John Mason MEI Keele June 2012

1 / 24

# 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.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' John Mason MEI Keele June 2012' - gus

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
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

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 (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 (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

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°
• 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)