Cuisenaire squares

I was wanting my Year 4 class to think and talk about the way odd and even numbers add, so I borrowed the numicon from Year 2.

When I was about to give it back, I noticed some square trays for Cuisenaire rods still in their bag.

Great! So we used them for exploring number patterns.

But I liked the frames a lot; I felt there must  be more to them. So, when Justus and Anabel had finished their work, I asked them to fill the hundred-frame with threes. Unfortunately it wasn't possible, so they added a one to try to cover up the nasty gap.

I, being the kind of teacher that assesses students' efforts carefully, was not fooled. "There's a one in the corner," I said. They must have a growth mindset, because they admitted it was true. "Can't you move the threes round so you don't have that white one?" I asked.

"No, there'll always be a one," said Justus. 

"How do you know?" I asked.

"Because a hundred isn't in the three times table," he said.

So, naturally, I tweeted about it later, and Miles Berry kindly replied:

@Simon_Gregg how do they know there'll always be a gap?
— Miles Berry (@mberry) January 15, 2015

I was beginning to feel a lesson idea coming on, so at home I played with nrich's Cuisenaire environment, this time with fours instead of threes:

Will there always be a gap with fours, @mberry? A hundred is divisible by four... pic.twitter.com/22bASoKkdK
— Simon Gregg (@Simon_Gregg) January 16, 2015

And he tweeted back with this wonderful visual proof:

@Simon_Gregg Yes. Nice question. pic.twitter.com/vGQfvfcIUB
— Miles Berry (@mberry) January 17, 2015

This proves that it's impossible: if you count up the four colours there aren't 25 of each. And yet every 4-rod has to cover one of each colour.

so there would need to be equal numbers of each colour for the square to be filled with them.

It's like the mutilated chess board problem. And a bit like the solution to the impossible peg solitaire problem.

So, all sorts of thoughts come now. Is there enough here for the students themselves to be presented with the rods and frames and asked to come up with their own lines of inquiry?

What about if there were two colours? Which combinations could fill and which couldn't?

What abut the question of where that last white one goes? Can  it go anywhere?

I check with Miles Berry's method... 

In the diagram below, each rod must cover an orange, a yellow and a pale green square. But there are 34 orange squares and only 33 of the yellows and pale greens. So the spare white one must be on a red squares, which can either be in the position on the left or in a position in the middle. Superimposing the two, it must be in one of the positions on the right:

This is so neat. I want to share it with the kids, or at least a simpler case. But then, no, my real hope is that there's enough in the materials that they can explore themselves, not follow. The kids could explore this from another entry point: trying with simple cases:

possible on the side?

Finding which combinations work and how, then documenting it would give plenty of arithmetic in a meaningful context, and hopefully develop those exploration muscles, more important.

Here's another question: if there are ones left over, how many could there be? That is, how many so that for example the ones are isolated?

I do this thing, you see, that perhaps we're not meant to do. I start with the materials and think, where could that go? Usually we're meant to start with the curriculum and work down to materials. But as I have a little more leeway than some, I can kind of make it up as I go along. Find wild places to explore. And for me, learning to explore is a lot of what the curriculum should be.

What do you think? Would it work as a lesson? Can you see other avenues that might be explored?

_______________________________________________________________________________

Later edit: interesting developents

Intuition and slow thinking

This post is going to be more questions than answers...

Some things have been going through my head. There's Kassia's post, Is There Room For Math That Isn't Hard? Also, a conversation on Twitter, one strand in a bigger conversation about intuition in maths learning. Things can get a bit abstract when you're down to 140 characters including names, but there were a lot of great points. Here's one definition of intuition that Kristin posted that I liked:

"your insights and intuitions as a native speaker..."

Somehow, it links for me too with a moment in our maths classes in Year 4 this term. I'd read a really interesting post, Making Sense, on Tracy's blog. I had all the Year 4s and I showed them this question:

I asked them to write their thoughts on their whiteboards. All of them, all of them, gave me a numerical answer! That really surprised me. I thought lots would, but all? I showed the classes the video on Tracy's blog afterwards, and very briefly talked about how some questions don't have answers.

Somehow, these things link, in my mind at least, because we need a solid base of intuitions about maths - partly what we call "number sense" - that helps us to deal with both meaningful and meaningless questions, and to tell the difference!

I also reached down Guy Claxton's brilliant book Hare Brain, Tortoise Mind from the bookshelf.

Claxton says there are three processing speeds in the brain. The fastest, faster than thinking, is the kind of response we have when we skid on ice and just do the right thing. It's the sort of processing a concert pianist or an Olympic fencer has to do. Then there's thinking itself, deliberation, which he calls d-mode. But "below this, there is another mental register that proceeds more slowly still. It is often less purposeful and clear-cut, more playful, leisurely or dreamy."

It maybe helps to look at deliberation, the familiar kind of thinking, first. Claxton lists some of its features:

D-mode

1. is much more interested in finding answers and solutions than in examining the questions.
2. treats perception as unproblematic.
3. sees conscious articulate understanding as the essential basis for action, and thought as the essential problem-solving tool.
4. values explanation over observation
5. likes explanations and plans that are 'reasonable' and justifiable, rather than intuitive.
6. seeks and prefers clarity, and neither likes nor values confusion.
7. operates with a sense of urgency and impatience.
8. is purposeful and effortful rather than playful.
9. is precise.
10. relies on language that appears to be literal and explicit
11. works with concepts and generalisations
12. must operate at the rates at which language can be received, produced, and processed.
13. works well when tackling problems which can be treated as an assemblage of nameable parts.

So, d-mode is how we operate in maths lessons. You could even see mathematics as the place in which it shines most brilliantly.

But what of the slower thinking?

There is evidently a place for it. Here's Henri Poincaré:

"Most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works at a hard question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half-hour, as before, nothing is found, and then all of a sudden the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted and the rest has given back to the mind its force and freshness."

(Claxton also gives lots of experimental results from cognitive psychology that demonstrate the effect of slow thinking. I'm glad he does this because words like intuition can sound unscientific, which they're evidently not.)

How to descend from this abstractness then? Is there a place for encouraging slow thinking and intuition in the primary classroom?

A few tentative answers. One: when you ask children what they notice, the pace slows down. There's time for a bit of pondering. Developing this as a regular part of lessons, and the respectful listening and responding that goes with it, allows half-formed and ill-expressed ideas space to breathe and develop.

Two: games. I got the classes playing Daniel Finkel's Prime Climb three times this term. There was no "teaching", apart from, briefly, how to play the game. But I feel that time when students aren't thinking, "I must learn this," is precious. Their hare brain's can be off duty. The games weren't physically slow. Lots of the kids were standing up! But... I hadn't "taught" anything. Slow in that way.

Maybe there's not time for slow thinking in your class. I understand. There's more pressure than ever to pack the learning in, to get the results. And we know ultimately, results will lead to jobs...

So, is there time to slow down?
Is it worth it?
If there is, and it is, what are good ways to do it?
Does it link with number sense?
Does it link with intuition?
Does this help with my meaningless number question?

Do you have any answers? Or more questions?

UPDATE - July 2015
I was really pleased when Gracia, towards the end of the year, came up with this question in class:

She knew it linked back to that "how old is the shepherd?" question we'd looked at before. Still, some people were not getting it.  But some were now. As Gracia put it, all that information distracts you; it's like a magic show.

I recently watched Jordan Ellenberg talking about this kind of thing in another guise. I liked what he said:

Also, in The Joy of X by Steven Strogatz:

Other classic word problems are expressly designed to trick their victims by misdirection, like a magician’s sleight of hand. The phrasing of the question sets a trap. If you answer by instinct, you’ll probably fall for it.
Try this one. Suppose three men can paint three fences in three hours. How long would it take one man to paint one fence?
It’s tempting to blurt out “one hour.” The words themselves nudge you that way. The drumbeat in the first sentence — three men, three fences, three hours — catches your attention by establishing a rhythm, so when the next sentence repeats the pattern with one man, one fence, hours, it’s hard to resist filling in the blank with “one.” The parallel construction suggests an answer that’s linguistically right but mathematically wrong.
The correct answer is three hours.
If you visualize the problem — mentally picture three men painting three fences and all finishing after three hours, just as the problem states — the right answer becomes clear. For all three fences to be done after three hours, each man must have spent three hours on his.
The undistracted reasoning that this problem requires is one of the most valuable things about word problems. They force us to pause and think, often in unfamiliar ways. They give us practice in being mindful.

Knowledge

I touched on knowledge in the last post. I was intimating some kind of process a bit like this, starting at the bottom with exploration:

and ending up with more exploration at the top.

Knowledge was something the Greek philosophers were keen to find, to develop, to pass on. Real knowledge for them was not simply believing the truth, but also having reason to believe it.

So many of our beliefs are not well-founded; so much of what we are told at school we believe because we are told, not because of experience or evidence or conclusiveness. Real knowledge is a precious thing.

So, it seems to me, respecting the genesis of this founded belief is really important. We have all sorts of utilitarian ideas about why schooling, or a particular kind of schooling is good. It's good because we can participate in the economy, we can be internationally competitive. As the new English National Curriculum for maths has it, "it is essential to everyday life, critical to science, technology and engineering, and necessary for financial literacy and most forms of employment." True as that may be, maths, like all subjects, can be something else, an opening of the eyes, a body of undertandings, of knowledge. Finance doesn't know. Employment doesn't know. How can they guide us? How can they weigh opinions, find truth? Maths, like no other subject, can give a sense of what certain knowledge really feels like. We need that. 

Have you ever read Plato's account of Socrates defence ("The Apology")? I recommend it. It's a really essential read, short, fascinating and pivotal. In it, Socrates describes his enemy-making quest to find what people really know:

Accordingly I went to one who had the reputation of wisdom, and observed to him - his name I need not mention; he was a politician whom I selected for examination - and the result was as follows: When I began to talk with him, I could not help thinking that he was not really wise, although he was thought wise by many, and wiser still by himself; and I went and tried to explain to him that he thought himself wise, but was not really wise; and the consequence was that he hated me, and his enmity was shared by several who were present and heard me. So I left him, saying to myself, as I went away: Well, although I do not suppose that either of us knows anything really beautiful and good, I am better off than he is - for he knows nothing, and thinks that he knows. I neither know nor think that I know. In this latter particular, then, I seem to have slightly the advantage of him. Then I went to another, who had still higher philosophical pretensions, and my conclusion was exactly the same. I made another enemy of him, and of many others besides him.

Read this too, from Jonathan Haidt's great book The Happiness Hypothesis:

In philosophy classes, I often came across the idea that the world is an illusion. I never really knew what that meant, although it sounded deep. But after two decades studying moral psychology, I think I finally get it. The anthropologist Clifford Geertz wrote that “man is an animal suspended in webs of significance that he himself has spun.” That is, the world we live in is not really one made of rocks, trees, and physical objects; it is a world of insults, opportunities, status symbols, betrayals, saints, and sinners. All of these are human creations which, though real in their own way, are not real in the way that rocks and trees are real. These human creations are like fairies in J. M. Barrie’s Peter Pan: They exist only if you believe in them. They are the Matrix (from the movie of that name); they are a consensual hallucination.

This is why it's great that we can find knowledge that's as firm as our direct knowledge of rocks and trees. And young children have this as they explore the world. Papert, points out how Piaget's "genetic epistemology" made us much more conscious of how much young children are learning:

While we can “see” that children learn words, it is not quite as easy to see that they are learning mathematics at a similar or greater rate. But this is precisely what has been shown by Piaget’s life-long study of the genesis of knowledge in children. One of the more subtle consequences of his discoveries is the revelation that adults fail to appreciate the extent and the nature of what children are learning, because knowledge structures we take for granted have rendered much of that learning invisible. We see this most clearly in what have come to be known as Piagetian “conservations”.

(It's interesting re-reading Seymour Papert's Mindstorms, because I find things in there that I say. Now I wonder, did I get that from Papert? Or was there just a similar take on things?)

School can and should be about "understanding rather than turning the handle" (as someone has written on their Twitter profile). There will have to be some handle-turning for sure, but let's keep pride of place for the understanding, the knowledge.

A short conversation

I found this conversation thought-provoking.

1. Alison Kiddle ‏@ajk_44  Nov 30

   Random thought: how does the stuff done in maths lessons relate to what a professional mathematician thinks of as mathematics?
2. Sue de Pomerai ‏@suedepom  Nov 30

   @ajk_44 @ColinTheMathmo not very much!
3. Colin Wright ‏@ColinTheMathmo  Nov 30

   .@suedepom @ajk_44 Indeed. Does it matter? Should we change that?
4. Alison Kiddle ‏@ajk_44  Nov 30

   @ColinTheMathmo @suedepom I do think it's confusing that we use the word "maths" for Early Years number as well as calculus and beyond.
5. Mary Pardoe ‏@PardoeMary  Nov 30

   @ajk_44 @ColinTheMathmo @suedepom what's wrong with 'elementary', 'higher' and 'advanced'?
6. Colin Wright ‏@ColinTheMathmo  Nov 30

   .@PardoeMary What is done in university and beyond is not simply a more advanced version of what happens in school. CC: @ajk_44 @suedepom
7. Alison Kiddle ‏@ajk_44  Nov 30

   @ColinTheMathmo @PardoeMary yes, see the popular misconception that university maths is "doing big sums"
8. Mary Pardoe ‏@PardoeMary  Nov 30

   @ajk_44 @ColinTheMathmo where is the evidence for that misconception?
9. Colin Wright ‏@ColinTheMathmo  Nov 30

   .@PardoeMary My experience is that generally people think "advanced maths" means more complicated sums. CC: @ajk_44
10. Simon Gregg ‏@Simon_Gregg  Nov 30

    @ColinTheMathmo @PardoeMary @ajk_44 But generalise: do young children not write, read, produce art, because there is an adult version? 1/2
11. Simon Gregg ‏@Simon_Gregg  Nov 30

    @ColinTheMathmo @PardoeMary @ajk_44 2/2 and they are doing more than sums - reasoning, making connections, exploring, making generalisatins

Colin Wright‏@ColinTheMathmo

.@Simon_Gregg If what you say is true, why do all the adults I talk to think that maths is just about sums? CC: @PardoeMary @ajk_44

I'm not experienced in university maths (I did science), but from what I do know I want to say that there is a continuity, and not just out of respect for the efforts of young children.

Of course, Colin is correct: there are an awful lot of sums in schools, hopefully less than there were when we were at school, but still, many people do think that sums are the thing. And it must get boring getting thought of as a difficult sum doer!

But, but... there is another side to school maths, and one that I would like to see take over as people's picture of school maths.

Take this Kindergarten Interlude that Joe Schwartz posted on the other day.

Here are children in a comfortable situation, getting to know number, its constancy, its patternability, its many possibilities. I like how Joe says, he is there to "poke, push and experiment".

Now this is well-trodden ground, but children are building up their knowledge here, step by step, tentatively at times, in leaps at other times. Sometimes the knowledge will be certain: however you arrange three bears, they will always be three. Other times it will be a hunch: you need four bears before you can have a pattern.

Or take Kristin's post on Articulating Claims in Math, The elementary kids are doing "sums" here, but they're doing a lot more, they're making generalisations about the way sums work:

Now you might argue that proof involves more than this. That it involves formal generalised description. Or that it must be a shared and socially verified knowledge.

But I would say the exploration comes first, then the hunches, then more exploration, then the knowledge, then the sharing.

Consider Polya's treatment of Nicomachus's theorem. How would you first discover this? Polya writes:

Mathematics presented with rigor is a systematic deductive science, but mathematics in the making is an experimental inductive science.

And that is what is happening in many excellent maths classrooms, and could be happening in even more. I like Polya's example, not least because the nine year olds in my class had a go at exploring this last year.

A different kind of activity comes before proof. You get a sense of it in this video:

And proof is not about formal language either. There's the famous Bhaskara proof:

Bhaskara's only word: "See!"

As Keith Devlin says about symbols, "They no more are algebra than a page of musical notation is music."

So, I want to make the case, here as I do in my teaching, that there is a continuity, there is in the best classrooms a relationship with what kids are doing and what mathematicians do.

What do you think?

The list goes on

Another item for the list:

XI. Computing

When I was a kid at school I got into programming in Basic. It was mathematical in ways that maths lessons weren't. For instance, the only time I ever asked a teacher how to do something that hadn't been taught was when I needed a bit of maths for a program I was writing.

"How do I stop this weather data looking so spiky?"
"You need a moving average."

When I did my teacher training at the Roehampton Institute back in the early 80s, computing was as you know in its early days, not quite as early as when I was at school (there was a screen for instance!) but early days:

But there was a book I really enjoyed, and found captivating, Seymour Papert's Mindstorms. The idea of using Logo in the classroom to program a turtle was a really inspiring one, where you really needed to get to get to grips with maths to get a lot of things done. This short video from the time gives you some idea of what was starting:

Over the years, I've used Logo a lot. You know the kind of thing: forward 10 right 90...
I used the Microworlds  environment quite a bit. Now I use Scratch. It can have too many knobs on, too many attractive features, and I think the simple maths of Logo is the best; it continues to be a great way of really needing a lot of maths to achieve an end. I never seem to get quite far enough with it to get on to the wealth of exciting possibilities, but I think this might be changing with the new emphasis on coding in the computing curriculum. I'm using code.org's great resources and courses with my kids, and encouraging the whole school to develop this kind of coding, and although the finely-grained steps of the courses are mainly convergent, I think they provide a great preparation for wonderful divergent work.

It's this sort of blank slate tool that I like best of all. I use other packages that aren't blank slate (Education City, mymaths.co.uk) and there are lots of brilliant websites for developing skills, but the tools I like best are the powerful ones where it's down to the students to make something of them. (Geogebra is another brilliant example, one I used today for instance to get the students to generate lots of different kinds of hexagons.)

Here's a great tribute to Papert:

Even more...

As I said, the UK government wants "harder sums". It wants rigour. Picture phalanxes of Roman centurions - very comfortable with their Roman numbers - marching rigorously up very straight roads, their shields held close together. Nothing gets past them.

My strategy for conquest is different. I've made a list of some of its components. Like Caesar's Gaul, so far it's got three parts:

First part:

I. Pick subjects that give power.

II. Find the subjects where kids can be creative,

III. Go into history and biography

Second part:

IV. Enthusiasm

V. Discussion

VI. Presentation

Third part:

VII. Climb up and down the ladder of abstraction 

- § - § - § - § - § - § - § - § - 

But now I'm going to add another part:

VIII. Estimation

I asked my friend Charlie who works as an engineer what maths he thought really needed to be taught at school. Estimation, he said, you need that all the time. And luckily it's become more accessible and more engaging than ever with Andrew Stadel's estimation180.com . I've used this with my Y4 class last year, and we're going to start again next week. What's so great about it, is that the kids are interested in the estimation and they like the challenge. They especially like it when there's a video "reveal" at the end.

I have this idea that if we like it, we'll start creating estimation challenges in the Year 4 classrooms, maybe begin an estimation blog, perhaps begin to find estimations to do at home too, photo or video.After that we get other classes to have a go.  Nothing too ambitious. First we take Manhattan, then we take Berlin.

My trial challenge was not a complete success, but it's helped me to get the measure of what's involved:



Anyway, here's Mr Stadel talking about what he does:



IX. The real world

To be honest, this is something I know I don't do enough of. Using real things, real places, things you might find at home. We've just been looking at reading scales, and for the first time we've got the classes to look for dials and scales at home this week. The range is amazing: weighing scales, a barometer, pressure gauges on pumps, a metronome, the rpm and the speedometer in car, a clock... and some things from an aeroplane: how level you are and speed. There is just so much to talk about.

In fact I can't stick to the real world. We did a bit of not-so-real world with our creation of meters to measure things not normally measured. The idea was to create a bit more attachment to our dial by investing more in it than usual.

It gave us a good chance to talk about what kind of units you might invent, as well as looking what the un-numbered marks represented. It also meant we could spend a bit more time on dials and still be doing fresh things.

X. Modelling!

This as Turtle Gunn Toms says in a comment on Graham Fletcher's excellent post on modelling, means taking a situation and mathematising it.

It's another thing I really don't do enough of. Probably none of us do enough of it! The ideal is a situation where you have a question, you put numbers to it, out comes some kind of answer.

It's a lot harder to find good examples for the primary / elementary classroom. I'd like to have a collection of this kind of mathematical modelling question.

Here's one I'm thinking of trying soon: I was talking to the kids: "Here I am, standing in the middle of the room..." and it occurred to me, "Where  exactly is the centre of the room? How would you work that out?" I said my thoughts out loud of course.

Not a very natural question perhaps, but I'd be interested to see how the class go about answering that. Some estimation first of all of course...

The ladder of abstraction

I'm interrupting my list to write about this. It's sort of in reply to a tweet from Tracy Johnston Zager, though I suspect I haven't answered the question in exactly the way she intended:

@mpershan @MathMinds @Zakchamp Maybe @Simon_Gregg could chime in about how he handles/introduces manipulatives?
— Tracy Johnston Zager (@TracyZager) September 12, 2014

No, actually this chime can be part of the list.

7. Climb up and down the ladder of abstraction

I've been reading Roy Peter Clark's entertaining and instructive book Writing Tools: 50 Essential Strategies for Every Writer over the summer (link to free podcasts of this on iTunes). One of the 50 strategies is "climb up and down the ladder of abstraction". Here's a snippet:

The ladder of abstraction remains one of the most useful models of thinking and writing ever invented. Popularized by S. I. Hayakawa in his 1939 book Language in Action, the ladder has been adopted and adapted in hundreds of ways to help people ponder language and express meaning.
The easiest way to make sense of this tool is to begin with its name: the ladder of abstraction. That name contains two nouns. The first is ladder, a specific tool you can see, hold with your hands, and climb. It involves the senses. You can do things with it. Put it against a tree to rescue your cat Voodoo.
The bottom of the ladder rests on concrete language. Concrete is hard, which is why when you fall off the ladder from a high place, you might break your foot. Your right foot. The one with the spider tattoo.
The second noun is abstraction. You can’t eat it or smell it or measure it. It is not easy to use as a case study. It appeals not to the senses, but to the intellect. It is an idea that cries out for exemplification.

Here's Hayakawa's illustration of his ladder:

This fits very well with the way I like to teach. Say in maths, I want the kids to use stuff they can touch or be physically part of. To take an example that I think works really well, I'll get them using the Cuisenaire rods to make their own patterns, and then I want them to take the numbers inherent in those patterns, then beyond that, the relationship between the numbers, and if possible to abstract that relationship into algebraic form. I'm quite prescriptive about what I want them to do, but hopefully the limitations are creative, because I want them to be creative.

If you haven't read Richard Feynman on why there is no science education in Brazil, do. He was astonished at how little teachers went down the ladder again from abstract to concrete.

"I didn't see how they were going to learn anything from that. Here he was talking about moments of inertia, but there was no discussion about how hard it is to push a door open when you put heavy weights on the outside, compared to when you put them near the hinge – nothing!"

bench by Zaha Zahid

The architect Zaha Hadid is designing the new maths gallery for the London Science Museum, a building I spent a lot of time in as a kid. She has a maths degree, and she uses maths as a place to find new abstract forms for her buildings. She was also inspired by the abstract mathematical paintings of Kazimir Malevich.

Black square by Kazimir Malevich

So, she's using abstract mathematical forms and ideas, and, quite literally, making them concrete:

"When I came to do architecture people said you must know how to add. There is that aspect to maths, of course. But there is another that was of interest to me and that was abstract thinking, and that was when I realised how important that degree was."

Zaha Hadid's design for the Mathematics Gallery

To me, that's thought-provoking.