Promoting Mathematical Thinking Attending to the Role of

  • Slides: 32
Download presentation
Promoting Mathematical Thinking Attending to the Role of Attention when Teaching Mathematics John Mason

Promoting Mathematical Thinking Attending to the Role of Attention when Teaching Mathematics John Mason Korean Maths Education Society Seoul Nov 3 2012 1 The Open University Maths Dept University of Oxford Dept of Education

Seeing & Believeing Say What You See 2

Seeing & Believeing Say What You See 2

Necker Cube v v v 3 What catches your attention? Say What You See

Necker Cube v v v 3 What catches your attention? Say What You See Can you prepare so that when the direction changes you see the cube appropriately?

Present or Absent? 4

Present or Absent? 4

Attention (Will) in Mathematics v v v Holding Wholes (gazing) Discerning Details Recognising Relationships

Attention (Will) in Mathematics v v v 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: but gazing but discerning details but recognising relationships but perceiving properties Communication will be difficult! discerning details recognising relationships perceiving properties reasoning … 5 Students:

What’s The Difference? – = First, add one to each First, add one to

What’s The Difference? – = First, add one to each First, add one to the larger and subtract one from the smaller 6 What then would be the difference? What could be varied?

Put your hand up when you can see … v v v 7 Something

Put your hand up when you can see … v v v 7 Something that is 3/5 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 see … Something that is 1/4 –

Put your hand up when you can see … Something that is 1/4 – 1/5 of 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? 8

Chord Expansion What is the phenomenon? What catches your attention? 9

Chord Expansion What is the phenomenon? What catches your attention? 9

Exercises for Practice v v 10 Imagine a page of exercises in your textbook

Exercises for Practice v v 10 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 v In a selection ‘game’ you start at the left and count

Counting Out v 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 10 … If that object is eliminated, you start again from the ‘next’. 11 Which object is the last one left? How do you know? Justify your conjectures Generalise!

Slogan v 12 A lesson without the opportunity for learners to generalise (mathematically) …

Slogan v 12 A lesson without the opportunity for learners to generalise (mathematically) … …is not a mathematics lesson!

Attention Attractors v v v Invariance in the midst of change Change in the

Attention Attractors v v v 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 v v 13 Dimensions of possible variation Range of permissible change Example Space

Follow-Up v v v v 14 Thinking Mathematically (in Korean!!) Questions & Prompts (ATM

Follow-Up v v v v 14 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/jhm 3 … go to presentations

Thinking Mathematically 15

Thinking Mathematically 15

16

16

Task Design & Use Content Potential Structure of a Topic 3 Only’s Task Activity

Task Design & Use Content Potential Structure of a Topic 3 Only’s Task Activity Actions Inner & Outer Balance 7 phases Theme s. Powers Interaction Teacher 6 Modes Questioning 18 Re-flection & Pro-flection Peers Roles Effectiveness of actions

Teacher Focus Teacher-Mathematics interaction Language/technical terms Enactive Obstacles Origins Affective Obstacles Cognitive Obstacles: common

Teacher Focus Teacher-Mathematics interaction Language/technical terms Enactive Obstacles Origins Affective Obstacles Cognitive Obstacles: common errors, … 19 Teacher-Student interaction Student-Mathematics interaction Examples, Images & Representations Applications & Uses Methods & Procedures

Actions v v v v 20 Right-multiplying by an inverse. . . Making a

Actions v v v v 20 Right-multiplying by an inverse. . . Making a substitution Differentiating Iterating Reading a graph Invoking a definition …

Themes v v 21 Doing & Undoing Invariance in the midst of change Freedom

Themes v v 21 Doing & Undoing Invariance in the midst of change Freedom & Constraint Restricting & Extending

Powers v v v 22 Imagining & Expressing Specialising & Generalising (Stressing & Ignoring)

Powers v v v 22 Imagining & Expressing Specialising & Generalising (Stressing & Ignoring) Conjecturing & Convincing (Re)-Presenting in different modes Organising & Characterising

Inner & Outer Aspects v Outer – What task actually initiates explicitly v Inner

Inner & Outer Aspects v Outer – What task actually initiates explicitly v Inner – – 23 What mathematical concepts underpinned What mathematical themes encountered What mathematical powers invoked What personal propensities brought to awareness

Challenge v Appropriate Challenge: – – 24 Not too great Not too little Scope

Challenge v Appropriate Challenge: – – 24 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 Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices

Structure of a Topic Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices 25

Three Only’s Language Patterns & prior Skills Imagery/Senseof/Awareness; Connections Root Questions predispositions Different Contexts

Three Only’s Language Patterns & prior Skills Imagery/Senseof/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise; dispositions 26 Techniques & Incantations Emotion ur vio ha Be Aw ar en es s Standard Confusions & Obstacles Only Emotion is Harnessable Only Awareness is Educable Only Behaviour is Trainable

Seven Phases Getting Started Getting Involved Initiating Mulling Keeping Going Sustaining Insight Being Sceptical

Seven Phases Getting Started Getting Involved Initiating Mulling Keeping Going Sustaining Insight Being Sceptical Contemplating 27 Concluding

Six Modes of Interaction Initiating Expounding Explaining Exploring Examining Exercising Expressing 28 Sustaining Concluding

Six Modes of Interaction Initiating Expounding Explaining Exploring Examining Exercising Expressing 28 Sustaining Concluding

Initiating Activity v v v Silent Start Particular (to general); General (via particular) Semi-general

Initiating Activity v v v 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) 29

Sustaining Activity v v v 30 Questions & Prompts Directed–Prompted–Spontaneous Scaffolding & Fading Energising

Sustaining Activity v v v 30 Questions & Prompts Directed–Prompted–Spontaneous Scaffolding & Fading Energising (praising-challenging) Conjecturing Sharing progress/findings

Concluding Activity v v v Conjectures with evidence Accounts that others can understand Reflecting

Concluding Activity v v v Conjectures with evidence Accounts that others can understand Reflecting on effective & ineffective actions – Aspcts of inner task (dispositions, …) v 31 Imagining acting differently in the future

Balanced Activity Affordances Intended & Enacted goals Means Current State 32 Outer Tasks Attunements

Balanced Activity Affordances Intended & Enacted goals Means Current State 32 Outer Tasks Attunements Inner Task Resources Implicit goals Ends Resources Constraints Means Current State Tasks