1. MiGen: Intelligent Support for Mathematical Generalisation INVESTIGATORS Richard Noss Alex Poulovassilis George Magoulas Celia Hoyles Niall Winters TEACHERS and TEACHER EDUCATORS Paul Clifford Peter Tang Teresa Smart Dietmar Kuchemann RESEARCHERS Darren Pearce Sergio Gutti é rez Ken Kahn Manolis Mavrikis Eirini Geraniou PHD STUDENT Mihaela Cocea OTHER PROJECT MEMBERS Dave Pratt John Mason Lulu Healy Jose Valente John Mason (consultant)

2. OUTLINE  Aims of the project  A brief demo of the current system  Initial results from trials with students  A teacher’s perspective  Hands-on activity  Discussion

3. AIMS  to co-design, build and evaluate, with teachers and teacher educators, a mutually supportive pedagogical and technical environment for improving 11-14 year-old students’ learning of mathematical generalisation.

4.  Most students can identify patterns, but this does not lead to articulation of generality  Algebra is viewed as an endpoint  Problems often encourage pragmatic approaches Research shows that:

5. We want to..  develop a pedagogical and technical environment to improve 11-14 year old students’ learning of mathematical generalisation comprising: sequenced and progressive activities within a prototype microworld – the eXpresser – designed to promote the learning of mathematical generalisation through model- construction; an intelligent tool, the eGeneraliser, which will be providing personalized feedback to students when they are tackling generalisation tasks and will be adapted to individual student’s learning trajectories; an intelligent tool for learners and teachers, the eCollaborator, through which students will be able to communicate with each other to view, compare and critique their constructions and ideas; also providing important information to the teacher.

6. The ShapeBuilder mockup ShapeBuilder is a first tool we’ve developed and used with students in order to inform the design of the eXpresser.

7. The Pond-Tiling Activity Someone wants to know the number of square tiles needed to surround a rectangular swimming pool with one layer of tiles. You don’t know the size of their swimming pool, so you need to tell them a rule for coming up with the number of tiles they need to surround it.

8. Initial Trials with Students DATESTUDENTSSCHOOLSHAPEBUILDERACTIVITIES 26/11/072Trinity – LeamV 0.78Pond-Tiling 06/12/072Trinity – LeamV 0.89Pond-Tiling 12/12/073Bridge – HackneyV 0.91Fam + Pond-Tiling 20/12/071Trinity – LeamV 0.92Fam + Pond-Tiling 18/01/082LKLV 0.93Fam + Pond-Tiling 30/01/082LKLV 0.93Familiarisation 06/02/082LKLV 0.93Pond-Tiling 20/02/081LKLV 0.95Pond-Tiling (L-shape) 26/02/085Bridge - HackneyV 0.96Fam + Pond-Tiling 27/02/082LKLV 0.97Pond-Tiling (L-shape) 05/03/082Bridge - HackneyV 0.98Pond-Tiling 24 sessions

9. Initial Results STUDENTS 1.Importance of familiarisation (appendix)appendix 2.Degrees of generality (snapshots)snapshots 3.The system supports their articulation process (snapshots)snapshots 4.“Messing Up” is effective (video)video 5.Importance of Collaboration (audio)audio

10. Initial Results INDIVIDUAL LEARNERS 1.Some students need constant encouragement and feedback 2.Telling a story about a task can engage students 3.Some students lose track of their thoughts and their goals 4.More time and repetition to familiarise is needed 5.Identify different prompts to help students reach a general rule

11. Initial Results TEACHERS 1.Importance of the teacher’s presence and support so possible difficulties in a real classroom. 2.The system could inform the teacher of the progress of all students in a classroom distinguished in predefined ways

12. A teacher’s perspective  There is a “richness” in the pond-tiling task compared to other tasks  A teacher-led activity discourages students to develop their own strategies  ICT allows students a deeper understanding of the general case  Students aim at getting a “correct” answer and are reluctant to explore  The system allows students to “try things out” and make mistakes  The system allows students to explain and justify their actions, discuss their ideas with other students and find equivalences  The challenge is to develop the system for classroom use

13. OVERVIEW OF INITIAL RESULTS STUDENTS 1.Importance of familiarisation 2.Degrees of generality 3.The system supports their articulation process 4.“Messing Up” is effective 5.Importance of Collaboration INDIVIDUAL LEARNERS 1.Some students need constant encouragement and feedback 2.Telling a story about a task can engage students 3.Some students lose track of their thoughts and their goals 4.More time and repetition to familiarise is needed 5.Necessity of different prompts to help students reach a general rule TEACHERS 1.Importance of the teacher’s presence and support 2.The system could inform the teacher for the progress of all students in a classroom

14. Hands-On  Would you like to try it?

15. Discussion  Please tell us how you might use the system in the classroom. Maybe through encouraging students to collaborate and share their constructions around this task. Would we need different tasks and/or different prompts, scaffolds or extensions for differently attaining students? What tasks might you design and for whom? Would you like to keep in touch with us or try out new versions? Please give us your feedback now or later by email to migen@lkl.ac.uk THANK YOU FOR YOUR ATTENTION AND HELP

16. APPENDIX 1: Familiarisation Tasks

17. Snapshots 2. Degrees of generality: Construction (specific example of shape)  specific expression  use of variables  general expression

18. Snapshots 2. Degrees of generality: Construction (specific example of shape)  specific expression  use of variables  general expression

19. Snapshots 3. The system supports their articulation process

20. a ‘messed-up’ construction Researcher: What would it [the width of the pond] be if it was half? Student:5 Researcher: So, now that it is 5, how many [tiles] do you think he [the owner of the pool] needs ? Student: The width plus... 6. I think. Teacher: You made this one, half as big? Student: I think I've done this one wrong. Video shown of a student’s messed-up construction.

21. Two students discussing their rules: Importance of Collaboration Meli: we did the… like you did… the height of the swimming pool plus two and then the width of the swimming pool plus two. And then I did… Maria: that wouldn’t work… Researcher: Say that again… Maria: If you did the height of the swimming pool plus two and then the width of the swimming pool plus two… you don’t… you don’t need the width of the swimming pool plus two… because otherwise you would have like… Meli: No, I know, but it does work. I don’t know. I thought that, but it actually does work if we make the shape…somehow. Researcher: Why wouldn’t it work? Meli: Because if you do… Maria: you know the height of the swimming pool plus two which it would be the end bits here which would be already the end bits of that…. And then you’ve got the width of the swimming pool plus two, which it would just go away… with the… height. Meli: That’s what I thought… and I don’t actually understand how it works, like… Maria: if you, if you like… Researcher: why… why would it do that? Meli: because… if you make the shape… I know what you mean… if you like make the shape and then you do… look…hold on.

22. Using ShapeBuilder to show their way of thinking and support their arguments Importance of Collaboration