The Conceptual and Architectural Design of a System Supporting Exploratory Learning of Mathematics Generalisation Darren Pearce, Alex Poulovassilis ECTEL09, October 09

Richard Noss Alex Poulovassilis George Magoulas Celia Hoyles Niall Winters Ken Kahn Sergio Gutierrez-Santos Manolis Mavrikis Darren Pearce Eirini Geraniou Mihaela Cocea Boon Liang Chua Institute of Education & Birkbeck College

Outline of the talk Project aims and challenges MiGen context and tools Conceptual model System architecture Architectural proof-of-concept Conclusions and future work

Aims to co-design, build and evaluate –with teachers and teacher educators –a pedagogical and technical environment for improving year-old students learning of mathematical generalisation seeing the general through the particular is a powerful way to introduce generalisation, and forms the basis of our research we are adopting a constructionist approach

Aims MiGens eXpresser tool allows students to create and manipulate patterns and algebraic expressions, and explore the relationships between them students are asked to construct generalised patterns, to derive corresponding expressions, to test out their patterns and expressions on new problem instances, to compare their constructions with other students ones our aim is to support students exploratory construction while also fostering progressive knowledge building

Challenges little work so far in supporting students in a constructionist context conversely, considerable guidance is required to ensure learning in such contexts feedback needs to be provided to students during their construction process, not just at the end since students will be undertaking exploratory rather than structured tasks, teachers need to be assisted in monitoring students activities and progress by appropriate visualisation and notification tools

Interdisciplinarity research challenges are both pedagogical and technological e.g. –modelling the domain, tasks, learners –identifying information necessary for provision of effective feedback to learners and to the teacher –designing learner feedback, developing and trialling appropriate intelligent technologies –designing, developing trialling appropriate visualisation techniques for teacher feedback –designing and developing an appropriate system architecture

Interdisciplinarity MiGen research team is diverse: maths education, AI in education, computer science this paper focuses on a subset of the above challenges, primarily: –designing and developing an appropriate system architecture: extensible, scalable, client-server, multiple concurrent users (learners & teachers) and also aspects of: –modelling the domain, tasks, learners –designing, developing trialling appropriate visualisation techniques for teacher feedback

MiGen Context Deployed in classrooms in schools Students working individually, in pairs, or in groups Real-time provision of feedback to students and teachers: –aiming to assist the teacher in focusing her attention across the class and informing her interventions (not to replace the teacher) Provision also of historical information about students activities and progress to teachers

MiGen Tools MiGen system comprises a number of tools: –eXpresser – a mathematical microworld –Discussion tool –Task design tool –eGeneraliser –Teacher assistance tools: currently Classroom Dynamics (CD) and Student Tracking (ST) –Activity Design and Management tool (possibly)

Iterative Research Methodology 1.Pedagogical and technical research 2.Requirements elicitation within domains of maths generalisation and intelligent support 3.Requirements analysis, in collaboration with students, teachers and teacher educators 4.Development of activity sequences and tasks 5.Technical design and implementation 6.Technical&pedagogical environment evaluation 7.Analysis of evaluation results; elicitation of pedagogical and technical outcomes; iteration

Conceptual Model This work falls under item 3 earlier: –Requirements analysis, in collaboration with students, teachers and teacher educators The CM currently comprises 4 overlapping subsections; we present three in the paper and refer to a recent project deliverable for the fourth Further iterative development of the CM is occurring as we move towards V1 of the full MiGen system, and will continue

(i) Users and Learner Models

(ii) Students Constructions

(iii) Tasks, Activities, Learning Objectives and Landmarks

MiGen System Architecture

Architectural Proof-of-Concept forms basis for implementation of full architecture (now almost completed) MiGen schools context lightweight approach Lightweight RPC/SOAP or REST Many advantages to REST: –More scalable –More cacheable –Easier to performance-tune we have used Restlet, a lightweight Java REST framework

Architectural Proof-of-Concept aim of the proof-of-concept implementation was to develop sufficient server and client infrastructure to demonstrate fulfilment of the systems architectural requirements the server side manages the data resources, registration of new clients, handling of messages posted by clients, notifying clients of updated data resources the client side provides access to server-side data resources, determines any changes made to these, provides presentation functionality

MiGen Sequence Diagram

Two functional prototype visualisations for the CD & ST tools

Conclusions and Future Work Conceptual model: –derived after extensive requirements elicitation –significant step towards development of a common vocabulary across the disciplines Architecture: simple, modular, scalable Proof-of-concept implementation: significant step towards first full version of the system Continuing iterative development, aiming for full system ready to be tested with students and teachers in November 2009