Alternative epistemologies for algebraic generalisation

1. Alternative epistemologies for algebraic generalisation Richard Noss, Celia Hoyles, Eirini Geraniou and Manolis Mavrikis London Knowledge Lab Institute of Education - University of London

2. an intelligent exploratory learning environment for supporting the construction of mathematical generalisations & their expression as rules

3. UK Curriculum for grade 4/5

4. Level 4 Level 4/5

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'.

6. 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?

7. The MiGEN team 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) Richard Noss (IOE) Alex Poulovassilis (BBK) Celia Hoyles (IOE) George Magoulas (BBK) School A (Hackney) School B (Leamington Spa) School C (Islington) School D (Alton) Teachers Teacher Educators Stakeholders Advisors John Mason (Open Univ.) Lulu Healy (Univ. UNIBAN, Sau Paulo, Brazil)

8. The eXpresser microworld

9. MiGen System • The MiGen system comprises • a microworld (the eXpresser) • intelligent Support to encourage generalisation through • adaptive feedback to students • support for teachers to monitor students’ progress and suggest strategies • tools to support collaboration • group tasks for sharing & discussing constructions and rules with each other and with teachers

10. 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

11. play swf

13. sharing process & outputs

14. see providing intelligent support to students and teachers Can this be ‘messed up’?

15. migenproject.wordpress.com

16. spare slides

17. 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 mutually supportive model and underlying rule construction objects & operations to construct patterns rules built on the basis of patterns’ building blocks and their transformations

18. 3. workon a particular case while `keeping an eye’ on the general operationalising the distinction between constants and variables fostering explicit expression of implicit relationships between quantities within a pattern motivatingexplicit expression of implicit relationships between variables in the model

19. 4. enabling assessment of theequivalence of rules • replacement • simple manipulation & collecting ‘like’ terms • relating rule and figure number • subsequent validation by animation