1

Theoretical Background

2

Task Design & Use

7 phases

Potential

Structure of a Topic

Interaction

Effectiveness of actions

Inner & Outer

Teacher

Roles

3 Only’

s

Balance

Activity

Re-flection

&Pro-

flection

Content Task

ActionsThemesPowers

Questioning

Peers

6 Modes

3

Teacher FocusTeacher-Student

interaction

Student-Mathematics interaction

Teacher-Mathematics interaction

Cognitive Obstacles:

common errors, …

Applications & Uses

Methods & Procedures

Language/technical terms

Origins

Examples, Images & RepresentationsEnactive

Obstacles

Affective Obstacles

4

Actions

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

5

Themes

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

6

Powers

Imagining & Expressing Specialising & Generalising (Stressing &

Ignoring) Conjecturing & Convincing (Re)-Presenting in different modes Organising & Characterising

7

Inner & Outer Aspects

Outer– What task actually initiates explicitly

Inner– What mathematical concepts underpinned– What mathematical themes encountered– What mathematical powers invoked– What personal propensities brought to awareness

8

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

9

Structure of a TopicImageryAwareness (cognition)

Will

Body (enaction)

Emotions (affect)

HabitsPractices

10

Three Only’sLanguage Patterns

& prior SkillsImagery/Sense-of/

Awareness; Connections

Different Contexts in which likely to arise;

dispositions

Techniques & Incantations

Root Questionspredispositions

Standard Confusions

& Obstacles

Only Behaviour is TrainableOnly Emotion is Harnessable

Only Awareness is Educable

Behaviour

Emotion

Aware

ness

11

PhasesGetting StartedGetting InvolvedMulling

Keeping Going

Insight

Being ScepticalContemplating

Initiating

Sustaining

Concluding

12

Six Modes of Interaction

ExpoundingExplaini

ng

ExploringExaminin

g

ExercisingExpressi

ng

Initiating Sustaining Concluding

13

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)

14

Sustaining Activity

Questions & Prompts Directed–Prompted–Spontaneous

Scaffolding & Fading Energising (praising-challenging) Conjecturing Sharing progress/findings

15

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

16

Balanced ActivityAffordances Constraints Attunements

Ends

Means

CurrentState

OuterTaskIntended

& Enactedgoals

Resources Tasks

Ends

Means

CurrentState

InnerTask Implicit

goals

Resources Tasks

17

Expounding

Explaining

Exploring

Examining

Exercising

Expressing

18

Teacher

Student

ContentExpounding

Student

Content

TeacherExamining

Student

ContentTeacher

Expressing

Teacher

StudentExplaining

Content

Student

ContentTeacher

Exploring

Student

Content

TeacherExercising

19

ActivityGoals, Aims,

Desires, Intentions

Tasks(as imagined,

enacted,experienced,

…)

Resources:(physical, affective cognitive, attentive)

Initial State

Affordances– Constraints–Requirements

(Gibson)

20

Potential

Most it could be

What builds on it(where it is going)

Math’l & Ped’cessence

Least it can be What it builds on

(previous experiences)Affordances– Constraints–

Requirements Directed–Prompted–SpontaneousScaffolding & Fading (Brown et al)

ZPD (Vygotsky)

(Gibson)

21

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

22

Frameworks ( 틀 )

Doing – Talking – Recording

Enactive– Iconic– Symbolic

See – Experience – Master

23

Follow-Up 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