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 M i GEN 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

M i Gen System The MiGen system comprises 1.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 play swf

Students’ Constructions Number Of Students Students’ Derived Rules

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 construction  objects & operations to construct patterns 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