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