Eirini Geraniou

Design requirements, student perception indicators and validation metrics for intelligent exploratory learning environments

The new forms of interaction afforded by innovative technology and open-ended environments provide promising opportunities for exploratory learning. Exploratory environments, however, require appropriate support to lead to meaningful learning outcomes. This paper focuses on the design and validation of intelligent exploratory environments. The goal is twofold: requirements that guide the operationalisation of pedagogical strategies to computer-based support and methodology for the validation of the system. As designers, we need to understand what kind of interaction is conducive to learning and aligned with the theoretical principles behind exploratory learning. We summarise this in the form of three requirements—rare interruption of interaction, co-location of feedback and support towards specific goals. Additionally, developing intelligent systems requires many resources and a long time to build. To facilitate their evaluation, we define three indicators— helpfulness, repetitiveness and comprehension—of students’ perception of the intelligent system and three metrics—relevance, coverage, and scope—which allow the identification of design or implementation problems at various phases of the development. The paper provides a case study with a mathematical microworld that demonstrates how the three requirements are taken into account in the design of the user-facing components of the system and outline the methodology for formative validation of the intelligent support.

Issues in the Design of an Environment to Support the Learning of Mathematical Generalisation

Expressing generality, recognising and analysing patterns and articulating structure is a complex task and one that is invariably problematic for students. Nonetheless, very few systems exist that support learners explicitly in the process of mathematical generalisation. We have addressed this by developing a novel environment that supports users in their reasoning and problem-solving of generalisation tasks. We have followed a stakeholder-centred design process, integrating feedback and reflections from twenty-four children, five teachers and a variety of other stakeholders. This paper focuses on several inter-related design issues that have been informed by this iterative process and demonstrates how the system can be used for a typical generalisation task to foster an appreciation of generality and indeed algebra.

Towards a constructionist approach to mathematical generalisation

{e.geraniou, m.mavrikis , c.hoyles , r.noss}@ioe.ac.uk Accommodating the idea of mathematical generalisation, recognising and analysing pattern and articulating structure, especially expressed in algebraic forms, are key aspects of mathematical thinking and have long been elusive ideas for students. Although students are often able to identify and predict patterns (Mason 2002), they are unable to articulate a general pattern or relationship in natural language or, even more demanding, in algebraic symbolism (Hoyles and K

Patterns of Collaboration: Towards Learning Mathematics in the Era of the Semantic Web

With current digital technologies there are a number of networked computer-based tools that provide ways for users, be they learners or teachers, to collaborate in tackling visual representations of mathematics, both algebraic and geometric. For learners, there are various ways of collaborating that can occur while the learners are tackling mathematical problems. In this chapter we use selected outcomes from recent innovative research on this aspect of learning and teaching mathematics with digital technologies to review the patterns of collaboration that can occur in terms of teacher and learner experience. Given that such patterns of collaboration are via current digital technologies, this chapter goes on to offer a view on the likely impact on the cyberlearning of mathematics of progress towards the next generation of Web technologies that seeks to make use of ideas related to the web of data and the semantic web. Such impact is likely to be in terms of enhancing the learning applications of digital technologies, improving ways of administrating the educational programmes that they support, and potentially enabling teachers to maintain involvement in technological development and use over the longer-term.

Supporting Reasoning and Problem-Solving in Mathematical Generalisation with Dependency Graphs

We present a brief description of the design of a diagram- based system that supports the development of thinking about mathematical generalisation. Within the software, the user constructs a dependency graph that explicitly shows the relationships between components of a task. Using this dependency graph, the user manipulates graphical visualisations of component attributes which helps them move from the specific case to the general rule. These visualisations provide the user with an intermediate representation of generality and facilitate movement between the specific details of the task, the appropriate generalisations, verbal descriptions of their understanding and various algebraic representations of the solutions.

Exploring the link between task features and generalisation

Research on students’ expression of generality in pattern-generalising problems has widely documented their difficulties in articulating and representing the functional rule in words or in algebraic notation (Hoyles and Küchemann 2001). But recent studies seem to attribute the difficulties mainly to the way the patterns are depicted in such problems (Stalo et al. 2006; Lannin, Barker and Townsend 2006). Apart from these findings, little else is known about what and how other features of the problems might affect students’ abilities to express generality. In this study, we speculate that students’ performance in pattern-generalising problems might be influenced by the five task features described below.

Similarity-Based Grouping to Support Teachers on Collaborative Activities in an Exploratory Mathematical Microworld

This paper describes a computer-based tool that helps teachers group their students for collaborative activities in the context of secondary school math teaching, the challenge being to organize groups of students based on their recent work, so that their collaboration results in meaningful interactions. The complexity of the exploratory learning tasks is such that the teachers would require too long a time to create meaningful groups without the tool. This paper describes the design of the tool, the algorithms and metrics used for generating the groups, and the pedagogical context in which the tool was designed. This paper reports on the evaluation of the tool. Its recommendations are found to be equivalent to human experts, and the time required is under a few seconds for the usual classroom size. Some possibilities to extend the work to other learning domains, such as programming, are finally considered.