PowerPoint Presentation

Attending to the Role of Attentionwhen Teaching Mathematics

John MasonKorean Maths Education SocietySeoulNov 3 2012

The Open UniversityMaths Dept

University of OxfordDept of Education

Promoting Mathematical Thinking

#1Seeing & Believeing

Say What You See#Where is your attention?What is invariant depends on relationship with background2Necker CubeWhat catches your attention?Say What You SeeCan you prepare so that when the direction changes you see the cube appropriately?

#Controlling how you attend is not so easy3Present or Absent?

#Seeking relationships4Why do students not always hear what the teacher says?Attention (Will) in MathematicsHolding Wholes (gazing)Discerning DetailsRecognising Relationships (in the situation)Perceiving PropertiesReasoning on the basis of agreed properties discerning detailsgazingbutrecognising relationshipsdiscerning detailsbutrecognising relationshipsperceiving propertiesbutreasoning perceiving propertiesbutCommunication will be difficult!Students:Teacher:#Whats The Difference?=First, add one to eachFirst, add one to the larger and subtract one from the smallerWhat then would be the difference?What then would be the difference?What could be varied?#6Notice that we do not need to know the numbers. We can work with generalityHuman beings start from the general and then specialise (Davydov)Dont want students attending to the specifics but to the property!Arithmetic as the study of actions on numbers.Put your hand up when you can see Something that is 3/5 of something elseSomething that is 2/5 of something elseSomething that is 2/3 of something elseSomething that is 5/3 of something elseWhat other fraction-actions can you see?How did your attention shift?#Generalise!7Put your hand up when you can see Something that is 1/4 1/5of something elseWhat did you have to do with your attention?Can you generalise?Did you look for something that is 1/4 of something elseand forsomething that is 1/5 of the same thing?

#Generalise!8Chord ExpansionWhat is the phenomenon?

What catches your attention?#Exercises for PracticeImagine a page of exercises in your textbookWhat is invariant and what is changing?What are your students attending to?Is that what you want them to attend to?#Counting OutIn 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?ABCDE123459876If that object is eliminated, you start again from the next. Which object is the last one left?10How do you know?Justify your conjecturesGeneralise!#How do you know? Generalise!11SloganA lesson without the opportunity for learners to generalise (mathematically) is not a mathematics lesson!#Attention AttractorsInvariance in the midst of changeChange in the midst of invariancePrinciple 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 rangeDimensions of possible variationRange of permissible changeExample Space#Follow-UpThinking 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 papersj.h.mason @ open.ac.ukmcs.open.ac.uk/jhm3 go to presentations#Thinking Mathematically

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#Task Design & Use7 phasesPotentialStructure of a TopicInteractionEffectiveness of actionsInner & OuterTeacherRoles3 OnlysBalanceActivityRe-flection&Pro-flectionContentTaskActionsThemesPowersQuestioningPeers6 Modes#ActionsRight-multiplying by an inverse ...Making a substitutionDifferentiatingIteratingReading a graphInvoking a definition

#PowersImagining & ExpressingSpecialising & Generalising (Stressing & Ignoring)Conjecturing & Convincing(Re)-Presenting in different modesOrganising & Characterising#ChallengeAppropriate Challenge:Not too greatNot too littleScope depends on student trust of teacherScope depends on teacher support of mathematical thinking not simply getting answers

#Six Modes of InteractionExpoundingExplainingExploringExaminingExercisingExpressingInitiatingSustainingConcluding#