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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-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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slide1

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
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
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
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 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
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

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 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
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
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 stance1
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
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 stance2
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
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
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
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
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
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
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

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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
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
Variation
  • A lesson without the opportunity for students to generalise …

… mathematically, is not a mathematics lesson.

  • What was varied …
    • By me?
    • By you?
structure of the psyche

Awareness (cognition)

Imagery

Will

Emotions (affect)

Body (enaction)

HabitsPractices

Structure of the Psyche
meta reflection1
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
To Follow Up

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

Presentations

Applets

Developing Thinking in Geometry

[email protected]

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