Folding fractions

Fraction Fortress was good for building up familiarity with fractions. But Year 4 will need more soon. Last year we used Cuisenaire rods and made fraction flags. What will we do this year? Estelle mentioned fraction faces. That's a nice idea.

And then I saw somewhere (where?) the idea of folding a square in different ways to make different fractions. I like that. Different ways of getting a half. A quarter. That seems worthwhile. And gives a space for children to explore in.

I saw this great page on folding fractions too by Rachel Thomas at Plus Magazine. It shows a way of folding a third that young children are not going to discover.

Thirds are usually a bit of a surprise to 8 year olds. They don't get to see them very much; so it's worth meeting them in various ways, including this one, even if they don't see why this is a third. It's easy to fold too, just three folds.

 And you get a ⅜ thrown in free too. And that other fraction, whatever it is (that would be a good one for older kids).

I thought I'd try a fifth too,. I started by halving the paper like this, and then folding over a quarter:

- - - That's neat - the orange part is a half!

Anyway, do that four times and you get:

And in the middle is a fifth. There's a neat visual demonstration of that. It would be worth seeing what else the children notice about that last picture. What do they wonder too? (There are some good fractions here for older children to work out too.)

That seems like a good way to spend half an hour.

Another idea: fill up those Cuisenaire rod squares in different colours to show different fractions.

The hundred would do halves, quarters, fifths well. The 36 square would be good for halves, quarters, thirds, sixths. And so on.

One and a half small lessons

I've been trying counting circles in my Year 4 class. Mostly it's gone fairly smoothly, but interestingly when I tried it with jumps of one and a half a lot of the children found it a bit tricky.

We got there, but it was interesting to see there was a fair amount of faltering. I thought, I need to return to this. Maybe just on a plain number line.

Then I saw something interesting on twitter. It led to Kassia Omohundro Wedekind's blog post on  a similar thing. She had used "brownies" and half-brownies to help the same age children make connections.

It didn't provoke quite the same ideas in my mind, but it provoked ideas. Whole biscuits and half-biscuits would create an interesting pattern as they were laid down. Perhaps we could see what was noticed?

I think, @kassiaowedekind @ekazemi, I'll do a Biscuit Buildup like this, 1½ at a time. & then go back to number line pic.twitter.com/fMThvfedm3
— Simon Gregg (@Simon_Gregg) March 4, 2015

Well, it was just too jam-packed this week to fit this in. But I did have 15 minutes with the other Year 4 class, and I thought I'd give it a try. It's introduced here as a decimal activity, but I'd definitely started it off as a fraction one!


Once again, there were some hesitations and mistakes among the correct moves. Which tells us that it's worthwhile doing this. I'll recommend that both classes spend a little time counting in one-and-a-halves in a number of different ways.

The Hare and the Tortoise

In the Year 4 and Year 5 classes all read The Hare and the Tortoise.

After a little work exploring metaphor, I read it to my class. Mentioning that Aesop wrote his fables to be about people as well as animals, as metaphors, I asked what questions might come from the story.

This was really quite hard. The children are usually quite willing to volunteer questions, but this one they found hard. We did, however, come up with:

All good questions. I think we could have had a good discussion with any of them. My one has behind it the idea of fixed mindset vs growth mindset. I'll try not to add questions myself next time - this time they voted to go with mine.

So we got into a circle and passed the egg to someone holding their palm out. I tried not to steer the discussion too much. I want the children of course to articulate their real thoughts not what they're "supposed" to think.

Someone asked what the word "ability" meant - and a few children explained it.

James talked about being good at climbing. I asked why he was good at climbing. Was he always going to be like that because of the way he was born?

We had lots of good answers about this. Someone said James had strong muscles. James said he had lots of trees in his garden and enjoyed climbing them.

Marie said that if practice is important - if you do something lots you get good at it. Samyak said that enjoying something makes you want to do it lots. Rod said that having a hobby is like this. Marie asked what I like doing. Someone else asked what everyone liked doing in their own time, and we finished by going round the circle saying.

It wasn't a particularly "conclusive" discussion, but I'm happy to be establishing the procedure, and as I said, don't want to be having a predetermined answer as a goal. The goal is the thought, articulation, listening, respect, responding, clarification of ideas...

Next time, I'm going to suggest, we might look at Aesop's North Wind and the Sun. Here the metaphor might be a bit clearer. I wonder. I also wonder what questions might come up?

Squares in Rectangles

As it's half term, I've had a chance to think about some lessons a bit more carefully. Here's one:

Gordon Hamilton's Squaring and Subtracting is a great activity, and just right for 8 and 9 year olds. It gets them subtracting, and adding, but more than that reasoning about number relationships in a geometric context that means minimal explanation is necessary, language or symbols have a very small part.

I know that we want children to articulate their mathematical thinking, but it seems to me that part of building up number sense, or intuition about numbers is also working without symbols or language for a while (obviously the numbers themselves are symbols, but you know what I mean).

Another nice thing about these rectangles is they give their own feedback - a bit like with a jigsaw puzzle, where if you squeeze the wrong piece in, the rest doesn't work.


I used the worksheet PDF last time, which was great for giving different difficulties to different people.

The only problem I found was that it was over a bit too quickly! It felt too much like a race, and I want to slow it down.

So this time I think we should get out the Cuisenaire rods. Make squares that fit together first of all:

They could do this in small groups or pairs. Just keep on growing your squares. Can you limit the sizes?
Then, perhaps, can you make a rectangle out of squares?

Then I think it's time to come together, and look at one example. Say,

Adapted from here.

And it's going to be, "What do you notice?' Think on your own. Tell a partner. Someone tell all of us.

And together we can fill it out, step by step.

Then, in pairs, have a go at solving lots, one of these:

(I think also, that making one of these in Cuisenaire rods would be worthwhile too. Just to be sure everyone has a really solid feel for what's going on.)

Then, I think is the time to use the PDF - with perhaps a choice of how hard you go?

And to finish, coming together for the story of finding squares packed in a square. This one is the smallest possible with all the squares different:

As mathematicians were so pleased with this, it's been made in all sorts of materials:

Source

And here it is in wool:

Source

And here it is as a cupboard!

Practical Pedagogies 2015

We're having a two-day conference at my school, the International School of Toulouse, in October. If you can, come. It will be good.

Click on the image to go to the conference website. You can click through to see the programme. It's really varied - a whole range of approaches to teaching across Primary and Secondary that teachers want to share!

It's being coordinated by Russel Tarr, our impressive history teacher. Russel also runs the brilliant history site activehistory.co.uk and also classtools.net!

We had a coference like this back in 2012, "Practical Learning Technologies in the Classroom", focused on technology in particular. I ran a session with Estelle on blogging in primary and got my first taste of this kind of thing. The whole event was great!

The conference owes something to the "teachmeet" model - which is where teachers meet and have short 7 minute presentations on something that works well for them and that they want to share.

For instance, at the teachmeet at BETT this year I did a 7-minute micropresentation on getting the students contributing voluntarily to the class blog.

This year I've put down to jointly run three workshops. I'm so pleased to do this jointly, partly because it's with three brilliant colleagues, partly because it means more sharing of and reflecting on good ideas in the run-up to the conference. And because there's ten months to go, there's space for more experimentation, reading, thinking for us before we run the sessions.

Valuing talk in the classroom

I'm doing this one with E. We began to talk about our session and what we're going to explore when we were at BETT, and we still need to talk more about this, but there are some directions I'm wanting to explore.
One is dialogic teaching. Have a look at Ilana Horn's summary of the book Beyond Best Practice to get some idea of this: responsiveness to what children say.
The next is the Thinking Together project. I want to start with their way of getting groups to define the rules for how they work together and move on from there: Are these useful rules for discussion? (pdf)
And then there's Pie Corbett's Talk for writing - giving children spoken models that they can imitate and improvise from.
[Edit: here I must add in a photo from my friend Mick and his EAL group discussing friendship:

]

So, a lot to be exploring!

Big questions & philosophy with children

This one I'm doing with R. Last year I did regular "Big Question" times with my class. But I need to see what other people are doing in this area - and the main approach is Philosophy for Children (P4C). This will give you an idea of some of the key principles of P4C. But central is a kind of pattern: a stimulus (perhaps a story), children posing questions, knowing how to select the ones that are big, ie  not just specific to the stimulus, and debatable, not just a question of finding information. The children then select a question to discuss, get in a circle and begin. Hold the egg when it's your turn to speak, pass it to someone cupping their hands when you've finished.
For me, I want to make it a bit wider than what is these days considered philosophy, to include psychology, cultural studies, anthropology and the like - matters which in the past were within the ambit of philosophy.

Next week we're going to try reading/telling The Hare and the Tortoise to our classes and see what questions come up from this.

Students’ creativity in maths in Primary and Secondary

With J - great to have a session spanning Primary and Secondary. "Designing tasks that have a divergent component. Everyone can do something different, at some point. Everyone can interpret the task in some way that’s their own. Creating space so that students can show their own approaches to whatever maths is being studied. This can be just the way different strategies are acknowledged for a piece of arithmetic." We'll kick off with a practical activity which allows participants to go off in their own direction - then share our experience. Really want to look at the importance of talk in this too!

RT @MathsinP5: Best Maths visualization this half term! pic.twitter.com/0FbFrIX9kL
— Jim Noble (@teachmaths) October 14, 2014

The problem with running sessions though is that I'll miss so many other brilliant sessions!

Developments

Since my last post about Cuisenaire Squares there have been developments.

One is that I did the lesson. I managed to resist temptation, and allow the students to explore on their own without any eye-glazing attempt to explain a proof or whatever. The only major intervention was to say that the ill-starred attempt of some of the boys to find a mathematical pattern in Minecraft Creeper faces with Cuisenaire rods wasn't going to work.

And it was worthwhile. There was all sorts of initiative (which is what I was wanting) and reflection following the square-filling, which led off in all sorts of directions.

The second development is that I've been thinking about Miles Berry's proof, and backsliding a little.
The problem that was nagging at me, was that his way of filling the hundred-square so that it can be (almost) filled with 4--rods with four different colours

or suchlike

isn't the only way.
There's going to be a kind of repeating latin square. For instance, taking this square:

and repeating it will make e a 10 x 10 square like the one below. Like Miles' square it has more of one colour than the other colours, so the 4-rods wouldn't work with it.

 But you could find a latin square like this:

which would give this satisfying 10 x 10 square, with twenty five of each:

So, such a colouring wouldn't disprove that the square can be filled by 4-rods.

I'm really not sure I've got to the bottom of this though, and would value someone else's thought.

The proof still stands for the position of the lone 1-rod in a hundred-square filled with 3-rods though:

and that's because there's only basically one way of doing the 3 x 3 latin square.

And the last development is that Helen J Williams has had a go at the same activity with her maths club too, with good results:

@Simon_Gregg @CuisenaireCo "see this gap it's because 30x3=90 &3x3=9 so you'll still have 1 left over." pic.twitter.com/xU7hS0VXWb
— Helen J Williams (@helenjwc) January 25, 2015

Fantastic!

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.