How do the Chinese do it? Chapter 8 - A vision of much more

Last night I attended an e-seminar with Alexandre Borovik.  I struggled to follow much of it as the conversation was heavily accented and the sound quality poor.  But then, in the last few minutes of the hour, something tremendous.  I opened a draft of his book:
Shadows of the Truth: Metamathematics of Elementary Mathematics
and in it, muddled among many other paths of thought, I could see that he has this incredible insight into the deeper concepts of mathematics that only children and highly eduated pure mathematicians can see.  Insights of which I have only begun to scratch the surface in my teaching.

Thank you Alexandre.

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? Part 7: The Last Chapter?

After the initial blast (maybe 6 lessons) of starter questions of this type, I simply come back to this starter activity whenever I feel like it for one-off questions.  It's worked well for me right up throught topics like surds and complex numbers.

There's nothing whizz bang or whistles and bells about it.  Students aren't inspired as they walk into the room, there's usually a bit of grumbling (oh no not this again) but their engagement builds gradually as they have to think for themselves mathematically.

It's effortless to set up (just 4 letters and 2 numbers on the board). I like the fact that I can spend my time listening to students and helping them give voice to their personal journeys into understanding primitive structures for mathematics rather than on class management. 

This task rests easily with there being a little off task chat and it taking some students a minute or two to properly engage.  As I've said before I've not taught in schools where there have been high standards of behaviour and I have never had the luxury of expecting all my students to come to my classes bright-eyed, bushy-tailed and well behaved.  I've had to learn to take my students from being badly behaved to being fully engaged and therefore on task through the way I teach and the type of task I use rather than through demanding that they behave well before I start to teach.

It's important to understand that this task will become more powerful for you as you use it with more classes.  You will gradually become aware of a wider and wider variety of structures that students use to support their thinking and as you become aware of them you become better at spotting what it is that a students is struggling to explain and at helping them to express it clearly to their peers. I think you will be surprised how your own thinking expands and takes you in the direction of confidently using flexibly the wide range of structures the teachers in Liping Ma's book used for calculation.

Is this the last post on this topic?  Maybe not.  Maybe you or I will come back with new ideas stimulated by this task.  I hope so.  Thanks to all who have asked questions.  Please keep them coming!

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? Part 6

I always ask students to describe their thinking on these problems.  To compare and contrast the structures they are using. 

Sometimes they work individually and I pick people at random to talk to the class, somtimes they work in pairs or groups. 

Then I'll 'go at little crazy'.  We'll do an SDPQ with two 1cm lines.   Or two lines with obviously different but unmarked lengths.  It sounds bizarre but the conversations are wonderful.  Structural insights are everywhere.  The students are really thinking hard.  By now they know I'm serious when I say this is all about the journey and that I'm only a tiny bit interested in the answers.

When they're ready we'll move on to algebra.  We can use letters, constants, linear term, quadratic terms and so on.  Remember learned tricks only get half marks.  They need to be able to explain structurally what's going on.  They will need to take those structures they've developed with numbers and transfer them to the algebra.  It's powerful, it's challenging, it's rewarding and its deep.  You should expect the unexpected, namely that when students describe their thinking you will be hearing things you have never heard before and will need to go away and think about.  Don't worry - if you get stuck you'll probably find the student who came up with it has had deeper thoughts themselves after a day or two.  Those who say something innovative often think about it a lot after the lesson.

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? Part 5

If I was teaching a lesson on long multiplication or long division, I would set an SDPQ starter which contained one of the toughest questions they would face.  It was fascinating to see them trying to wrestle out an answer.  The method I was teaching that lesson would then become closely contextualised into their existing thinking as this existing thinking was now active, rather than being developed somewhere separate in their mind. 

They didn't object to doing a lesson on long multiplication or division even if they could do the initial example, because by now they understood the validity and power of having multiple methods and of revisiting structural methods.  But, of course, in general I found they needed far less teaching on basic operations, because we were covering it very effectively through these starters.

One structural lesson I would never miss would be counting the squares in rectancles which leads to grouping them in hundreds tens and unit in a visual representation of the grid method for multiplication, because this so beautifully scaffolds the expansion of quadratic brackets and more.

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? part 4

S,D,P,Q  7,7012

Try it - it's hard!  Students will rapidly write:
Sum 7019
Difference 7005, -7005   (I like to ask them to fully justify that -7005, remember that the trick:  it's the negative of the other difference, which they will have discovered by now, only gets them half a mark)

The multiplication will keep some busy a bit longer.  Many will naturally split the 7000 and the 12.  It's nice to get them to explain and notice what they are doing here too.

But the divisions?  Nasty!  The obvious division is hard enough but the reciprocal?  They will wrestle with it.

I often give full credit for a numerical answer which is very nearly right and has a correct method and a lot of effort behind it.  It helps to emphasise the aspects of their work which I value and to keep them engaged.
They will more deeply consider their thinking about reciprocals because it is challenged. 

The answers aren't the point yet. It's all about the method and their thinking.  If they get that right then the answers will come and the methods won't be forgotten so rapidly as they are when we teach them abstracted algorithms.  Because they have discovered their method for themselves they will have the confidence to believe they can do so again as and when they need to.

One nice aspect of this pair of numbers is that they encourage students to estimate.  One of the answers is going to be about 1000 isn't it.  Does that mean the other will be close to 1/1000?  Can we be more accurate than that?  When students get the hang of writing divisions as fractions (which is a really useful skill), change the rules - allow only mixed numbers or allow only decimals to at least 3dp.  It's your classroom - you can have different rules for different students at different times provided the underlying theme is fair - that all students are being stretched.

Choose another couple of examples like this yourself.  That takes us to about an hour of classroom time doesn't it?  What have the students learned in this time?  What have you the teacher discovered?

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? 30 minutes in. Reciprocals in Calculation.

SDPQ 3,9
SDPQ 2,5
SDPQ 1,7
Each is a 5-10 minute starter.
Watch as you and your class discover and explore the fundamental structures of mathematical calculation.
One of the most surprising things which reliably emerged with my classes was their insight into reciprocals.

They're always looking for short cuts so they rapidly spot the links in the results for division:
4 and 1/4
3 and 1/3
2/5 and 5/2
7 and 1/7
They're reciprocal pairs.  Even though they only get half a mark for a second quotient calculated in this way, they still spot the pattern.  So my students have a strong familiarity with reciprocal pairs and an insight into their relevance for multiplication and division from early in year 7 (age 11) when I first teach them.  I'm sure that younger students could cope with this too.

It's so easy and natural but I've never seen this understanding in other classrooms. Usually in the UK reciprocals are taught as a disconnected entity at GCSE level or are found in algorithms, again without their connected context.  Yet a structural understanding of them is clearly there in the Chinese classrooms described by Liping Ma.  Chinese teachers use them fluently and flexibly for calculation.

Of course I'm not saying that the Chinese teachers taught this in the way I did.  I'm just trying to convince teachers that it is possible to replicate significant aspects of the Chinese teaching strategies with classes of students and teachers who have never thought structurally about calculations before.  Oh and I always taught in tough schools.  No docile classes for me.  And I found that students were more settled if I taught them in this way.  It suited them to focus on structures rather than to learn recipes.  They felt more in control of what they were doing and their education became more relevant to them.  They weren't good at 'learning recipes'.

There's more to come....

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? More insights and a first quick starter.

You can do maths by:
- Picturing structures and using them to support your numerical calculations
- Using learned results
- Using processes (algorithms) which were at one time derived from mathematical structures (often by someone else).
Most people use a mixtures of these strategies, rapidly switching between them depending on the particular calculation.

Mathematical structures were clearly more evident in the Chinese classrooms than in most western classrooms.  We spend a great deal of time teachign our students processes (algorithms) which were written down long ago and most mathematics teachers are not good at demanding students give structural justifications for their answers. 

So given that both teachers and students tend to be lacking these structures, how do we 'put them back in' to mathematical thinking?  As a teacher who's unsure of them with students who don't want to bother to try, where do we begin?  If you're thinking along these lines then the following may work for you as it did for me.

What is the exercise? - It's just a quick starter I use often.
Write on the board: SDPQ 2, 8   (it doesn't have to be 2&8 but you always need two quantities and it makes sense to start with small integers).
Tell the students that when they see SDPQ they have to write down:
Sum
Difference
Product
Quotient
then they have to try to add, take, multiply and divide the numbers.
But there's a catch.  They only get a mark for an answer if they can draw or fully explain their answer.  No tricks allowed!  If they get the righ answer but have used a method they can't fully justify they only get half a mark.  =D

Some students will quickly write down four answers.  Have they found them all?  Are they sure?
Students will come to understand that addition and multiplication are commutative.  You can use this word as it will make sense to them as giving vocabulary to something they can clearly see.

What about difference?  Is that commutative?  Division clearly isn't (4 and 0.25 are both possible answers).  But can we have a difference of -6 or is difference always positive?  I tend to deliberately allow this to be a moot point.  We usually have a vote on it and the class splits.  Why does it matter?  We can decide.  It's only vocabularly.  I tend to give a mark for 6 and a mark for -6 but I don't clearly define what difference is.  I want my students to learn to fight for what they instinctively feel is correct.  I want them to experience how arbitrary mathematical vocabulary is.  Once we decide that answer that fight is curtailed so we never do.  I'm making a point.  "Who cares what anyone else says? - If it's not true to you don't accept it."

Then we mark the exercise out of 6.  To get a full mark a student has to be able to draw a picture of the calculation they did.  Can they?  If they can't but they got the right answer it's only half a mark remember.  Being a bit of a pedant I also knock off half a mark if they didn't write down the words - sum, difference, product and quotient because I find that unless there's marks in it they don't bother and if one doesn't bother they all stop and that's a shame.

What structures did they use?  Did they use number lines for addition and subtraction?  Did they use money?  For division did they use splitting (8 circles split into two groups, 2 cakes split into 8 parts) or chunking (how many 2s in 8, how many 8s in 2) or did they use another structure?  How did they 'see' multiplication?  Cherish variety, cherish the student who tries to explain but can't quite.  Take time to make sure that everyone can see that you value all attempts students make to express their own thinking and to innovate.

Enough for today.  We've covered one 5-10 minute starter so far.  There will be more and they will be similar but each will challenge and develop students' and teachers' structural thinking.

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more

How do the Chinese do it? Introduction

International comparative studies clearly show that students in China make far better progress with mathematics than students in most other countries.

Why is this?

Many authors have written about the social and societal differences between China and the West.  While parental and societal expectations may well have some impact I am convinced the best insights lie in detailed analysis of the way in which mathematics itself is taught.

As far as I am aware there is only one substantial contemporary comparative study of the teaching of mathematics in China and a western culture (the US) and that is the work done by LiPing Ma (as written up in her book:
Knowing and Teaching Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States.
(please - if others know of more relevant studies do post me to them in comments to this blog).

Her book details her study, which showed the wide ranging, secure and versatile strategies Chinese teachers had for handling arithmetic strategies in contrast to the brittle and limited mathematical strategies US teachers had at their disposal.

What struck me was that I and my students discovered and developed arithmetic techniques similar to those found in China when I taught using particular strategies which I developed as a teacher.

So, as soon as I get the time, I'll try to write up the strategies I used with my students.  It was all very simple really and other teachers could easily experiment with the same techniques.  I hope it will lead to some interesting discussion.

How do the Chinese do it?  Links to my other blogs in this series.

1. Introduction

2. More insights and a first quick starter

3. 30 minutes in.  Reciprocals in calculation.

4. Part 4

5. Part 5

6. Part 6

7. Part 7: The Last Chapter?

8. Chapter 8: A vision of much more