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:

@Simon_Gregg @estelleash awesome. There's a Mars task called patterns of Prague. https://t.co/5P6XJmJc8j

— Martin Joyce (@martinsean) June 5, 2016

And Danny Brown, when he saw what we were up to, tweeted:

@Simon_Gregg brings to mind this from 'approaching arithmetic algebraically" MT163 @ATMMathematics @journaleditor6 pic.twitter.com/cEmqd5ttpt

— danny brown (@dannytybrown) June 7, 2016

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!

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?

First encounter with Cuisenaire

We're looking after V who's six years old tonight. I'd made little maths books with him before, asking him what he'd like to put inside, and making a few suggestions too.

This weekend I brought plenty of Cuisenaire rods home and put them on the table loose in a tray.

"What are these?"

"I use them for teaching maths at school."

"What do you do with them?"

"You can do anything... but you have to sit down."

He laughed and sat down, and then first of all tried balancing the rods on their ends. Then he picked up a white rod and said, "This is a one." Just like that.

Then he grabbed a blue rod and asked what that was. I asked him to have a guess. He thought maybe twelve.

He said the red one was a two, and then lined some whites and a red up on top of the blue. "It's a nine," he said.

I was really amazed at the speed this was going at, with very little input from me, except to say that I didn't like the plastic ones as much as the wooden ones.

And then he made the staircase, starting at ten, and searching for the colours he needed:

When he'd done that he decided it was time to build up again:

Then he begin to fill in the space:

"It's a table!" he said, and added legs:

And that brought us to dinner time. I'm pretty impressed by all this. How much V did in that one short session of play!

Liping Ma

I've been reading Liping Ma's 1999 Knowing and Teaching Elementary Mathematics, very relevant to the moment, for teachers in (or from!) England as much as teachers in the States. It compares the pedagogical subject knowledge of teachers in the US and in China. And finds that the Chinese teachers have got a lot more of it.

This is relevant to the English because there's a lot of similarity between English and American teachers, and the UK government is looking towards Shanghai for inspiration, and promoting a "mastery" curriculum. There is of course the PISA-envy we feel towards Shanghai and Singapore, and along with it plenty of doubts and caveats.

But, whatever your starting point as an elementary maths educator, there's a lot of good research and a lot to think about in this book. Here's two of my takeaways:

1. Depth of teacher knowledge

One of the things she does is look at how teachers would teach four scenarios. For instance subtraction. She looked at how teachers responded to this question:

Let’s spend some time thinking about one particular topic that you may work with when you teach: subtraction with regrouping. Look at these questions: 

 How would you approach these problems if you were teaching second grade? What would you say pupils would need to understand or be able to do before they could start learning subtraction with regrouping?

She found that lots of US teachers talked about "borrowing", while most Chinese teachers talked about regrouping, with lots talking about multiple ways of regrouping.

Have a look at Liping Ma talking about this:

She found that the Chinese teachers had a much better grasp of where this scenario fits within the whole curriculum. This subtraction questions for instance fitted in here:

She gives a detailed description of what "profound understanding of fundamental mathematics" is. It's deep, broad and thorough.

"I define understanding a topic with depth as connecting it with more conceptually powerful ideas on the subject... understanding a topic with breadth, on the other hand , is to connect it with those of similar or less conceptual power." p121

I like this description of understanding and its importance. Though most of the book discusses the teaching of arithmetic, beyond the arithmetic is a deeper, algebraic understanding:

"There is a misunderstanding of arithmetic in this country. Many people think arithmetic equals computational skills, and that’s it. Arithmetic has a theoretical core and there is intellectual depth to it. That arithmetic can serve as a good foundation for students to learn algebra or other advanced math." (Interview in NY Times)

2. How teachers learn

My second takeaway is about how the Chinese teachers, who had a lot less school maths themselves, developed their profound understanding of fundamental mathematics.

They learnt mathematics:

1. studying the materials intensively;
2. from colleagues;
3. from students;
4. from doing mathematics.

The second jumped out. These teachers taught exclusively maths, and had plenty of time to talk about it with colleagues. I love these quotes on page 137:

"...I liked to listen to Xie and other teachers discussing how to solve a problem... I was so impressed that they could use seemingly very simple ideas to solve very complicated problems. It was from them that I started to see the beauty and power of mathematics." 

"In my teaching research group I am the oldest one... yet I learn a lot from my younger colleagues. They are usually more open-minded than I am..." 

"I think a teaching research group is always helpful because you need to be stimulated by someone else when trying to have a better understanding of something."

In my career the person who coordinates a subject in a school rarely gets extra time to do it, and usually doesn't have the time or the opportunity to talk over their subject on a regular basis. They don't get to visit classrooms, they don't get to discuss lessons. I plan with my partner teacher in the other Year 4 class, and that's a great time to share ideas, but the pace is fast, we don't have a lot of time for reflection.

I loved seeing how Kristin and teachers at  Richard A. Shields Elementary School have given us lots of pointers about looking at our teaching together, including spending time looking at the mathematics and how it can be approached.

But this sort of great work is so rare. How do we move to a situation where teachers are talking to each other about lessons, reflecting on how to improve?

I was also struck that the Chinese teachers said they learnt from students too. So, the teacher is standing at the front, using a textbook, orchestrating the lesson - there seems still in their lessons to be a lot of space for students' ideas. Liping Ma presents a techer (p138) wh says, "Sometimes the way of solving a problem proposed by a student is one I've never thought about..." and gives the example of when he was teaching the are of this shape with algebra:

A student puts their hand up, and suggests drawing a rectangle round the shape like this:

The teacher "caught" the idea and ran with it in the lesson.

-§-

The book throws up a lot of questions about how we as teachers deepen our knowledge. I haven't got the answers, but I think they're definitely worth thinking about...

Come September

I'm going to be teaching younger children next September. Back from Year 4 down to Year 1 again - 5 and 6 year olds! Except it's not going to be called Year 1 anymore - it's going to be called Kindergarten 3. K3. (Early Years Foundation Stage 1 and 2 are being renamed Kindergarten 1 and Kindergarten 2.) And it's not going to be like Year 1 was last time; it's going to be much more play-based. There's a plan to have the first hour of the day outside too - possibly over in the woods sometime. That's really new, and a great new challenge! I've not really taught in an Early Years style play-based way before so it's going to be a great learning curve for me. And Estelle is going from Year 2 to K2, so we're heading off to a 3-day course on play-based learning in Prague in June.

I've asked if we can order stacks of Cuisenaire rods for the K3 classes - I really hope we can, there's so much I want to try out with young children (have a look at Sonya Post's lovely series of posts for the kind of thing I'm thinking of).

I'm still rereading Madeleine Goutard's Mathematics and Children, and I'm really looking forward to trying some of her ideas out with younger children.

I've posted about her work before, for instance in my post Can students ask - and answer - vast abstract questions, without being taught? Madeleine Goutard on Free and Conquering Minds and Cuisenaire rods. 

What she seems to do, to link in children's play (particularly with Cuisenaire rods) and provoked-but-independent mathematical writing with the discovery, contemplation and exploration of generalisations in arithmetic is just what I need, we need.

 I want to post a longer selection from her. If you prefer, just skim and read my bolded parts:

But when a new notion is being introduced or its new signs used, it is wise to wait until these appear in the spontaneous work of children before preparing systematic exercises. In fact this appearance is the signal that the notion is being integrated. In addition, by so doing, children are inventing their own exercises on this new subject and showing that they understand it. If we use their inventions in the formula­tion of the first exercises we help children understand at once what the rules of the game are. 

Hence it is of paramount importance for the teacher to examine carefully children's inventions if he wishes to give his class a tailored education, well-adjusted to the level of awareness reached by them. Instead of satisfying themselves with marking and correcting, the teachers could on the one hand draw from such inventions many ideas useful in expanding their teaching, and on the other leave to the children themselves the correction and appreciation of their work or that of their fellows. Whenever the free compositions are not considered at the conscious level they tend to degenerate into repetitions and do not show from one series to the next the progress that we have been able to present above. Free expression is usually considered a luxury that one should encourage but not as the main stream of the work, which remains entirely the exclusive monopoly of the teacher. 

For the experienced teacher all the lessons can have their starting point in the children's suggestions. When the children's own thoughts are driving the lessons forward a natural rhythm is evolved in which the various phases of the activity are harmoniously articulated. The challenge is thrown open, ideas are pooled, individual suggestions are gathered, a collective examination of the various proposals is carried out, followed by constructive criticism and evaluation which lead to fruitful awarenesses and to objective estimations which in their turn serve as the starting point for new projects, new realisations, new exercises, and so on. 

In my experience the children's creations only become brilliant when the teacher is deeply involved in them and can give them the right response. Personally I have read a large number of children's productions and have learnt a great deal from them. Children do not tire of composing if they are in an appreciative environment where their works are recognised and valued. They learn to appreciate the work of their fellows and to enter into dialogue with reality and with their contemporaries.

During a series of seminars conducted at a University I made students work in the following direction: I would take some expressions from the spontaneous productions of their own pupils and we would enquire into what made them of interest to us and what could be done with them. They would then take these suggestions to their class, experiment with them and report on what they found as a result of this conscious effort of analysis. Discussion followed the experiments. 

Here are some of these expressions (a number of which appearing uninteresting at first sight) as well as a summary of the follow-ups.
A girl of 6 had written 8+8+8+8-(7+7+7+7)=4. From this expression followed proposals to alter the number of 8's and of 7's; then the difference between the two numbers chosen (8 and 6, 9 and 5... ). Using their rods the class found that the given expression and the following were equivalent: 

4 ×8 – 4×7=4.
(8 – 7) + (8 – 7) + (8 – 7) + (8 – 7)=4
4×(8– 7)=4×1 

I'm getting more of a sense of how Goutard worked, the toing and froing between child and manipulatives, child and symbols, child and teacher, teacher and children. It's something I recognise from when a sequence of lessons goes well - like our recent work on Beatriz and Maria's fraction squares.

I wonder how that class found the equivalence of the expressions with the rods? Something like this maybe?

Goutard's method is a precursor to many of the "dialogic" approaches that we see in the #MTBoS. She's spotting the work of one student that will bring the whole class on. In using the student's work she's sure that it's something that arises out of the students' current understanding, and she's also emphasising that this progress is in their hands, is a joint enterprise between class and teacher. She's making the work of the class a collaborative inquiry. In my mind it's something like this: