Big sigh. All I want to do is be a maths teacher and to contribute to the maths teaching profession. I just want to carry on with my blog about maths education.

But there is a need to interject.

He's really doing it isn't he?

http://www.telegraph.co.uk/education/8227535/Michael-Gove-my-revolution-for-culture-in-classroom.html described, in December, his intention to launch a cultural revolution on teaching in England.

And, with his plausible lies about 'increasing teacher freedom' and Red Ofsted Guard already in place he's off and running. Every day there are more edicts, each more disconnected and ludicrous than the first.

Here are today's: http://www.dailymail.co.uk/news/article-1390358/Weak-teachers-removed-classrooms-just-term-heads-new-sacking-powers.html#comments It takes a while to sort out failing teacheres because, er, we sort them out.....failing situations can happen to anyone during their career. Problems at home, a personality clash at school, changes from outside, stressful and bullying inspections. How many great teachers have each of us seen go down?

Failing, contentious and problematic teaching situations occur for all sorts of reasons, usually there are multiple contributory sources which it takes a while to unravel and put right. There aren't thousands and thousands of teachers who clearly are failing teachers Mr Gove, this is a fantasy of your imagination. There weren't millions of enemies of the state in China either, but the cultural revolution demanded they exist so people had to invent reasons to blame others to avoid the beatings themselves.

At consultations groups Conservative MPs happily tell us that the days of policy in education being generated by those in education are over. Mathematics education policy will now be gerated in accordance with the voting preferences of average members of the middle class public.

WHY? HOW ON EARTH DO THEY HAVE SUCH LITTLE INSIGHT INTO HOW GOVERNMENTS ACTUALLY WORKS?

I sometimes wonder, do they really believe that Maggie Thatcher pulled her policies out of the air? Do they not understand that she liaised with a substantial professional and business community of experts who understood what she was doing much better than she did? Who had analysed the consequences and saw how horrific they would be but had also analysed the consquences of doing nothing..... Who wrote and consulted through both the main press and the IOD and the journals and so on.

This government is nothing like that. Honestly for those of you who are not involved in the upper echelons of education please believe me that there is no link between what those who are respected in education are advising and what Gove is actually doing. There are just people floating round desparately trying to limit the damage and make good from bad despite the horrors of it all.

This is surely just a nightmare. I'll wake up soon. Please. This blog post wasn't needed. We can go back to undertanding the possibilities broadband brings for enhancing education in the 21st century rather than suddenly having to ditch all that and focus on sacking teachers of subjects students find relevant and forcing reluctant 14-16 years old from tough backgrounds through a 1950s grammar school education.

Please reassure me this isn't real.

Please.

## Wednesday, 25 May 2011

## Thursday, 19 May 2011

### 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?

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

4. Part
4

5. Part
5

6. Part
6

## Thursday, 5 May 2011

### 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....

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

4. Part
4

5. Part
5

6. Part
6

## Monday, 2 May 2011

### 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.

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?

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.

- 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

4. Part
4

5. Part
5

6. Part
6

## Sunday, 1 May 2011

### 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.

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

4. Part
4

5. Part
5

6. Part
6

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