Friday, 14 April 2017

Helen & Mike's Domino workshop

Here's my report on one of many lovely sessions at this year's ATM Conference, which I went to for the first time this year.

I've picked it because of the way people with all sorts of depths of mathematical knowledge, working with all sorts of ages, worked with each other. There was no hint of haste or one-upmanship, no competition or even comparison; just a lot of good humour and concentration, and a lot of maths.

As Helen tweeted: One very important guiding principle of @ATMMathematics: "Any possibility of intimidating with mathematical expertise is to be avoided"

Helen and Mike got us working in pairs. Minisha and I worked together. The first thing we had to do was, from a set of blank dominoes, make a set up to 3-3.
We were asked how we did it. Minisha was thinking about whether they would make one big loop. Mike and Helen meanwhile were going round chatting to people about what they were up to.
Other people were trying other things.
Then we moved on to triominoes. We were to construct a 3-3-3 set.
Mike was on hand to answer our question, almost before we'd asked it: was the 1-2-3 domino the same as the 1-3-2 domino? He said we could do it either way, but his bought set had them just one of them. Minisha and I noticed the 0-0-0 set had only one triomino, the 1-1-1 set had four and the 2-2-2 set had ten. It was looking like we were adding triangle numbers, making a kind of pyramid of triangles. I wanted to make shelves so that we could show the pyramid. Looking around... the water glasses...
We organised our pyramid differently to the way we'd created the triominoes. As you moved towards the bottom left there were more ones in the triomino, towards the bottom right more threes, the bottom back had more blanks. And as you moved up you moved towards two-ness. Helen was coming round with her notebook, writing down things that people said.
Mike wondered why we hadn't put the 0-0-0 on the top, so that the second layer would have the additional 1-1-1 set triominoes, and the third layer the additional 2-2-2 set triominoes, etc. I think Minisha and I were both pleased with the way we'd done it, and other people liked the 3D-ness of it.
Helen and Mike stopped us after a while and Helen read out ("re-proposed") some of the things she'd heard said, including my thing about two-ness. She just left a pause after reading them... making me feel like I needed to say more to everyone, which I did. Someone asked how we'd show the next-sized set with our organisation. I said we couldn't. John on our table wasn't happy with that. 'Well, we'd need to go into four dimmensions,' I said. 'We couldn't do it with the glasses.'

I can't remember what Helen and Mike asked us to do next. I think we all had things we wanted to check out. Maybe they told us all to get on with it.

I had a quick look round.
Could the 20 3-3-3 triominoes be fitted, with symmetry onto an icosohedron?
Could the icosohedron's edges have the same totals?
What is the formula for finding the nth tetrahedral number?
Could you make a game with L-shaped dominoes?
Minisha and I decided to reorder our layers in the way Mike had mentioned, seeing how that would look.
I remembered a way that tetrahedral numbers were put together to make a cuboid, tried to make it with Cuisenaire rods and failed, googled it and found what I needed.
Six of them fit in the cuboid. 

Barbara came over and started looking for a good way of proving that the sets would always follow this pattern, looking at each triangular layer. I peeled off and for a while contemplated the diagonal in Pascal's triangle that has these numbers in.
But I hit the buffers at this point.

What Mike and Helen managed to do was not only, or primarily, show us a fertile area for maths exploration that could be approached by any age, but also to show us a kind of lesson, where they, as teachers, are present more to gently orchestrate a group of people exploring related interesting questions. They watched and listened, enjoyed what people were doing, asked questions, called us all together to share a few things, then sent us off again. Some of you will recognise this pattern. If you're fortunate, you'll recognise it from your own classrooms.

Monday, 27 March 2017

Beyond Answers

Most mathematics curriculums give a place to the "process skills" of the subject. When the first English national curriculum was unveiled in 1989, its "Attainment Target 1" was "Using and Applying Mathematics"
"Pupils should use number, algebra and measures in practical tasks, in real-life problems and to investigate within mathematics itself."
Successive incarnations of the national curriculum had similar sections or statements. Problem was, we tended not to turn to this bit. It was the more particular 'add two two-digit numbers'-type statements that we turned to on a week-by-week basis. Why? Maybe the process skills are, when expressed this way, too general, too all-pervasive; it's easier to turn to the more specific, to the 'know the names for various kinds of triangles' sort of requirement.

But the process skills are where the mathematics really happens!

In the US, the Common Core State Standards for mathematical practice are:
  • Make sense of problems and persevere in solving them.
  • Reason abstractly and quantitatively.
  • Construct viable arguments and critique the reasoning of others.
  • Model with mathematics.
  • Use appropriate tools strategically.
  • Attend to precision.
  • Look for and make use of structure.
I suspect there will be the same tendency to pass quickly to the more week-by-week standards.  And yet these skills are the core of what it is to work mathematically. What to do? How to make 'reasoning abstractly and quantitatively' less nebulous-sounding?

Enter Mike Flynn's book Beyond Answers.

Mike devotes a chapter to each of the mathematical practices. He begins each with a personal experience. This is useful in a number of ways. One, it's good to get to know the author of a book you're reading. It somehow makes the exchange more on the level, more real. Two, he gives an analogy for the practice from his own experience. I like analogies; they work well for me, and I guess they do for other people too. They ground the practices in our everyday non-mathematical experience.

For instance, in the introduction Mike talks about getting lost in Boston, a town he wasn't that familiar with, and how his brother helped him to learn to navigate the city without helping him too much, by asking powerful guiding questions:

I like the metaphor in lots of ways. For one, mathematics is like a city, with links criss-crossing it in all sorts of ways. Also, navigating is a primal kind of function for us, one we share with animals, whether hunting or fleeing, patrolling or harvesting. I blogged about the link for me once. The book is about how we help students to navigate for themselves, developing skills to find their way round, to become at home in the world of mathematics.

Another thing I like about the book lots is that it gives us a lot of classroom vignettes that exemplify teachers enabling their young students to develop the standards. It's an approach that's worked well in a number of great recent teacher education books, because we learn by concrete example more than by definition.

There can often be just one adult in the classroom, me, the teacher, and though we learn so much from the students themselves, how do we get to learn from other adults? Well, these kind of vignettes let us peek in at key moments, listen to an actual conversation, catch how adroitly the teacher puts themselves to one side while she allows ideas to come from the students rather than her.

There’s a consistent emphasis on student voice in the book. ‘The benefit of taking extra time to discuss their strategies is that it allows ideas to come from the students and not from me. That means they have opportunities to hear mathematical arguments from their peers and to critique their reasoning (MP3). It also helps to create ownership of the ideas and changes the power structure in the classroom by showing that we all contribute to the learning in math class’ (p123).

Here's part of a vignette from chapter 3, Construct Viable Arguments and Critique the Reasoning of Others:

It’s important that we think about the big picture of what we’re doing in mathematics lessons, and how it links ultimately with what we do in the adult world, and Mike takes time to spell out the essentials of each mathematical practice.

In the chapter on MP 4, Model with Mathematics, Mike discusses one of the essentials of modelling, the process of abstraction.

Abstraction is an essential tool way beyond the maths curriculum, as I’ve mentioned in a post before, but Mike helps us see how it’s there in so many of the simple tasks we give young children.

He gives the example of a first grade class who are asked to make representations of who sits at their tables (p84). They show it in lots of ways:
The students are learning to ignore, for this task, the myriad other important features of the situation, the appearance and personalities, how they’re relating and feeling, what their position is in relation to each other, and just to focus on number and one category, boy/girl. This narrow focus, ignoring most of the context, is something young students learn through practice.

You can see how this process works with adult mathematical modelling, how for instance a traffic planner ignores all sorts of context in traffic, car colour and make for instance, to mathematise the situation, focusing on numerical data such as number of cars and speeds. Once she’s represented the structure of the network of roads and the numbers she needs, she doesn’t for a while need to think about them as roads or cars as such. After calculating possible solutions, she can then translate them back into road locations, numbers of lanes, traffic capacities and suchlike, and ultimately actual road construction can begin.

Young children learn to do essentially the same thing. Mathematising is there all the way. When we count a handful of pebbles, for a short while we attend less to colour and texture, to shape and size. Or, if we sort by shape, we might ignore all the other features.

The three-act task, as Mike says, is a really useful tool for helping students to mathematise. Graham Fletcher’s great collection is a great introduction to this, and one I’ve just been sharing with teachers in Qatar.

In Chapter 5 (Use mathematical tools strategically) Mike lists examples of tools in five categories: supplies, manipulatives, representation tools, digital tools, and mathematical tools. I was particularly interested in the discussion of the last, also called ‘thinking tools’. As he writes (p111), ‘A large part of our work with MP5 developing and supporting our students’ metacognition.’ We’ve all had students that when asked, ‘How did you know that?’ answer, ‘With my brain’! Helping students to recognise how they’re actually thinking, in detail, is such a wonderful part of what we do, and Mike gives attention to this, with plenty of great examples and vignettes.
I loved ‘Are there any other foxy shapes?’ in chapter 6 (Attend to precision). On p133-6 three’s a great vignette of how children play a game sorting shapes according to some property they’ve chosen. (This would make a great development from a Which One Doesn’t Belong!) One student’s category is ‘foxy shapes’ and the Socratic way in which the teacher shifts the children from a sort of holistic recognition of a fox-faced shape to themselves giving it a precise definition of essential properties is a delight.

I realised that I'm probably an MP 7 guy (of course all of them are important!) - someone that spends a lot of time in class getting students to explore and search for structure. I'd not seen such a complete list of the sorts of mathematical claims young students might make; is that kind of thing out there widely, because it should be? I do less problem-solving (learning to do more!) but focus on this lots.
When you add one to a number the result is the next counting number.
I've merely touched on a few features of the book. There's so much more in there. It would be an excellent introduction to what's important in mathematics education for any beginning teacher of five to eight year olds. And an excellent tool for reflection for an experienced one too, bringing the focus back again and again to students' reasoning and discussion.

As Mike says, 'Enjoy these moments with your students and cherish the opportunities to learn alongside them, for this work is as engaging for us as it is for the students.'

Wednesday, 22 February 2017

I close my eyes and see...

I wonder if you could do this?

When you've read this sentence, could you close your eyes and, in your mind's eye, see some birds standing somewhere (and when you've done that, open your eyes and read on).

A question:

How many birds were there? Was it a definite number?
(I'm interested. If you reply in the comments, I'd really like to know.)

I got interested in this question after I read this by Borges:
"I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer–not-nine, not-eight, not-seven, not-six, not-five, etc.–is inconceivable. Ergo, God exists."
Now, I'm not really into the metaphysical dimension here I'm guessing that Borges' was playing. What interests me is the visualising and the idea of an indefinite number.

For one, we don't ask children to visualise very much, and this seems like another way to approach thinking, including number and shape.

Another thing is the idea of the indefinite nature of the interior image and its slightness and malleability, its sometimes fleetingness, vagueness.

One of the great pluses about number talks is that we're asking students not just what's 'out there' but what's in there too; How did they see those dots? How did they do that calculation? How are they sure? This kind of introspection is useful, and not just in mathematics.

I asked Sam and Pam to imagine birds on a wire. Then I asked how many there were. Pam said she saw seven, but wasn't sure if that was how many were in the first mental image she'd seen. Sam said it was a number between ten and thirty, but he didn't know the number.

I found these caveats intriguing. Down into the place where numbers aren't definite.

I asked Sarah and Lana. Sarah went on to write a great blog post about it. She says:
If, in my minds eye, I see more birds than I can subitize, can I ever truly count them in their original form? Can I capture them? Or will they always be a “clump” somewhere between 10 and 15? When I try to count them, do I change them? By assigning them a number, do I bring them into existence?
She asked her husband, who said:
4 pigeons.
There were birds and their existence was switching around. There wasn't a set amount; they were flicking between a few and a couple. I picked four when you said 'how many?' because I knew if I didn't answer you, you would say, 'I need you to tell me a number'.
I love the candour of this reply, admitting to the process by which a thing whose existence is switching around becomes a definite number.

Lana said:

Lana asked her students, who gave quite definite answers:
I've asked a few pairs of my students too. With each pair I asked them to visualise a square first, and then a circle inside the square. They do that straightforwardly, drawing them on whiteboards. Then:


The next pair didn't draw the circles inside the squares. They said they'd seen:


D, on my right went on to talk about a quintillion!

I'm struck by how the pairs give similar answers. Small numbers, big numbers, many really big numbers.

I wonder how much what they say reflects any original image? I know that as a boy I wasn't particularly concerned about  being truthful. I remember my teacher asking me about a picture on the wall that I'd painted, of a deer under a tree on a hill, 'Did you see it somewhere, or imagine it?' I thought, 'What does he want me to answer?' and said 'I saw it somewhere,' because I thought that was what was required. Actually I'd imagined it. It was only in my teens that I began to discover the pleasure of talking about things as they really are, the pleasure of sharing real thoughts and experiences with all their ambiguities and questions. 

In addition, the confabulation of children is delightful and very fruitful. A four year old girl I don't know comes up to me in the playgound. 'Would you like a sweetie?' 'Okay,' I say. She hands me a stone which I pretend to gobble down gratefully, and walks away.

How many birds did you see?

Friday, 10 February 2017


Late in March I'm running five workshops for elementary/primary teachers in Qatar on using technology in teaching maths.

I'd love it if you would have a look at my skeleton plan and share any ideas that come to mind!

Saturday, 21 January 2017

Cardinality, ordinality and developments with the Cuisenaire rods in K3

A few posts ago, I talked about asking children to use a particular manipulative, thinking of the Cuisenaire rods in particular. I got some great replies, most of which emphasised asking children to select the appropriate tools is an important part of the problem-solving process. I've been pondering this lots, which is why I haven't replied to the replies. I'm also reading Mike Flynn's brilliant new book Beyond Answers, which outlines how the CCSS Standards for Mathematical Practice can be brought to life in K-2 classrooms (I hope to blog about the book soon).

I'm trying to make problem-solving more and more part of my class, and I want to have more lessons where the children select the right tool, whether it be cubes, number line, hundred square, ten frames or whatever to help the solve problems.

But I've also got the provisional conviction (if such a thing can exist!) that the Cuisenaire rods, used right, can give young children an environment to explore in a more open-ended way, and give them a really robust and flexible number sense. It's a conviction I want to test through my reading and thinking, and also in practice.

There's a few things the rods do really well. One is being solid colourful things that children enjoy making things with. That's a great starting point.

Another is that, in some ways, they bypass counting.

Now counting is important and I've been putting more emphasis on counting collections this year as well as choral counting.

But counting is complicated. As Young Children's Mathematics by Carpenter et al summarise it,
  • There's an ordered sequence of counting numbers, and numbers are always assigned to items in a collection in the same order starting with one.
  • The one-to-one principle. Exactly one number from the counting sequence is assigned to each item in the collection.
  • The cardinal principle. The last number in the counting sequence assigned to the collection represents the number of objects in the collection.
And when it comes to counting for addition or subtraction there are added complications.

Alf Coles contrasts this way of knowing numbers, cardinality, with one that isn't based on a set of objects counted, ordinality, which is represented as a teaching approach in Gattegno's use of Cuisenaire rods:
"One clear hypothesis to emerge is that students’ awareness of ordinality may be distinct from awareness of cardinality and, in terms of developing skills needed for success in mathematics, that ordinality is the more significant."
I see it as, once the different lengths become familiar, children can think about addition and subtraction without having to count. Like here, you can see that the pink plus the white are the same length as the yellow. You get to see pretty quickly that if you switched the white and the pink they'd still equal the yellow in length. And lots more besides. You can hold the whole relationship in your head, without any smaller units distracting.
What I've been able to do is help the class to gradually build up a familiarity with this, and say to the students, 'you go and make something now, and write it down.' And what's really exciting, they're starting to explore patterns in this, starting to systematise what they're exploring.

There were some great developments this week. Here's T looking at how if you repeatedly add a white rod you move up through all the different lengths in turn:
C had a variation on this:
 F was exploring the fact that something equals itself, enjoying the tautology of it:
D knew that he could generate lots of trains equal in length to the orange rod, just by creating the now-familiar staircase that children are making again and again:
 M was looking at what you have to add end-to-end to a pink rod to move up through all the lengths of rod:
I asked if anyone wanted to be videoed reading reading what they'd written - and some did.
The ability to set out all the information systematically is a great skill in maths, aside from any breakthrough that it helps you make. I hadn't asked for this and of course not everyone was doing it. And a few children were getting muddled.
Or needing to go back to the rods and fix what they'd written:
But luckily my TA and I were able to get round to everyone, and I think all of us are getting the basic idea, and that all is one of the my main considerations for when I'm happy to point to new developments and try new things.

One of the things we did move on to this week was based on E's work:
E was measuring these trains with a long line of white rods and counting them up. That's hard to read; it says:
o + b = 17w
o + B = 19w
o + o = 20w

It might look too simple to be a development, but this implicit measuring of the rods by another rod both connects with cardinality and leads onto lots of other things. It can lead to measuring with other rods, and crucially to talking about the rods as a number rather than simply as a colour, things that we'll be doing soon.

So the next day we looked at E's work and I asked the class to do similar things. And off they went:
Interesting with the bigger ones - a few children getting tangled with the troublesome teens - twelve, thirteen, fourteen, fifteen - it's a tongue-twister, and so easy to miscount at this age.

One group wanted to try it with ten oranges!
(We're going to have to come up with a convention to distinguish o from 0.)
We looked at one creation as a class:
I asked the class how we'd write this, and everyone wrote their way on little whiteboards. I chose a few to come to the whiteboard and share their ways:
We also looked at another lovely bit of systematisation, this time from A.
She was measuring all the rods with the whites. We're about ready with this to start talking about the rods "as numbers". A little pinch of cardinality, and our ordinality has new wings...

------------------added a little later---------------------