Thursday, 12 January 2017

Reasoning with Which One Doesn't Belong

I do a 15-minute Which One Doesn't Belong session most weeks. I've blogged about WODB before, and I regularly tweet about it. Last year my Grade 3s were getting really expert. This year, my K3 (5,6 year olds) are all really into them.

Over Christmas my copies of Christopher Danielson's book and teacher's guide arrived. I've been enjoying the teacher's guide; there's a lot of background in there, and a lot of useful advice. I really recommend it. Christopher took the book through its paces with children of  all different ages, and despite the simplicity of the basic idea, there's a lot to think about in its execution.

Christopher looks at development as a geometer using the Van Hiele model:
A lot of my students' thinking is, not surprisingly, at Level 0. But there's also a fair bit of Level 1 emerging. And at Level 1 with WODB you start to get some great reasoning.

Today I used this one that I'd made:
All but one of the 22 students had at least one thing to say:
Christopher suggests using this kind of recording as a reminder, and spur to further thought:
"Simply writing a key word (square) or phrase (all angles the same) or sketching a quick diagram, or circling key features of the shape are all quick ways to maintain visible reminders of the unfolding conversation for everyone to access." (p30)
And it's making me think that I should keep these up, because there's a lot of things that need returning to and developing, and who knows at what pace and at what moment thoughts will come to people?

I made some low-res video of the session. Here's a few moments. In this one you can see one of the "it's like" observations, and also one that describes a property (the red one is the only one with one round bit).


Some of the "it's like" statements need further investigation:


Sometimes an observation of a property leads to a discussion about what exactly that property is, the sort of reasoning I really want to develop in my students:


So maybe we need to pin up that second sketch too, and come back to it?

Tweeting about moving from "it's like" to properties, I got some good advice from Christopher and from David Butler:
We also talked about when to introduce vocabulary. I'd be interested in ideas about how to develop this. How could this kind of lesson be extended in another lesson, maybe in smaller groups or individually? What, in all this is worth developing? What I think would really help my class is to  encourage everyone somehow to look for properties and reason about them. Ideas?

Tuesday, 10 January 2017

My maths autobiography

This is a brief response to chapter one of Tracy Johnston Zager's wonderful Becoming the Math Teacher You Wish You'd Had. It's not the typical maths learning experience she describes there, but it's mine.

Paddington Green Primary School was a chaotic kind of place. Once, at least, a supply teachers left crying within the hour. And there were a lot of supply teachers. I don't remember a lot about it (except about Ivor Cutler). But I know the maths teaching was practically non-existent. Sometimes loads of sums on the board.

But my mum loved maths. Unlike her brothers she'd not been allowed to go to university, but she did train in electrical engineering at the BBC's college in Evesham.

She didn't 'teach me maths'. We did play a lot of games though.

I said these words about my mum at her funeral:
And yes, we played chess. And battleships, and boxes, and draughts. And backgammon. And Chinese chess, and Go. And Mah-Jong and Hanafuda. Endless games of Totopoly and Cluedo. With Peter too. Monopoly we weren’t so keen on, but we played it when Tad came home. Then there was Diplomacy (you didn’t like the garish board) and of course Scrabble. There was something mathematical in all this, the best kind of mathematical.
When I was in my teens we used to get a magazine called Games and Puzzles. We tried out lots of the games, and often the puzzles. My mum had a lot more patience for them than I did.

My secondary school maths was probably a bit better than the average. I got my 'O'-level, got my 'A'-level. But given the average wasn't so hot, I was never A. really excited, B. awed, C. exploring for myself.

No, there was some exploring. When I was 14 (in 1974) there was the possibility of computing in the maths classroom, outside of lesson times. There were two teleprinters. One of them was connected to a telephone. You could dial the City of London Polytechnic mainframe, there'd be a wrrr-weee-wbong-wbong noise and you'd put the receiver into a wooden box with a receiver-shaped insides, and the teleprinter was connected. About half a dozen of us became obsessed. We were writing programs in BASIC, learning it all by trial and error, no books, no teacher. We stayed in at breaktimes, and after school until about 8pm when the caretaker would come in shocked to find us still there and shout at us to get out of the building.  I wrote things like a program to play a race game. You printed out the track, rewound the paper and then ran the race. You could accelerate or decelerate either left-right or up-down by one each time.

While one person was on the connected teleprinter the other was on the other, typing their program. You then printed out as punched tape ready to feed in to the online teleprinter when it was your turn.
File:Punched tape.jpg
punched tape - source
 There was always a bug when you ran the program; I'd go home scrutinising the lines of code on the 36 bus. It was really compulsive, the adrenaline of the hunt, if that doesn't sound too overblown. I still have some of this stuff. Here's a page of the program for that game:
The first of two pages of the program to run the race game
The game:
The race (coloured in after)
I wrote a program to play Monopoly (!), one to print out a graph of weather data we'd collected on our balcony at home (I had to ask my maths teacher how to smooth the graph out - the only time I asked any teacher how to do something I wanted to  do! - he knew too: I needed a moving average), there was also a game where spaceships were in 3D space, accelerating and decelerating and shooting at each other (I worked out that I needed what I later learnt was Pythagoras' theorem).

Because it was a bit obsessive and I wanted to give more time to other things, I gave it up when I was 17. But I learnt so much from it.

I did 'Natural Sciences' at Cambridge, mainly biological kinds of things, with some history and philosophy of science thrown in. There were some maths lectures, on the basis of the statistics we were using, but I didn't really keep up. One thing I did do however, I read Martin Gardner's pieces in the Scientific American, then bought his books. That was the kind of maths I actually enjoyed.

So when I began teaching in primary, I had a fairly strong feel about what the real thing is: play, invention, the thrill of the hunt. I'd also been given Seymour Papert's 'Mindstorms' (pdf) to read at teacher training college, which, taking kids' free exploration of the Logo programming language as its theme, strengthened what I felt already. All this got worn down by all sorts of pressures, tests, National Curriculums, Numeracy Strategies, and most of all by having virtually no training or real ongoing professional development as a teacher.

Of course, all that has changed now, with the #MTBoS...

Thanks, Tracy.

Sunday, 27 November 2016

Mandating the materials

Kim was kind enough to comment on my last post. Here's part of what she said
I've been working with tape diagrams for several years now, so I do feel like I know them (though I can always learn more), but it's true that they were foisted upon me by EngageNY and so it wasn't a voluntary process. I guess one question I have about that as a teacher, though, is: is there ever a time when we should make a particular tool or model mandatory because we are trying to help kids become familiar with it, so that later they can have a choice about whether to use it or not? My leaning has always been toward not mandating any tool or model ... toward always leaving it open to the child to explore and choose the representation that makes sense to him/her. But when I started working with younger kids, it seemed like it might be necessary to "mandate" certain models (like the tape diagram, or the number bond, or the "quick tens" drawings) for at least a couple of days as a way to lay a foundation. I wonder deeply about this, because it goes against my instincts to mandate, but then I think that maybe as a 4th grade teacher I was just benefitting from the groundwork that my colleagues had already laid through some of their "today we're all going to try this model" work. Would love your thoughts on this.
It's a really interesting comment, and I've been pondering it through the weekend.

It's been a delight to work with Graham Fletcher's 3-Act tasks over the last few years. Is there anyone who doesn't know these yet? Just in casre, let me tell you, it's an excellent approach to problem-based learning, where, in "Act 1" children watch a scenario, and then are asked to notice things and ask questions about it. Hopefully, and usually, there's a "how many?" question that comes up naturally, just the right one to pursue. Children can then estimate an answer to the question, and think about what additional information they need. Then, in Act 2, they're given some supplementary information that should help them to get started on calculating. While they're doing this, they can select the resources, the materials that will help them best, whether it be pencil and paper, number line, hundred square, snap cubes or whatever.When they've had plenty of time to struggle with the question and come up with answers, there's Act 3, which shows them the answer being revealed.

So, to repeat Kim's question, is there another type of lesson, where students simply get to know and explore what a material or a representation can do?

I  would say a definite yes to this. Especially with younger learners, I want them to spend time getting to know number lines, ten frames, counting, snap cubes, counters and of course Cuisenaire rods, just to see what they can do. I still want there to be a degree of openness in the task; I never want students to just follow instructions, but the challenge can be to achieve certain things, or explore possibilities with the manipulative.

Take a recent lesson with number lines. We'd had a really interesting discussion about where numbers should go on a line, and I then had a great lead in to an idea I'd seen on Kristin's blog, children themselves placing numbers onto an empty strip of tape (a literal one this time).
And we're regularly doing this with the Cuisenaire rods. Just on Thursday, the task was to find different ways of making a "train" of two rods that is the same length as the orange rod.
I've gone into more detail about what we've been doing in our Cuisenaire lessons here; one things's for sure - I've definitely mandated their use. I want my students to be really familiar with them and use them for exploring how numbers work, for asking questions and investigating. And hopefully to have a real "feel" for numbers because they've navigated the model in lots of ways.

A couple of examples. During the Thursday lesson, M, who knows I like his questions and observations, called me over. He had something to say.  He said making the same-length train with the blue and the white rod was "fussy". I'm puzzled, and, after a bit of questioning, he asks a neighbour about it, in Spanish. No, it's not "fussy", it's "easy"! I ask if I can video him talking about it:
I love it when students bring up something we can explore further, in this case how easy making a set of equivalent trains is when the white rod is part of one of them.

Earlier on, in free play with the rods, T had made a German flag.
Something must have really intrigued T about this, because he then started trying to make same-colour trains in other colours, adding a white rod at the end if necessary.
I'm looking forward to sharing this with the class again and exploring this idea together.

Now, this isn't problem-solving in the way we most often talk about it in maths lessons, but this sort of inquiry is, to me, worth sharing with the class and pursuing together. 

As we build up an understanding of the way numbers work like this, I'm expecting that the children will be flexible thinkers with numbers, because they've seen how they work. I'm hoping that, eventually they won't need to pull the Cuisenaire rods out every time, because of the number sense they've developed through them. There will be new things we explore where we'll need them again, so they will always be a place for them. But there will also be the "mastery of structure" as Goutard calls it, that means students carry all sorts of acquired understandings with them mentally.

But I'm interested in Kim's instinct about mandating. How would it respond to the kind of work I'm talking about?

Wednesday, 23 November 2016

Cuisenaire around the world

There is a story about Nasrudin:
Terribly afraid one dark night, Mulla Nasrudin travelled with a sword in one hand and a dagger in the other. He had been told that these were a sure means of protection. On the way he was met by a robber, who took his donkey and saddlebags full of valuable books. The next day, as he was bemoaning his fate in the teahouse, someone asked: 'But why did you let him get away with your possessions, Mulla? Did you not have the means to deter him?' 'IF my hands had not been full' said the Mulla, 'it would have been a different story.'
I thought about this when I read  Kim Van Duzer's candid blog post about trying to use tape diagrams (bar modelss) with the class and the lesson going wrong. A big part of the problem, as I see it, was that the method had been dropped on teachers from above and Kim didn't really have a feel for the way teaching with the tape diagrams evolves from early beginnings and a liking for what they can do.

Some people suggested Cuisenaire rods and Kim took up there suggestion:
I'm not trying to say that tape models are bad and Cuisenaire rods good. More that, like Nasrudin, when someone else gets us using an unfamiliar tool, and we don't know it and don't particularly like it, we're not going to use it  with the required subtlety and skill, and the students are not going to benefit. It helps if the process has been more voluntary, and we've built up our liking and skill with the tool ourselves.

Which is why thousands of Cuisenaire sets languish unloved at the back of cupboards. They were promoted in a similar way to Kim's diagrams (and Nasrudin's sword) and without the understanding and enjoyment of them they're pretty useless, may indeed if you're told to use them stop you doing something better that does make sense to you.

Having said that, yesterday was an amazing day for friends voluntarily using the rods, with understanding and pleasure, some of them just beginning their Cuisenaire journey others trying new things.

In our staff meeting at the International School of Toulouse we all stood in a circle and shared things that have gone well recently. Estelle, who every day is trying all sorts of new and wonderful things with her K2 children (4yos), had the rods and square frames out:
Amanda told us how her Grade 1s had been making and verifying Hundred Faces. Isobel shared how they had adapted the Hundred Face idea to be about signs:

And that very same yesterday Kristin was trying out the rods with Ks and 3rd Grade!
Kristin's careful planning, brilliant collaboration and enthusiasm for the students taking centre-stage make me all the more excited to see how she uses this tool!

 Over in Adelaide meanwhile, David Butler had a set arrive:
And yesterday he got the people at One Hundred Factorial puzzling with them:
Meanwhile in Maine, Sarah Caban's Hundred Face posters were up:

(Read her blog posts about this here and here.)

And if that wasn't already an plenitude of surfeits, over in Winnipeg Geneviève Sprenger published a storify about how she's adapted the Hundred Face idea to be about evolving monsters:

My friends, it gives me a lot of pleasure seeing educators and teachers having a go at these things, not because they have to , but because they can imagine good things happening and know how to guide others to the same kind of curious, open and reflective approach!

Saturday, 19 November 2016

'You're an idiot, and we don't trust you'

I listened to a radio program about traffic, and it made me think about teaching. I think you might see the analogy...
He took me on my first day down a little rural road, and I was a bit puzzled about why he was taking me here. It was a sunny day, and, being Friesland, there were lots of cows, Friesian cows everywhere looking over this fence, and cowpats on the street.And he said, 'Did you see that sign back there?' and I said, 'No.' It was a standard triangular European warning sign with a cow on it. And he said, 'What does that mean?' 'I suppose it means beware of cows.' He said, 'No, no, you can see them, you can smell them, you can hear them, you can just about reach out your hands and touch them! You would have to be completely sensorily deprived not to be aware there are cows here. That sign says, 'You're an idiot, and we don't trust you.' Now he said, 'First rule of safe engineering: never treat drivers as idiots. Use their intelligence to respond to the surroundings.'
That's Ben Hamilton-Baillie, talking about Hans Monderman, pioneer of the 'Shared Space' approach to urban planning, on a 30-minute BBC radio program Thinking Streets.
The streets beneath our feet are getting smart. Pavements are melting into the roads and traffic lights are disappearing. Inspired by the work of scientists and engineers in Holland and Japan, this is a revolution in urban design. Part of it is a movement known as 'Shared Space', which promises to dramatically change the way cities look and how we experience them. In Thinking Streets, Angela Saini asks if all these ideas really fulfill the promise of making us all safer, happier and more efficient?
The idea is: cars go fast, too fast, because they have their own exclusive rule-bound space. Drivers don't need to think, or think they don't need to. But when the traffic lights, barriers, road markings, curbs are taken away and tarmac is replaced with paved brick, drivers have to become aware of the space they're in and what else is going on in it. They slow down.

I've had experience of this. Toulouse has been changing some its most beautiful spots, getting rid of curbs, paving the road, making it hard for the driver to see where the road begins and ends. Driving through, I slowed down. I was no longer in my narrow rat-run, I was having to become conscious of the space. As a pedestrian, I enjoy being there much more. 
Place de la Daurade, Toulouse
Image source: Mairie de Toulouse
So, in teaching maths, in emphasising the algorithmic - 'This is what we do; you don't need to think about it too much, just follow the method and it will come out right" - we undervalue both the intelligence of our learners, and the complexity of the real world. We privilege speed over understanding. We should expect understanding. Can we open up spaces, take the road markings away, and get students to think about the space they're in rather than rush them on?

Certainly it's a delight to see what happens when we trust children's intelligence. Just yesterday, there was a delightful moment as children struggled to make sense of how numbers fit on a number line. We slowed down, we discussed, we had different answers.
They had good reasons for their choices, and they expressed them well. Actually, as we'll hopefully confirm in a lesson like Kristin's number lines lesson, all of them were correct: 5 goes in all three places once the numbers are spaced out.

So, removing the road markings, making the space more confusing, trusting the learner...

Wednesday, 9 November 2016


John Golden's tweet made me think about pattern blocks and holes again:

There's a nice kind of arithmetic with pattern blocks. For instance, these two houses have the same shape and size:
(Picture created on mathtoybox)
Which tells us that the area of the square is equal to the area of the two rhombuses.

A dodecagon like this
can be made a lot of different ways (try it!) but they will all have an area equivalent to six squares and twelve triangles.

Which brings me to holes. Using these two bits of knowledge, we can say what area this hole in the dodecagon must have.
We've got the twelve triangles. We've got the equivalent of one square (in the form of the two thin rhombuses). So the spiky shape in the middle must be... five squares big. (In this case you can also see how five rhombuses and two squares would fill the space.)
I'm thinking how the triangle family is big (the triangle, the blue rhombus, the red trapezium/trapezoid and the hexagon) - great for all kinds of fraction work and substitution. But the square has only two in its family. Not so interesting, comparison-wise. Plus, it's not immediately apparent that the thin rhombus is half the size of the square.

So, thinking to enlarge the family, if the square is twice the thin rhombus, what would three times the thin rhombus look like? I used holes to find some. Here they are, coloured in non-pattern-block colours:
 The pink S is just the shape of three thin rhombuses next to each other.
I like that purple S-shaped one!

The fuchsia one is the most obvious: a square and a thin rhombus joined.

[edit] Here's a claim. All those 1½-square holes are concave. I think there is no convex pattern-blocks-compatible convex shape (ie with unit-length sides, pattern block angles).

I'm thinking about alternative pattern blocks you see. I recently bought some deci-blocks:
They extend the triangle even further, in interesting ways;
Christopher Danielson is thinking of other ways to create a beautiful new set of 21st century pattern blocks:
What I'm wondering here though, is, if there was a family for the thin rhombus, how would that look? Or should we just go for bigger members of the square family, with a domino, triomino, hexomino?

And then I'm thinking the arithmetic of holes could be a good one for the older years/grades of primary/elementary or even beyond. These three dodecagons all have the same size hole. What size is it?
Then, getting a little harder, what size hole would this hole be?
And what about the two holes in this?

By the way, if you want to use my images yourself, go ahead.

Sunday, 16 October 2016

How we've begun using Cuisenaire rods

Kassia Wedekind tweeted:
Knowing Kassia as a writer and blogger with a lot of understanding about young children's explorations of mathematics, I'm a little daunted trying to answer. But I'll have a go anyway.
First of all I'm impressed with just open exploration. My K3 class show no sign of losing enthusiasm for it. At first, when I asked them, they had no idea that the rods were linked with numbers or measurements, but there is so much implicit maths in what they do, and just doing it extends their understanding of it. It's an aesthetic enterprise as much or more than a mathematical one, but that seems right. After all, mature mathematicians often describe their motivation as a kind of aesthetic pleasure they find in it.

I document a lot by taking photos. This makes it easier to break up our creations at the end of a session, because they are not "lost", but more importantly we look again at what we've made, and I refer back to particular ones as starting points for new departures.

Students need a lot of rods. This is true for any material you use where students like either a large scale or want to continue a pattern. The little sets that people usually buy are not enough.

One of the things the teacher can then do is to say, did you see the way Ana made that wall? Do you think you could all make a wall of some kind?

And they have been so creative, there's enough for many sessions of this kind of return. You can see it right from our first meeting.

For instance, there's been lots of rectangles and surrounded rectangles:
I've used Cuisenaire trays in sessions after this:
Other things that have appeared are plans, maps really, of roads and carparks, models of houses with walls and chairs of different sizes. All of which could be followed up. Helen Williams suggested using different-sized rods to represent the three bears as the story is told, with students choosing their three sizes of rods, and holding the right one up at the right time. There could be three different chairs made, beds...

Using narrative seems is a great way in. We've been reading and writing stories about rockets, aliens and going to the moon, and T created several beautiful rocket images, which also showed one of the first examples of a staircase. So it got returned to twice. Once for a, "let's all try to make that kind of pattern in some way" session, and once for a "show a way to get to the moon" session!
We've just been making some faces within squares with sides as long as the orange rod - which was challenging but also delightful.

Alongside this, we play games. Rods behind backs, can you take out the red rod? Together and with partners. Or, here's the sandwich with something missing, what  needs to go in to finish it off.

And we're just beginning writing.

As you can imagine, there's been some great opportunities for sitting with students and listening to them talk about what they're doing. But here I think I've go a lot more to learn form Kassia and her colleagues and their students... 

Wednesday, 5 October 2016


All our individual work in K3 has been with manipulatives and orally so far, with lots of play and games. Today I tried them with a bit of recording in their books.

"What trains can you make that are the same length as the yellow rod?"

Following Madeleine Goutard's lead, I'm leaning towards - and today I asked for - writing rather than drawing as a way of recording, which I would have gone for more in the past, but I was happy to see the drawing too. The rationale here is that symbols are a quicker and simpler way of recording and so they can allow you to think further.

We talked through using the first letter for the rods:

Impressively, just about everyone got it. (One child who doesn't have much English didn't quite understand what I was asking for. I did sit with him for a while and do one example, but he followed up by drawing around rods in various patterns.)

It was interesting to see the different ways the children demarcated their different trains. We should look together at how they managed that.

Although I'd shown them the + sign before, when on the carpet at first with their whiteboards I asked them to write down a train I showed, that wasn't how children recorded, so we went with a list of letters. One boy did add the +s. He wrote


I suggested that to show p+w was the same as y, he wrote


but this was evidently confusing as he then switched to


Maybe I should have just left it!

Where would you go next with this?

I think they need to do something similar a few more times, look at each other's pages, see how to show what they want to say really clearly.

Any thoughts, suggestions? I'm feeling my way through this...