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.
source
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
Teacher
Teacher
Teacher
Student
Planning the procedure
Teacher
Teacher
Student
Student
Drawing conclusions
Teacher
Student
Student
Student

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.

Sunday, 23 July 2017

Looking back, looking forward - a few thoughts

I should take a moment to look back at how mathematics teaching in my K3 (5/6 year olds) class went this last year.

First some background: there was roughly 45 minutes of maths daily. I was so pleased that Annaïg teaching in the other K3 class was happy to plan so much together, and that the two classes worked on so much together.
One of the big things this year - you'll know this if you've seen my tweets - was using Cuisenaire rods extensively to explore equations. (See my Cuisenaire-related tweets from the year.) Cuisenaire-related work took about one third of the time. We started from just play, moved to making equal-length trains, then to writing these down using the initial letters of the colours, then on to writing numbers, with orange as ten.

It was good enough that I definitely want to return to it and follow a similar pattern with our K3s next year.

There was a lot of space for the students own creations and explorations. I was keen to keep a sense of agency, and tried to respond to any initiative. This was as important, I feel, as the exact direction we went in. That sense of 'this is an inquiry we're following because B started us off with this; let's see where it goes' is something I really want to nurture again, and even more so, next year.

My overall sense was of enthusiasm and enjoyment. I loved it when kids were bringing in drawing s of Cuisenaire staircases from home, and sets of equations that they'd been writing. But there were two students who at particular points said that they didn't like school! I asked X why and she said she what she liked best was sitting on the sofa in her pyjamas with biscuits and hot chocolate and watching TV. I could identify with that. Y told me she just wanted to play. That too I get. I'm going to try to make it more play-based next year, and work with smaller groups to make our learning more chatty and sociable, and let me listen in on thinking more.

A guiding thought was that I wanted to move forward as a whole class and didn't want anyone to feel boggled or left out at any point. There were two students who were my touchstone for this, Y and Z. Y often didn't really focus on what we were talking about as a whole class, although she was fine when I sat with her and we took things slowly. Z had lots to say and again benefited from having me close by; he found the writing down bit hard, knowing how to write letters or numbers or signs. It was these two (and maybe my own lack of nerve) that stopped me going further with equations with fractions in, even though more than half the class were comfortable with them. I feel like I did the right thing here, even though we didn't get the same astonishing progress that Gattegno and Goutard reported.

Next year...

  • I'm working with Marie as the teacher in the other class. I hope we'll work really closely; we're planning to do lessons at the same time, and have whole year group lessons that ensure we're sharing our best ideas with all the children.
  • I want us to use big maths journals, to keep all the photos of student creations as documentation, to allow the adults to scribe thoughts alongside these, and for students to add their written work into. And to look back and reflect on more.
  • We must include some things we hardly touched on at all (!) like time, and more of things that were under-explored, like measurement and 3D shape.
  • I want to include the parents in our learning a lot more.
  • Our two teaching assistants will play a full part in this.
  • We'll do 'number of the day' as a little ritual. Doing the 100th day was a big success, and the size of the numbers is just right for this age group, of all age groups. I'm hoping when we get back our new magnetic hundred squares will be hanging from the walls above the whiteboards.

Friday, 14 July 2017

Going Sideways

A problem with the metaphor of 'progress' in learning is that the 'journey' becomes roughly linear:
If that's extended to an individual lesson, students will be making 'progress' through the lesson. They won't all make as much 'progress' as each other.
Some of them have shot forwards, others are tarrying back nearer where they started.

And what to do with them then, in the next lesson? Put those three shoot-aheaders in a separate group? Ask them to hang around for a bit? Teach them and hope the tarriers will keep up?

This is a real challenge for us all, and I don't claim to have the answer. But I do recommend Going Sideways.
Take a detour, a road less travelled, follow a student's deviation, make room for the unfamiliar embodiment, for variation and investigation. After the number lines, try hopscotch.
Read a story about a hundred ants.
Look at a strange picture:
Solve an unusual problem.
Because maths isn't just forwards, it's sideways too. Maybe it's like this:
Or perhaps it's like this:

But whatever it's like, it's not in a straight line. So when the students go sideways,
make sure to show the ones on the left what the ones on the right did. And the ones on the right should see what the ones on the left did too.

Friday, 19 May 2017

Grids: inquiry and risk-taking

DW says he was playing noughts and crosses in the car, and then wrote numbers on the paper. He brought it in to me, a 10 cm square:
It lay on my desk for a day or two, then we showed it to the class. I asked them what they noticed about it. We wondered about the 17. We counted it through. Then I did what I often do when someone originates something - I asked the children, on similar-sized pieces of paper, to make their own, but different grids. There was a variety, lots with the grids continuing on the reverse side:
I was impressed with how careful the children had been, lots of them choosing to use rulers (I hadn't mentioned them). It seemed like the few that still needed to practice writing their numbers could get on with that, and the others could decide on a layout, a pattern, decide how far to go.

It took a few days for it to really sink into my head that all the children seemed to be enjoying doing this. There was an element of play, of choice. Grids could be large or small. They could even be different, like Bianca's even number one. In the Primary Years Program of the International Baccalaureate, inquiry is one of the key learner attributes we're looking for. For 5 and 6 year olds, this playful exploration is inquiry. There was a constraint, it should be a grid, but otherwise the field was open.

I felt like we needed to be celebratory, high-spirited in this play, so next time, I had large paper squares.

Another important learner attribute in the PYP is being a risk taker. When we looked at what had been done so far, I took a little more time on the ones that had been more different - the ones where the numbers weren't simply in counting order. If people were going to go new places, divergence would help. 'You can start in a different place, or from a different number, or try a different pattern of numbers.'
 A lot of children wanted to get up to big numbers. It took more than one session. So, while next day some finished, I sat with the others in front of the whiteboard. I drew a 4x4 grid and invited ways of filling it in. There were these:
We all contributed to the "random" one.
Then I asked the students to draw a 4x4 grid on little whiteboards and fill it in somehow:
These last two amazed me, capturing something I'd wanted to explore with the students. I'd just read What comes after nine? by  in the latest Mathematics Teaching issue, which looks at how this kind of table gives an enormous amount of naming power to students. Modifying it slightly, I put it together with 3 others to make a #wodb:
which we duly answered:
With the big grids, I was really impressed that the students who are sometimes the least confident in maths kept going on their own and went much further with numbers than I'd seen them go before. And of course there's the variety. (I wonder if our hopscotch work helped with this?) They're up on the wall now:
And the other K3 class is getting going too:
We've given them a middle-sized piece of paper and asked them to make one more grid at home.

And the K2 children have caught the grid bug: