Monday, 28 December 2015

How we write numbers

I'm thinking to start off with this image*:
And the questions are - of course - What do you notice? What do you wonder? 

I don't know which way it will go then.

But I know there are lots of great ways to go from this image. I'm hoping for some observations that are historical, and perhaps some mathematical ones.

One area I'd like to explore - historically and mathematically is the changeover from Roman numerals to Hindu-Arabic numerals. It could show us how a change in the way we think brings about changes in society. It could show us the importance of cultures outside Europe. And there's some potential great exploration of different number systems and numbers in different bases.

Already F has been writing some equations in Roman numerals, and R has been talking about how our system comes from India.

I like what Graeme Anshaw has done with this:

I think we'd probably want to get our heads round how Roman numerals work. Here's a good way:
Arithmetic wasn't so easy for the Romans, so they used a counting board or abacus:
It would be interesting to make, and try to use one of these. It looks like they work like a Japanese soroban:
And then, how did they do this on a counter with jetons?

Here we could investigate other ways of representing numbers from different cultures, and at different times.

Now for Hindu-Arabic numerals...
File:Bakhshali numerals 2.jpg
How did they make that journey, from India to the Muslim world, to Europe? Lots of stories in there, and I'm pleased there's such a strong cross-cultural element.

I've just been reading Keith Devlin's Man of Numbers, about how Fibonacci introduced them into Europe. There's lots worth finding out in that story too.
And thanks to Tracy, I have a nice little picture book about him to read and contemplate with the class:

Part of why this all seems more do-able to me now, is partly that I've seen there is lots of maths that can be explored by the students, rather than simply being told about. Reading Madeleine Goutard on our base system, and other bases, I see more clearly how we could represent them:

from Madeleine Goutard, Mathematics and Children
The bottom one is perhaps how three-toed sloths might represent numbers, in base six.
Three-toed sloth (source: wikimedia)
How would those numbers in base six be written?

How about other bases? They could be represented by Dienes' blocks too:
Only problem is, even though there's loads of sets gathering dust in maths cupboards everywhere, they're not used any more, and not sold in the catalogs. Maybe I'll make some...

So, lots of directions to explore from that first image.

And then there's that dress, with that mysterious pattern of numbers.

*Calculating-Table by Gregor ReischMargarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th. Source

Monday, 21 December 2015

The student’s active attitude towards Mathematics

As we're just about to begin the IB primary years programme, with its emphasis on inquiry-led learning, and as it is anyway one of my preoccupations at the moment, now is a good time to ponder what inquiry might mean.

For me it's part of a set of ways of thinking that make up an active attitude.

Once, taking my class out on a museum visit, I was struck by how the students moved around town, how they crossed the road. They seemed not to be actively looking around much at all, not to be be checking the road was safe. It was almost a kind of trance, maybe talking as they walked, but relying on the teacher to make sure they were safe. It occurred to me that this was a metaphor for a lot of learning in school, a kind of passive reliance on the teacher to do the work. Mirrored by the teacher's expectation of being the authority.

I've just read Danny Brown's great post The student’s passive attitude towards Mathematics and his [sic] activities which is all about this kind of passivity.

So, inquiry for me is part of an active state of mind. Think Sherlock Holmes rather than Inspector Lestrade ("wholly conventional, lacking in imagination, and normally out of his depth"). What does Holmes do? He notices things. He wonders things. He, perhaps implicitly, asks himself questions. He goes out, looking for further clues. There is something of the hunter noticing the traces of his prey in him, perhaps going back to the roots of detective fiction in stories like The Three Princes of Serendip. The alertness of these detectives... the opposite of trance. The present case is not routine, there is not a procedure to follow that will bring sure results. It must be interrogated, understood. There may need to be experiment.

So how do we help our students to this kind of attitude?

That must be the subject of a whole book, and I've posted thoughts around this question already. But here are today's thoughts.

I came across the booklet Cognitive Activation in Maths and it seemed to be getting to the same thing (Cognitive activation = "starting to think actively"?). Here are some thoughts:

Another viewpoint: the idea of "open". An open-ended problem is one where pupils can follow something up in different ways. Dan Meyer, among much else, talks about open-middle tasks. And then in a unit of inquiry in the PYP there's meant to be space for open-beginning inquiry. What does open mean? To me, it means that students can do some choosing:

Where could I explore in this place the teacher has shown us?
How can I get to the place the teacher has shown us?
Where would I like to go?

That last one is a hard one to ask students to do, and for the teacher to manage. It's one I'd like to find ways into.

All of these seem ways of encouraging students to be active, not passive, in their learning. Alongside this there are all sorts of number talk, estimation and quick image activities that ask students how they see things, how would they explain things. They are asked to give not the answer, but their answer.

I like this table in Making the PYP Happen (though I'm not sure whether this is how mathematical practices are really changing - I'd like it to be):

"Instruction built on what students know, what they want to know, and how they might best find out".

It would connect  the motivation to play, explore and create that children are allowed to exercise in Foundation Stage (K) to the wonderful explorations and results of mathematics that have come to us across the centuries. 

As for manipulatives, they don't just make mathematics understandable for students, they hand over some control, they allow students to think with their fingers, to experiment, to explore, to create. Take for instance make a pattern lessons that we do regularly.


I'll be reflecting on this idea of being active next term, how best to encourage students to follow up their spoken or unspoken questions. 

This post has been hopelessly general, but I'd be interested in your ideas about how to allow and encourage an active and inquiring attitude.

Saturday, 19 December 2015

The Greeks, Geometry and Proof

As we begin the PYP, we're trying things out. We've got some initial ideas of our own, but we're looking for ideas from the students that we can follow up on. We want them to be genuinely following an inquiry that they're interested in, or better still, that they have originated. But what follows are some initial ideas that could serve as a series of six or so lessons.

Rosy and Rachel, my colleagues in Year 5, are planning a unit on myths, focussing on the Greeks. I'm so pleased that they're using Hugh Lupton and Daniel Morden's retelling of The Odyssey as a starting point. A few years back my class got a lot out of that book (see some products on that year's blog). If I did it again, I'd definitely use Peter Worley's The If Odyssey to get to some great philosophical themes in the Odyssey too.

But Rosy was thinking whether it might be a thing to link in with the geometry that the Greeks did in ancient times. She asked me about it, and here  are my thoughts about where I might start. I've not done these lessons before, but now I want to!

As I say, I haven't done this sequence before. What I'm trying to lead to is the idea of proof. I've pared down Euclid, to try and find the simplest possible starting points. But maybe you can think of an easier or better way of doing it? Maybe you've had some experience with this? Please post a comment!

I think I might introduce the topic with this image, asking, of course, What do you notice? What do you wonder? I might also tell the story of the Delian Problem.
An English illustration of a translation of an
Arabic translation of the ancient Greek Elements by Euclid

Lesson 1

I think I would introduce students to the tools the Greeks used for geometry – get the compasses and rulers out and make something artistic with circles.

Lesson 2

I would make things in GeoGebra too. Let’s do a little exploring of this brilliant tool!

(I like this too: )

Lesson 3

I might start by showing this image:

And asking “What do you notice?” + “What do you wonder?”

Will someone say something to the effect that some of those angles are the same? Perhaps…

You could then write this up as eg. “Maria’s Claim”.

Lesson 4

Return to Maria’s proof. Could anyone say how her claim might be proved?

Show them Euclid’s book (I’ll lend it to you).

The Greeks developed the idea of proving something, of being completely sure.

Each proof in the book rests on other ones, like bricks in a house, going back down to the things Euclid said were obvious. (We worked on proving a little last year - example)

Here’s an example.

Look at it for a while together & see if anyone can see what’s going on in this proof.

(You could tell them that ∴ means “therefore”.)

Basically, the same thing is added to the two opposite angles giving the same sum = the angles on a straight line, 180°, so the two opposite angles must be the same. Hopefully some of the children will be able to see this and explain it. It really is best if most of the thinking can come from them.

It might be best for them to actually play with physical angles, arranging them – maybe they could do a video of how they know it’s true? Fraction circles would do the job well.

Lesson 5

What do children notice / wonder about this one?
 Hopefully, eg, that the two red angles are equal.

Perhaps they could have a big copy on paper to cut up & see if they can show that this is so.

(Could make a poster, or video, or just demo to class)

Lesson 6

Look at this image:

What do you notice? What do you wonder?

Here, there’s the idea that the angles inside a triangle add up to the angles on a straight line, 180°.

How might the students go towards proving this? What they’ve done with the previous two images might help. Another route would be to get everyone to make a different triangle on A4 paper. Colour the three corner angles different colours. Photocopy it so there are three copies altogether.

They could then cut the three out and arrange the triangles next to each other. If they put the three different-colour corners together, they should line up on a straight line. 

Tuesday, 15 December 2015

Rows of apples

I don't do word problems very often. But I liked what was happening over in this great 4th grade Learning Lab lesson described by Kristin. So I adapted the less-numbered prompt and asked my class what they noticed. 

They were indignant (maybe because they've been made wary now):
J's "It's a lie!" was vehement.

I disagreed: this situation, though it is made up, could be a true one. And we do get some information from the description. I asked if anyone could draw an apple display that might be possible:
That one at the top is my counter-example.

So now for some numbers. The class seemed relieved by their appearance! What questions could we ask?
I wanted to get them working individually at this point, so off we went straight away. "You can use cubes or Cuisenaire rods if that helps you. Show me your answer, and what you thought to get to that answer."

There was a great range of approaches. Here's a selection:
This student was the only one who just added the numbers together.
I suggested he got out 56 cubes and arranged them in four rows...
I was so pleased she chose to do this!
This student was so excited that he'd got this.
But when asked, he said his method was to estimate, and then tinker a bit to get the right answer. 
Halving 56 took this student a lot of time though!
I think someone told her in the end!
I only saw this at the end! He'd multiplied by four! I explained that there were meant to be 56 apples altogether. Maybe because his first language is not English, he'd not quite understood? I need to ask. I told him I thought what he had achieved was good even so!
I like the fractions added at the end!
There were quite a few of these halving and then halving again ones.
It wasn't a strategy I'd talked about, so it's great to see it coming up!

Again, I love K's link with fractions
We'd been writing a few fraction equations (whatever they wanted!) earlier in the lesson, and K had noticed a pattern ("Can it go on the claims board?"):
He's used that pattern
So, two people set off in the wrong direction. One student, whose work isn't scanned here, couldn't explain how she got to 14 and I suspect that maybe she'd simply copied the answer. I'll need to sit with her next time!

It definitely felt like starting with the less-numbered situation gave students a chance to understand it before any number grabbing happened. We got to talk about what information was actually there too. I loved seeing all the approaches, and scanning them means I can show them tomorrow and compare different strategies. We'll showcase K's claim too!

Sunday, 13 December 2015

Teachers as inquirers together: the PYP and mathematics

My school is now taking up the International Baccalaureate Primary Years Programme (PYP) for all learning including maths. The philosophy of the PYP is to make the students into "inquirers, thinkers, communicators, risk takers, knowledgeable, principled, caring, open-minded, well-balanced, and reflective."

Heavy on abstract nouns, eh? But a nice pick. Inquiry is especially central.

Now, as Dan Meyer recently tweeted, inquiry can mean a lot of different things to a lot of people.

But (and I think I've got another blog post to write on this) it does mean something. It means, for one, that the students should be questioning, and that their questions should count for something, maybe a lot.

And, if we extend it to teachers, they can be questioning, asking how they can develop too.

In England there's been exchanges with teachers in Shanghai, in an attempt to see how they get such a good ranking on the PISA tables. One of the commonly remarked on differences is that teachers in Shanghai "have more time to...  have teaching research groups, and they can observe other teachers teaching in the classroom. They can engage themselves in a lot of professional development activities."  (link)

I'm also impressed with this, from the PYP document Making the PYP happen (PDF link p84):

How mathematics practices are changing

Structured, purposeful inquiry is the main approach to teaching and learning mathematics in the PYP. However, it is recognized that many educational innovations (or, more accurately, educational reworkings) suffer from the advocacy of a narrow, exclusive approach. The PYP represents an approach to teaching that is broad and inclusive in that it provides a context within which a wide variety of teaching strategies and styles can be accommodated, provided that they are driven by a spirit of inquiry and a clear sense of purpose.
The degree of change needed to teach mathematics in this way will depend on the individual teacher. For those teachers who have grown weary of imposed change for which they see little point, it should be stressed that teachers are not expected to discard years of hard-earned skill and experience in favour of someone else’s ideas on good teaching. It is suggested, rather, that teachers engage in reflection on their own practice, both individually and in collaboration with colleagues, with a view to sharing ideas and strengths, and with the primary aim of improving their teaching to improve student learning. In doing so, they will be modelling the skills and attitudes that have been identified as essential for students. 
So, what's good for the students is good for the teachers. Rather than a curriculum that is imposed, there should be space to interrogate our practice, and to learn from each other's ideas.

What might this look like? Well, I was very struck by Elham Kazemi's idea of teacher time-outs. Even if we don't go for this particular idea, there's something about the spirit of collaboration in this video that is really constructive. As Elham says: “not to watch and evaluate each other, but to problem-solve and inquire into teaching, and, most important, student thinking, together”.

Of course, I'm part of a lot of online collaboration as part of the wonderful #MTBoS maths teacher blog-O-sphere, but face-to-face is good too.

And in fact I've had some great experiences thinking together with colleagues this year, working with Estelle on valuing talk in the classroom, and with Rosie on philosophy and big questions, visiting each other's classrooms and trying things out. Although I already have great planning sessions with Julie, my colleague in Year 4, I'd really like to spend more time with colleagues in other year groups seeing together what inquiry could mean in practical terms in maths lessons.

Friday, 11 December 2015

¾ of a hexagon

M is an enthusiast. He is busy writing a piece based on Saving Private Ryan at the moment. He asks to stay in at break times to do it ("No, I need a coffee!") and is always busy on some project.

So, after our fraction work yesterday, finding a third of a hexagon chocolate cake in various ways:
M was the first one to come up with the idea of splitting the triangles for a more interesting third
M took one home. He brought it in again this morning with some careful explanation written on it. While the others were getting on with it, I asked him to explain his work. 

I know, I took over a bit! Once I'd finished dividing the hexagon up for him, we coloured it in together on the touchscreen, the way he'd shown. He just needed help with the first line, then he got the idea, and I went off to see how everyone else was doing.

I love the way he just keeps on with his explanation. I wonder whether I did the right thing now. Perhaps rather than helping him to draw his idea, I should have asked him how he would divide a triangle into quarters?
But M was pleased with the results, and really pleased that his prediction that the coloured parts would make hexagons was correct.
I think I have a bit of a weakness for beautiful things rather than hard thoughts. But I wonder, should I do it differently next time?

Thursday, 10 December 2015

Making sense

I mentioned before this question about the age of the teacher:
Last year all 30 children in my Year 4 (Gr 3) class gave me a numerical answer to it.

This response seems so amazing to me that I was determined to do it again this year.

But first of all, yesterday, we watched the famous Asch conformity experiment:
It's now a classic psychology experiment, as Zimbardo says, and I was interested in what the class would make of it.
The experiment is so simple, that children this age can appreciate it, and the design really appealed to the class. We talked about what the results might mean

We talked about what might motivate people to just say what everyone else is saying rather than what they could plainly see. Most people thought they wouldn't do the same thing in that situation.

We talked more, and discussed how sometimes you really need to just go along with what other people are saying, even if you don't want to. Like when you want to play a game together at break time. You don't always get to play the game the others want to. But you want to play together.

We didn't really get to a conclusion, and I didn't add too much to what the class said. I would prefer the students had the chance to ponder this themselves and think their own thoughts about it.

So today I asked about the age of the teacher. I said, think about the question; if you want to, write something down. And... almost everyone wrote "30"!

Here they are:

Afterwards, I told them that I didn't think what they'd written was reasonable. They hadn't got the information to write a number down. Why should there be a link between the number of tables and chairs and the age of the teacher?

While the children had been answering on their whiteboards I'd overheard J saying, "But it's not true", before he wrote down 30, so I asked him now what he'd meant by that. He said he hadn't thought the question made sense. "Why did you write 30 then?" I asked. "Because I saw everyone else doing it and thought I should write something." Quite a few people sat up at this. There was some laughter of recognition. M said immediately, "It's just like that experiment with the lines."

I asked if others had thought like J who had thought that the question didn't give them enough information to have an answer. About six put their hand up. One said they didn't want to leave their whiteboard empty, and subtraction and division gave numbers that were small, and multiplication gave an answer that was too big.

As before, I told them that most people answer like this when they're given this kind of question, even older children. As a debrief, I showed them Robert Kaplinksy's video, How Old is the Shepherd?
They liked this, and I think it made them feel a bit better.

What do you think? Is there a place for this kind of mini-lesson?

Saturday, 5 December 2015

Fractions of a square

I liked this tweet:
and saw there was at least one ingenious way to solve it:

I looked at it like this:
and it reminded me of this:
(The yellow square is one fifth of the big square. There's a proof here.)
So, I could see here that the diagonal was divided into thirds. After a bit of thrashing around, I could also see that the green triangle must be ⅙ of the whole square.

I like the way the square is divided so simply up into fractions. There are halves and quarters:
and all sorts of other possibilities:

Oh - that would make a good Which One Doesn't Belong?
It strikes me that it would make a good shape for students to divide up in different ways like this.
In the same way that they have here:

And then there's the one fifth square...