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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