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# John Mason MEI Keele June 2012 PowerPoint PPT Presentation

The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Transformations of the Number-Line an exploration of the use of the power of mental imagery and shifts of attention. John Mason MEI Keele June 2012. Challenge.

John Mason MEI Keele June 2012

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#### Presentation Transcript

The Open University

Maths Dept

University of Oxford

Dept of Education

Promoting Mathematical Thinking

## Transformations of the Number-Linean exploration of the use of the power of mental imageryand shifts of attention

John Mason

MEIKeele

June 2012

### Challenge

• By the end of this session you will have engaged in the mathematucal and psychological actions necessary to solve and explain to someone else how to resolve:

• Given a rotation through 180° of a number line about a specified point, followed by a scaling by a given factor from some other given point, to find a single point that can serve as the centre of rotation AND the centre of scaling, when this is possible.

### Assumptions

That which enables action

–> (mathematical) Actions

–> (mathematical) Experience

–> (mathematical) Awareness

This requires initial engagement in activity

• But …One thing we don’t seem to learn from experience …

• is that we don’t often learn from experience alone

• In order to learn from experience it is often necessary to withdraw from the activity-action

and to reflect on, even reconstructthe action

### My Focus Today

• The use of mental imagery

• Shifts from Manipulating to Getting-a-sense-of to Articulating and Symbolising

• All within a conjecturing atmosphere

• What you get from today will be what you notice yourself doing … ‘how you use yourself’

### Imagine a Number-Line (T1)

• Imagine a copy on acetate, sitting on top

• Imagine translating the acetate number-line by 7 to the right:

• Where does 3 end up?

• Where does -2 end up?

• Generalise

Notation:

T7 translates by 7 to the right

T7(x) =

x + 7

Notation:

Tt translates by t

Tt(x) =

x + t

### Reflexive Stance

• How did you work it out?

• Lots of examples?

• Recognising a Relationship in the particular?

• Perceiving a Property in its generality?

### Two Birds

Two birds, close-yoked companionsBoth clasp the same tree;One eats of the sweet fruit,The other looks on without eating.

[Rg Veda]

### Imagine a Number-Line (T2)

• Imagine a copy on acetate, sitting on top

• Imagine translating the acetate number-line by 7 to the right;

• Now translate the acetate number-line to the left by 4;

• Where does 3 end up?

• Where does -2 end up?

• Generalise

T-4 o T7 translates by 7 and then by -4

Does order matter?

Ts Tt = Tt Ts = Ts+ t

### Reflection

• How did you work it out?

• Lots of examples?

• Recognising a Relationship in the particular?

• Perceiving a Property in its generality?

• How fully do you understand and appreciate what you have done?

• Could you reconstruct the sequence?

• Could you explain it to someone else?

• Could you do it without using a diagram?

### Imagine a Number-Line (R1)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

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5

6

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8

9

10

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12

• Imagine a copy on acetate, sitting on top

• Imagine rotating the acetate number-line through 180° about the point 0:

• Where does 3 end up?

• Where does -2 end up?

• Generalise

Notation:

R0 (x) = –x

### Reflexive Stance

• How did you work it out?

• Lots of examples?

• Structurally?

• Recognising a Relationship in the particular?

• Perceiving a Property in its generality?

### Imagine a Number-Line (R2)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

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11

12

• Imagine a copy on acetate, sitting on top

• Imagine rotating the acetate number-line through 180° about the point 5:

• Where does 3 end up?

• Where does -2 end up?

• Generalise

Notation:

R5 (x) =

Notation:

Ra(x) =

5 – (x – 5)

a – (x – a)

Expressing relationships in general

Tracking Arithmetic

### Reflexive Stance

• How did you work it out?

• Lots of examples?

• Recognising a Relationship in the particular?

• Perceiving a Property in its generality?

Working from examples

Many examples?

One generic example?

John Wallis 1616 - 1703

David Hilbert 1862-1943

### Imagine a Number-Line (R3)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

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12

• Imagine a copy on acetate, sitting on top

• Imagine rotating the acetate number-line through 180° about the point 5;

• Now rotate that about the point where 2 was originally

• Where does 3 end up?

• Where does -2 end up?

• How are these results related?

• Generalise!

Given a succession of rotations about various points, when is there a single point that can act as the centre for all of them?

Ra(x) = 2a – x

Rb(Ra(x)) =

Rb(2a – x)

= 2b – (2a – x)

= 2(b–a) + x

= T2(b–a)(x)

### Reflective Stance

• How fully do I understand?

• Write down a pair of reflections in different points whose composite in one order is T6

• What is the composite in the other order?

• Write down another such pair

• And another

• What action am I going to suggest you undertake now?

• Express a generality!

• What needs further work?

• Could you reconstruct the sequence?

• Could you explain it to someone else?

• Could you do it without using a diagram?

### Imagine a Number-Line (R3a)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

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8

9

10

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12

• Imagine a copy on acetate, sitting on top

• Imagine rotating the acetate number-line through 180° about the point 5;

• Now rotate that about the point where 2 now is.

• Where does 3 end up?

• Where does -2 end up?

• Generalise

Q5 rotates about where 5 currently is

Qb(Qa(x)) =

RR (b)(Ra(x))

a

Qa(x) = Ra(x) = 2a – x

= R2a–b(Ra(x))

= 2(2a–b) - (2a – x)

= 2(a – b) + x

Any resonances?

= T2(a–b)(x)

### Meta Reflection

• What mathematical actions have you carried out?

• What cognitive actions have you carried out?

• Holding wholes (gazing)

• Discerning Details

• Recognising Relationships in the particular

• Perceiving Properties (generalities)

• Reasoning on the basis of agreed properties (expressing generality; reasoning with symbols)

• What affectual shifts have you noticed?

• Surprise?

• Doubt/Confusion?

• Desire?

• Shift from ‘easy!’ or ‘boring’ to intrigue?

### Imagine a Number-Line (S1)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

7

8

9

10

11

12

• Imagine a copy on acetate, sitting on top

• Imagine stretching the acetate number-line by a factor of 3/2 with 0 fixed:

• Where does 3 end up?

• Where does -2 end up?

• Generalise

Notation:

S3/2(x : 0) scales from 0 by the factor 3/2

• S3/2(x : 0) =

3x/2

Suggestive:

• Sσ(x : a) scales from a by the factor σ

• Sσ(x : a) =

σ(x–a) + a

### Imagine a Number-Line (S2)

-12

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

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12

• Imagine a copy on acetate, sitting on top

• Imagine scaling the acetate number-line from 2 by the factor of 3/2;

• Now scale the acetate number-line from where -1 was originally, by a factor of 4/5;

• Where does 3 end up?

• Where does -2 end up?

• Generalise

• What about a succession of scalings about different points: when is there a single centre of scaling with the same overall effect??

• What about a succession of scalings each about the current position of a named point?

### Review

• How did we start?

• Imagining a number line

• What actions did we carry out?

• Translating the numberline: Ta(x)

• Rotating the numberline through 180°

• Scaling the numberline

• from painted points: Sσ(x : a)

For exploration

• from current points: Uσ(x : a)

• How all the formula relate to each other

• ### Variation

• A lesson without the opportunity for students to generalise …

… mathematically, is not a mathematics lesson.

• What was varied …

• By me?

• By you?

Awareness (cognition)

Imagery

Will

Emotions (affect)

Body (enaction)

HabitsPractices

### Meta-Reflection

• Was there something that struck you,that perhaps you would like to work on or develop?

• Imagine yourself as vividly as possiblein the place where you would do that work,working on it

• Perhaps in a classroom acting in some fresh manner

• Perhaps when preparing a lesson

• Using your mental imagery to place yourself in the future, to pre-pare, is the single core technique that human beings have developed.

• Researching Your Own Practice Using the Discipline of Noticing (Routledge Falmer 2002)