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The Micro-Features of Mathematical Tasks

The Open University Maths Dept. University of Oxford Dept of Education. Promoting Mathematical Thinking. The Micro-Features of Mathematical Tasks. Anne Watson & John Mason Nottingham Feb 9 2012. Our design methods. Task audience teachers & novice teachers; teacher educators

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The Micro-Features of Mathematical Tasks

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

  2. 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]

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

  4. 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

  5. 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 ...

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

  7. 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?

  8. Linearity • Plot countdowns as graphs • Predictable issues

  9. 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

  10. 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)

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

  12. 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

  13. 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)

  14. 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’?

  15. 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

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