through spatial means, often with physical objects;
  • hardly ever symbolic (for instance, using the names and written symbols for numbers);
  • often something that happens for a few minutes and then it's over for now;
  • not about trying to remember anything;
  • difficult for us to see, or recognise as mathematics.
We may not even recognise that we're not recognising it. It's a little like the way the substance of what becomes a tree enters the tree - not only are the roots underground, but the tiny root hairs where the uptake happens are hardly visible to us even when we dig. And, that's not all: most of what becomes trunk, branch, leaf, flower, fruit comes from the air, entering the tree through tiny holes in the leaves. It certainly doesn't arrive as wood in any way! And what does enter the tree - the carbon dioxide, the water, the minerals - doesn't enter in any obvious way - it enters through a million invisible doorways.

What are the tiny mathematical doorways for young children?

I came across an interesting list in an interview with author Grace Ling:

'I think the biggest challenge was to get out of the mind frame that “math is numbers.” I kept thinking it had to be kids counting, but after many talks with Marlene Kliman, a senior scientist and math specialist at TERC, she really opened my eyes to how we use math without even knowing it — sorting, sharing, comparing, finding, waiting.'
I was particularly struck by a couple of those items. Finding for instance - how does that link to mathematics? When I tweeted her that as a question, Grace answered, 'I mentioned finding because in “What Will Fit?” Olivia finds something to fit her basket.'
What Will Fit? - one of Grace Lin's Storytelling Math books
Yes, and finding because:
She has set herself a task;
She has set herself constraints;
She has a way of measuring whether what she finds will fit the constraints.
This is especially mathematical in my view, because she has her own inquiry that she is following through on.

In this case, the finding has to do with the size of the pumpkin - that it fits in the basket. There are no numbers involved. Here it's continuous magnitudes that are important, and these manifest themselves by a kind of comparison - does it fit in the basket? (My post on continuous magnitudes is here.)

Do we recognise this kind of fitting as mathematics?
I like blogging about this, because it helps me to get my thinking clearer - to focus in on the mathematics. Fitting in the pumpkin-in-basket case is about filling. Filling seems to link closely with the play schema of enclosing. There's a boundary and you put things inside it. With filling, we often want to completely fill up to the boundary, to fit in as much as possible. Sometimes, we dispense with the boundary, and just try to cover the space without leaving any gaps.

We do a lot of filling in our classes:
Filling a peg board
Filling containers with water
Filling 20 cm square trays with square tiles
Filling space without gaps with magnetic Polydron triangles
Filling triangular holes with pattern blocks
Filling a square tray with Tangram pieces
Filling a chessboard with glass pebbles
Filling a Numicon board
What are some of the qualities of the mathematics here? 
  • There's a rigidity in the frame, just as there was to the basket that had to fit the pumpkin, or sometimes in the way the pieces fit together - that is: there are constraints;
  • There is also freedom - the space can often be filled in a variety of ways. Take this last image of the Numicon. The student chose to try and fill with the light blue "2" pieces and the orange "1" pieces (we talked about them as 2s and 1s) until these ran out. Another time she tried with other pieces. She's also thinking a little about symmetry;
  • There's often a kind of beauty to the finished product;
  • The activity is often quite abstract - it doesn't have to link with a narrative. Children get used to abstraction;
  • The activity is about equivalence, equality - all the parts add up to the whole; and the different ways of filling are equivalent to each other;
  • There are discrete or continuous magnitudes involved - for instance the number of holes in the Numicon pieces or the space they occupy;
  • As well as the final product, there's a process, and the process could be different for the same end product - for instance in the way pegs are added to a pegboard: some students go round the edge first, some start in the middle, some fill randomly. There's time during the process for conversation, and comment on what's being done;
  • Where there's a boundary, there's usually a clear end point - when it can be seen there's no more space. The product, or a photo of it, is an object that can be celebrated, discussed and reflected on. Is there a pattern, symmetry? How is the student's work developing?
At the moment, it's hard for me to do the conversation part much. Most of my children have English as second or third language. Sometimes I'm talking to Spanish speakers in French. But a lot is communicated about the students' intentions in the choices they make during the fitting and filling.

There's probably a lot more to this than I've listed, but already that's quite a lot. A look at the overarching concepts referred to in the Diploma Program (for the 16-18 year olds) of the International Baccalaureate shows surprising links. Or, perhaps they shouldn't be surprising, since the more conceptual we get, the more generality:
  • approximation, 
  • change,
  • equivalence,
  • generalisation,
  • modelling,
  • patterns,
  • quantity,
  • relationships,
  • representation,
  • space,
  • systems,
  • validity
Which ones might crop up in filling?

The ones that jump out to me are equivalence, patterns, quantity and space.

Children aren't necessarily articulating anything about these yet, but they are nevertheless thinking mathematically as they construct examples, thinking for instance implicitly about equivalence. When we ask 5 year olds to make this more explicit, the background they've had of experiencing equal areas filled makes this a small step:
Equal-area Cuisenaire rods
Equal area pattern blocks

Tuesday, 20 October 2020

Positioning and enclosing

I often find it hard to fit the play and creation of my students into an overall map of development. Is this not something that's possible? Perhaps we have to trust that a rich ecosystem of experiences, play and creation will provide the environment for all the learning that needs to take place? In all this, can we identify thresholds, turning points, key moments? I don't know. Perhaps it's best to try and document and ponder certain things students are giving time to, having success with?

Here's a kind of success story. Though I'm not sure if I can discern a story arc in it, we could call it a months-long high point:

One of my 3 year old students is finding it hard to get used to life in school. One of her 'safe spaces' is to sit and paint or draw a picture. Here are some of her drawings:

She takes her time with them, often watching what's happening in the classroom from the vantage point of her creation. This one above is an early example. To me it's fascinating: how she has divided up the paper into sections with the orange lines and then experimented with dots in lines, dots in circles, circle parts, dots in groups, scribbly lines. 

She has a stencil here and she's adding areas of black using the stencil, on top of the orange and black lines and areas that she's already laid down.

There's a discipline in her investigations. This one has stuck to roughly oval areas of colour each separated from the other.

Here she has created her circular areas with dramatic movements of the pen, and then has carefully filled them with red.

More wild movement and careful colouring. I wish I could ask her about her pictures but we only share a few words of English. I'm left to interpret them myself, and, who knows? I'm likely to be missing things. But I see a concern with positioning elements, with spacing them out on the paper. Also a balance of wild and careful, of pen stroke and filled colour.

This one surprised me - hands! She'd done this after I'd been cutting out outlines of my hands to record ways that we could hold up three fingers.
She and another student caught the offcuts as I cut and made sure they went in the bin! I didn't imagine she would take the idea into her drawings.

How do I "assess" this? What development is there?

I think the answer is that I'm in no hurry for any development. This exploration seems so beautiful, so complete as it is. I wish it was all in one book. I'd love to be able to talk about the images. I'm glad I've got photos of many of them, before she folded them so that they'd fit in her bag to take home.

How do we analyse work like this? I'm not sure. The play schemas give us some small handle:
  • Transporting
  • Enveloping
  • Enclosing
  • Trajectory
  • Rotation
  • Connecting
  • Positioning
  • Transforming
Which do you see in these drawings? I see a lot of concern with positioning, and some with enclosing. Add in a little rotation and connecting. But this hasn't taken me a lot further. It does help me to see how  drawing links with other learning, opens my eyes to possible connections.

I'm pretty sure there will be development. I don't expect it to follow an obvious path - I can't predict how it will go. Will she keep that fine balance between the wild scribble and the careful positioning and filling? Will she continue to take her time on each piece? I hope so. 

Thursday, 24 September 2020

The Gardener and the Carpenter

I've just discovered the work of Alison Gopnik, and it's very interesting. These two paragraphs from a review of her book The Gardener and the Carpenter jumped out at me:


n 2011, a team of psychologists did an experiment with some preschool children. The scientists gave the children a toy made of many plastic tubes, each with a different function: one squeaked, one lit up, one made music and the final tube had a hidden mirror. With half the children, an experimenter came into the room and bumped – apparently accidentally – into the tube that squeaked. “Oops!” she said. With the other children, the scientist acted more deliberately, like a teacher. “Oh look at my neat toy! Let me show you how it works,” she said while purposely pressing the beeper. The children were then left alone to play with the toy.

In the “accidental” group, the children freely played with the toy in various random ways. Through experimenting, they discovered all the different functions of the tubes: the light, the music, the mirror. The other group, the children who had been deliberately taught how to use the toy by the teacher, played with it in a much more limited and repetitive way. They squeaked the beeper over and over again, never discovering all the other things the toy could do.

For us teachers, this is momentous. Just by the act of 'being a teacher', in the sense of demonstrating something, we can close something down.

Here's another similar experiment from a talk by Alison Gopnik. (I suggest you just watch to the end of the part about the Thingamibob experiment and variations, or maybe carry on a bit to listen to implications.)

Two things come out of this: the impressive causal reasoning of three and four year old children, and how the stance of the adult ('clueless' or knowledgeable) influences whether children bring their powers to bear on the subject.

I'm still trying to work out what this means exactly for us teachers, especially us teachers of young children, but teachers of all ages - and I'd be interested in your thoughts.

So - I haven't read the book yet - I think the distinction between the gardener and the carpenter, is that a gardener creates conditions for the shaping of the garden (the plants themselves will create the garden), whereas the carpenter does all the work on the wood, shaping it themselves.

This reminds me of Socrates, who said his mother was a midwife and his father was a sculptor, and that he aimed in his conversations to be more like the midwife, to help his conversation partners to bring to birth their own ideas.

There's this thing called Socratic Ignorance. Partly it seems to be a genuine understanding of the limits of our knowledge, partly a device for getting back to the 'clueless' way of operating in a conversation. For instance this, edited for brevity from the beginning of the Meno dialogue:

Meno: Can you tell me, Socrates, whether virtue is acquired by teaching or by practice; or if neither by teaching nor practice, then whether it comes to man by nature, or in what other way?

Socrates (edited to keep the quote short):  ...And I myself, Meno, confess with shame that I know literally nothing about virtue...

Meno: No, Indeed. But are you in earnest, Socrates, in saying that you do not know what virtue is? And am I to carry back this report of you to Thessaly?

Socrates: Not only that, my dear boy, but you may say further that I have never known of any one else who did, in my judgment.

So, lots to think about.

I recommend this TED talk by Alison Gopnik too:

Friday, 12 June 2020

Mathematics Inquiry

The Power of Inquiry

We're reading Kath Murdoch's great The Power of Inquiry, my friends-and-colleagues Rachel and Estelle and I. We've had some really nice Zoom meetings chatting over chapters.

The Primary Years Programme of the International Baccalaureate, which our school follows, gives a central place to inquiry - and yet, and yet, on the ground, old habits die hard. The book challenges us to think how inquiry works in practice, what we can do to make it more of a reality in the classroom.

We've jumped about. We're reading Chapter 3 at the moment - 'Beyond Topics'.

On page 40, Kath Murdoch lists features that characterise most journeys of inquiry:
  • They are generally driven by questions - both teacher and student generated.
  • They require active research/investigation.
  • They most often seek to connect learning with students real-life experiences.
  • They are as much about process as they are about content - and content is conceptual.
  • Students experience connected learning episodes - one task is clearly linked to the next rather than being simply an 'activity'.
  • The learning is responsively planned (rather than fully mapped ahead in detail).
  • Aspects of the learning tasks / assessments are co-constructed with students.
  • The planning is emergent - the details of the process unfolds rather than being pre-determined.
One of the things to strike me was that these describe well how I see lessons that involve thinking mathematically.

(For the third point, about real-life experiences, I have a qualification to make: mathematics, as well as describing many real-life situations, is an abstract study in its own right - a little like some art and music are. It's not always, or even mostly, going to be about real life.)

Kath Murdoch gives some examples of Mathematics inquiry questions:
  • How do we measure time?
  • What is long?
  • What makes a pattern?
...and many more.

These are big questions, but on a much smaller scale, small moments in the mathematics lesson can also show the same features. I want to write briefly about some examples, in this case with young learners in playful, child-initiated contexts.
Here are some of our PK children (4 and 5 years old)  making a kind of carpet. They've put lots of square magnetic Polydron together, and now they're adding a triangle border. It was initiated, and completed by the children without any input from the teacher.

They haven't stated a question in words, but play like this is purposeful, and you could say it's investigating an implicit question. Something like, 'Is it possible to make a really big rug out of squares and then round it off with a border of triangles?'

If I was to extend this with the children, I might want to show the class the image of what was created and invite comment. I might ask if other sizes could be made. What is the smallest rug like this that's possible? How about if there was some pattern to the colours? What would that look like? I might keep the image up somewhere for reference. And wait and see if the inquiry took off again. Connected learning episodes allow students to tune in, to make variations on a theme, create a base of shared experience for discussion.

In one of our K3 classes (5 and 6 year olds) recently, a boy, I'll call him Y, was making a similar pattern in an odd moment, this time out of pattern blocks. He often chooses to create patterns with manipulatives. I sat with him. He doesn't generally want to chat much or even answer questions; he sometimes has his own monologue about what he's doing. I knew I wouldn't be able to 'steer' his play much, but still I wanted to be there, maybe throw something into the mix. Here's what he made:
I admired his square and how he'd surrounded it with a border of triangles (might this be a good way into investigating perimeter with older students?)
Y then broke that, and started making a bigger one. Then he made a bigger one still.

This seems like the essence of a lot of mathematical inquiry. First of all his attention is on a whole - the square surrounded by triangles. It's quite a satisfying whole. Then he wants to see how that extends. Does it just work for his particular case, or is there a general pattern? I got the iPad  out and recorded what he'd done. I also started making the ones he'd broken so we could see them all together.

Concepts here:
  • We don't just look at individual cases, we look for patterns or regularities in the situation.
  • We find ways of keeping the pattern as well as the individual case in mind.
As a teacher, in a playful child-initiated situation, I want a light touch. I want to understand his interest, the unspoken 'question' he's answering for himself, and do what I can to enhance that inquiry, including if necessary, staying at am's length!

I did direct Y's attention to the pattern of numbers of triangles in the whole family afterwards: 4,8,12,16,20... but I don't think he was that keen on me 'directing' anything!  I didn't push it, knowing that if we valued his 'question', his beginning, this was something that could be a focus later.
I enthused about his work a little and posted it on Seesaw. A few weeks later, I popped into his classroom and saw that he was making the same patterns. And another time, he made it with Polydron.
It was obviously something that Y was getting satisfaction returning to.

This process - of actively returning, of varying, of building on previous experience - is so much part of the process of mathematics - whether it's child-directed play, or teacher-led investigation. I would love all our teaching to centre on and respond to this process of inquiry, whether it be with a big explicit question, or as in Y's case, a small implicit question.

Thursday, 27 February 2020


Plato famously defined knowledge as 'justified true belief'. There have been critiques of this, but it seems like a good first approximation. The 'justified' part is important. If I roll two dice and say, 'It's going to be a seven!' and then a seven is rolled, because I have no reason for believing that a seven will be rolled, I did not know that it was going to be a seven.

Something similar happens with misconceptions. If someone has a misconception they have a reason to know correctly.

For instance, we might say there's a "public misconceptions about antibiotic use" - people perhaps think that antibiotics will help with a virus. But they have good reason to know this is not true - it is widely held by pretty well all reliable sources, and plenty of easily-available sources at that. They are adults and have had lots of opportunity for learning this.

But, I often see educators talking about children's misconceptions. Often it's children who have no reason to have had much experience of the matter they're supposedly misconceiving about. Many of their conceptions will not be our mature ones. But they are not mis-conceptions. They have no reason to have our more expert ideas yet.

Some things will just be 'Terra Incognita' - as yet unknown land. The way European explorers marked parts of Australia or America while only parts of their coast had been explored. Like Australia here:
It seems incorrect to call these misconceptions, as if those cartographers somehow had access to our modern knowledge but hadn't properly absorbed it. It's more just... not knowing yet. The map-maker gives the Terra Incognita a fairly random coast - perhaps they should have made it blurred or just drawn the parts of the coast they knew. But that's another skill again - to be able to judge and show the extent and character of your knowledge and certainty about something.

It's not just a question of semantics. 'Misconceptions' often seems to have a note of frustration in it: 'They had all the information they needed, but then they've gone and adopted this weird idea that will probably be difficult to correct.' Perhaps there's also a hint of satisfaction in there, that I am the one with the correct conception of things.

If however it's a question of territory that's still unfamiliar, where learners need more exploration of a familiar domain or an introduction to a new domain - then the onus is on us teachers to make that new exploration as achievable and instructive as possible.

Thursday, 20 February 2020

Natural powers

I'm reading Designing and Using Mathematical Tasks by John Mason and Sue Johnston-Wilder, and liking it lots.

One thing I like is the discussion of natural powers. On page 74 they write:
The kinds of powers that are relevant for learning mathematics are ones that learners will have demonstrated by the time they arrive at school. Learners have innate ability to emphasise or stress some features and to ignore others, enabling them to discern similarity and difference in many subtle ways. They can also specialise by recognising particular instances of generalisations, and they can generalise from a few specific cases. In addition, they can imagine things and express what they imagine in words, actions or pictures, together with labels or symbols.
That's quite general itself, so I wanted briefly here to 'specialise', to think of a specific case, and how these powers might be evident in it. Taking one of my favourite routines, Which one doesn't belong?, how do these natural powers appear?

Let's look at this record of discussion with a Grade 2 (=UK Year 3) class:

Stressing and Ignoring
The image itself contains myriad possibilities for comparison, for finding the odd one out. As children contribute, they're stressing just one feature of the shapes, and ignoring the others for the moment. For instance, take that observation in the bottom right. In Triangle D, the coloured shapes are next to each other symmetrically. All other features have been momentarily blanked out or screened off. The student considers only the symmetry of the coloured shapes. They then have the challenge of flipping out of their focus, and hearing each other, focusing in with them on each new feature in turn.

Specialising and Generalising
Finding an odd one out involves finding a group whose elements share the same general property. When Patrik identifies the bottom left (C) as being different, he's seeing a general feature of the others that the coloured areas 'touch'. For his definition, he's including touching corner-to-corner as touching. Are students specialising from the generalities they identify? Perhaps as they check that the general property they notice is in fact exemplified by three of the triangles and not by another. It might also be interesting as follow-up to ask them to make other triangles which fulfill the general condition, or which don't.

Distinguishing and Connecting
Like the same-different routine, where the teacher asks of an image 'What's the same? What's different?', here I think the seed of an idea might well begin with noticing two images being the same in some way, or different in some way, and then checking the others. So, perhaps Manu looked at A and C and saw that, for both of them, their coloured shapes were not equal area. He compared with the others. D jumps out as having equal-area triangles. Antonio was talking about B, saying that he thought, but wasn't sure, that the two pink areas have equal areas. Manu thought not, which gave him three triangles without coloured shapses being equal. Only in D were they clearly equal.

Imagining and Expressing
Santi commented that D could be folded up to make a solid shape. I often find children imagine transformations with these shape-based images. They might imagine moving part of the shape round, or chopping part off, or filling in a concavity. They might imagine, as in this case, folding the shape. This makes me think, as transformations come up more than we seem to teach them, maybe we should be teaching them more. Anyway, in this case, Santi went off and found four Polydron triangles, connected them up like D and then folded them - and behold  - a tetrahedron! I'd taught most of these children two years before, so one student, B, and I recalled when she had made the same shape two years ago and been impressed that it could unfold into a different net than the one she started with.
About expressing, all of these students, almost all of them not first-language English speakers, found ways of explaining their idea. Sometimes the idea might be subtle or even opaque at first, but I've found that it's worth trying to listen and hear what they're saying, even if it seems at first not to be correct. (For more on that, see this post.) Hearing themselves and each other articulate their ideas is both a reminder or introduction to many mathematical perspectives and a celebration of bringing natural powers to bear!

Conjecturing and Convincing
Often children's contributions have an element of conjecture about them. They might not have double-checked; on closer inspection, one of the four images might have the property they thought it didn't have. So there's sometimes a little risk in making contributions. They are making claims. It's good to get used to doing this in this bite-sized way. They may need to justify the claim, again in a small way. 'You say B is not diagonal or straight. What do you mean by that?' Sometimes, there's a claim that could be debated and investigated further. One such is the claims made about triangle B by Antonio and Manu. Antonio claimed at first that the two pink triangles were equal-sized. He then changed his mind. Manu claimed they were different sizes. I chose not to spend time on this then, but with other classes we've investigated a similar claim further (blogged here).

Organising and Characterising
In a sense, when we play this game, we are mentally putting three shapes together and separating another. Sometimes students bend the rules a bit, and want to tell me when two fit in one group and two in another. For instance where Gustaw says that A and D do not have stripes. The students are sorting, finding a taxonomy for the four images in front of them.


Looking at these natural powers helps to anatomise some of the ways in which students are thinking when they perform this task, and in fact many tasks. It also helps to explain why I like WODB so much. On this occasion, every member of the class contributed. When I stopped, I had to promise the ones who still wanted to speak again that, even though the board was full, I would hear them while we got on with our next task, which was to construct quarters at our tables using a frame and tiles.

Saturday, 16 November 2019


One of the great things about my STEAM coach role this year is that I get time to research new approaches to familiar units of inquiry. Last year I taught the Grade 3 Structures unit, but felt I wanted to explore the ideas of load and stress a little more scientifically.

The Structures unit of inquiry is part of the How the World Works transdisciplinary theme, with these as the lines of inquiry:
• Properties of building materials
• Considerations to take into account when building a structure
• Structures used for building
• Relationship between structure, design and function.

Why are we seeing so many circles, triangles and squares in structures? Which one is strongest in which ways?

One idea I came across this year was the simple idea of using a sheet of A4 paper to construct three simple towers, one triangular, one square and one circular in cross section. How many books would these hold up?

We made predictions first:
The G3 students really got stuck into this investigation. Here, in G3P:

The results were similar in the two classes. With twelve groups getting similar results, we think the results are quite reliable.
And then, the big question is why? Why is the cylinder tower so much stronger? We don't know the answer, but here are some of our ideas. Maybe more than one of them is correct?

The following week, a mother of one of our Grade 3 students, Mrs N, who's a stress engineer, came into school and talked to us briefly about what being a stress engineer involves.
We also looked at some transparent plastic being pushed or pulled between two polarised filters.
Rainbow patterns start to appear. And the more the plastic is stressed, the more rainbows there are.
In pairs, we wrote about what we noticed and what we wondered.
We shared some of our observations and questions together:
Mrs N showed us a sponge, which you can squash. Engineers call this squashing 'compression'.
She pulled a hair tie too, to illustrate stretching, or as engineers call it, 'tension'.
She also drew a rectangle on a pool noodle:
When the pool noodle was bent, we could see what was happening with the rectangle:
This has started to give us more idea about what was going on inside those three towers.

We asked Mrs N about becoming an engineer. What do you need to be good at? She said that she had always enjoyed maths. She was also really curious about how things work and used to like to take things to pieces!

Mrs N wrote later to give her impression of the lesson:
"They were very clever and they could imagine what was happening in the sample without need of anybody to tell them. 
I am amazed by the learning methodology at the school, letting kids think and solve the problems by themselves with just a guidance of the teacher instead of teachers/adults lecturing them.  I am sure this has a positive effect on their learning."
There are still questions though. What is it that makes the cylinder pillar stronger? Would several smaller pillars be smaller than one? What if the pillars were covering exactly the same area? If we have time, we may investigate further.

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.