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...