Theoretical Background

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Presentation transcript:

Theoretical Background

Task Design & Use Activity Re-flection & Pro-flection Content Task Potential Actions Structure of a Topic Inner & Outer Effectiveness of actions Themes Powers 3 Only’s Balance Interaction 7 phases Teacher Peers 6 Modes Roles Questioning

Teacher Focus Teacher-Student interaction Teacher-Mathematics interaction Student-Mathematics interaction Language/technical terms Examples, Images & Representations Enactive Obstacles Origins Applications & Uses Affective Obstacles Cognitive Obstacles: common errors, … Methods & Procedures

Actions Right-multiplying by an inverse ... Making a substitution Differentiating Iterating Reading a graph Invoking a definition …

Themes Doing & Undoing Invariance in the midst of change Freedom & Constraint Restricting & Extending

Powers Imagining & Expressing Specialising & Generalising (Stressing & Ignoring) Conjecturing & Convincing (Re)-Presenting in different modes Organising & Characterising

Inner & Outer Aspects Outer Inner What task actually initiates explicitly Inner What mathematical concepts underpinned What mathematical themes encountered What mathematical powers invoked What personal propensities brought to awareness

Challenge Appropriate Challenge: Not too great Not too little Scope depends on student trust of teacher Scope depends on teacher support of mathematical thinking not simply getting answers

Structure of a Topic Awareness (cognition) Imagery Will Emotions (affect) Body (enaction) Habits Practices

Three Only’s Awareness Behaviour Emotion Only Emotion is Harnessable Language Patterns & prior Skills Imagery/Sense-of/Awareness; Connections Root Questions predispositions Different Contexts in which likely to arise; dispositions Standard Confusions & Obstacles Techniques & Incantations Emotion Awareness Behaviour Only Emotion is Harnessable Only Awareness is Educable Only Behaviour is Trainable

Phases Initiating Sustaining Concluding Getting Started Getting Involved Mulling Sustaining Keeping Going Insight Being Sceptical Concluding Contemplating

Six Modes of Interaction Initiating Sustaining Concluding Expounding Explaining Exploring Examining Exercising Expressing

Initiating Activity Silent Start Particular (to general); General (via particular) Semi-general (via particular to general) Worked example Use/Application/Context Specific-Unspecific Manipulating: Material objects (eg cards, counters, …) Mental images (diagrams, phenomena) Symbols (familiar & unfamiliar)

Sustaining Activity Questions & Prompts Directed–Prompted–Spontaneous Scaffolding & Fading Energising (praising-challenging) Conjecturing Sharing progress/findings

Concluding Activity Conjectures with evidence Accounts that others can understand Reflecting on effective & ineffective actions Aspcts of inner task (dispositions, …) Imagining acting differently in the future

Balanced Activity Affordances Constraints Attunements Outer Task Inner Task Intended & Enacted goals Implicit goals Ends Ends Resources Tasks Resources Tasks Means Means Current State Current State

Expounding Explaining Exploring Examining Exercising Expressing

Student Content Teacher Exercising Teacher Student Content Expounding Student Content Teacher Exploring Teacher Student Explaining Content Student Content Teacher Examining Student Content Teacher Expressing

Activity Goals, Aims, Desires, Intentions Resources: (physical, affective cognitive, attentive) Tasks (as imagined, enacted, experienced, …) Initial State Affordances– Constraints–Requirements (Gibson)

Potential (Gibson) What builds on it (where it is going) Most it could be Math’l & Ped’c essence Role of reflection in achieving affordances Least it can be What it builds on (previous experiences) Affordances– Constraints–Requirements (Gibson) Directed–Prompted–Spontaneous Scaffolding & Fading (Brown et al) ZPD (Vygotsky)

Thinking Mathematically CME Do-Talk-Record (See–Say–Record) See-Experience-Master Manipulating–Getting-a-sense-of–Artculating Enactive–Iconic–Symbolic Directed–Prompted–Spontaneous Stuck!: Use of Mathematical Powers Mathematical Themes (and heuristics) Inner & Outer Tasks

Frameworks (틀) Enactive– Iconic– Symbolic Doing – Talking – Recording See – Experience – Master

Follow-Up j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3 Designing & Using Mathematical Tasks (Tarquin/QED) Thinking Mathematically (Pearson) Developing Thinking in Algebra, Geometry, Statistics (Sage) Fundamental Constructs in Mathematics Education (RoutledgeFalmer) Mathematics Teaching Practice: a guide for university and college lecturers (Horwood Publishing) Mathematics as a Constructive Activity (Erlbaum) Questions & Prompts for Mathematical Thinking (ATM) Thinkers (ATM) Learning & Doing Mathematics (Tarquin) Researching Your Own Practice Using The Discipline Of Noticing (RoutledgeFalmer) j.h.mason @ open.ac.uk mcs.open.ac.uk/jhm3