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

2 Seeing & Believeing Say What You See

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

4 Present or Absent?

5 Why do students not always ‘hear’ what the teacher says? Attention (Will) in Mathematics  Holding Wholes (gazing)  Discerning Details  Recognising Relationships (in the situation)  Perceiving Properties  Reasoning on the basis of agreed properties discerning details gazing but recognising relationshipsdiscerning details but recognising relationshipsperceiving properties but reasoning … perceiving properties but Communication will be difficult! Students:Teacher:

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

7 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?

8 Put your hand up when you can see … Something that is 1/4 – 1/5 of something else What did you have to do with your attention? Can you generalise? Did you look for something that is 1/4 of something else and for something that is 1/5 of the same thing?

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

10 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?

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

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

13 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 variation Range of permissible change  Example Space

14 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  open.ac.uk  mcs.open.ac.uk/jhm3 … go to presentations

15 Thinking Mathematically

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17 내일 오전 10시 더 생생한 이야기를 들으실 수 있습니다. 지금 복도에서 사전등록 접수중

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

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

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

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

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

23 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

24 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

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

26 Three Only’s Language Patterns & prior Skills Imagery/Sense- of/Awareness; Connections Different Contexts in which likely to arise; dispositions Techniques & Incantations Root Questions predispositions Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable BehaviourBehaviour EmotionEmotion AwarenessAwareness

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

28 Six Modes of Interaction Expounding Explaining Exploring Examining Exercising Expressing InitiatingSustainingConcluding

29 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)

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

31 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

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