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11
The Micro-Features of The Micro-Features of Mathematical TasksMathematical Tasks
Anne Watson & John MasonAnne Watson & John Mason
NottinghamNottingham
Feb 9 2012Feb 9 2012
The Open UniversityThe Open UniversityMaths DeptMaths Dept University of OxfordUniversity of Oxford
Dept of EducationDept of Education
Promoting Mathematical ThinkingPromoting Mathematical Thinking
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Our design methodsOur design methods Task audienceTask audience
– teachers & novice teachers; teacher teachers & novice teachers; teacher educatorseducators
Task purposesTask purposes– to bring to awareness important to bring to awareness important
mathematical and pedagogical constructsmathematical and pedagogical constructs– to offer task-types rather than particular tasksto offer task-types rather than particular tasks– to promote deep consideration of conceptsto promote deep consideration of concepts– to provide current experience of ways of to provide current experience of ways of
working mathematicallyworking mathematically– to articulate effective mathematical actionsto articulate effective mathematical actions
What?: What?: NOT roll out materials but methods of working NOT roll out materials but methods of working
[and possible tasks][and possible tasks]
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Generic Task structuresGeneric Task structures
Unusual features (e.g. inter-rootal distance) …Unusual features (e.g. inter-rootal distance) … Another & anotherAnother & another Imagine a …Imagine a … Silent lessonSilent lesson Make up one which ........Make up one which ........ Change .... so that .....Change .... so that .....
Generic strategies for teachers to add to their repertoire (Q&P; Thinkers)
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Generic Task StucturesGeneric Task Stuctures
Microtasks are generic strategies applied in Microtasks are generic strategies applied in particular conceptual fields, e.g.particular conceptual fields, e.g.
• teacher asks students to compare ....teacher asks students to compare ....• teacher asks students to invent a teacher asks students to invent a
representation of …representation of … (generic tactics which become didactic in a (generic tactics which become didactic in a
conceptual context)conceptual context)
Sequences of microtasks which direct Sequences of microtasks which direct development of a network of conceptual development of a network of conceptual ideasideasCf. (Hypothetical) learning TrajectoriesCf. (Hypothetical) learning TrajectoriesCf. Spiral CurriculumCf. Spiral Curriculum
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Recent example of task construction Recent example of task construction and sequencingand sequencing
AudienceAudience: primary mathematics teacher educators’ : primary mathematics teacher educators’ residential workshopresidential workshop
PurposePurpose: : – to bring to awareness important mathematical to bring to awareness important mathematical
and pedagogical constructsand pedagogical constructs– to promote deep consideration of conceptsto promote deep consideration of concepts– to provide current experience of ways of working to provide current experience of ways of working
mathematicallymathematically ChoicesChoices: to bring to the surface or to dig deep to : to bring to the surface or to dig deep to
find ...find ...
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Count downCount down
Count down from 10 to -10Count down from 10 to -10 How many numbersHow many numbers Why?Why?
What next? Number or linearity?What next? Number or linearity?
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NumberNumber
Count down from 101 in steps of 1 1/10Count down from 101 in steps of 1 1/10 Count down from 46 in steps of 1 1/5Count down from 46 in steps of 1 1/5
Make up some like Make up some like thisthis
Predictable Predictable issues?issues?
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LinearityLinearity
Plot countdowns as graphsPlot countdowns as graphs Predictable issuesPredictable issues
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Design issues relevant to this Design issues relevant to this sequencesequence
Who?Who? What purpose?What purpose? Context, resources, timeContext, resources, time My understanding of underlying conceptual issuesMy understanding of underlying conceptual issues Organised direct (and directed) experience of Organised direct (and directed) experience of
thesethese Dig deep rather than bring to the surface!Dig deep rather than bring to the surface! Sequencing tasks to address the same issues Sequencing tasks to address the same issues
again and againagain and again Importance of associated ‘pedagogical’ context Importance of associated ‘pedagogical’ context
and strategiesand strategies
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Genesis of ideas: the story of Genesis of ideas: the story of elasticselastics
Ulla Runesson using elastic to vary the ‘whole’ in Ulla Runesson using elastic to vary the ‘whole’ in fractions; fractions; dimension of variationdimension of variation
Problem with seeing multiplication only as Problem with seeing multiplication only as repeated addition or arrays (Nuffield study)repeated addition or arrays (Nuffield study)
Need a model for scaling (BEAM elastic; PGCE Need a model for scaling (BEAM elastic; PGCE Cabri)Cabri)
John works repeatedly with a range of audiences John works repeatedly with a range of audiences exploring dimensions of variation, making kit, exploring dimensions of variation, making kit, and observing their actions and commentsand observing their actions and comments
Microtask sequence: expansion; contraction; Microtask sequence: expansion; contraction; expansion and related contraction e.g. 3/2 and expansion and related contraction e.g. 3/2 and 2/3; invariance & (MGA)2/3; invariance & (MGA)
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Embedded Q&PEmbedded Q&P
CompareCompare Same/differentSame/different RepresentationsRepresentations ExemplificationExemplification VariationVariation Construct meeting constraintsConstruct meeting constraints
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PrinciplesPrinciples
Need for raw material for Need for raw material for empirical conjecture empirical conjecture and-or and-or structural relationships structural relationships(from experience, observation, pattern) (from experience, observation, pattern) conjectureconjecture
Strong relation between inferrable relations and Strong relation between inferrable relations and underlying conceptual structureunderlying conceptual structure
Attention directed to structural relationships as Attention directed to structural relationships as propertiesproperties
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Reflecting on Designing Bigger TasksReflecting on Designing Bigger Tasks
Macro worlds for classroom exploration need Macro worlds for classroom exploration need appropriate pedagogy appropriate pedagogy
Need for a strong relation between likely Need for a strong relation between likely actions, inferrable (?) relations, and underlying actions, inferrable (?) relations, and underlying conceptual structure (Witch Hat)conceptual structure (Witch Hat)
Realisable mathematical potential (Christmas Realisable mathematical potential (Christmas decorations)decorations)
Maths has to be worthwhile and necessaryMaths has to be worthwhile and necessary Reasoning about relationships; not empirical Reasoning about relationships; not empirical
fiddlingfiddling Purpose and utility (Ainley and Pratt)Purpose and utility (Ainley and Pratt) Vertical mathematisation (FI)Vertical mathematisation (FI)
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Our idealsOur ideals
Pick up others’ task and push-probe mathematical Pick up others’ task and push-probe mathematical potential ourselves: Dudeney goat-tetheringpotential ourselves: Dudeney goat-tethering
Influence/develop student mathematical Influence/develop student mathematical reasoning and/or conceptual understandingreasoning and/or conceptual understanding
Maths has to be necessary to some further end, Maths has to be necessary to some further end, yet all can get startedyet all can get started
Vary the initial level of complexity and generality Vary the initial level of complexity and generality so as not to create dependencyso as not to create dependency
Non-Tasks: chord-slopeNon-Tasks: chord-slope If … is varied, what will attention be drawn to?If … is varied, what will attention be drawn to? Are the mathematical affordancesAre the mathematical affordances
‘ ‘worth the caloriesworth the calories’?’?
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Follow-UpFollow-Up
Questions & Prompts (Primary & secondary Questions & Prompts (Primary & secondary versions)versions)
ThinkersThinkers Mathematucs as a Constructive ActivityMathematucs as a Constructive Activity Thinking MathematicallyThinking Mathematically Design & Use of Mathematical TasksDesign & Use of Mathematical Tasks Teaching Mathematics: Action and AwarenessTeaching Mathematics: Action and Awareness Contact: j.h.mason @ open.ac.ukContact: j.h.mason @ open.ac.uk