Alternative epistemologies for algebraic generalisation

Richard Noss, Celia Hoyles, Eirini Geraniou and Manolis Mavrikis

London Knowledge Lab

Institute of Education - University of London

an intelligent exploratory learning environment

for supporting the construction of mathematical generalisations & their expression as rules

UK Curriculum for grade 4/5

Level 4

Level 4/5

an epistemological obstacle

teachers’ epistemology the special case is a way of thinking about

the general case• How many here? How many there? How many

in general? Count, recognize pattern, apply the pattern to any 'given number'.

an epistemological obstacle

students’ epistemology the answer is the number of tiles – i’ll count

them! the teacher says ‘any number’. OK, 6. oh alright, 7!!!

what is this other thing i’m supposed to do and why am I supposed to be doing it?

Teachers

Teacher Educators

Stakeholders

Advisors

John Mason (Open Univ.)

Lulu Healy (Univ. UNIBAN, Sau Paulo, Brazil)

Teachers

Teacher Educators

Stakeholders

Advisors

John Mason (Open Univ.)

Lulu Healy (Univ. UNIBAN, Sau Paulo, Brazil)

School A (Hackney)School B (Leamington

Spa)School C (Islington)

School D (Alton)

School A (Hackney)School B (Leamington

Spa)School C (Islington)

School D (Alton)

The MiGEN team

Richard Noss (IOE)Alex Poulovassilis (BBK)Celia Hoyles (IOE)George Magoulas (BBK)

Richard Noss (IOE)Alex Poulovassilis (BBK)Celia Hoyles (IOE)George Magoulas (BBK)

Eirini Geraniou (IOE)

Sergio Gutierrez (BBK)

Ken Kahn (IOE)

Manolis Mavrikis (IOE)

Darren Pearce (BBK)

Niall Winters (IOE)

PhD students

Mihaela Cocea (BBK)

Boon Chua Liang (IOE)

Eirini Geraniou (IOE)

Sergio Gutierrez (BBK)

Ken Kahn (IOE)

Manolis Mavrikis (IOE)

Darren Pearce (BBK)

Niall Winters (IOE)

PhD students

Mihaela Cocea (BBK)

Boon Chua Liang (IOE)

The eXpresser microworld

MiGen System

The MiGen system comprises1. a microworld (the eXpresser)

2. intelligent Support to encourage generalisation through adaptive feedback to students support for teachers to monitor students’ progress

and suggest strategies

3. tools to support collaboration group tasks for sharing & discussing constructions

and rules with each other and with teachers

some responses to the epistemological challenge dynamically presented tasks ‘unlocked numbers’ as a model for for

constants and variables synchronous view of the general case

alongside any actions on special cases the big idea - any epistemological rupture

between competing epistemologies is made explicit and manipulable

play swf

Students’Constructions

Number Of

Students

Students’ Derived Rules

8

2

3

1

1

1

sharing process & outputs

see

Can this be ‘messed up’?

providing intelligent support to students and teachers

migenproject.wordpress.com

spare slides

1. scaffolds  to construct general models • provision of a rationale for generality through animated task

presentation• allowing student-controlled validation of constructions and rules by

animation

2. mutually supportive model and underlying rule construction1. objects & operations to construct patterns

2. rules built on the basis of patterns’ building blocks and their transformations  

3. work on a particular case while `keeping an eye’ on the general

a. operationalising the distinction between constants and variables

b. fostering explicit expression of implicit relationships between quantities within a pattern

c. motivating explicit expression of implicit relationships between variables in the model

4. enabling assessment of the equivalence of rules

• replacement• simple manipulation & collecting ‘like’ terms• relating rule and figure number • subsequent validation by animation