Candia Morgan

The Interpretative Nature of Teachers' Assessment of Students' Mathematics: Issues for Equity

This paper discusses fairness and equity in assessment of mathematics. The increased importance of teachers’ interpretative judgments of students’ performance in highstakes assessments and in the classroom has prompted this exploration. Following a substantial theoretical overview of the field, the issues are illustrated by two studies that took place in the context of a reformed mathematics curriculum in England. One study is of teachers’ informal classroom assessment practices; the other is of their interpretation and evaluation of students’ formal written mathematical texts (i.e., responses to mathematics problems). Results from both studies found that broadly similar work could be interpreted differently by different teachers. The formation of teachers’ views of students and evaluation of their mathematical attainments appeared to be influenced by surface features of students’ work and behavior and by individual teachers’ prior expectations. We discuss and critique some approaches to improving the quality and equity of teachers’ assessment.

Word, Definitions and Concepts in Discourses of Mathematics, Teaching and Learning

National Numeracy Strategy (NNS) guidance appears to characterise mathematical language as a set of specialist words with unambiguous definitions, yet analysis of the classroom transcript suggests that at least some mathematical concepts cannot be captured by such definitions. This paper explores the notion of definition within mathematics, considering both school mathematics and the field of academic mathematics research. Extracts of texts from these domains are analysed using tools drawn from systemic functional linguistics. The more advanced texts show definitions to be constructed and used in creative and purposeful ways. This contrasts with the one-to-one word-concept relationship apparent in the NNS guidance and in a text for ‘Intermediate Level’ secondary students. This finding raises the question of the extent to which the linguistic models implicit in the texts for less advanced school students and their teachers allow students to learn more powerful aspects of mathematics.

Teacher as examiner: The case of mathematics coursework

Abstract Recent developments in curriculum and assessment in the UK have led to increased involvement of teachers in high‐stakes summative assessment of their own students. Case studies of experienced secondary school mathematics teachers reading and assessing General Certificate of Secondary Education (GCSE) coursework texts show that they experience tensions between their various roles and responsibilities as teachers and as examiners. Moreover, different teachers appear to resolve these tensions in different ways, adopting various positions in relation both to the content of the written coursework texts and to the student‐authors of the texts. Variations in teachers’ approaches to reading and assessing mathematics coursework may lead to differences in the ranks or grades allocated and, even where they do not, the meanings of the grades given by different teachers may be substantially different. Through examining teachers’ assessment practices, questions are raised both about the validity of such assess...

MATHEMATICS AND HUMAN ACTIVITY: REPRESENTATION IN MATHEMATICAL WRITING

The prevailing image of mathematical writing is formal and impersonal, reflecting an absolutist view of the nature of mathematics. This paper challenges this image. Analysis of texts written by research mathematicians shows that they include different kinds of representations of the human activity of doing mathematics. Genres of school mathematics writing are discussed in the light of this analysis and it is suggested that knowledge of the ways in which mathematicians include their problem solving activity in their writing might be used to support students as they learn to write reports of their own investigative work.

Applied Linguistics and Mathematics Education: More than Words and Numbers

The preceding set of papers has explored various aspects of the role of language in mathematics education. The papers reflect the work of individual contributors. An important part of our collaboration, however, has been the conversation between us. This paper reflects on aspects of that conversation, as we draw together some of the themes that have emerged during our work. In particular, we discuss some of the implications of our analyses for theory, policy, practice and inter-disciplinarity in mathematics education and applied linguistics.

Communicating Experience of 3D Space: Mathematical and Everyday Discourse

In this article we consider data arising from student-teacher-researcher interactions taking place in the context of an experimental teaching program making use of multiple modes of communication and representation to explore three-dimensional (3D) shape. As teachers/researchers attempted to support student use of a logo-like formal language for constructing 3D trajectories and figures in a computer microworld, a system of gestures emerged. Observations of multimodal classroom communication suggest that teachers/researchers and students used similar words and gestures to represent different types of movement. We discuss possible sources of these differences, contrasting formal mathematical and everyday systems of representation of 3D space. More generally, we argue that understanding the structures of everyday discourse and their relationships to the structures of specialized mathematical discourse can provide insight into student interactions.

Representation in Computational Environments: Epistemological and Social Distance

Computational environments have the potential to provide new representational resources and new ways of supporting teaching and learning of mathematics. In this paper, we seek to characterize relationships between the representations offered by particular technologies and other representations commonly available in the classroom context, using the notion of ‘distance’. Distance between representations in different media may be epistemological, affecting the nature of the mathematical concepts available to students, or may be social, affecting pedagogic relationships in the classroom and the ease with which the technology may be adopted in particular classroom or national contexts. We illustrate these notions through examples taken from cross-experimentation of computational environments in national contexts different from those in which they were developed. Implications for the design and dissemination of computational environments for use in learning mathematics are discussed.

Making sense of curriculum innovation and mathematics teacher identity

In this chapter, professional identity is explored in relation to the problem of how official curriculum frameworks organise, influence and shape professional concepts, choices and processes through their texts and other resources. The Key Stage 3 National Strategy for England, a framework of curriculum guidance and professional development for lower secondary mathematics teachers, is taken as an example. The way in which the guidance constructs the nature of mathematics teaching and the professional identity of the mathematics teacher is analyzed. The chapter discusses how alternative and even contradictory mathematics teaching approaches can be brought into accommodation with the official framework, and suggests that the successful implementation of the National Strategy is strongly related to its openness to alternative readings, incorporating potentially oppositional voices. Implications for the nature of professional curriculum practice and development are drawn.

Studying Discourse Implies Studying Equity

Understanding discourse as the linguistic moment of a social practice entails that studying discourse involves studying the nature of the social practice and the social structures within which it functions as well as studying communicative interactions. Issues of equity inevitably arise because discourse involves the production and maintenance of relationships and worldviews. This chapter suggests that more substantial attention at the level of social structure would enhance our understanding of the reasons underpinning particular classroom practices and text level characteristics as well as our ability to address the question of who is advantaged or disadvantaged by particular forms of discursive and pedagogic practice. Bernstein’s theory of pedagogic discourse is proposed as one approach to make connections between interactions, practices and social structure. Discussion of pedagogy in multilingual classrooms highlights the ways that language practices contribute to the exercise of power.

Investigating changes in high-stakes mathematics examinations: a discursive approach

ABSTRACT This article focuses on the theoretical-methodological question of how to identify reform-induced changes in school mathematics. The issue arose in our project The Evolution of the Discourse of School Mathematics (EDSM), in which we studied transformations in high-stakes examinations taken by students in England at the end of compulsory schooling. We have adopted a conceptualisation that draws on social semiotics and on a communicational approach, according to which school mathematics can be thought of as a discourse. Methods of comparing examinations of different years developed on the basis of this definition enable identification of subtle disparities that are nevertheless significant enough to make an important difference in students’ vision of mathematics, in their performance and, eventually, in their ability to cope with problems that can benefit from the use of mathematics. In this article, we present these methods and argue that they have wider application for comparative studies of school mathematics.