Promoting Mathematical Thinking The MicroFeatures of Mathematical Tasks

Promoting Mathematical Thinking The Micro-Features of Mathematical Tasks Anne Watson & John Mason Nottingham Feb 9 2012 The Open University Maths Dept 1 University of Oxford Dept of Education

Our design methods /Task audience – teachers & novice teachers; teacher educators /Task purposes – to bring to awareness important mathematical and pedagogical constructs – to offer task-types rather than particular tasks – to promote deep consideration of concepts – to provide current experience of ways of working mathematically – to articulate effective mathematical actions /What? : /NOT roll out materials but methods of working [and possible tasks] 2

Generic Task structures /Unusual features (e. g. inter-rootal distance) … /Another & another Generic strategies for teachers to /Imagine a … add to their repertoire (Q&P; Thinkers) /Silent lesson /Make up one which. . . . /Change. . so that. . . 3

Generic Task Stuctures /Microtasks are generic strategies applied in particular conceptual fields, e. g. • teacher asks students to compare. . • teacher asks students to invent a representation of … /(generic tactics which become didactic in a conceptual context) Sequences of microtasks which direct development of a network of conceptual ideas Cf. (Hypothetical) learning Trajectories Cf. Spiral Curriculum 4

Recent example of task construction and sequencing /Audience: primary mathematics teacher educators’ residential workshop /Purpose: – to bring to awareness important mathematical and pedagogical constructs – to promote deep consideration of concepts – to provide current experience of ways of working mathematically /Choices: to bring to the surface or to dig deep to find. . . 5

Count down /Count down from 10 to -10 /How many numbers /Why? What next? Number or linearity? 6

Number /Count down from 101 in steps of 1 1/10 /Count down from 46 in steps of 1 1/5 Make up some like this Predictable issues? 7

Linearity /Plot countdowns as graphs /Predictable issues 8

Design issues relevant to this sequence /Who? /What purpose? /Context, resources, time /My understanding of underlying conceptual issues /Organised direct (and directed) experience of these /Dig deep rather than bring to the surface! /Sequencing tasks to address the same issues again and again /Importance of associated ‘pedagogical’ context and strategies 9

Genesis of ideas: the story of elastics /Ulla Runesson using elastic to vary the ‘whole’ in fractions; dimension of variation /Problem with seeing multiplication only as repeated addition or arrays (Nuffield study) /Need a model for scaling (BEAM elastic; PGCE Cabri) /John works repeatedly with a range of audiences exploring dimensions of variation, making kit, and observing their actions and comments /Microtask sequence: expansion; contraction; expansion and related contraction e. g. 3/2 and 2/3; invariance & (MGA) 10

Embedded Q&P /Compare /Same/different /Representations /Exemplification /Variation /Construct 11 meeting constraints

Principles /Need for raw material for empirical conjecture and-or structural relationships (from experience, observation, pattern) conjecture /Strong relation between inferrable relations and underlying conceptual structure /Attention directed to structural relationships as properties 12

Reflecting on Designing Bigger Tasks /Macro worlds for classroom exploration need appropriate pedagogy /Need for a strong relation between likely actions, inferrable (? ) relations, and underlying conceptual structure (Witch Hat) /Realisable mathematical potential (Christmas decorations) /Maths has to be worthwhile and necessary /Reasoning about relationships; not empirical fiddling /Purpose and utility (Ainley and Pratt) /Vertical mathematisation (FI) 13

Our ideals /Pick up others’ task and push-probe mathematical potential ourselves: Dudeney goat-tethering /Influence/develop student mathematical reasoning and/or conceptual understanding /Maths has to be necessary to some further end, yet all can get started /Vary the initial level of complexity and generality so as not to create dependency /Non-Tasks: chord-slope /If … is varied, what will attention be drawn to? /Are the mathematical affordances ‘worth the calories’? 14

Follow-Up /Questions & Prompts (Primary & secondary versions) /Thinkers /Mathematucs as a Constructive Activity /Thinking Mathematically /Design & Use of Mathematical Tasks /Teaching Mathematics: Action and Awareness /Contact: j. h. mason @ open. ac. uk 15