Some big ideas in arithmetic with Cuisenaire rods

This is a very quick run through of just some of the approaches I'm using with my classes of 8 and 9 year olds. (So quick in fact, it seems like driving down the high street and only just seeing which shops are there. But maybe there's a place for that.) There are a few slips in my narration, but I'll let it stand as it is.

A few years back I made a short video, How I Teach Using Cuisenaire Rods. There weren't actually that many ways of teaching in there, although I was beginning to use the rods more and more.

Since then I've made a lot of progress in my thinking about them and my ways of using them, thanks in huge part to other people.

Caroline Ainsworth's work, based on her Madeleine Goutard's teaching approaches with Cuisenaire rods have been really useful.

Saying Yes to Students' Ideas

Miguel's question overflow pic.twitter.com/nUM5rWj9RJ

— Simon Gregg (@Simon_Gregg) January 11, 2016

-o-

Dan Finkel's TED talk is online:  Five Principles of Extraordinary Math Teaching

I was especially struck by "Say Yes to Students' Ideas". 

-o-

One of my students lent me one of his Toto books. This was my favourite page:

Toto is asking lots of questions. Why do birds fly? Why do we dream at night? Why do you have to work all the time? Why is the sky blue? Dad brushes the questions off. "Are my questions bothering you Daddy?' says Toto.  "Not at all," says Dad, "if you don't ask questions you won't learn anything!"

I've always liked questions. In the last few years I've harvested them, and built them into the timetable. This last year, there are more than ever. There's still further to go, but it's getting good.

Just recently I've started a Window of Wonder because I want to signal that questions are very welcome, and to keep them there to remember to answer some of them.

We've also got a wall of wonder where the students can write questions that come up. And often I just jot them on the whiteboard as we're working. Sometimes they do get lost, because they usually come up when you're in the middle of something else, but I'm making a big effort to keep most of them. And the students know it. Miguel, my Chief Questioner, said we should put The Question Classroom up on the door. Sometimes I'm asking for questions - I ask "What do you wonder" about all sorts of images, equations, statements. But more often they'll just come up. We're looking at a map of Paddington (because we're learning about the discovery of antibiotics - that came from a question too - and because I grew up there!) and Ines asks, "How do they make a map?" Too big a question for now, and too tangential, but a great question!

We were talking about something else when Armand asked his question, "Is zero odd or even?" Some students were desperate to answer straight away. But I wanted to keep it back, give it more time. So it went up on the window, and only later did we return to it. 

As I knew a lot of people already had clear ideas about it we let them say that it was even, then I asked them to justify their answer. This was the interesting bit. There were two story justifications:

and some number pattern justifications:

 Rhea acknowledged that zero did seem different however!

 We also have a Mathematical Claims Board. At the moment it's got claims that are generally accepted:

 but I'm hoping for some claims that might be more debatable!

I'm lucky that my school has just begun the IB Primary Years Programme. One of its learner profile words is Inquirer. Inquiry is central. This legitimates learning that begins with students ideas, their questions. I can give time to questions like these that came up when we were talking about millimetres last term:

Miguel keeps reminding me that I said I'd get on to these questions. Which is good because it's a great story...

Art for Maths' Sake

Four years ago my Year 5 class made factor trees. We inverted the usual upside-down tree:

We made the prime factors into flowers, and we added a bit of treeishness and colour:

There was a whole forests of factor trees. Someone had the idea that, as 1 is not prime, but not like other composite numbers, it could be a bird:

It was something I've repeated with my Year 4 classes in the last few years, sometimes making the trees in different media:

There's always a tension for me with maths-and-arty activities. They can be not as artistic as an art lesson, and not as mathsy as a maths lesson. And with so much real mathematical and artistic exploration to do, we don't want to do anything that's less than best. 

But sometimes, taking a bit more time with appearances, can make things clearer. Sometimes it can let you re-approach something that needs revisiting in a new form. And as in this case, sometimes it gives you something that doesn't look plain to leave up on the wall for a while. We went further with it, we used it as an opportunity to share what we'd been learning with the younger classes that passed by our display.

 

The immediate aim of this kind of event is that the students will have to revisit their learning as explainers, finding their own words for what they have created. There is of course the chance that the trees might intrigue some of the listeners too!

Something else - an unusual thing - happened with these trees recently. Two artists Charlie Youle and Bevis Martin came across our trees on a trawl through children's maths images, and made sculptures based on them! 

a "thought flower"

I've posted about their exhibition here.

meeting the artists

Quadrilateral Sets - the lesson

In my last post I shared an idea for a lesson. So, how did it go?

First of all, I checked that we understood the idea of a set. Showing this,

students responded that it was a set of animals. When I asked them to be more specific, they observed that it was a set of rainforest animals. I tried another, this time with some of the girls' names, and again it seemed to be straightforward; the students at first said 'girls' and then 'girls in our class'. So onto the sets of shapes:

1. Students look in pairs at what the members of each set have in common . What are its defining features?

 

We paired up and got looking and annotating.

2. Students share what they think

While this was going on, Estelle, Laura and I circulated and asked children to talk to us about what they had noticed. It was good to hear them attributing some of the observations to their partner, and set us up for asking for hands up if they could say something their partner had noticed. Almost everyone did this, while I scribed some of the observations on the whiteboard. 

I was pleased when K said, "E saw something really cool..." and made sure I emphasised not just the listening but also the appreciation K was giving.

There was plenty noticed. Some things I'd hoped for didn't come up; no-one pointed out the parallel lines in the shapes in the bottom right. No-one saw the features of kites in the top right. But there was lots else. A lot of students commented on the fact that the shapes could be split into triangles. And quite a few talked about how some shapes were like squares that were distorted.

That was quite a lot of articulating our thinking together, so I called it a day, and we played a few games of Aggression to finish off (which involves thinking of its own kind).

Today I returned to it. Rather than try to clarify or introduced terms, I decided to just go with what the students had noticed.

One of the things was that the four-sided shapes could be split into triangles. We looked at that using Geogebra.

"Do you think all four-sided shapes can be split like this?' - everyone agreed with

 this generalisation.

The other thing that had stood out was their way of seeing parallelograms as "twisted", or "tilted" squares. Again using GeoGebra, I showed how a square could be viewed in a 3D space and by changing the point of view seen as different shapes.

It was as if, I said, we were turning a square in the sunlight and seeing different shadows. Someone asked if we could do that, so at the beginning of break time we took down some squares and spent a couple of minutes drawing shadows before the other classes came down:

I've not really thought  of teaching projective geometry before, but the lesson gave me the impression that the students understood the idea more readily than they did the categorisation of quadrilaterals. Perhaps something to return to?

3. After we've got a thorough feel for the four sets, we do the normal Which One Doesn't Belong?

Knowing that the picture was more complex than normal, I didn't push this for too long; in fact I stopped after four observations:

And there's "squares and rectangles! "No rectangle." Strawberries and fruit.

Where next? Afterwards I talked with Estelle about how the lesson went. We both thought the quality of the students' listening and reporting to the class while they were working in pairs was great. How to build on that, and get them listening even better and building on each others ideas, that will be part of our project. I'm looking forward to reading, adopting and adapting some of the thinking moves suggested in Making Thinking Visible.

It's given me a picture of which ideas about these shapes are within reach of and appealed to the students. Aside from the projective geometry and splitting into triangles, which I'm not sure how I would approach, the top left set seemed the most approachable, with the idea of right angles featuring. I wonder if we should return to Mondrian with this lovely Mondrian geometry tool, and use the idea of perpendicular lines to construct digital Mondrians?

What would you do next?

Quadrilateral Sets - Which One Doesn't Belong?

Teaching 2D shape, I often go for hexagons rather than quadrilaterals. There's less history, less convention, more space for student thinking rather than finding out what things are called. Have a look at Christopher Danielson's video about using hexagons for proof.

But Kristin's latest post about a Which One Doesn't Belong image (again there's a Christopher Danielson element), made me think that this might be a viable way to approach quadrilaterals:

Which one doesn't belong? (Diamond edition)

cc @MathMinds pic.twitter.com/VmMh8rakXg

— Christopher (@Trianglemancsd) February 9, 2016

This was really to address that we call the rhombus a diamond, without getting all vocabulary-ish in the process.

It made me wonder...

Could we make a WODB about sets of shapes?

The aim: look at categories, and the idea of inclusion and exclusion based on a criterion, starting with not vocabulary but the students observations and thinking.

So I've rustled up four sets of quadrilaterals:

I'm planning to do this tomorrow in three stages:

1. Students look in pairs at what the members of each set have in common . What are its defining features?
2. Students share what they think.
3. Then, after we've got a thorough feel for the four sets, we do the normal Which One Doesn't Belong?

I don't know how it's going to work. Am I putting too much in? First of all looking for what's the same, then looking for what's different?

Or will it be a feast of thinking and generalisation-making?

I'm encouraged that everyone's got a good grasp of how WODBs work, and can make their own, justifying the inclusion of each corner.

and I was really pleased when a student came in with her own home-made WODB this morning, and wanted to run it with the class (she's got another one, but one a day is enough):

So, if the class were ever stoked up for this challenge, it's now.

Just in case dealing with sets is too much of a stretch, or in case students go down the vocabulary road, I'm going to keep in the wings a few statements from the Always, Sometimes, Never list, and we'll switch to these if it's what's needed:

• A square is a rectangle.
• A rectangle is a square.
• A rectangle has two long sides and two short sides.
• All four sides of a square have the same length.
• Parallelograms are slanted rectangles.

--o--

There's another dimension to the lesson. I'm looking, with Estelle, at how to design talk that helps learning. We're reading Making Thinking Visible, and I'm really impressed by the tone and content of the book, and before I'm even on to the book's thinking moves, I want to try to use its analysis as a guideline, to set up talk that around different kinds of thinking:

Naturally, there's

• Observing
 Closely 
and
 Describing
 What’s
 There 
• Building
 Explanations 
and 
Interpretations 
• Reasoning 
with 
Evidence

But I'm also keen to get the students' listening going with 

• Considering 
Different 
Viewpoints 
and 
Perspectives

I'm hoping that the pair work will give lots of opportunity for this.  I'm going to ask them to pay especial attention to what their partner says, and note this down, so that they can report back on this.

Estelle's coming in to observe the lesson, and I'm hoping she'll pick up on ways the students are talking to each other that are helping their thinking.

I've just started a system for making sure everyone works with (almost) everyone at some point. Each student had to choose a 'Bucharest partner', then a 'Bangalore partner', then a 'London partner' and so on and note down all these 'Place Partners' on a list. The result (after me going through the choices) is a matrix that looks like this (with the names taken off):

(There's a few gaps (black) but I'll try and pair those up another time.) It's important that the students can and do work with everyone in the class, not just he ones that they get on with. So tomorrow it will be the London partners.

After the paired work looking at the sets, we'll come together to share and look at the WODB. I think they might need steering through this.

The L-shaped room II

We did the lessons I'd planned on area.

Yesterday we went with Graham's Paper Cut.

I think I got this out of sequence: most students found it hard.  Half the class looked pretty stumped (at that point I should I think have given them the pdf to help, but I wanted to see what the other ones would do without it!) About a quarter of the class, give or take, got to an answer.

B did really well - she made a squared paper version of the large sheet, and one of the small sheet, and could see how the big one was twice the little one. I love how she goes off her own way!

I think I know what went wrong:

1. Although we'd done multiplication as area, this was area as multiplication as areas, a step removed.
2. As one of the students said, if the little paper was taken out of the corner it would have been easier to think about. The hole seemed to really puzzle some students.

This is no criticism of the task; just of me getting the order wrong.

I've just started to read Making Thinking Visible, and among lots of really good thinking about thinking, there's a great list of ways of thinking that lead to understanding:

• Observing
 Closely 
and
 Describing
 What’s
 There 
• Building
 Explanations 
and 
Interpretations 
• Reasoning 
with 
Evidence
• Making
 Connections 
• Considering 
Different 
Viewpoints 
and 
Perspectives 
• Capturing
the 
Heart 
and
 Forming 
Conclusions
• Wondering
 and 
Asking 
Questions

We did the first plenty (though actually, as the authors make clear, there's not really an order) but that fourth one, making connections was not strong enough. There weren't enough landmarks, for both the reasons I give above. (Making connections is interesting: there can only be connections if there's enough previous learning; but the previous learning can't be too close in time or content, otherwise it's not really a connection that's being made, just a repetition. The new has to be partly the same, partly different.)

So, as we were two steps ahead, today I went back two steps, and then forward from there.

First of all we drew round books, estimated the area of the front cover, and then measured with hundreds, tens, and smaller Cuisenaire rods. Back to something easy.

Then we went down to look at the teachers study room, for quick orientation. Then we came back and I showed them my video (not shot very well I know!) -

When I asked them what they wondered, it was very obvious (the book measuring was too close!).  They wanted to know how many carpet tiles in the whole room. We looked at this:

and people saw very quickly what the other sides must be, so we got down to working the area out. Some just used hundreds and tens:

Some drew:

And there we had to leave it. I headed off to Amsterdam. There wasn't time for John's great idea of choosing your own L shape. I could still do the staff room. This could be something for next week.

I benefited a lot from the comments in the planning post. Nina's comment has reminded me again that I need to have more knowledge of and be more sensitive to the sequence and level of difficulty of tasks, and although I got it wrong (they would maybe now be ready for Paper Cut), I'm aware that I just need to have a much more detailed appreciation of the order It's especially true when I'm using someone else's lesson; with my own ideas or more familiar lessons I've got more of a feel for sequence.

Joe's stress on seeing the room was spot on, and Graham's idea for a video, though mine is evidently not at all exciting, gave them a really thorough sense that it was area we were looking at by focusing down on tiles (still one student measured round the perimeter with rods). 

While I'm away they'll be looking at some triangle areas on real and virtual geoboards.

I also feel that, to make connections with other lessons they'll do, the classes could do with a better understanding of dimensions, of area as two dimension:

And I have another thing for them next week, something I saw when I arrived in Amsterdam Airport, which seems like a next step...