Malcolm Swan

Challenging minds? Students’ perceptions of computer-based World Class Tests of problem solving

World Class Arena is a British government initiative to assess and develop the skills of gifted and talented children. Part of the strategy is to use computer-based tests. Students attempt tasks that require them to engage in higher-order problem solving, often in interactive, realistic, contexts. This study reports observations and interviews in schools. Students found tasks engaging and motivating, despite the unfamiliarity of the problem types and the challenging nature of the items. Students had no problems working with computers. They were sometimes distracted by attractive graphics, and sometimes used poor heuristics when attempting tasks. The study provides evidence that a computer environment can provide new ways to assess the problem solving skills of highly able students.

The impact of a professional development programme on the practices and beliefs of numeracy teachers

This article describes some outcomes of a nine‐month design‐based research study into the professional development of 24 numeracy teachers with post‐16 learners. Teachers analysed research‐based principles for teaching, and engaged in a design‐research process by testing and refining teaching activities to embody these principles. Data from questionnaires, interviews and classroom observations suggest that many of the teachers’ practices and beliefs were profoundly affected. Practices became less transmission‐oriented and teachers began to create new, collaborative learning environments where students were challenged to confront difficulties and take on more active classroom roles. Changes in beliefs regarding mathematics, learning and teaching are also described and discussed.

The Roles of Modelling in Learning Mathematics

This chapter illustrates how Applications and Modelling promote the learning of mathematics by developing the student’s mathematical language and her use of tools, and by developing the learner’s capacity to ask and answer questions in, with, and about mathematics.

Assessing Mathematical Thinking Via FLAG

Teachers of undergraduate mathematics face a range of problems which include an increasing number of less well qualified students, and increasing academic diversity in the student population. Students face courses which are radically different from mathematics courses encountered in school; they often face assessment systems which are ill-aligned to course goals, and which use assessment methods likely to encourage a surface rather than a deep approach to learning. The paper describes materials developed by the MARS group for the US National Institute for Science Education for use on their FLAG Web site. Collections of assessment materials can be downloaded free of charge which assess a range of valuable process skills in mathematics — proof, reasoning from evidence, estimation, creating measures, and fault finding and remediation. They are designed to offer a wide repertoire of assessment styles, as part of a process of encouraging a broadening of teaching and learning styles.

Teachers’ attempts to integrate research‐based principles into the teaching of numeracy with post‐16 learners

This paper describes some outcomes of a nine‐month design‐based research study into the professional development of 24 numeracy teachers with post‐16 learners. The teachers were encouraged to integrate eight research‐based teaching principles into their classroom practices as they implemented a set of discussion‐based mathematics learning resources. A mixed methods approach was used to evaluate the outcomes, including interviews and classroom observation. The paper examines teachers’ perceptions of the project, the reasons they found some pedagogical principles more difficult to incorporate than others, and the factors that enabled and impeded their use of the learning resources. In particular, it is noted that the principles that teachers considered to be most important were not the ones that they were observed using most effectively. The paper concludes by considering the implications of the research for initial and continuing teacher education.

Designing Tasks that Challenge Values, Beliefs and Practices: A Model for the Professional Development of Practicing Teachers

In this chapter I describe a model for the professional development of practicing secondary and adult education mathematics teachers. In particular, I describe specific tasks that challenge their existing beliefs and practices. These tasks mirror the kinds of task we expect teachers to use with students; they expose existing ways of thinking, produce tension and discussion by observing contrasting practices, and learning is accommodated by reflecting on practical classroom experiences.

Improving the Design and Balance of Mathematical Assessment

In recent years, considerable attention has been paid to improving the range and balance of activities that take place in mathematics classrooms. More emphasis is being given to the tackling of practical, extended tasks, which develop higher order skills such as the ability to formulate a question in mathematical terms, select appropriate techniques, and interpret and communicate mathematical results. These aspects are now widely recognized as important and are described in documents around the world (MSEB, 1989, 1990; NCTM, 1989; CITMS, 1982; DES, 1989; NCC, 1989).

Improving the Alignment Between Values, Principles and Classroom Realities

The curricular reforms described in this book are wide-ranging and are driven by many external factors and value systems. They usually begin with a vision of ‘how things should be’, but as we have seen, their implementation is often a travesty of their aims. In this chapter I begin with a synthesis of the values exhibited in curricula across the world, then go on to analyse the kinds of classroom activity that are implied when these are taken seriously. This process will be illustrated through a specific case—a national consultation in England that attempted to elicit, prioritise and exemplify apparently competing values held by mathematics educators. I argue that the misalignment of the intended and enacted curriculum is at least partly due to the almost universal lack of vivid exemplification in curriculum specifications and consequent reductive interpretations of them by their users. An argument is thus made for a serious systematic design-research effort into the production of beautiful examples that illustrate and effectively communicate our core values to the key educational stakeholders.

Formative Assessment Lessons

Formative assessment is the process by which teachers and students gather evidence of learning and then use this to adapt the way that they teach and learn in the classroom. In this paper, we describe a design-research project in which we integrated formative assessment into mathematics classroom materials. We outline two examples of formative assessment lessons, one concept-based and the other problem-solving, highlighting the important roles within them of pre-assessment, formative feedback questions, and sample work for students to critique.

Design and Development for Large-Scale Improvement

This chapter describes the Shell Centre team’s “engineering research” approach to the improvement of practice through researched-based design and development of tools for teaching and learning mathematics, for professional development and for supporting large-scale change. The contributions of projects over the past 35 years to the development of design principles and tactics are outlined and illustrated. The roles of tasks of different kinds in learning and assessment are explained, with particular reference to the design of tests, and of formative assessment lessons for concept-development and problem solving. The chapter concludes with a look at the barriers to turning success at classroom level into large-scale change—and how this challenge can be tackled.