Seeking authenticity in high stakes mathematics assessment

This article derives from a scrutiny of over 100 national secondary mathematics examination papers in England, conducted as part of the Evaluating Mathematics Pathways project 2007-2010 by a team of eight researchers. The focus in this article is of the extent to which mathematics assessment items reflect and represent the current curriculum drive for increased mathematical applications in the curriculum. We show that whilst mathematics is represented as a human activity in the examinations, peopling assessment items may serve actually only to disguise the routinised calculations and procedural reasoning that largely remains the focus of the assessments, with the effect that classroom mathematics remains unchanged. We suggest that there are more opportunities for assessment items to illustrate mathematics in use, and we draw attention to ways of assessing mathematics that allow these opportunities to be taken

Widening and increasing post-16 mathematics participation: pathways, pedagogies and politics

This paper explores the potential impact of a national pilot initiative in England aimed at increasing and widening participation in advanced mathematical study through the creation of a new qualification for 16 to 18 year-olds. This proposed qualification pathway - Use of Mathematics - sits in parallel with long-established, traditional advanced level qualifications; what we call ‘traditional Mathematics’ herein. Traditional Mathematics is typically required for entry to mathematically demanding undergraduate programmes. The structure, pedagogy and assessment of Use of Mathematics is designed to better prepare students in the application of mathematics and its development has surfaced some of the tensions between academic/pure and vocational/applied mathematics. Here we explore what Use of Mathematics offers but we also consider some of the objections to its introduction in order to explore aspects of the knowledge-politics of mathematics education. Our evaluation of this curriculum innovation raises important issues for the mathematics education community as countries seek to increase the numbers of people that are well-prepared to apply mathematics in science and technology-based higher education courses and work places

Mathematical knowledge for teaching probem solving: lessons from lesson study

Although the importance of mathematical problem solving is now widely recognised, relatively little attention has been given to the conceptualisation of mathematical processes such as representing, analysing, interpreting and communicating. The construct of Mathematical Knowledge for Teaching (Hill, Ball & Schilling, 2008) is generally interpreted in terms of mathematical content, and in this paper we describe our initial attempts to broaden MKT to include mathematical process knowledge (MPK) and pedagogical process knowledge (PPK). We draw on data from a problem-solving-focused lesson-study project to highlight and exemplify aspects of the teachers’ PPK and the implications of this for our developing conceptualisation of the mathematical knowledge needed for teaching problem solving

Mathematical knowledge for teaching probem solving: lessons from lesson study

Although the importance of mathematical problem solving is now widely recognised, relatively little attention has been given to the conceptualisation of mathematical processes such as representing, analysing, interpreting and communicating. The construct of Mathematical Knowledge for Teaching (Hill, Ball & Schilling, 2008) is generally interpreted in terms of mathematical content, and in this paper we describe our initial attempts to broaden MKT to include mathematical process knowledge (MPK) and pedagogical process knowledge (PPK). We draw on data from a problem-solving-focused lesson-study project to highlight and exemplify aspects of the teachers’ PPK and the implications of this for our developing conceptualisation of the mathematical knowledge needed for teaching problem solving

Professional learning through the collaborative design of problem-solving lessons

This article analyses lesson study as a mode of professional learning, focused on the development of mathematical problem solving processes, using the lens of cultural-historical activity theory. In particular, we draw attention to two activity systems, the classroom system and the lesson-study system, and the importance of making artefacts instrumental in both. We conceptualise the lesson plan as a boundary object and use this to illustrate how professional learning takes place through the introduction of carefully designed artefacts that draw on teachers’ professional knowledge of potential student approaches, and to the nature of progression in problem-solving processes. We identify the roles of instrumentalisation and instrumentation in supporting professional learning as these artefacts are prepared for use before a lesson and as they are again used as catalysts for reflection in post-lesson discussions. These artefacts are seen to effectively facilitate the socially situated learning of all participants. We conclude that the design of artefacts as boundary objects that support teaching and professional learning in their respective activity systems may be fundamental to the success of lesson study as a collaborative venture

Mathematics students’ aspirations for higher education: class, ethnicity, gender and interpretative repertoire styles

This paper reports how students talk about their aspirations in regard to higher education (HE) and their mathematics, what ‘repertoires’ they use to mediate this discourse, and how students’ predominant ‘repertoire style’ relates to their cultural background. Our analyses draw on an interview sample (n=40) of students selected because they are ‘on the cusp’ of participation or non‐participation in mathematically demanding programmes in further and higher education. The interviews explored the students’ aspirations for their future in general and HE in particular, influences on these choices, and the place of mathematics in these. Thematic analysis revealed four interpretative repertoires commonly in use, which we call ‘becoming successful’, ‘personal satisfaction’, ‘vocational’, and ‘idealist’ repertoires. Most of the sample was found to use a single, predominant repertoire, which we call their repertoire ‘style’: what is more, this style is found to be strongly related to background factors independently obtained. The implications for policy and practice are discussed