Expressing generality, recognising and analysing pattern and articulating structure are at the core of mathematical thinking and scientific enquiry. These ideas are notoriously elusive for students, who routinely find themselves unable to understand what mathematics is about . There have been attempts to foster an appreciation of generality, e.g by finding ways for students to construct their own mathematical models. Some studies have also attempted to exploit the potential of collaborative learning. However, despite some successes, difficulties remain, and these tend to coalesce around the need for intensive, timely and appropriate pedagogic support.
The project responds to this need, by developing an intelligent learning environment for improving 11-14 year-old students' learning of mathematical generalisation. We are a collaborative research team of social, educational and computer scientists, at the London Knowledge Lab, an interdisciplinary facility shared between the Institute of Education and Birkbeck . We work in close collaboratiion with networks of teachers/educators coordinated by the London Mathematics Centre at IOE, the London regional hub of the National Centre for Excellence in Teaching of Mathematics .
Our methodology is characterised by iterative cycles of design, testing and revision of a pedagogical and technical environment consisting of:
- i. sequenced and progressive activities undertaken within a prototype microworld – the eXpresser – designed to promote the learning of mathematical generalisation through model-construction;
- ii. an intelligent component, the eGeneraliser , which will provide personalised feedback during the process of model analysis and generalisation;
- iii. a second intelligent component, the eCollaborator , which will foster cumulative knowledge construction through the sharing and discussion of models, aiming to develop and sustain an effective online community.
In the latter stages of the project, we will extend the eXpresser to include a range of alternative representations, e.g. graphs and equations, and will enhance the eGen and eCollab so that they provide these extensions with intelligent learner support. We will evaluate, formatively and summatively , our environment and scale it up from a design experiment to a sustainable learning community.
The project will take place in 4 phases. Phase 1 (6 months) will develop the first version of the eXpresser, research suitable computational techniques for the eGen and eCollab, design sequenced learning activities and instruments for their formative assessment with teachers & educators, and identify the types of data and instruments needed for their collection during the piloting of the eXpresser (including data to test the functionality of the eXpresser alongside data to assess student responses). Phase 2 (1 year) will pilot the eXpresser with a small team of teachers and a group of Year 8 and 9 students on a variety of modelling tasks, focussing on identifying trajectories of construction, the students' collaborations and interactions, and analysing the collected data in order to characterise the different pathways taken, the kinds of generalisations made etc. The eXpresser will be enhanced iteratively in the light of the feedback from students and teachers and results of the data analysis, and first versions of the eGen and the eCollab, will be developed. Phase 3 (1 year) will pilot the complete system with teachers in their Year 8/9 classrooms, analysing the data collected from this usage, iteratively enhancing the three components of the system, and prototyping and piloting an extended eXpresser microworld, new activity sequences that use it, and extensions to the eGen and eCollab. Phase 4 (9 months) will undertake summative evaluation of the overall pedagogical and technical environment alongside evaluation of the potential to scale up the design research and prototyping to a wider group of teachers and learners.