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John Mason MEI Keele June 2012

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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

- 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 180Â° 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.

That which enables action

- Tasks â€“> (mathematical) Activity
â€“> (mathematical) Actions

â€“> (mathematical) Experience

â€“> (mathematical) Awareness

This requires initial engagement in activity

- But â€¦One thing we donâ€™t seem to learn from experience â€¦
- is that we donâ€™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

- 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 â€¦ â€˜how you use yourselfâ€™

- 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

- How did you work it out?
- Lots of examples?
- Recognising a Relationship in the particular?
- Perceiving a Property in its generality?

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 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

- How did you work it out?
- Already familiar or expected?
- 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?

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- Imagine a copy on acetate, sitting on top
- Imagine rotating the acetate number-line through 180Â° about the point 0:
- Where does 3 end up?
- Where does -2 end up?
- Generalise

Notation:

R0 rotates about 0

R0 (x) = â€“x

- How did you work it out?
- Lots of examples?
- Structurally?
- Recognising a Relationship in the particular?
- Perceiving a Property in its generality?

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- Imagine a copy on acetate, sitting on top
- Imagine rotating the acetate number-line through 180Â° about the point 5:
- Where does 3 end up?
- Where does -2 end up?
- Generalise

Notation:

R5 rotates about 5

R5 (x) =

Notation:

Ra rotates about a

Ra(x) =

5 â€“ (x â€“ 5)

a â€“ (x â€“ a)

Expressing relationships in general

Tracking Arithmetic

- 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

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- Imagine a copy on acetate, sitting on top
- Imagine rotating the acetate number-line through 180Â° 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 â€“ x

Rb(Ra(x)) =

Rb(2a â€“ x)

= 2b â€“ (2a â€“ x)

= 2(bâ€“a) + x

= T2(bâ€“a)(x)

- 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?

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- Imagine a copy on acetate, sitting on top
- Imagine rotating the acetate number-line through 180Â° 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 â€“ x

= R2aâ€“b(Ra(x))

= 2(2aâ€“b) - (2a â€“ x)

= 2(a â€“ b) + x

Any resonances?

= T2(aâ€“b)(x)

- 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 â€˜easy!â€™ or â€˜boringâ€™ to intrigue?

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- 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:

- SÏƒ(x : a) scales from a by the factor Ïƒ
- SÏƒ(x : a) =

Ïƒ(xâ€“a) + a

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- 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?

- How did we start?
- Imagining a number line

- What actions did we carry out?
- Translating the numberline: Ta(x)
- Rotating the numberline through 180Â°
- about painted points: Ra(x)
- about current points: Qa(x)

- Scaling the numberline
- from painted points: SÏƒ(x : a)

For exploration

- from current points: UÏƒ(x : a)

- A lesson without the opportunity for students to generalise â€¦
â€¦ mathematically, is not a mathematics lesson.

- What was varied â€¦
- By me?
- By you?

Awareness (cognition)

Imagery

Will

Emotions (affect)

Body (enaction)

HabitsPractices

- 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)

http://mcs.open.ac.uk/jhm3

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