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.
The value of any subject on a crowded, contested and compulsory curriculum is, or at least should be, open to debate. More importantly, when different subjects are prioritized over others, the justification for the ranking of such subjects should also be submitted to sustained enquiry. Mathematics enjoys a prestigious place in the English National Curriculum, elsewhere, and perhaps everywhere. There have been several responses, over the last few years, to the question of not only why pupils should be compelled to learn mathematics but also for how long this should continue. This article takes a broad and critical look, from a largely philosophical standpoint, at some of the dominant themes that underlie the most recurrent claims that have been made in defence of the privileged place of mathematics in school.
This paper draws on videotapes of mathematics lessons prepared and conducted by pre-service elementary teachers towards the end of their initial training at one university. The aim was to locate ways in which they drew on their knowledge of mathematics and mathematics pedagogy in their teaching. A grounded approach to data analysis led to the identification of a ‘knowledge quartet’, with four broad dimensions, or ‘units’, through which mathematics-related knowledge of these beginning teachers could be observed in practice. We term the four units: foundation, transformation, connection and contingency. This paper describes how each of these units is characterised and analyses one of the videotaped lessons, showing how each dimension of the quartet can be identified in the lesson. We claim that the quartet can be used as a framework for lesson observation and for mathematics teaching development.
This paper is a continuation and extension of ongoing work, investigating the mathematics subject matter knowledge of prospective primary school teachers, which has recently become a high profile issue in the UK and elsewhere. In earlier work, a grounded approach to data analysis led to the identification of a knowledge 'quartet', with four broad dimensions which we are calling foundation, transformation, connection and contingency. Subject knowledge could be evidenced in practice through these dimensions. This most recent development builds upon the original work. It provides a case study of a trainee who has been invited to reflect upon her lesson a few hours after teaching it. In talking freely about her lesson and also by responding to an interviewer who had previously viewed and analysed a video recording of the lesson, further insights were gained.
In this paper we describe a framework for the identification and discussion of prospective elementary school teachers' mathematics content knowledge as evidenced in their teaching. This framework ‘the knowledge quartet’ emerged from intensive scrutiny of 24 videotaped lessons. Application of the ‘quartet’ in lesson observation is illustrated with reference to a particular lesson taught by one trainee teacher.
We begin by asking a simple question: To what extent can art education be related to mathematics education? One reason for asking this is that there is, on the one hand, a significant body of claims that assert that mathematics is an art, and, on the other, work in art that has a mathematical basis. Observations of these kinds are not trivial. They have significant implications for the teaching of these areas of the curriculum in at least two ways. First, there is the methodological issue of the extent to which we should teach mathematics and art separately, and second, the teleological question of why they appear in the curriculum at all. So the relationship between the nature of mathematics and art, perceived or real, bears down on questions of the individuation and the justification of these disciplines, or, in other words, upon pedagogy and purpose. Although in principle both the pedagogy and the purpose of any discipline are distinct, there are important connections between them, as we shall draw out.
Creativity in education has traditionally had its home in the arts, yet descriptions of mathematics lessons in terms of ‘creativity’, and its cognates, often creep into the discourse of many educationalists. Sometimes these descriptions apply only to the kind of teaching involved. Recently, for example, one group of authors advise trainee teachers that they ‘... need to exercise ... professional skills to make constructive, creative use of [their] time while still adhering to the good practice principles of direct, interactive teaching in mental, oral and written mathematics’ (Mooney et al., 2001, p. 21, my emphasis). However, it is often the nature of children’s mathematical activity itself that is described in this way. Murray (2000) outlines the way in which a young pupil, who initially seems to believe that multiplication terminates at 10, is led by her teacher to extend multiples of 5 to 13 3 5. The pupil gures this out by assembling her knowledge of 10 3 5 and 3 3 5. It may seem innocuous to describe the deriving of a fact, in the way that Murray does, as the ‘creation’ of ‘new knowledge’. Yet, even if it is more succinct to call what is variously derived, extended, gured out or assembled a ‘creation’, it certainly adds no additional explanatory force. There is something even more strained in another description of a familiar mathematical activity as ‘creative arithmetic’ (Craft, 2000, p. 77). Clearly, there is no irony intended as there is in the expression ‘creative accountancy’. In seeking and recording different pairs of numbers that sum to a given total, pupils are not suspected of cooking the books. What is suggested, however, is that making totals is tantamount to making mathematics. Craft is not alone in this. Rousham (1995) gives an example of what he calls ‘creative activity’ with calculators. This ‘essentially ... consists of saying to the class’, he writes, ‘something like: “The answer is 36; what is the question?’ ” so that they are challenged to ‘make’ 36 but in as many different possible ways as they can’ (pp. 98–99). Rousham is careful to place ‘make’ in snigger quotes to indicate that he is not committed to its usual sense; nonetheless he still acknowledges it as a link in his description. In the hands of another group of writers this ‘creativity-in-the-making’ is
The mathematics subject matter knowledge of primary school teachers has in recent years become a high profile issue in the UK and beyond. There is statistical evidence (e.g. Rowland, Martyn, Barber and Heal, 2001) that secure knowledge of mathematics topics including and extending beyond those encountered in the primary curriculum is associated with more competent mathematics teaching in the case of pre-service (‘trainee’) primary teachers. Likewise, weak subject knowledge is associated with less competent teaching of the subject. This paper reports on a videotape study of 25 mathematics lessons prepared and conducted by trainee teachers. The aim was to identify ways in which their subject knowledge, or the lack of it, was evident in their teaching.
Whenever one justification for learning mathematics is questioned, educationists have usually been ready to offer an alternative rationale. Often this involves presenting (or re-presenting) the subject in a new light. This paper will consider the implications of trying to invoke creativity in mathematics. We shall argue that 'creativity' is a complex notion and should only be applied to mathematics with caution and attention to meaning.
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.
