Janet's post about the Pimlico schools and the Hirsch curriculum model (here) again raises the commonly-cited dichotomy between knowledge-based and skills-based approaches.** I argue that the primary focus in teaching and learning in all Key Stages should be neither. It should be individual pupil development**.

I recently amused my eight year old granddaughter with this conundrum that I remembered from my childhood.

'Think of a number between 1 and 10 (but don't tell me what it is).

Double it.

Add 6

Halve it

Take away the number you first thought of.

The answer is 3 isn't it?'

As expected, the response was, "Wow granddad, how did you know that?"

When she repeated the exercise with different starting numbers she soon discovered that the answer was always 3, so she then asked me why this was the case.

I then gave her a much simpler conundrum.

'Think of a number'.

Add 1.

Take away the number you first thought of.

What's the answer ?'

She immediately saw that only it was 1 but that it would always be 1, whatever number you started with. She also realised that this was the same sort of problem as the first one, but was unable to generalise why both examples worked.

So is this conundrum a matter of knowledge or skills? Clearly it is neither. There was no knowledge deficit that stopped her understanding why the conundrum always worked. There was no skills deficit either. She could add, multiply, divide and subtract perfectly well.

I see this as clear confirmation of Piagetian developmental stages. My granddaughter is well established within the concrete stage. She can't be expected to understand in general how such conundrums work until she begins the transition to the formal stage. I would place this conundrum firmly on the border between concrete and formal. A well established formal thinker would explain the conundrum by resorting to simple algebra. She can easily understand it as a concrete problem as in the second example but cannot generalise how it works.

**My argument is that the primary focus of all children's education should be helping children progress through the Piagetian stages starting with consolidation of the one they are in.** In Y7 most pupils will have established themselves within the concrete stage with varying degrees of confidence. KS3 and KS4 should be all about promoting development through the stages, with special reference to the concrete/formal transition that is essential (or used to be) for accessing higher level GCSE grades and progression to A level.

The power of Piagetian stage theory that underpins its relevance to curriculum and teaching methods is that it doesn't matter in which subject context the child develops fastest because the cognitive gains transfer across to all subjects, all learning and all problem solving. Therefore schools can exploit children's enthusiasm for their 'favourite subject' to the benefit of deeper understanding in all subjects.

This is the basis of Shayer and Adey's 'Cognitive Acceleration through Science Education' (CASE) programme, which I have experience of. The same principles underpin CAME (maths) and other applications in the humanities. The approach also transfers into pre-school and infant teaching where the key transition is from pre-concrete to concrete, which is mainly about understanding various conservations.

All this is explained with examples from various teachers and schools in their book,

Adey P & Shayer M (Edited) (2002), Learning Intelligence, Open University Press

However the real brilliance of Shayer and Adey that has still not resulted in the deserved degree of recognition, is to combine the stage theory of Piaget with the 'social learning' theories of Vygotsky. This synthesis is absolutely central to CASE, CAME etc.

I am not a maths teacher so I am taking a risk here. I hope the following generates plenty of comment and possible criticism from maths teachers. This is how I would propose to use in a teaching context, the conundrum my granddaughter and I played with.

1. Give the conundrum to the whole class from the front.

2. Let them enjoy it and play with it.

3. Divide the class into groups of 5 or 6 (not smaller) and ask them to discuss it with a view to explaining why the answer is always 3. This is peer to peer teaching within the 'Zone of Proximal Development' (ZPD), both key features of the Vygotsky approach. The teacher provides further support and 'scaffolding' by moving from group to group and injecting suggestions and questions into the discussions without ever 'telling answers'.

4. Feedback and class discussion.

5. Teacher from the front to the whole class again: ask pupils to devise their own novel conundrums and try them out with the partner sitting next to them.

6. The teacher can then explain how these conundrums are the basis of many simple card tricks. She may even be able to demonstrate one. Shayer and Adey call this process 'bridging' - making lateral connections.

This lesson is not about knowledge or skills but is pure developmentalism. I don't think 'skills' is the right word at all to describe cognitive growth. Skills imply repetitive training that is certainly appropriate to design, technology, art, music. PE etc alongside developmental approaches. A repetitive 'training' approach will not promote cognitive development where it is most needed, which is in the area of the basic 'skills' (ugh!) of English and maths, that are best taught developmentally.

I will conclude with another example. Ask the class (any age including A Level) to use their calculators to work out, 'twelve divided by four times three'. Help them by writing it on the board as a fraction with 12 as the numerator (top) and 4 x 3 as the denominator (bottom).

Various answers will be immediately forthcoming, very few being the correct answer of 1.

Is there a 'knowledge' deficit here? Is there a 'skills' deficit? Well, yes in the sense that most pupils of all ages do not 'know', or lack 'skill' in how to use a calculator, but this just confuses the deeper issue, which is rooted in the development of a 'properties of numbers' concept.

The pupils that make the mistakes do not understand the processing of numbers sufficiently deeply at a formal, mathematical level and do not realise, for mathematical (not arithmetical) reasons, that the order in which the processes are completed in the calculator, matters.

Little can be more 'basic' as a numeracy 'skill' than learning how to use a calculator, but this is not a skill at all. It is a matter of mathematical understanding. This is why it is so important to teach maths developmentally to all pupils of all abilities. It**is 'education' not 'training' that all pupils need and are entitled to receive. For me that is the essential 'comprehensive promise'.**

I recently amused my eight year old granddaughter with this conundrum that I remembered from my childhood.

'Think of a number between 1 and 10 (but don't tell me what it is).

Double it.

Add 6

Halve it

Take away the number you first thought of.

The answer is 3 isn't it?'

As expected, the response was, "Wow granddad, how did you know that?"

When she repeated the exercise with different starting numbers she soon discovered that the answer was always 3, so she then asked me why this was the case.

I then gave her a much simpler conundrum.

'Think of a number'.

Add 1.

Take away the number you first thought of.

What's the answer ?'

She immediately saw that only it was 1 but that it would always be 1, whatever number you started with. She also realised that this was the same sort of problem as the first one, but was unable to generalise why both examples worked.

So is this conundrum a matter of knowledge or skills? Clearly it is neither. There was no knowledge deficit that stopped her understanding why the conundrum always worked. There was no skills deficit either. She could add, multiply, divide and subtract perfectly well.

I see this as clear confirmation of Piagetian developmental stages. My granddaughter is well established within the concrete stage. She can't be expected to understand in general how such conundrums work until she begins the transition to the formal stage. I would place this conundrum firmly on the border between concrete and formal. A well established formal thinker would explain the conundrum by resorting to simple algebra. She can easily understand it as a concrete problem as in the second example but cannot generalise how it works.

The power of Piagetian stage theory that underpins its relevance to curriculum and teaching methods is that it doesn't matter in which subject context the child develops fastest because the cognitive gains transfer across to all subjects, all learning and all problem solving. Therefore schools can exploit children's enthusiasm for their 'favourite subject' to the benefit of deeper understanding in all subjects.

This is the basis of Shayer and Adey's 'Cognitive Acceleration through Science Education' (CASE) programme, which I have experience of. The same principles underpin CAME (maths) and other applications in the humanities. The approach also transfers into pre-school and infant teaching where the key transition is from pre-concrete to concrete, which is mainly about understanding various conservations.

All this is explained with examples from various teachers and schools in their book,

Adey P & Shayer M (Edited) (2002), Learning Intelligence, Open University Press

However the real brilliance of Shayer and Adey that has still not resulted in the deserved degree of recognition, is to combine the stage theory of Piaget with the 'social learning' theories of Vygotsky. This synthesis is absolutely central to CASE, CAME etc.

I am not a maths teacher so I am taking a risk here. I hope the following generates plenty of comment and possible criticism from maths teachers. This is how I would propose to use in a teaching context, the conundrum my granddaughter and I played with.

1. Give the conundrum to the whole class from the front.

2. Let them enjoy it and play with it.

3. Divide the class into groups of 5 or 6 (not smaller) and ask them to discuss it with a view to explaining why the answer is always 3. This is peer to peer teaching within the 'Zone of Proximal Development' (ZPD), both key features of the Vygotsky approach. The teacher provides further support and 'scaffolding' by moving from group to group and injecting suggestions and questions into the discussions without ever 'telling answers'.

4. Feedback and class discussion.

5. Teacher from the front to the whole class again: ask pupils to devise their own novel conundrums and try them out with the partner sitting next to them.

6. The teacher can then explain how these conundrums are the basis of many simple card tricks. She may even be able to demonstrate one. Shayer and Adey call this process 'bridging' - making lateral connections.

This lesson is not about knowledge or skills but is pure developmentalism. I don't think 'skills' is the right word at all to describe cognitive growth. Skills imply repetitive training that is certainly appropriate to design, technology, art, music. PE etc alongside developmental approaches. A repetitive 'training' approach will not promote cognitive development where it is most needed, which is in the area of the basic 'skills' (ugh!) of English and maths, that are best taught developmentally.

I will conclude with another example. Ask the class (any age including A Level) to use their calculators to work out, 'twelve divided by four times three'. Help them by writing it on the board as a fraction with 12 as the numerator (top) and 4 x 3 as the denominator (bottom).

Various answers will be immediately forthcoming, very few being the correct answer of 1.

Is there a 'knowledge' deficit here? Is there a 'skills' deficit? Well, yes in the sense that most pupils of all ages do not 'know', or lack 'skill' in how to use a calculator, but this just confuses the deeper issue, which is rooted in the development of a 'properties of numbers' concept.

The pupils that make the mistakes do not understand the processing of numbers sufficiently deeply at a formal, mathematical level and do not realise, for mathematical (not arithmetical) reasons, that the order in which the processes are completed in the calculator, matters.

Little can be more 'basic' as a numeracy 'skill' than learning how to use a calculator, but this is not a skill at all. It is a matter of mathematical understanding. This is why it is so important to teach maths developmentally to all pupils of all abilities. It

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