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 Say What You See
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 • 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? 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 … • 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 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 What is the phenomenon? What catches your attention?
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 • 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 • A lesson without the opportunity for learners to generalise (mathematically) … • is not a mathematics lesson!
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 • 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
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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-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 • Right-multiplying by an inverse ... • Making a substitution • Differentiating • Iterating • Reading a graph • Invoking a definition • …
Themes • Doing & Undoing • Invariance in the midst of change • Freedom & Constraint • Restricting & Extending
Powers • Imagining & Expressing • Specialising & Generalising (Stressing & Ignoring) • Conjecturing & Convincing • (Re)-Presenting in different modes • Organising & Characterising
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 • 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
Imagery Awareness (cognition) Will Emotions (affect) Body (enaction) HabitsPractices Structure of a Topic
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 Getting Started Initiating Getting Involved Mulling Sustaining Keeping Going Insight Being Sceptical Concluding Contemplating
Six Modes of Interaction Initiating Sustaining Concluding Expounding Explaining Exploring Examining Exercising Expressing
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 • Questions & Prompts • Directed–Prompted–SpontaneousScaffolding & Fading • Energising (praising-challenging) • Conjecturing • Sharing progress/findings
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 Affordances Constraints Attunements OuterTask InnerTask Intended& Enacted goals Implicit goals Ends Ends Tasks Tasks Resources Resources Means Means CurrentState CurrentState