Tim Rowland

Mathematical Knowledge in Teaching

Cumulative research within a number of traditions has shown that effective teaching calls for distinctive, identifiable forms of subject-related knowledge and thinking, yet the significance and complexity of such knowledge is not well represented in professional debate and policymaking. This is a particularly pressing issue within mathematics education, given world-wide aspirations to improve quality of teaching and learning in the face of widespread difficulties in recruiting teachers who are conventionally well-qualified in mathematics and confident in the subject. This book, the outcome of two years of collaborative effort, brings together a team of experts in the field of mathematics teacher knowledge to produce an authoritative, ‘state of the art’ exposition and critical commentary on this important and topical domain, including reports of original research in the field. It offers constructive and helpful ways of conceptualising mathematics teacher knowledge in its cultural context, as well as a range of theorised tools to support its improvement.

Developing primary mathematics teaching : reflecting on practice with the Knowledge Quartet

This book helps readers to become better, more confident teachers of mathematics by enabling them to focus critically on what they know and what they do in the classroom. Building on their close observation of primary mathematics classrooms, the authors provide those starting out in the teaching profession with a four-stage framework which acts as a tool of support for developing their teaching: -Making sense of foundation knowledge – focusing on what teachers know about mathematics -Transforming knowledge – representing mathematics to learners through examples, analogies, illustrations, and demonstrations -Connection – helping learners to make sense of mathematics through understanding how ideas and concepts are linked to each other -Contingency – what to do when the unexpected happens Each chapter includes practical activities, lesson descriptions, and extracts of classroom transcripts to help teachers reflect on effective practice. Video versions of these lessons are also available on a companion website.

PRIMARY TEACHER TRAINEES’ MATHEMATICS SUBJECT KNOWLEDGE AND CLASSROOM PERFORMANCE

UK government agencies have recently set in train requirements to ‘crack down’ on primary [elementary] school teacher trainees whose own knowledge of mathematics is weak. The responsibility to identify and support (or fail) them currently rests with training providers (mainly university schools of education). We describe one approach to this process, presenting some findings concerning what trainees find difficult and how their knowledge is related to their teaching competence. We flesh out these findings with a case study of a mathematically-strong trainee whose path to qualification was less than smooth.

Hedges in Mathematics Talk: Linguistic Pointers to Uncertainty.

Analysis of transcripts of interviews with children aged 10 to 12, focused on a mathematical task designed to provoke prediction and generalisation, reveals a category of words (called hedges) associated with uncertainty. It is argued that these words — examples includeabout, around, maybe, think — are frequently deployed as a ‘Shield’ against accusation of error. The analysis draws on linguistic frameworks for categorising types of hedge, and for a theoretical account of how they might succeed in conveying uncertainty to listeners.

Contingency in the Mathematics Classroom: Opportunities Taken and Opportunities Missed

We describe and analyze three episodes from mathematics classrooms. In each case, the teacher was confronted by a “contingent” situation that they had not anticipated or planned for yet that offered interesting and fruitful learning possibilities if pursued. In two cases, we analyze the teacher’s response; in the third, we speculate how they might have responded. In each case, we propose that the teacher’s ability to capitalize on these contingent situations is underpinned by their knowledge and awareness of the mathematical potential of the unexpected opportunity and by an interest in, and commitment to, mathematical enquiry.RésuméNous présentons une description et une analyse de trois situations provenant de cours de mathématiques. Dans chacun des cas, l’enseignant a dû faire face à un événement « imprévu », donc une situation qui ne faisait pas partie du programme de la leçon, mais qui offrait des possibilités intéressantes à exploiter pour l’apprentissage. Pour deux de ces cas, nous analysons la réaction de l’enseignant, et pour le troisième cas nous imaginons comment l’enseignant aurait pu réagir. Dans les trois cas, nous estimons que sa capacité de tirer profit de telles situations dépend de son habileté à reconnâıtre le potentiel mathématique des occasions imprévues, et de son intérêt et de sa curiosité pour l’investigation mathématique.

EXAMPLES, GENERALISATION AND PROOF

The interplay between generalisations and particular instances—examples—is an essential feature of mathematics teaching and learning. In this paper, we bring together our experiences of personal an...

The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers’ Mathematics Knowledge

In this chapter we present some findings from a study which evaluated the effectiveness of one classroom-based approach to the development of elementary mathematics teaching. This approach drew on earlier research into teachers’ mathematical content knowledge at the University of Cambridge, when a framework for the analysis of mathematics teaching - the Knowledge Quartet - was developed. The chapter begins with a rationale for our focus on teachers’ content knowledge in action in the classroom and a brief description of the study which led to the development of the Knowledge Quartet. It then proceeds to a report of the longitudinal study in which this framework was used to identify and develop a group of beginning teachers’ mathematics content knowledge for teaching.

‘Well Maybe Not Exactly, but It’s Around Fifty Basically?’: Vague Language in Mathematics Classrooms

It may come as something of a surprise to find a mathematician (albeit in the guise of a mathematics educator) writing about vagueness, since it is commonly supposed that precision is the hallmark of mathematics. Such a point of view is reflected in the landmark 1982 Report of the Committee of Inquiry into the Teaching of Mathematics in Schools (the Cockcroft Report), which asserted that, ‘mathematics provides a means of communication which is powerful, concise and unambiguous’ (Department of Education and Science 1982, p. 1), and proposed the communicative power of mathematics as a ‘principal reason’ for teaching it. There was refreshing novelty in such a claim, which seemed to be justifying the place of mathematics in the curriculum in much the same way that one might justify the learning of a foreign language, and it did much to promote and sustain interest in the place of language in the teaching and learning of mathematics. Such a view of mathematics is in contrast, however, with that expressed in a contemporary pamphlet issued by the Association of Teachers of Mathematics (ATM 1980, pp. 17–18), whose authors argued that: Because it is a tolerant medium, everyday language is necessarily ambiguous. /…/ Now, mathematising is also a form of action in the world. And its expressions, however carefully defined, have to retain a fundamental tolerance /…/ Because it is a tolerant medium, mathematics is also necessarily an ambiguous one.

Analysing secondary mathematics teaching with the Knowledge Quartet

This paper describes how the Knowledge Quartet (KQ), which was developed with mathematics teachers in primary schools, has been tested in a secondary mathematics context. Aspects of this research are illustrated with reference to a lesson on completing the square. First we exemplify the mapping of episodes in the lesson to the KQ, then we report how one of these episodes, concerning the choice of examples, was subsequently used in a secondary mathematics PGCE teaching session.

`Creative mathematics' – real or rhetoric?

The notion of creativity has its natural home in the fine arts, where the artist literally creates something that can be perceived by the senses. The products of mathematical activity are clearly not of this kind, yet some distinguished mathematicians have claimed that mathematics offers considerable scope for creativity. The title of the book under review, and some claims to be found in it, suggest that creativity can indeed be associated with mathematics, and that young children may experience it in the classroom. We suggest that the word ‘creative’ is being used in rather different senses in these different contexts, yet the meanings associated with the arts, say, are in danger of being applied to mathematical situations for rhetorical purposes.