The quality of primary and secondary school mathematics teaching is generally agreed to depend crucially on the subject-related knowledge of the teacher. However, there is increasing recognition that effective teaching calls for distinctive forms of subject-related knowledge and thinking. Thus, established ways of conceptualizing, developing and assessing mathematical knowledge for teaching may be less than adequate. These are important issues for policy and practice because of longstanding difficulties in recruiting teachers who are confident and conventionally well-qualified in mathematics, and because of rising concern that teaching of the subject has not adapted sufficiently. The issues to be examined in Mathematical Knowledge in Teaching are of considerable significance in addressing global aspirations to raise standards of teaching and learning in mathematics by developing more effective approaches to characterizing, assessing and developing mathematical knowledge for teaching.

Now in its fourth edition, the bestselling text Mathematical Knowledge for Primary Teachers provides trainee teachers with clear information about the fundamental mathematical ideas taught in primary schools. With rigorous and comprehensive coverage of all the mathematical knowledge primary teachers need, the text goes beyond rules and routines to help readers deepen their understanding of mathematical ideas and increase their confidence in teaching these ideas. Fully updated to incorporate recommendations of the Williams review, new sections are included covering talk for learning in mathematics, with an emphasis placed on the language and vocabulary used in arithmetic contexts. Throughout the book, knowledge is linked to the TDA standards for Qualified Teacher Status, and features include: ‘Check’ questions to test the reader’s understanding ‘Challenges’, to increase teachers’ confidence and stretch their mathematical abilities ‘Links with the classroom’ to emphasise the relevance of ideas to the classroom context Straightforward coverage from theory to practice for all aspects of the Mathematics framework. The book is accompanied by e-resources, which contain further visual activities and support, designed to scaffold and support the reader’s own understanding. Essential reading for all practising and trainee primary teachers, this book is ideal for those who wish to increase their mathematical understanding and confidence in presenting mathematics in the classroom.

In the first BACOMET volume different perspectives on issues concerning teacher education in mathematics were presented (B. Christiansen, A. G. Howson and M. Otte, Perspectives on Mathematics Education, Reidel, Dordrecht, 1986). Underlying all of them was the fundamental problem area of the relationships between mathematical knowledge and the teaching and learning processes. The subsequent project BACOMET 2, whose outcomes are presented in this book, continued this work, especially by focusing on the genesis of mathematical knowledge in the classroom. The book developed over the period 1985-9 through several meetings, much discussion and considerable writing and redrafting. Our major concern was to try to analyse what we considered to be the most significant aspects of the relationships in order to enable mathematics educators to be better able to handle the kinds of complex issues facing all mathematics educators as we approach the end of the twentieth century. With access to mathematics education widening all the time, with a multi tude of new materials and resources being available each year, with complex cultural and social interactions creating a fluctuating context of education, with all manner of technology becoming more and more significant, and with both informal education (through media of different kinds) and non formal education (courses of training etc. ) growing apace, the nature of formal mathematical education is increasingly needing analysis.

A perennial discussion about teacher development is the optimal content background for teachers. In recent years, that discussion has taken center stage in the work of mathematics education researchers, mathematicians, mathematics professional developers, and mathematics education policymakers. Much of the existing and prior work in this area has been directed toward mathematical knowledge for teaching at the elementary level. The work described in this volume takes a sometimes-neglected approach, focusing on the dynamic nature of mathematical understanding rather than on a stable description of mathematical knowledge, and on mathematics for secondary teaching rather than mathematics for teaching at the elementary level. The work reported in Mathematical Understanding for Secondary Teaching: A Framework and Classroom-Based Situations is a practice-based response to the question of what mathematical understandings secondary teachers could productively use in their teaching. For each of more than 50 events, our team of almost 50 mathematics educators who were experienced mathematics teachers developed descriptions of the mathematics that teachers could use-each of those descriptions (consisting of the event and the mathematics related to the event) is what we call a Situation. We developed our Framework for Mathematical Understanding for Secondary Teaching (MUST) based on an analysis of our entire set of Situations. We call the work practice-based because the MUST framework is based on actual events that we witnessed in our observations of secondary mathematics practice. Groups of mathematics teachers can use this volume to enhance their own understandings of secondary mathematics. School leaders and professional developers in secondary mathematics will find our MUST Framework and Situations useful as they work with teachers in enhancing and deepening their understanding of secondary mathematics. Mathematics teacher educators and mathematicians who teach mathematics to prospective and in-service secondary teachers will be able to couch their mathematical discussions in the Situations-examples that arise from secondary mathematics classrooms. They will be able to use this volume as they design courses and programs that enhance mathematics from the perspectives identified in the MUST framework. Policymakers and researchers can use our MUST framework as they consider the mathematics background needed by teachers.

Now is a time of great interest in mathematics education. Student performance, curriculum, and teacher education are the subjects of much scrutiny and debate. Studies on the mathematical knowledge of prospective and practicing U. S. teachers suggest ways to improve their mathematical educations. It is often assumed that because the topics covered in K-12 mathematics are so basic, they should be easy to teach. However, research in mathematics education has shown that to teach well,substantial mathematical understanding is necessary--even to teach whole-number arithmetic. Prospective teachers need a solid understanding of mathematics so that they can teach it as a coherent, reasoned activity and communicate its elegance and power. This volume gathers and reports current thinkingon curriculum and policy issues affecting the mathematical education of teachers. It considers two general themes: (1) the intellectual substance in school mathematics; and (2) the special nature of the mathematical knowledge needed for teaching. The underlying study was funded by a grant from the U.S. Department of Education. The mathematical knowledge needed for teaching is quite different from that required by students pursuing other mathematics-related professions. Material here is gearedtoward stimulating efforts on individual campuses to improve programs for prospective teachers. This report contains general recommendations for all grades and extensive discussions of the specific mathematical knowledge required for teaching elementary, middle, and high-school grades, respectively.It is also designed to marshal efforts in the mathematical sciences community to back important national initiatives to improve mathematics education and to expand professional development opportunities. The book will be an important resource for mathematics faculty and other parties involved in the mathematical education of teachers. Information for our distributors: This series is published in cooperation with the Mathematical Association of America.

First published in 1994. This book and its companion volume, Mathematics, Education and Philosophy: An International Perspective are edited collections. Instead of the sharply focused concerns of the research monograph, the books offer a panorama of complementary and forward-looking perspectives. They illustrate the breadth of theoretical and philosophical perspectives that can fruitfully be brough to bear on the mathematics and education. The empathise of this book is on epistemological issues, encompassing multiple perspectives on the learning of mathematics, as well as broader philosophical reflections on the genesis of knowledge. It explores constructivist and social theories of learning and discusses the rile of the computer in light of these theories.

While the importance of knowledge for effective instruction has long been acknowledged, and the concept and structure of mathematics knowledge for teaching are far from being new, the process of such knowledge acquisition and improvement remains underexplored empirically and theoretically. The difficulty can well associate with the fact that different education systems embody different values for what mathematics teachers need to learn and how they can be assisted to develop their knowledge. To improve this situation with needed consideration about a system context and policies, How Chinese Acquire and Improve Mathematics Knowledge for Teaching takes a unique approach to present new research that views knowledge acquisition and improvement as part of teachers’ life-long professional learning process in China. The book includes such chapters that can help readers to make possible connections of teachers’ mathematical knowledge for teaching in China with educational policies and program structures for mathematics teacher education in that system context. How Chinese Acquire and Improve Mathematics Knowledge for Teaching brings invaluable inspirations and insights to mathematics educators and teacher educators who wish to help teachers improve their knowledge, and to researchers who study this important topic beyond a static knowledge conception.

Now in its fifth edition, the best-selling text Mathematical Knowledge for Primary Teachers provides trainee teachers with clear information about the fundamental mathematical ideas taught in primary schools. With rigorous and comprehensive coverage of all the mathematical knowledge primary teachers need, the text goes beyond rules and routines to help readers deepen their understanding of mathematical ideas and increase their confidence in teaching these ideas. The book has been updated to incorporate changes in the National Curriculum and the associated tests. In addition, Chapter 1 has been expanded to discuss mathematical understanding in the light of the challenges posed by the current changes. These include the re-introduction of traditional calculation methods for multiplication and division, the early coverage of abstract fractions calculations and much more. Features include: ■ ‘Check’ questions to test the reader’s understanding ■ ‘Challenges’ to increase teachers’ confidence and stretch their mathematical abilities ■ ‘Links with the classroom’ to emphasise the relevance of ideas to the classroom context ■ Straightforward coverage from theory to practice for all aspects of the Mathematics Framework. The book is accompanied by eResources which contains further visual activities and support, designed to scaffold and support the reader’s own understanding. Essential reading for all practising and trainee primary teachers, this book is ideal for those who wish to increase their mathematical understanding and confidence in presenting mathematics in the classroom.

What mathematics is entailed in knowing to act in a moment? Is tacit, rhetorical knowledge significant in mathematics education? What is the role of intuitive models in understanding, learning and teaching mathematics? Are there differences between elementary and advanced mathematical thinking? Why can't students prove? What are the characteristics of teachers' ways of knowing? This book focuses on various types of knowledge that are significant for learning and teaching mathematics. The first part defines, discusses and contrasts psychological, philosophical and didactical issues related to various types of knowledge involved in the learning of mathematics. The second part describes ideas about forms of mathematical knowledge that are important for teachers to know and ways of implementing such ideas in preservice and in-service education. The chapters provide a wide overview of current thinking about mathematics learning and teaching which is of interest for researchers in mathematics education and mathematics educators. Topics covered include the role of intuition in mathematics learning and teaching, the growth from elementary to advanced mathematical thinking, the significance of genres and rhetoric for the learning of mathematics and the characterization of teachers' ways of knowing.

Globally, mathematics and science education faces three crucial challenges: an increasing need for mathematics and science graduates; a declining enrolment of school graduates into university studies in these disciplines; and the varying quality of school teaching in these areas. Alongside these challenges, internationally more and more non-specialists are teaching mathematics and science at both primary and secondary levels, and research evidence has revealed how gaps and limitations in teachers’ content understandings can lead to classroom practices that present barriers to students’ learning. This book addresses these issues by investigating how teachers’ content knowledge interacts with their pedagogies across diverse contexts and perspectives. This knowledge-practice nexus is examined across mathematics and science teaching, traversing schooling phases and countries, with an emphasis on contexts of disadvantage. These features push the boundaries of research into teachers’ content knowledge. The book’s combination of mathematics and science enriches each discipline for the reader, and contributes to our understandings of student attainment by examining the nature of specialised content knowledge needed for competent teaching within and across the two domains. Exploring Mathematics and Science Teachers’ Knowledge will be key reading for researchers, doctoral students and postgraduates with a focus on Mathematics, Science and teacher knowledge research.