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Transformations of the Number-Line

an exploration of the use of the power of mental imagery

and shifts of attention

John MasonMEI

KeeleJune 2012The Open University

Maths Dept University of OxfordDept of Education

Promoting Mathematical Thinking

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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.

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Assumptions Tasks –> (mathematical) Activity –> (mathematical) Actions –> (mathematical) Experience –> (mathematical) Awareness

That which

enables action

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 reconstruct the action

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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’

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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 rightT7(x) = Notation:

Tt translates by tTt(x) =

x + 7

x + t

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Reflexive Stance How did you work it out?

– Lots of examples?– Recognising a Relationship in the particular?– Perceiving a Property in its generality?

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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]

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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

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Reflection 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 Number-Line (R1)

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 rotates about 0R0 (x) = –x

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

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Reflexive Stance 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 Number-Line (R2)

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 rotates about 5 R5 (x) =

Notation:Ra rotates about a Ra (x) = 5 – (x – 5) a – (x – a)

Expressing relationships in generalTracking Arithmetic

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

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Reflexive Stance How did you work it out?

– Lots of examples?– Recognising a Relationship in the particular?– Perceiving a Property in its generality?

Many examples?

John Wallis 1616 - 1703

David Hilbert 1862-1943

One generic example?

Working from examples

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Imagine a Number-Line (R3)

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?

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

Ra(x) = 2a – x

Rb(Ra(x)) = = 2b – (2a – x)

Rb(2a – x)

= 2(b–a) + x= T2(b–a)(x)

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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?

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Imagine a Number-Line (R3a)

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 isQb(Qa(x)) = Qa(x) = Ra(x) = 2a –

x= R2a–b(Ra(x)) = 2(2a–b) - (2a – x) = 2(a – b) + x = T2(a–b)(x)

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

RR (b)(Ra(x)) a

Any resonances?

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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?

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Imagine a Number-Line (S1)

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 3/2S3/2(x : 0) =

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

3x/2 Suggestive:Sσ(x : a) scales from a by the factor σSσ(x : a) = σ(x–a) + a

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Imagine a Number-Line (S2)

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?

-1 1 2 3 4 5 6 7 8 9 10

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-2-3-4-5-6-7-8-9-10

-11

0-12

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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°

… about painted points: Ra(x)… about current points: Qa(x)

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

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Variation A lesson without the opportunity for students to

generalise …… mathematically, is not a mathematics lesson.

What was varied …– By me?– By you?

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Structure of the Psyche

ImageryAwareness (cognition)

Will

Body (enaction)

Emotions (affect)

HabitsPractices

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Meta-Reflection Was there something that struck you,

that perhaps you would like to work on or develop? Imagine yourself as vividly as possible

in 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)

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To Follow Up

http://mcs.open.ac.uk/jhm3Presentations

AppletsDeveloping Thinking in Geometry

j.h.mason@open.ac.uk