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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.

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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


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

  • Tasks –> (mathematical) Activity

    –> (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?

    • 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?


Imagine a Number-Line (R1)

-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 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 0

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

6

7

8

9

10

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 rotates about 5

R5 (x) =

Notation:

Ra rotates about a

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

6

7

8

9

10

11

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

7

8

9

10

11

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

4

5

6

7

8

9

10

11

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°

      • 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


  • 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

    Structure of the Psyche


    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)


    To Follow Up

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

    Presentations

    Applets

    Developing Thinking in Geometry

    [email protected]


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