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---------------------
Gattegno:


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).

video

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

video

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:

video

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:
video
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

holes

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.