MiGen: Intelligent Support for Mathematical Generalisation

INVESTIGATORSRichard NossAlex PoulovassilisGeorge MagoulasCelia HoylesNiall Winters

TEACHERS and TEACHER EDUCATORSPaul CliffordPeter TangTeresa SmartDietmar Kuchemann

RESEARCHERSDarren PearceSergio GuttiérezKen KahnManolis MavrikisEirini Geraniou

PHD STUDENTMihaela Cocea

OTHER PROJECT MEMBERSDave PrattJohn MasonLulu HealyJose ValenteJohn Mason (consultant)

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

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.

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:

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.

The ShapeBuilder mockup

ShapeBuilder is a first tool we’ve developed and used with students in order to inform the design

of the eXpresser.

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.

Initial Trials with Students

DATE STUDENTS SCHOOL SHAPEBUILDER ACTIVITIES

26/11/07 2 Trinity – Leam V 0.78 Pond-Tiling

06/12/07 2 Trinity – Leam V 0.89 Pond-Tiling

12/12/07 3 Bridge – Hackney V 0.91 Fam + Pond-Tiling

20/12/07 1 Trinity – Leam V 0.92 Fam + Pond-Tiling

18/01/08 2 LKL V 0.93 Fam + Pond-Tiling

30/01/08 2 LKL V 0.93 Familiarisation

06/02/08 2 LKL V 0.93 Pond-Tiling

20/02/08 1 LKL V 0.95 Pond-Tiling (L-shape)

26/02/08 5 Bridge - Hackney V 0.96 Fam + Pond-Tiling

27/02/08 2 LKL V 0.97 Pond-Tiling (L-shape)

05/03/08 2 Bridge - Hackney V 0.98 Pond-Tiling

24 sessions

Initial Results

STUDENTS

1. Importance of familiarisation (appendix)

2. Degrees of generality (snapshots)

3. The system supports their articulation process (snapshots)

4. “Messing Up” is effective (video)

5. Importance of Collaboration (audio)

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

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

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

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

Hands-On

Would you like to try it?

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

APPENDIX 1: Familiarisation Tasks

Snapshots

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

Snapshots

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

Snapshots

3. The system supports their articulation process

a ‘messed-up’ construction

Researcher: What would it [the width of the pond] be if it was half?Student:5Researcher: 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.

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.

Using ShapeBuilder to show their way of thinking and support their arguments

Importance of Collaboration