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.

The Prime Climb numbers are going up around the classroom too! Spot the 'deliberate mistake' @MathforLove pic.twitter.com/GUnKsPhqye

— Simon Gregg (@Simon_Gregg) November 25, 2014

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!

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.

We were looking at a 'Which one doesn't belong?' image, where children choose one of the four images that are different in some way to the other three, and, importantly, say how it's different.

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.

Yes. I am wondering if he means that the "support" part isn't as "high" as on the house. So fun to try to understand his thinking.

— sarah caban (@csarahj) November 7, 2017

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 convex shape, the rest were concave. Justin Lanier queried it

Also, why do you say that the bottom right shape is convex?

— Justin Lanier (@j_lanier) September 17, 2017

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...

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: