Saturday, 30 July 2016

naturally ingenious combining

I'm rereading some of the chapters in Madeleine Goutard's Experiences With Numbers in Colour. Something jumped out at me. It was written in 1964, but in a way it's better suited to now, to 2016. The destiny of many if not most sets of Cuisenaire rods around in the 60s was to end up in the bottom of cupboards, along with the Dienes apparatus and Geoboards. What was not there, the ingredient that was needed to make the pedagogy work, was the idea and practice of number talks. Even now this is maybe not that widespread, but at least it's out there, with wonderful books like Intentional Talk leading the way. Now that we are using quick dot images, asking children to look at groups of dots and tell us how they see the total, we understand better the importance of what Goutard below calls "naturally ingenious combining".

I won't be using this particular part for a while as my five year olds will be doing a lot of playing and other things before we get to this stage. But you'll see what's happening. The class is motivated to explore something together, they are creative and playful in the way they find solutions. The emphasis is on doing and trying out rather than having remembered (although they evidently have a lot of experience with rectangles as products of two numbers).

The passage starts with Goutard introducing some rods:



She then introduced a black rod (7 white rods long)...
And not just flexibility of mental calculation. Flexibility in thinking. And also comfort with manipulating numbers. These days too it's easy enough to take photos of what the kids have made , get them up on the whiteboard, and come together to talk about the different representations together.

Friday, 8 July 2016

Meeting my new class

So, last week we all met our new classes for 45 minutes. Mine will be all five years old in September, what up until now we've called Year 1, but come September, we're calling Kindergarten 3 because 1. we're being less British, and 2. we're being much more play-based.

So, after an introduction ("Do you want to call me Mr Gregg, or Simon?") and a song, I introduced - of course - the Cuisenaire rods. I asked them what they thought they were for, and they said making pictures of various kinds, not a mention of maths. I said they were sometimes used for patterns and numbers, and one boy counted to twenty in response.

So - of course - I said, "Make anything you like out of them, pictures, patterns, whatever you like!"

And they did!
a blue-black pattern
We'll return to this, and explore lots more patterns.
a gate
A lot of people made things in 3D, like for instance this house:
a house with a chair
 I wonder what other-size chairs we can make...?
a face - we know where this leads
 And here's an interesting rectangle...
 Here is the beginning of lining things up, measuring them up against each other neatly. We're going to follow up with a lot of this.
a sun shining!

When I look at these pictures, and all the others, I see the kind of things the children are interested in. They like houses, and roads, car parks and castles. Some made abstract patterns of rectangles, or spelled out their name. And I see lots of starting points for further exploration.

(The next day a couple of the girls came to me with a home-made envelope with some nice messages and pictures inside, like this one of that first meeting. Those yellow trays on the tables are the ones with the Cuisenaire rods in.)

Tuesday, 7 June 2016

Patterns of Prague

I was in Prague for a great weekend course on play-based learning this weekend, with Estelle as we're both moving to Kindergarten in September. We also got to explore the beautiful city a bit in the evenings, and Estelle was very indulgent when I kept suddenly stopping in my tracks to snap the amazing paving patterns all around.
I was looking forward to trying these out with my Year 4s. I was not disappointed.
This one first, with us all in front of the whiteboard: What mathematical questions could you ask about it?
So, look for a while at it. How many squares in the black cross?
OK, 33. Look again, if you didn't get 33, and work it out to see that it's 33.
How did you see that?
A good crop of answers. So now on to some individual work. How many squares in the black star? And communicate how you work this out.







There were a few slips here and there, but the good thing is, everyone had a clear idea about the task, and everyone was trying to cut their own path through. It's what our quick image tasks are really good for - "it's over to you - find your own way through!"

And there were so many different ways! This seems so much healthier for students' adaptability and independence than the One-Ring-To-Rule-Them-All approach to calculation and algorithms. 

Martin Joyce made me aware that the Patterns of Prague were already a maths thing:
And Danny Brown, when he saw what we were up to, tweeted:
Here's more of that article:

There is a difference between counting and watching yourself counting. It is observing how you count, rather than just counting, which leads to statements about counting, such as 6 X 5 + 6 X 5, or l(w+ l)+w(l+l). An algebraic statement about the number of matches used to create an n by n square, or an I by w rectangle comes from finding a way of counting. A certain awareness is required to be able to count the matches, but a second level of awareness is needed to observe and articulate how that counting is being carried out. It is a double level of awareness - awareness of awareness which is required for you to be in a position to write an algebraic statement. John Mason has talked on several occasions about the following from the Rig Veda: 
     Two birds, dosed yoked companions,
     both clasp the same tree.
     One eats of the sweetfruit,
     the other looks on without eating. 
It is awareness of awareness which is involved in working algebraically. Arithmetic is concerned with the result there are 60 matches. Algebra is concerned with organising the counting, finding a structured way to get the result. To be able to count requires a way of counting, a way of structuring and organising the counting. To be able to count requires you to work algebraically. 
 Approaching Arithmetic Algebraically, Dave Hewitt,
Mathematics Teaching 163
So, these quick images foster independence, a focus on contemplation and communication as much as calculation, and algebraic thinking. Give one a try!

Sunday, 22 May 2016

Fractions... and Someone Help Me Out With My Lack of A Theory of Learning

Danny Brown asked which theory of learning I subscribe to, and I was a bit stumped. Not because it isn't in there somewhere. It's probably quite a good theory of learning. But I'm not exactly sure what it is. I'm not proud of that (really!); I think it would be a good thing to read up on and then place myself on the map. It's just not something I've done enough to really locate myself very accurately.

I thought I'd blog about our recent fractions work, so why not, in the absence of a theory of learning, throw in a few things I think direct me.

A fraction talk is always a good thing. There should be looking, and thinking and talking. Mistakes should be okay. Learning happens best when we can try things out, where there's not an authority we defer to, but we get stuck in and make of things what we can. So I note down "mistakes" with just my usual interested expression on my face, no eyebrow ironically raised (I can't do that anyway):

I think it's okay for the teacher to chip in too, with as much interaction as possible:

And then we can revisit what we said, and see if we can make more sense of things:
I think there should be lots of hands-on and play, especially for primary children. They are more alert, engaged and comfortable when they're playing, and usually everyone gets on with each other a lot better.
"Fraction Fortress" - great for getting familiar with the relative size of fractions
 
"Fraction Formula" - I maybe need to make fraction cards that aren't coloured for this.
 Another thing - what Zoltan Dienes called "multiple embodiment". We need to sweep through the territory in more than one direction, make links with more than one other area of learning (Liping Ma calls this breadth of learning). Some of these give depth too, in Liping Ma's terms - they connect to more important ideas on the subject. One of the connections it's useful to make is to the number line, that fractions aren't just parts of something, they are also numbers in their own right:
We had to say which two other fractions the new one fitted between.
Adding the fifths, on green card, to the washing line
Another idea that's in there is that students don't need to know or remember things all the time, sometimes they can just contemplate, notice, explore, ask questions. These are skills that will always be useful, so the trick is to find environments where there's plenty of real maths to notice and think about. I found one such environment in Ford Circles.
I didn't know about them until I saw this video


I didn't show the video to the class of course, but I wanted them to be in this unfamiliar place, and to feel comfortable, to learn again what it's like for things to come into focus after seeming completely unintelligible. And in terms of fractions, all sorts of things - that they slot between each other, that they continue to do so, that there are hierarchies and patterns in them, that they can be organised as a complete set...

Here's what they thought first of all:


Then after the half was slotted in half way between the 0 and the 1:
Impressive thinking from Kirill. We then saw the next slides step by step, and I showed them the Wolfram Demonstration, which allows you to show different numbers of circles and to zoom in and out. (You need to install the Wolfram CDF player to do this.) Things were really hotting up with the noticing:
Students need to move between whole-class conversations and individual and group work (otherwise how will you and they know what they think?) so we went off to make notes about what we'd noticed, and any questions we had. 
A number of students were bothered by the size of the cirlces. This was a productive red herring.
This representation made links between decimal and fraction notation



1/3  "not goodly slotted" - great thing to notice!
Another potential brick for the edifice of my theory is that we learn well by having an enviroment where we can assemble our own data and organise it in our own way, that gives feedback itself about how good a systematisation it is. It's often hard to find such environments, but the trusty Cuisenaire rods provide one here:







I'm thinking to ask students to place these on a number line next!
Some students wanted to draw them
J asked to do a bigger one!
Some students were puzzled by the fractions that were bigger than one - but puzzlement that comes from something you yourself have produced is no bad thing.

None of this amounts to much of a theory of learning of course, but hopefully the small components might fit in somewhere.

As for that theory can anyone - Danny? - point me in the right direction for some reading to help me?