Mathematics Teaching in the Early Years

An Investigation of Teachers' Subject Knowledge

Carol Aubrey

• Publisher: Routledge
• Year: 2013
• eBook ISBN:
  9781136367878
• Topic: Education
• Subtopic: Education General

Young children start school already able to do a surprising amount of mathematics. This book examines the nature and origin of subject knowledge and is based on information gathered from observing the interactions between teachers and their first-year pupils. It demonstrates the necessity of the classroom teacher to draw on many kinds of knowledge in order to deal with various issues surrounding classroom learning and teaching. Two important core areas are knowledge of lesson structure and of subject matter; this book address the area of subject matter and, as such, it should be of interest to classroom teachers and lecturers in education.

1 Teachers’ Subject Knowledge: Emerging Themes

Introduction

Over the last decade in Britain and in the United States, as noted by McNamara (1991), policy-makers have been promoting teachers' knowledge of subjects and the application of this subject knowledge in the classroom as a key element in the raising of standards in teaching. As policy-makers have sought to increase the effectiveness of subject teaching through educational reform, conceptual and empirical enquiry has contributed to a changing and developing research base concerning teachers' subject knowledge. Although in the early stages of development, new lines of enquiry are generating fresh debate about the nature and substance of teachers' knowledge and the way this knowledge is transformed in classroom teaching. It was this emergent field of enquiry which provided the impetus for the project which forms the basis for this book. In spite of the concern of policy-makers, however, it is too soon to make general statements about what teachers do know or should know about subject content and about its translation into forms accessible to young children. It is still difficult, at this point in time, to characterize or synthesize these new conceptual constructs and research findings, to indicate their scope or to anticipate future trends and developments.

Before reviewing the substantive and methodological issues it is necessary to make more explicit some of the underlying assumptions, understandings and beliefs concerning the teaching of subjects which have helped to shape knowledge production and utilization, to define legitimate problems and, thus, to influence the direction of research and the interpretation of data.

Aims

The aim of this chapter is to offer, first, a brief overview of the relevant political educational context and, second, to present a brief account of some of the main philosophical and conceptual issues which have contributed to the current debate concerning teachers' subject knowledge. Chapter 2 will then examine, in more detail, changing models and methods of investigating subject teaching. This will provide a context in which to locate the aims and objectives for the present project and set the scene for the consideration of substantive issues in Chapter 3.

Teachers’ Subject Knowledge: The Political-educational Context

Of the wide-ranging organizational and practical changes for education, brought about by the Education Reform Act (1988), the introduction of a centrally directed National Curriculum is of particular relevance to the discussion of teachers' subject knowledge. This designated three core subjects and seven foundation subjects which must be taught. The core subjects were mathematics, English and science. The foundation subjects were history, geography, technology, music, art, physical education and, at secondary stage, a modern language. (Welsh was a core subject for Welsh-speaking schools and a foundation subject in non-Welsh-speaking schools in Wales.) Religious education was included as part of the 'basic' curriculum, without being subject to the particular conditions attached to the core and foundation subjects.

In the decade before the introduction of the Education Reform Act (DES, 1988) a number of policy documents was issued by successive Secretaries of State which showed an increasing interest in the curriculum, for instance, the Organisa tion and Content of the 5-16 Curriculum (DES, 1984) and Science 5-16: A Statement of Policy (DES, 1985b). Her Majesty's Inspectors (HMI) also increased their output of curriculum papers of which the Curriculum from 5-16 (DES, 1985c) stands out as an important example. Whilst the White Paper Quality Schools (DES, 1985a) stated that the Government had no intention to introduce legislation redefining responsibility for the curriculum, the move towards a centrally controlled, subject-based curriculum had begun.

The White Paper Quality Teaching (DES, 1983) can be identified as a key document which expressed a parallel concern about student teachers' subject matter knowledge. Criteria set out in a circular (DES, 1989) from the Council for the Accreditation of Teacher Education (CATE), moreover, marked a move towards a greater emphasis in training on students' acquisition of subject matter knowledge and on its role in effective teaching. Student teachers were expected to have subject expertise in one or more areas of the curriculum and to receive tuition in the application of their specialist subject, or subjects, to the teaching and assessment of pupils.

Following the introduction of the National Curriculum concern about the adequacy of school teachers' subject knowledge was still being expressed, in particular, at the primary stage. Alexander, Rose and Woodhead (1992), in a discussion document commissioned by the Government, stated:

The resistance to subjects at the primary stage is no longer tenable. The subject is a necessary feature of the modern curriculum. It requires appropriate kinds of knowledge on the part of the teacher.

(Alexander, Rose and Woodhead, 1992, Summary, para 3.2)

In the follow-up to this report the Office for Standards in Education (OFSTED) 1993 concluded:

Improvement of teachers subject knowledge was widely acknowledged as of central importance if primary schools were to make the looked-for progress with teaching the National Curriculum.

(OFSTED, 1993, para 32, p. 16)

The National Curriculum Council (1993), too, recommended a number of changes for primary schools, including the greater use of single subject teaching and of subject teachers, the provision of further guidance and training with respect to teaching methods and subject knowledge, and the revision of initial teacher training which, currently, did not ensure mastery of subject knowledge across the full range of the National Curriculum or focus sharply enough on teaching skills. More recently the call came from the Department for Education (DFEE) Circular 14/93 The Initial Training of Primary Teachers for the strengthening of subject knowledge and practical teaching skills which equipped primary student teachers to teach effectively and which were the foundation for further professional development.

The need to strengthen the subject knowledge of class teachers and student teachers has become a recurrent theme. Whilst subject knowledge has been generally accepted as a fundamental component of teachers' knowledge policy-makers and researchers alike have not, until recently, focused so much emphasis on its development. The emerging emphasis on teachers' subject knowledge indicates a fundamental shift in thinking about teaching.

It is difficult, however, to consider policy and practice without first examining the implicit themes underlying the current educational reform movement with its goal to improve the academic quality of teaching and the new research emphasis on the role of subject knowledge in teaching. The next section will attempt to examine and to analyse the theme of subject knowledge in teaching as reflected in educational thinking.

Teachers’ Subject Knowledge: The Contribution of Philosophical and Conceptual Analysis

If teaching involves helping others to learn, then understanding the subject content to be taught is a fundamental requirement of teaching. Teachers' knowledge of subject matter provides an example of an essential category of teachers' knowledge which is neither new nor controversial but one which can be separated into a variety of distinct components for examination. One contribution that the philosophical method can make to the debate on teachers' subject knowledge is to re-examine generally held concepts or assumptions which may carry systematic distinctions needing to be made explicit in order to clarify distinct components and to identify ways of reconsidering these.

Activities which are typically associated with the philosophical method include conceptual and logical analysis of argument, examining assumptions, claims, linguistic intuitions about terms, their associations and implications. No clear distinction between philosophical enquiry and other types of investigation exists and discussion which draws on philosophical methods may well draw on other forms of knowledge and enquiry in the social sciences. Furthermore the philosophical method employs logical analysis of language or concepts which underpins all forms of discussion and writing and, hence, a formal statement that it is being used may not always be made. This means, as noted by Floden and Buchmann (1990), that philosophy can make an important contribution to arguments which are not obviously philosophical or even advanced by writers who are philosophers.

Debate concerning the role of teachers subject knowledge spans the twentieth century. A classic essay of John Dewey (1904-64) argued that knowledge of teaching subjects conferred an understanding of the ways the mind worked. For Dewey knowledge of subjects equated knowledge and enquiry, and knowledge of the educational process as he conceived it. Learning subjects was about learning:

The fundamental mental attitudes and operations — that indeed, particular scientific methods and classifications simply express and illustrate in their most concrete form that of which simple and common modes of thought-activity are capable when they work, under satisfactory conditions ...

(Dewey, 1904-64, pp. 161-2)

Dewey's argument was that subject knowledge provided knowledge of teaching method and, moreover, it was a resource which could be more systematically exploited. Some people were good teachers without studying pedagogy. A conclusion which might be drawn from this was that there may be little advantage to be gained from knowledge of teaching method. His argument hinged on a distinction inherent in his theory of learning and knowledge.

People's minds are driven by problems originating in concrete experience. These problems engendered enquiry, and actions resulting from this enquiry, provided the basis for resolving them. Hence, teachers using this 'scientific' method of thought, themselves, were able to, and should be able to teach in ways that supported pupils' own practice of enquiry. For Dewey this mode of so-called scientific thinking was the appropriate model for all thought. Teaching in ways which stimulated pupils' own practice of enquiry transformed teaching into science, thus, required deep knowledge of subjects which, in turn, embodied knowledge of the educational process.

Wilson (1975), a professional philosopher of education, considered that the meaning or logic of concepts used offered guidance for teaching. In attempting to identify what being a teacher logically required he suggested that the concept of being a teacher entailed that people acquired a taxonomy of characteristics. First, the teacher must have knowledge of subject matter in a way that is most useful for the pupils' learning. Whilst this might include relevant knowledge and facts, it required:

the idea of having a clear understanding of what it is to make progress in the subject — the type of reasoning involved, its logical structure, the marks of a 'good historian' (scientist, mathematician, etc.) and so forth.

(Wilson, 1975, p. 111)

Second, teaching others also involved demonstrating a serious care about, or commitment to share that knowledge. Third, teaching, in the sense of 'getting others to learn', required additional interpersonal knowledge.

Like Dewey, Wilson believed that preparation for teaching involved getting people to know their subjects and to care about them, to be serious and conceptually alert. In other words the acquisition of these three characteristics was a matter of teachers' knowledge and, hence, education. Whilst Wilson was interested in linguistic analysis, common meanings across time for such terms as, 'knowledge' and 'education', cannot be assumed. The important point to note, however, is that the distinctions which were being made allowed consideration of important meanings that were set in long-established usage. Furthermore, such distinctions, once made explicit, provided access to tacit assumptions that the words conveyed.

Enquiry about teaching, however, is seldom purely conceptual and has often combined philosophical argument concerning meanings, concepts and logic with empirical elements. Exemplifying this approach is the work of Gage (1978; 1985) who argued for the relevance of research on teaching through a blend of philosophical argument with empirical claim. By contrast to Dewey and Wilson, Gage argued that weaknesses in teacher preparation resulted from neglect of pedagogy. Moreover, he believed that further study of school subjects was of little value:

when the teacher might already know far more about that subject than he or she will ever need in teaching third-graders (9-year-olds) or even twelfthgraders (17-year-olds).

(Gage, 1985, p. 27)

His argument incorporated a number of empirical claims, for instance, that teacher preparation was inadequate in its attention to teaching method since teachers already knew much subject matter. This carried an implicit argument in favour of increasing the importance of teaching method and assumed that teachers needed to know only the subject matter content already learned in their own schooling or acquired in relevant undergraduate study. Moreover, it assumed that sufficient subject knowledge would allow the teacher to deliver the subject effectively and enable pupils to learn it. More convincing, however, was Gage's assertion that teachers needed to learn teaching method as well as subject knowledge, such as structuring coursework, planning a lesson and interacting appropriately with children. Past failure to address teaching technique adequately was attributable, in part, he asserted, to the lack of an appropriate research base. Teaching, however, was more than the appreciation of technique and, in fact, what Gage was advocating was instruction in empirically based skills, united to a view of teaching which included flexibility, judgment and intuition, with a recognition that teaching could not be reduced to mere technical formulae.

Research results, thus, provided a constructive starting point for Gage rather than a prescription for practice. The limitation of this argument lies in its assumption of the existence of a research base to provide or to determine worthwhile or valued kinds of pupil learning. Moreover, there is an additional assumption that the implementation of such teaching skills will lead inevitably to effective learning and that effective teaching, in terms of attainment outcomes, is necessarily good, or desirable teaching. Teaching is, thus, judged worthwhile by Gage in terms of gains on achievement tests. This view however, promotes a narrow and instrumental view of teaching which, it might be argued, distracts attention away from other, and equally important educational goals.

Buchmann (1982; 1984), too, used a combination of analysis of meaning and logic with reference made to empirical, educational research. It is significant to note, at this point, however, that neither Gage nor Buchmann based these claims about teacher preparation on investigations of actual teacher preparation programmes. By way of argument, in contrast, whilst Gage argued that teachers needed more instruction in teaching techniques because they already knew more than they needed to know about the subject they would teach, Buchmann argued that subject matter knowledge had been neglected and that professional preparation of teachers placed too much emphasis on classroom technique.

The fundamental basis for the disparity of views between Gage and Buchmann is revealed in Buchmann's discussion of what and how much teachers need to know about subject matter. Buchmann proceeded by a conceptual argument for the priority of subject matter knowledge. She pointed out that teaching ...

Table of contents

• Cover
• Dedication
• Title
• Copyright
• Contents
• List of Tables
• List of Figures
• Acknowledgments
• Summary
• 1 Teachers' Subject Knowledge: Emerging Themes
• 2 Changing Models and Methods of Investigating Teachers' Subject Knowledge
• 3 Knowing and Understanding Mathematics: Concerning a Theory of Instruction
• 4 The Construction and Early Learning of Mathematics in School and Out
• 5 Investigating the Mathematical Knowledge and Competences Which Young Children Bring into School
• 6 Reporting on Teachers' Classroom Practice
• 7 Teacher and Pupil Interactions in the Course of Mathematical Instruction
• 8 Teacher and Pupil Mathematical Subject Knowledge and the Processes of Instruction in Reception Classes
• 9 Towards a Deeper Understanding of Pedagogical Subject Development
• Appendix 1: Extracts from Mathematical Assessment Tasks
• Appendix 2: Copy of Field Notes Schedule
• Appendix 3: Copy of Teacher Interview Schedule for Phase Four
• Appendix 4: Extracts from Teacher Interviews
• References
• Index