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The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. Attending to the Role of Attention when Teaching Mathematics. John Mason Korean Maths Education Society Seoul Nov 3 2012. Seeing & Believeing. Say What You See. Necker Cube.

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attending to the role of attention when teaching mathematics

The Open University

Maths Dept

University of Oxford

Dept of Education

Promoting Mathematical Thinking

Attending to the Role of Attentionwhen Teaching Mathematics

John Mason

Korean Maths Education SocietySeoul

Nov 3 2012

seeing believeing
Seeing & Believeing

Say What You See

necker cube
Necker Cube
  • What catches your attention?
  • Say What You See
  • Can you prepare so that when the direction changes you see the cube appropriately?
attention will in mathematics
Attention (Will) in Mathematics
  • Holding Wholes (gazing)
  • Discerning Details
  • Recognising Relationships (in the situation)
  • Perceiving Properties
  • Reasoning on the basis of agreed properties

Why do students not always ‘hear’ what the teacher says?

Teacher:

Students:

but

gazing

discerning details

but

recognising relationships

discerning details

but

perceiving properties

recognising relationships

but

perceiving properties

reasoning …

Communication will be difficult!

what s the difference

=

What’s The Difference?

First, add one to each

What then would be the difference?

What then would be the difference?

First, add one to the larger and subtract one from the smaller

What could be varied?

put your hand up when you can see
Put your hand up when you can see …
  • Something that is 3/5 of something else
  • Something that is 2/5 of something else
  • Something that is 2/3 of something else
  • Something that is 5/3 of something else
  • What other fraction-actions can you see?

How did your attention shift?

put your hand up when you can see1
Put your hand up when you can see …

Something that is 1/4 – 1/5of something else

Did you look for something that is 1/4 of something else

and for

something that is 1/5 of the same thing?

What did you have to do with your attention?

Can you generalise?

chord expansion
Chord Expansion

What is the phenomenon?

What catches your attention?

exercises for practice
Exercises for Practice
  • Imagine a page of exercises in your textbook
  • What is invariant and what is changing?
  • What are your students attending to?
  • Is that what you want them to attend to?
counting out
Counting Out
  • In a selection ‘game’ you start at the left and count forwards and backwards until you get to a specified number (say 37). Which object will you end on?

A

B

C

D

E

1

2

3

4

5

9

8

7

6

How do you know?

Justify your conjectures

10

If that object is eliminated, you start again from the ‘next’. Which object is the last one left?

Generalise!

slogan
Slogan
  • A lesson without the opportunity for learners to generalise (mathematically) …
    • is not a mathematics lesson!
attention attractors
Attention Attractors
  • Invariance in the midst of change
  • Change in the midst of invariance
  • Principle of Variation:what is available to be learned is what varies within limited space and time(Ference Marton)
    • Becoming aware of what can change and over what range
  • Dimensions of possible variationRange of permissible change
  • Example Space
follow up
Follow-Up
  • Thinking Mathematically (in Korean!!)
  • Questions & Prompts (ATM Derby)
  • Designing & Using Mathematical Tasks (Tarquin)
  • Mathematics Teaching Practice: a guide for university lecturers (Horwood)
  • Counter-Examples in Calculus (College Press)
  • Various chapters and papers
  • j.h.mason @ open.ac.uk
  • mcs.open.ac.uk/jhm3 … go to presentations
slide17
내일오전 10시더 생생한이야기를들으실 수 있습니다.지금 복도에서사전등록 접수중
task design use
Task Design & Use

Re-flection&Pro-flection

Content

Task

Activity

Potential

Actions

Structure of a Topic

Inner & Outer

Effectiveness of actions

Themes

Powers

3 Only’s

Balance

Interaction

7 phases

Teacher

Peers

6 Modes

Roles

Questioning

teacher focus
Teacher Focus

Teacher-Student interaction

Teacher-Mathematics interaction

Student-Mathematics interaction

Language/technical terms

Examples, Images &

Representations

Enactive Obstacles

Origins

Applications & Uses

Affective Obstacles

Cognitive Obstacles:

common errors, …

Methods & Procedures

actions
Actions
  • Right-multiplying by an inverse ...
  • Making a substitution
  • Differentiating
  • Iterating
  • Reading a graph
  • Invoking a definition
themes
Themes
  • Doing & Undoing
  • Invariance in the midst of change
  • Freedom & Constraint
  • Restricting & Extending
powers
Powers
  • Imagining & Expressing
  • Specialising & Generalising (Stressing & Ignoring)
  • Conjecturing & Convincing
  • (Re)-Presenting in different modes
  • Organising & Characterising
inner outer aspects
Inner & Outer Aspects
  • Outer
    • What task actually initiates explicitly
  • Inner
    • What mathematical concepts underpinned
    • What mathematical themes encountered
    • What mathematical powers invoked
    • What personal propensities brought to awareness
challenge
Challenge
  • Appropriate Challenge:
    • Not too great
    • Not too little
    • Scope depends on student trust of teacher
    • Scope depends on teacher support of mathematical thinking not simply getting answers
structure of a topic

Imagery

Awareness (cognition)

Will

Emotions (affect)

Body (enaction)

HabitsPractices

Structure of a Topic
three only s
Three Only’s

Language Patterns& prior Skills

Imagery/Sense-of/Awareness; Connections

Root Questions

predispositions

Different Contexts in which likely to arise;dispositions

Techniques & Incantations

Standard Confusions & Obstacles

Emotion

Behaviour

Awareness

Only Emotion is Harnessable

Only Awareness is Educable

Only Behaviour is Trainable

seven phases
Seven Phases

Getting Started

Initiating

Getting Involved

Mulling

Sustaining

Keeping Going

Insight

Being Sceptical

Concluding

Contemplating

six modes of interaction
Six Modes of Interaction

Initiating

Sustaining

Concluding

Expounding

Explaining

Exploring

Examining

Exercising

Expressing

initiating activity
Initiating Activity
  • Silent Start
  • Particular (to general);General (via particular)Semi-general (via particular to general)
  • Worked example
  • Use/Application/Context
  • Specific-Unspecific
  • Manipulating:
    • Material objects (eg cards, counters, …)
    • Mental images (diagrams, phenomena)
    • Symbols (familiar & unfamiliar)
sustaining activity
Sustaining Activity
  • Questions & Prompts
  • Directed–Prompted–SpontaneousScaffolding & Fading
  • Energising (praising-challenging)
  • Conjecturing
  • Sharing progress/findings
concluding activity
Concluding Activity
  • Conjectures with evidence
  • Accounts that others can understand
  • Reflecting on effective & ineffective actions
    • Aspcts of inner task (dispositions, …)
  • Imagining acting differently in the future
balanced activity
Balanced Activity

Affordances

Constraints

Attunements

OuterTask

InnerTask

Intended& Enacted

goals

Implicit

goals

Ends

Ends

Tasks

Tasks

Resources

Resources

Means

Means

CurrentState

CurrentState