Lulu Healy

Software Tools for Geometrical Problem Solving: Potentials and Pitfalls

Dynamic geometry software provides tools for students to construct and experiment with geometrical objects and relationships. On the basis of their experimentation, students make conjectures that can be tested with the tools available. In this paper, we explore the role of software tools in geometry problem solving and how these tools, in interaction with activities that embed the goals of teachers and students, mediate the problem solving process. Through analysis of successful student responses, we show how dynamic software tools can not only scaffold the solution process but also help students move from argumentation to logical deduction. However, by reference to the work of less successful students, we illustrate how software tools that cannot be programmed to fit the goals of the students may prevent them from expressing their (correct) mathematical ideas and thus impede their problem solution.

Unfolding Meanings for Reflective Symmetry

In this paper, we analyse the processes through which students come to negotiate mathematical meanings for reflective symmetry. We describe a microworld, Turtle Mirrors, designed to provide tools to help students focus simultaneously on actions, visual relationships and symbolic representations. Through a detailed case study of one student‘s thinking-in-change, we examine how the interactions with her partner and with the machine support a fusion of spontaneous and scientific concepts. Other examples of students‘ work further illustrate how the microworld tools offer a means to supplement local understandings of symmetry with those with more explicit, mathematical formulations.

Interdependence and autonomy: aspects of groupwork with computers

Abstract This paper presents case studies of two groups of six pupils undertaking three research tasks involving mathematical ideas and incorporating work with computers. We attempt to characterize effective groupwork and analyze the importance of developing a synergy between pupil interdependence and pupil autonomy. We examine the interrelationship between task, group as a social system and the role of the computer in establishing good group practice and identify the need for a pupil-teacher role within such a practice.

If this is our mathematics, what are our stories?

This paper sets out to examine how narrative modes of thinking play a part in the claiming of mathematical territories as our own, in navigating mathematical landscapes and in conversing with the mathematical beings that inhabit them. We begin by exploring what constitutes the narrative mode, drawing principally on four characteristics identified by Bruner and considering how these characteristics manifest themselves in the activities of mathematicians. Using these characteristics, we then analyse a number of examples from our work with expressive technologies; we seek to identify the narrative in the interactions of the learners with different computational microworlds. By reflecting on the learners’ stories, we highlight how particular features, common across the microworlds—motion, colour, sound and the like—provided the basis for both the physical and psychological grounding of the behaviour of the mathematically constrained computational objects. In this way, students constructed and used narratives that involved situating mathematical activities in familiar contexts, whilst simultaneously expressing these activities in ways which—at least potentially—transcend the particularities of the story told.

Working with Teachers: Context and Culture

This chapter concerns collaborations between teacher educators and teachers in activities involving digital technologies in the teaching and learning of mathematics. In light of the complexity involved in introducing new artefacts into existing cultures of practices, we focus on our attempts to develop ways of working with teachers so that they can become active participants in designing practices and routines appropriate for the particularities of their own classrooms. Three case studies are presented, from three different countries, Norway, Greece and Brazil, each of which describes the participation of teachers in a process of communal design of mathematical tools and activities. Two theoretical notions, boundary objects and instrumental genesis, are employed in order interpret the case studies and to illuminate the challenges associated with involving teachers in considering when, how and why digital technologies might be used fruitfully in the teaching of mathematics.

The use of spreadsheets within the mathematics classroom

This paper examines the potential of the use of spreadsheets in the teaching and learning of mathematics. Consideration is given to the teachers role in structuring computer‐based learning environments so that pupils are encouraged to reflect on mathematical processes. Crucial aspects of the software which pupils need to learn in order to use the spreadsheet as a mathematical tool are identified, and a number of introductory activities aimed at addressing these aspects are included. Examples of spreadsheet activities related to different types of mathematical problems are also presented. Finally we include an example of pupils using a spreadsheet as a context for formalizing mathematical generalizations.

Students’ Developing Knowledge in a Subject Discipline: Insights from Combining Quantitative and Qualitative Methods

In this paper, we describe research that combined quantitative and qualitative methods in order to investigate how students’ mathematical explanations change over time and to identify factors underlying any changes. The quantitative methods used included tracing trends in hierarchically ordered categorical data, and multi‐level analyses of student scores to identify significant predictors of students’ progress. Qualitative methods used included interviews with selected students in schools identified from the multi‐level modelling as those in which students performed significantly better than would be predicted. By reference to these analyses as applied to one geometrical item, the paper points to how the mixing of methods shed light on trends in patterns of student response as well as on unexpected results, and led to reflexive testing of some initial assumptions about the development of mathematical reasoning.

Learning Mathematics in Groups with Computers: reflections on a research study

Abstract This paper traces a methodology for analysing groupwork with computers, together with some of the findings. The theoretical underpinning of the research is outlined along with how this theory informed the design of the study. In order to make explicit the research process, reflections on the project revolve around a detailed description of how the data were analysed and the phases through which the analysis progressed. This includes how decisions were made in order to cope with large qualitative data sets, the choice of appropriate quantitative techniques, and how qualitative and quantitative methods were combined. Advantages and limitations of this analysis are considered, and finally generalisations on the role of groupwork with computers in mathematics education are explored.

Understanding and Overcoming “Disadvantage” in Learning Mathematics

Past research has largely characterized disadvantage as an individual or social condition that somehow impedes mathematics learning, which has resulted in the further marginalization of individuals whose physical, racial, ethnic, linguistic and social identities are different from normative identities constructed by dominant social groups. Recent studies have begun to avoid equating difference with deficiency and instead seek to understand mathematics learning from the perspective of those whose identities contrast the construction of normal by dominant social groups. In this way of thinking, “understanding” disadvantage can be discussed as understanding social processes that disadvantage individuals. And, “overcoming” disadvantage can be explored by analyzing how learning scenarios and teaching practices can be more finely tuned to the needs of particular groups of learners, empowering them to demonstrate abilities beyond what is generally expected by dominant discourses. In this chapter, we consider theoretical and methodological perspectives associated with the search for a more inclusive mathematics education, and how they generally share a conceptualization of the role of the teacher as an active participant in researching and interpreting their students’ learning. Drawing from examples with a diverse range of learners including linguistic, racial and ethnic minorities, as well as deaf students, blind students, and those with specific difficulties with mathematics, we argue that by understanding the learning processes of such students we may better understand mathematics learning in general.

Towards a methodology for analysing collaboration and learning in computer-based groupwork

Abstract This paper presents a methodology for researching effective groupwork within computer environments, developed as part of the Groupwork with Computers Project. The research involves eight groups of six mixed-sex, mixed-achievement pupils, undertaking research tasks using both the Logo programming language and a database program. Our aims are to identify factors influencing effective computer-based groupwork in terms of both group outcome and individual learning. Two groups working on a Logo-based task are described to focus attention on how our methods of analysis address the relationship between group processes, individual progress and group outcome, and some emerging considerations are discussed.