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

2 Our design methods Task audience Task audience –teachers & novice teachers; teacher educators Task purposes 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?: What?: NOT roll out materials but methods of working [and possible tasks] NOT roll out materials but methods of working [and possible tasks]

3 Generic Task structures Unusual features (e.g. inter-rootal distance) … Unusual features (e.g. inter-rootal distance) … Another & another Another & another Imagine a … Imagine a … Silent lesson Silent 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)

4 Generic Task Stuctures Microtasks are generic strategies applied in particular conceptual fields, e.g. Microtasks are generic strategies applied in particular conceptual fields, e.g. teacher asks students to compare.... teacher asks students to compare.... teacher asks students to invent a representation of … teacher asks students to invent a representation of … (generic tactics which become didactic in a conceptual context) (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 Audience: primary mathematics teacher educators residential workshop Purpose: 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... Choices: to bring to the surface or to dig deep to find...

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

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

8 Linearity Plot countdowns as graphs Plot countdowns as graphs Predictable issues Predictable issues

9 Design issues relevant to this sequence Who? Who? What purpose? What purpose? Context, resources, time Context, resources, time My understanding of underlying conceptual issues My understanding of underlying conceptual issues Organised direct (and directed) experience of these Organised direct (and directed) experience of these Dig deep rather than bring to the surface! Dig deep rather than bring to the surface! Sequencing tasks to address the same issues again and again Sequencing tasks to address the same issues again and again Importance of associated pedagogical context and strategies 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 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) Problem with seeing multiplication only as repeated addition or arrays (Nuffield study) Need a model for scaling (BEAM elastic; PGCE Cabri) 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 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) Microtask sequence: expansion; contraction; expansion and related contraction e.g. 3/2 and 2/3; invariance & (MGA)

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

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

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

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

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