There was time for this

I pondered whether to do it, and decided Yes! We did make a start on our popup cuboids. We made the cubes.

 

 I decided that as there was a fair bit of arithmetic in it too, it covered a lot of bases. Working out the surface area, and the volume, usually involved a one digit X two digit calculation, and it was a chance to remind about grid multiplication in a context.

They would make a good book! I would also really like to make to get the class struggling with making non-cube cuboid popups. Doing this involves getting to grips with how a cuboid works (opposite faces being the same).

The children who finished quickly I steered towards making some cuboid pictures with pattern blocks:

This is something I'll maybe do more systematically next year (if I'm teaching around this age).

Popup cuboids

We've already done more than usual about cuboids, designing and making our fruit juice cartons, and then making them with cm cubes and using isometric and squared paper to represent others.

There were other things I would have done had time allowed. One is to create similar representations in pattern blocks, like those of Daniel Ruiz Aguilera, but simpler:

Tres vistes (3) pic.twitter.com/B00TAg45mj
— Daniel Ruiz Aguilera (@druizaguilera) May 9, 2014

Another was to use Paula Beardell Krieg's colourful isometric collage.

I've come up with another idea (after admiring lots of posts on Paula Beardell Krieg's Playful Bookbinding and Paper Works blog, and after making popup chicks for Easter cards), that I think is a good one: popup cuboids.

I'm hoping that other people can also see that there's a cuboid outlined in the middle of the fold. It's a 3cm x 2cm x 6cm cuboid (albeit with top and bottom faces made of air). Here it is, made with the NCTM isometric drawing tool:

(I particularly like this cuboid,
because it's central diagonal is a whole number.
But that's not for my students.)

So, if I decide there's no time this year, this can be a note to self as well as anyone else. Here's what I'll do.

I'll show another example with a cube. Ask students to:

1. make a smallish (< 8) cube out of interconnecting cm cubes;
2. ask them to fold squared paper along a central line (scoring with scissors first might help);
3. then make cuts away from that fold the size of the cube;
4. score the new folds they need to make;
5. fold in the cube;
6. make annotations about how many squares there are on each face, what the total number of squares covering the cube would be and how many little cubes are needed to make this cube (lots of just-right calculations here).

Then I'd display the results with the cubes and next show a cuboid like this and ask them whether it's possible to make a popup version of it. This is a little tricky because if all the dimensions are different you can't use that first central fold. Some experimentation would be essential. If someone finds the way - great! We'll go ahead and do roughly the same list of things I wrote for the cube.

Worksheets

I used to be the worksheet king. Making them clear, uncluttered, as simple as possible. They're all, hundreds of them, on the system at work. Here: I've found one from February 2005:

That must have been Year 6, ten, eleven year olds, I was teaching that year.

But these days I seem to be using them less and less. And the Abacus textbooks, with lots of sums and things in,

I've hardly touched them at all this year.

What is happening?

Partly it's that I'm more and more wanting the kids to be creative in their maths work, and neither the worksheets, nor the textbook seem to give enough space for this. With creative work, there's usually a stimulus at the beginning of the lesson (after we've done a bit of number circle or estimation or some starter or other) and then there are constraints (often the manipulatives we use are part of this) and a requirement that I give by speaking to the class, maybe an example. Then off you go...

I'm not against either worksheets or textbooks. It's just the way I've been moving. You can see why perhaps in my earlier posts. (You can see some of our work on the Year 4 blog.) And, besides, as well as my own ideas I've been getting so many from the great people in the Math-Twitter-Blog-o-sphere #mtbos that there's really not enough time to do half of the things I want to do.

So I was interested to read a post by Andrew Gael on different kinds of worksheets that approach the same task. I immediately wanted to do a worksheet-free lesson based on the same premise. But I also wanted to see what the worksheet would do for us. I chose the one I liked best, and passed it out without too much explanation to my Year 4s (8 and 9 year olds). I had to translate a little. I don't call it graph paper; I call it squared paper. But most of them could see the idea, though I think some them were a bit fazed by getting a worksheet with written instructions out of the blue like that. A few asked for a bit of supplementary explanation.

Quite a few found it straightforward, like this:

For one it was just so too easy, he went all one-dimensional:

Some got the side lengths wrong:

And one got a bit confused:

Aside from anything else, you can see it's a worthwhile activity just to draw a grid of a certain size (just as it's a worthwhile activity for young kids to create their own number line) and about half the class need a bit more practice at this. I think about a third of the class could do with repeating this activity and getting it right. So it's been informative. All of them could do with annotating their grid with a bit of explanation. Checking wouldn't be a bad thing either! Tomorrow.

What did we do next? More of an open-middle kind of thing. They cut rectangles form squared paper. Wrote how many squares on one side and their name on the other. Then they cut bites out.

We looked at them a bit together, and worked out the area of the original rectangles of some of them. We'll return to them, and the worksheet, tomorrow.

Anyway, what do you think? I've changed Andrew's question to fit my case. What are the advantages of the worksheet here? (Should I be using them more?) And how about the task afterwards? I'm interested in your thoughts.

POSTSCRIPT
The next day I gave everyone their work back to check, and change if need be. I also gave one of these orange sheets (with the orange taken out to make space for writing) to everyone.

We'd been talking about 16 X 5 type questions - and will talk more.

Most people seem to have got the idea; though I think I'll throw a few "draw me a rectangle with 45 squares on it"-type questions when we all have our whiteboards for a lesson starter.

Thank you, all you brilliant commenters! I've certainly got one or two things a lot clearer in my head about worksheets good and bad, and more besides.

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!