Sunday, 11 March 2018


I'm thinking about variation.

In Mr Barton's podcast of his conversation with John Mason and Anne Watson, Anne Watson says:
"Comparison is a hugely important idea throughout mathematics. When you have something to compare, you don't focus on the thing itself, you focus on connections, same, difference, relationships between the things, relationships within the things."
I like the way she keeps it simple:
"Sadly, I think variation has become Variation, with a capital V, and some teachers think there's a right way to do it and a wrong way to do it. There isn't: it's actually embedded in mathematics."
Back in 2006, Anne Watson and John Mason wrote a short article, Variation and Mathematical Structure in the Mathematics Teaching Journal.
It seems to me, beginning to think about his, that considering variation is useful right from the start of education.

When we present the classic shapes poster to 3 year olds, we start doing things with variation.
Children get the message, whether we intended it or not, that shapes with straight sides 'sit' with their sides, not their corners, at the bottom. Unless they're 'diamonds'! That rectangles and squares are different things. That triangles and pentagons and octagons are regular. It seems innocent enough, but teachers have some lessons later on undoing these impressions. This is why Christopher Danielson's Which One Doesn't Belong?s are so good. They consider the possible variation, including counter-examples, to help students think about the categories for themselves.
David Butler also has some great quadrilateral posters which reflect the real range of possibilities.
Similarly, when we always write 2+3=5, with the sum on the right, we're not doing justice to the possibilities with this notation. We should  sprinkle in some 5=3+2 versions too.

Often a series of examples leads to the possibility of certain generalisations. For instance, back in this post, I talked about how a set of differences can lead to  noticing about subtraction.
Indeed, it did lead to a conjecture:

The other day I saw this tweet from Duane Habecker:
Putting this series of subtractions next to each other invites noticing - oh look, you can carry on to negative numbers! - and - look! - when you subtract -1, it's just like adding 1. I wonder if it always works like that...?

You could see a 'number string' routine as a way of presenting examples to show the range of  possibilities and asking students to make links and comparisons between them. Take this number string in a 1st Grade (Year 2) lesson with Kristin Gray.
The number of examples is limited; the focus on responding to an unknown in different places in the equation. There's a lot of thought that goes into how the different examples relate to each other.

Another routine that I've seen and used just a little is What's the same? What's different? with two images; there are lots on Twitter with the #samediffmath hashtag. Here's one:

I've thought about variation lots when designing Which one doesn't belong?s, but not calling it variation. I've just had a piece on them, written with Jim Noble, in Mathematics Teaching. and I've got a folder of them.
The design of them is fascinating: how to elicit the most responses, how to allow particular relationships to be noticed. Here, on this Othello board, my theme is half. That's what's the same about them. I could have introduced one where it wasn't a half. So we're mainly looking to arrangement for the differences. Are the two halves 'the same'? How many are there in total? Which colour is 'on top'? And so on.

I want to become more conscious of how we present examples, how we vary them, how we display new relationships in them. I'm now going to be looking out for how variation crops up in various places, trying to make my awareness of it as a theme more systematic.

Friday, 9 March 2018

Être plutôt qu’avoir ?

After Le Maître est l'Enfant, Estelle and I watched another French education documentary tonight, Être plutôt qu’avoir ? -  'To Be rather than to have?' It was a look at education, especially in France, historically. I found the pictures of school before the 19th-century desks-in-rows era interesting. Like this from Pieter Bruegel The Elder.

And then a look at some of the ways that some educators have enriched education for primary-age children: through circle times, philosophy for children, forest school, Montessori classes. Lots to think about again: good to now and again, or even regularly, challenge the way we teach - how much is a product of the relatively short history of public education, and how much is a conscious approach to real education?

We saw Célestin Freinet working the vegetable patch with his students. And some of his 'pedagogical constants', a kind of manifesto:
  1. The child is of the same nature as us [adults].
  2. Being bigger does not necessarily mean being above others.
  3. A child's academic behavior is a function of his constitution, health, and physiological state.
  4. No one - neither the child nor the adult - likes to be commanded by authority.
  5. No one likes to align oneself, because to align oneself is to obey passively an external order.
  6. No one likes to be forced to do a certain job, even if this work does not displease him or her particularly. It is being forced that is paralyzing.
  7. Everyone likes to choose their job, even if this choice is not advantageous.
  8. No one likes to move mindlessly, to act like a robot, that is to do acts, to bend to thoughts that are prescribed in mechanisms in which he does not participate.
  9. We [the teachers] need to motivate the work.
  10. No more scholasticism.
  11. Everyone wants to succeed. Failure is inhibitory, destructive of progress and enthusiasm.
  12. It is not games that are natural to the child, but work.
  13. The normal path of [knowledge] acquisition is not observation, explanation and demonstration, the essential process of the School, but experimental trial and error, a natural and universal process.
  14. Memorization, which the School deals with in so many cases, is applicable and valuable only when it is truly in service of life.
  15. [Knowledge] acquisition does not take place as one sometimes believes, by the study of rules and laws, but by experience. To study these rules and laws in [language], in art, in mathematics, in science, is to place the cart before the horse.
  16. Intelligence is not, as scholasticism teaches, a specific faculty functioning as a closed circuit, independent of the other vital elements of the individual.
  17. The School only cultivates an abstract form of intelligence, which operates outside living reality, by means of words and ideas implanted by memorization.
  18. The child does not like to listen to an ex cathedra lesson.
  19. The child does not tire of doing work that is in line with his life, work which is, so to speak, functional for him.
  20. No one, neither child nor adult, likes control and punishment, which is always considered an attack on one's dignity, especially when exercised in public.
  21. Grades and rankings are always a mistake.
  22. Speak as little as possible.
  23. The child does not like the work of a herd to which the individual has to fold like a robot. He loves individual work or teamwork in a cooperative community.
  24. Order and discipline are needed in class.
  25. Punishments are always a mistake. They are humiliating for all and never achieve the desired goal. They are at best a last resort.
  26. The new life of the School presupposes school cooperation, that is, the management by its users, including the educator, of life and school work.
  27. Class overcrowding is always a pedagogical error.
  28. The current design of large school complexes results in the anonymity of teachers and pupils; It is, therefore, always an error and a hindrance.
  29. The democracy of tomorrow is being prepared by democracy at the School. An authoritarian regime at the School cannot be formative of democratic citizens.
  30. One can only educate in dignity. Respecting children, who must respect their masters, is one of the first conditions for the redemption of the School.
  31. The opposition of the pedagogical reaction, an element of the social and political reaction, is also a constant, with whom we shall have, alas! to reckon unless we are able to avoid or correct it ourselves.
  32. There is also a constant that justifies all our trial and error and authenticates our action: it is the optimistic hope in life.

Friday, 23 February 2018

Becoming the Math Teacher You Wish You'd Had

Tracy Johnston Zager's book, Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms is a - perhaps I should say the - book that I'd recommend to all math/maths teachers, and that includes all of us primary/elementary teachers. And for those of us working in inquriy-led IB PYP classrooms, the fit is perfect.

Tracy takes a range of things that real mathematicians do as her starting point. Chapter 7 for instance is titled, Mathematicians ask questions. She starts with a quote from Jo Boaler's book What's Math Got to Do with It?:
Peter Hilton, an algebraic topologist, has said: “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” Such work requires creativity, original thinking, and ingenuity. All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students do not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked. 
She gives examples of resources for encouraging questioning, like 101questions and Notice and Wonder.

My favourite part of the chapter is one of the dips into real (and yes, vibrant) classrooms, this time with Deborah Nichols' first and second grades (p152). The question had come up, 'Are shapes math?'
The first step was to find out what the students wondered about shape.

And here are their questions. What an amazing set:
I want to ask some of these students what they meant by some of these! That first one, 'How big can circles go?' - is that about the practical constraints on us creating circles? Or is it about how circles start to look straight when they're really big? Like us walking on our planet. Or is she asking about circles in space? Or something else? Is the question 'How round can a circle be?' related?

Also, 'Are shapes fragile?' Did the questioner mean can shapes be distorted easily? Like the way a triangle or a tetrahedron model made of edges is quite robust, but a square or cube can be deformed easily?

Anyhow, what the teacher did was to arrange a sequence of experiences, with shapes that allowed for there to be a real dialogue between these questions, the shapes themselves and what the teacher needed to be learnt. Without allowing the inquiry to go off in directions that wouldn't really answer the questions, the students' questions - and the answers that came - stayed to the fore. Bit by bit the students built up the knowledge and vocabulary they needed to answer the more mathematical sides of their questions. And they thoroughly covered the learning that's set down for those grades.

As Tracy writes:
Perhaps knowing that students' inquisitiveness leads to the same ideas that mathematicians study and standard-writers emphasize can help us feel less pressure to tell and cover and explain. If we allow students to ask, we will likely end up in the same place but with much more engaged, empowered students.

Sunday, 26 November 2017

Laura - and me

I've been really lucky with the teaching assistants I've worked with. Not least with Laura, who was my TA in Year 4 / Grade 3. She created such a buzz in the classroom, and was a great person to have around to talk things over with. She was wonderful with the students, incredibly energetic and self-directed, and especially a wonderful wizz with art and display.
She wasn't so keen on maths, but even with that she was prepared to take on more and more; running our Which one doesn't belong? and estimation starters, and really getting into the spirit of things with groups.

I'm saying this in the past tense because... she's gone off to train as a teacher! She's back in England and really enjoying the course.

I offered to cast an eye over her first assignment, and found myself in there! Have a read:
Simon often provided his class with a hook to quickly engage and inspire them at the beginning of his maths lessons. He presented challenges as puzzles, number problems: the Fibonacci rabbit sequence for example, which one doesn’t belong and estimation games.

I had the pleasure of working with Simon for many years. His students would brainstorm numbers, some of them becoming methodical in their thinking, starting to see patterns and numbers everywhere! They would explore growing patterns using matchsticks, cubes, and Cuisenaire rods. Simon always asked the children open-ended questions and encouraged his class to explain their reasoning. They were taught to understand mathematical relationships and to develop and prove their solutions to problems in front of the whole class [...]
Additionally, his class were regularly asked to reflect on their learning, empowering them to self-assess and go deeper into their understanding and learning. 
I became passionate about maths during my journey with Year 4. I am beginning to understand maths for the first time, a subject which I had previously misunderstood. I had only ever been exposed to rule-bound mathematics, did not understand, and so became that child who believed she was no good at maths. Simon taught me that, to understand maths, you need to be solving problems, that maths can be creative and fun. As a trainee teacher these are the values and strategies I want to develop and during my next placement.
I like the picture of my class that Laura paints. And that last paragraph especially makes me feel proud!

Thursday, 9 November 2017

a realisation

I've had a realisation. Or maybe just a clearer picture of something that is 'in the air' anyway, or I sort of knew anyway. I'll need to go round the houses a little to say what it is.
HA said the bottom left shape doesn't point up so much. It seemed to point up quite a lot to me, and I told him so. But he was insistent, so I wrote it anyway.
Sarah wondered what he meant, and it made me go and find him and ask him to explain it to me slowly. He drew a bit more:
and explained that, if I understood rightly, the bottom right corner of that bottom left shape points downwards, and that kind of makes the whole shape less upwards-pointing.

I thought it was interesting that underneath the apparently simple idea of pointing upwards, other interpretations or meanings could lie. It just took a bit of digging.

It put me in mind of another WODB adventure from back in September...

A couple of posts back, I blogged about a pentomino Which One Doesn't Belong? and possible answers to it.
I mentioned, in error, something about concave and convex. I said the bottom right was the only concave shape, the rest were concave. Justin Lanier queried it
and it led to all sort of treasure. How would you be more precise about the intuitive idea of  convexness or of concavity? How would you measure it?

At first Justin, and Vincent, suggested the number of squares you'd need to add to one of the pentominoes to make it convex. I wondered about the same idea but with triangles. I found out that the shape this would create is called the 'convex hull'. Vincent went on to work out how convex the pentominoes were by the criterion of how much of their convex hull they filled:
I had another idea. What if you looked at the 'compactness' of the shape, perhaps looking at what proportion of it is within a same-area circle?
Justin then pointed us to an article, Convexity and Gerrymandering, that talks about all sorts of measures of shape compactness that reflect four essentially distinct characteristics of shape: elongation, indentation, separation, and puncturedness. That really struck me: essentially distinct ideas lurking under the surface of the intuitive sense of compactness or convexness! Ideas which could be measured in very different ways. The way the authors of the article chose was to take pairs of random points within the shape and see what proportion of times lines between those pairs of points fall entirely within the shape.
Rod then got thinking about how to calculate this theoretically, and then wrote some code to work it out experimentally:
The really striking thing in all this to me is this: you have a vague intuitive sense of how convex a shape is. You start digging. You see there are many different aspects of convexness that you could have in mind. How deep are the indents? What's their area? How stretched out is the shape anyway? And all sorts of other questions. Look at Vincent's and Rod's convexness results: they're different, they order the pentominoes differently, because they're looking at different measures of what being convex is.

Do you see how this links with what HA said?

Students, even 5-year olds like HA, have intuitions about mathematics. They might not link to the ideas on our own map, or trajectory for our students. But that doesn't mean that they aren't describing something that could be mathematised in some precise way, measured numerically. We may just need to question more, see it from another perspective.

HA may have meant (and I'm still not sure) something that could be expressed as: if from the centre of each shape we take the mean direction of all the acute angles, the house points up, the star nowhere and the bottom right points up. Only the bottom left points up and off to the right.

As Sarah quoted in her recent post:
  • All students have mathematical ideas worth listening to and our job as teachers is to help students learn to develop and express these ideas clearly.
  • Through our questions, we seek to understand student’s thinking.
So my realisation? 

Well, hopefully it's come through what I've written. That there is a whole wealth of mathematical interpretation of initial intuitions, that young students have them too, that we need to question and think through different perspectives if we are to honour these. That it's a wonderful and creative thing...

Wednesday, 1 November 2017

Play and Manipulatives

I went to see a documentary in the cinema yesterday (thank you Estelle), a charming inside-the classroom child's-height study of a French Montessori classroom, Le Maître est l'Enfant (The Teacher is the Child):

I love all the Montessori materials, like these beautifully-made cylinders and the wonderful tuned bells behind. I love these sets of cylinders:
I love the Pink Tower and the Broad Stair:
And who wouldn't like this lot. I especially like the ovoid and ellipsoid:
If I could, I'd have all of these things in Kindergarten. But then, there's a whole wealth of materials and manipulatives out there. How to choose?

I've just been reading Matthew Oldridge's blog post (again, thank you Estelle!), Stay in Kindergarten, For Your Whole Life which in turn is a review, of sorts, of Mitchel Resnick’s Lifelong Kindergarten.

Resnick makes the useful distinction between 'playpen' and 'playground' play, which I think I understand. Think of Lego. You get the play where you're puting it together according to the instructions, then there's when you do whatever you like with a pile of Lego. I agree, both are important. And I'm also looking for ways to bridge the two, ways of getting more playground into the playpen particularly.

Thing is, there's a lot of learning for us teachers to do about how to use any materials, which ones lend themselves to 'playground'-style play, which ones have the right constraints to help with more structured (but still playful) learning. Materials don't have magic properties when it comes to structuring learning, they have to be used intelligently and responsively. There's a sad bit in this section of a video about Froebel's 'Gifts', another wonderful set of sets of manipulatives.
Teachers ended up using the gifts 'by rote', 'dictating their use to the four and five-year-olds in their care'!

Look too at this photo from around 1970. The teacher is getting all the students to build virtually the same tower.
I still think they enjoyed it, but the teacher could have also said, 'Could you all build me any kind of tower?' and she probably would have got lots of this kind of thing and more, and the students would have been challenged in a different way, maybe felt more ownership, and have learnt more from each other. The teacher would have noticed new things, new next steps to take...

And we all need to learn about ways to build in more options, more choice.
Sometimes I see materials being played with in a way that suggest new more-structured inquiry that I haven't thought of before. Like how this student keeps making circles and circle-related things out of the straws and connectors:
What would a whole set of circles look like? What could you build a circle into? What other simple curved shapes are possible?

Sometimes I see a place manipulatives that aren't on the market. These last few days I've been making some Truchet tiles. They're not very regular, and I think I either need to go bigger or use the laser cutter:
Playing with straws and connectors, I've realised we need different connectors that don't just connect at 90°:
I've been lucky to part of Kassia Wedekind's short course on mathematical play. Teachers of students from Kindergarten to University all reflecting on what makes maths playful. It's really interesting to compare notes on what goes well in terms of play. Often people choose materials that really lend themselves both to divergent play and to mathematics, like dominoes, Cuisenaire rods, pattern blocks and Polydron. We looked together at Sara VanDerWerf's blog post You Need a Play Table in Math Class. It's challenging me to think how I bridge the playground-playpen gap with thoughtful responses to students' play.

And in case you haven't seen it, you might like to watch Kassia's talk on play:

Friday, 15 September 2017

Pentominoes Which One Doesn't Belong?

I often ask my students to find all possible shapes made of five squares - the pentominoes. There are twelve of them. Another good task is sorting them into two groups according to some criterion.
I saw John Golden was getting students to do this.
And it made me think a Which One Doesn't Belong might be the thing to uncover other criteria for sorting. I made one with my wooden set at home.
What do you think? Which one is the odd one out and why?

Normally I don't think it's useful for me to propose answers, because what I value most of all in this task is the maths basic, creativity - looking for yourself and deciding for yourself how to compare them. It's this that makes me return to WODBs weekly.

But Christopher Danielson does something useful in his teacher guide to Which One Doesn't Belong? He gives the background to the shapes on each page of his shape book, and talks through likely responses.

You could run WODBs without knowing background. You could just record students' responses and it would still be useful. This is especially true if you're happy with uncertainty, thinking on your feet and coming back to things later; but it helps to have thought through the possibilities. You're more likely to understand what the students are seeing, and the significance of it.

So, let's touch on some of the responses you might have to this. I got lots on Twitter that helped me to see a lot more than I had before, thanks to Vincent Pantoloni, John Golden and his students, Becky Warren and Rod Bogart.
One of the nice things about using photos rather than drawings is that you get extra aspects you might not have been thinking about. In this case, the shapes are made out of straight strips of wood; you can see the joins. So you could sort them by how many strips are needed.

Symmetry: the ones on the right have no symmetry, the top left has one line of reflective symmetry, and the one on the bottom left has four lines of reflective symmetry, and order four rotational symmetry.

How many 'ends' are there? These are squares with only one neighbour. How many 'branches' are there? These start at squares with three or four neighbours. And is there a part where four squares are touching in a square? How long are any branches (or 'appendages', or 'limbs')? How long are the longest straight lines? How many pieces could you leave by removing one square?

Number of vertices. Number of side lengths. What size rectangle would it fit into?

Negative spaces: what shapes are left by the concave part of the shapes? In all but the top right, these concave shapes are isosceles triangles.

Convex/concave: The bottom right is the only convex shape; the others are concave.

[edit: Why did I write that - it's not true. Thank you Justin Lanier for noticing, and for helping me build something out of this. The bottom right is 'more convex' in some way; as Justin puts it, it has the 'least amount of notch'!

Perimeter: The bottom right pentomino doesn't have a perimeter of 12.

Orientation on the page/screen: the X is the only one lined up with the edges of the image.

Pentomino addition: You can think of pentominoes being made by adding a square to a tetromino.
The bottom left pentomino can't be made from the L-tetromino by adding a square. You could go back to trominoes and think which can be built from those.
I made another WODB with my classroom pentominoes:

'Seeing' the whole shape from inside it: If the shape were the shape of a room, is there a point you could stand in to see the whole room? Not for the W.

And there are no doubt plenty more. Maybe you could comment with other ways that you see?

It all goes to show that a simple image can be the starting point for students' own ideas - there are a lot to choose from, and they don't have to be second-guessing the teacher or the maker of the image.

Saturday, 9 September 2017

Arranging things

We've just finished our first week of the new year, me and my class of five-year olds.

I've been thinking about Graeme Anshaw's blog post where he asks, What type of maths inquiry do we employ the most and the least in our classrooms?

He outlines these different forms of inquiry:

Demonstrated Inquiry
Structured Inquiry
Guided Inquiry
Open Inquiry
Posing the question
Planning the procedure
Drawing conclusions

How do we find a place for open inquiry, where the students are asking the questions? More specifically, how do I do it, with my class, especially as most of the students don't have English as a first language?

For young children like mine, how they ask questions is often through their play. If I pile these up here, what would it look like? How could I arrange them more satisfyingly?

Here's a couple of students arranging wooden blocks in a line, then arranging frisbees and bats on top. 
You need lots of components to get good patterns going - we need more bats and frisbees! We've just got more magnetic Polydron and the building is impressive. Here's T asking how he can transform shapes, and what happens if he uses triangles to add star-points to other shapes?

Others were enjoying our new straws and connectors, starting with squares, building cubes, and then puting them together to produce a tall tower:
Arranging squares:
Cutting holes in folded paper:
Making balls and worms of play dough:
Creating train track networks:
and playing with Cuisenaire staircases:
EL made this last series of staircases. Actually she had to make it three times. The first time someone slid some rods into it, and it was too late to repair, the second time it was standing up and got knocked over during lunch time. I made sure there was time when she could get it finished and photographed.

I think it's important that everyone recognises that there is maths implicit in all these things. Many people still think of maths as needing to be about counting or numbers or sums, and feel anxious that these should happen. But these things are there in the physical things that the students are doing already! And also:
  • Spatial and geometrical awareness
  • Categorising and sorting
  • Creating patterns
  • Investigating the results of processes
There are also the social skills being exercised in much of the group and paired work. All of the IB PYP social skills are needed at various points:
  • Accepting responsibility
  • Respecting others
  • Cooperating
  • Resolving conflict
  • Group decision-making
  • Adopting a variety of group roles
Of course, it's great to connect all this making to language, to talking about what we've done and reflecting on it, and ultimately to symbols too. And to start to abstract from the particular. But I don't want to be hasty with this. A lot of the student's creations are so pregnant with mathematical possibilities that I want to (as Helen says) re-propose them to the class, perhaps along with what they said to me at the time. I'm also this year, annotating photographs of the students' creations with them.

And there will be time to start making connections to some of the big concepts in maths. Like the way Graeme starts with the big central idea, "Our base 10 number system evolved for a variety of reasons and led to place values that extend infinitely in both directions." What these big ideas are varies from list to list. I see Jo Boaler and her team have just developed some:
Others have come up with big overarching ideas that stretch right over the years. Like Mike Askew's:
In addition to considering this, in the IB PYP we work with transdisciplinary themes, that bring out the connections between traditional subject areas. Including maths of course.

So, I'm hoping, for instance, to connect, for our current unit of inquiry on transport systems, the train track building that's going on with some discrete maths. How places are linked, how are nodes linked by edges. Moving from very concrete things like trains and cars, onto networks, reasons for networks, connections. I tried this last week, asking students to make roads between every pair of two houses, and after a few asking them to guess how many roads there will be:

What I really want to keep though in all this, while trying to develop the big ideas, is the wonderful, individual, confident play and creativity.