Thinking Mathematically

How Humans Learn to Think Mathematically describes the development of mathematical thinking from the young child to the sophisticated adult. Professor David Tall reveals the reasons why mathematical concepts that make sense in one context may become problematic in another. For example, a child's experience of whole number arithmetic successively affects subsequent understanding of fractions, negative numbers, algebra, and the introduction of definitions and proof. Tall's explanations for these developments are accessible to a general audience while encouraging specialists to relate their areas of expertise to the full range of mathematical thinking. The book offers a comprehensive framework for understanding mathematical growth, from practical beginnings through theoretical developments, to the continuing evolution of mathematical thinking at the highest level.

Some students think mathematically. They have the curiosity to notice relationships, the confidence to ask why, and the knowledge to understand the answer. They are the lucky ones.Many others just "do" maths. They look at a question, think about how to answer it, answer it, and then move on.In this book, Craig Barton, maths teacher and best-selling author of How I wish I'd taught maths, offers an approach to help all our students think mathematically. It requires the careful sequencing of questions and examples, the role of the teacher, and the mathematical behaviour of our students. It has transformed his teaching. Drawing upon research into the Self-Explanation Effect, the Hypercorrection Effect and Variation Theory, together with two years of developing this approach with teachers and students around the world, Craig describes exactly what this looks like in the classroom.But be warned: not everyone agrees. Indeed, it is this very approach that led to Craig being labelled as "the most dangerous and clueless man in maths education". If that is not a recommendation to keep reading, I don't know what is.

Developing mathematical thinking is one of major aims of mathematics education. In mathematics education research, there are a number of researches which describe what it is and how we can observe in experimental research. However, teachers have difficulties developing it in the classrooms. This book is the result of lesson studies over the past 50 years. It describes three perspectives of mathematical thinking: Mathematical Attitude (Minds set), Mathematical Methods in General and Mathematical Ideas with Content and explains how to develop them in the classroom with illuminating examples.

This book is the first major study of advanced mathematical thinking as performed by mathematicians and taught to students in senior high school and university. Topics covered include the psychology of advanced mathematical thinking, the processes involved, mathematical creativity, proof, the role of definitions, symbols, and reflective abstraction. It is highly appropriate for the college professor in mathematics or the general mathematics educator.

In recent years national and international reports have been issued that speak of the sad state of the educational system in the United States and the desperate need for reform in teaching science and mathematics. Cognitive psychologists and mathematics and science educators have responded to this need by designing instructional programs that are more compatible with our knowledge of how people acquire, use, and retain knowledge. Many of the guiding principles that underlie these programs are presented in this volume such as teaching comprehension of scientific text through a problem-solving approach: problem planning and representation, selection of relevant information, and simultaneous monitoring of both the specifics of the problem and the mental processes being used to solve it.

Language can be simultaneously both a support and a hindrance to students’ learning of mathematics. When students have sufficient fluency in the mathematics register so that they can discuss their ideas, they become chiefs who are able to think mathematically. However, learning the mathematics register of an Indigenous language is not a simple exercise and involves many challenges not only for students, but also for their teachers and the wider community. Collaborating to Meet Language Challenges in Indigenous Mathematics Classrooms identifies some of the challenges—political, mathematical, community based, and pedagogical— to the mathematics register, faced by an Indigenous school, in this case a Mäori immersion school. It also details the solutions created by the collaboration of teachers, researchers and community members.

This book shows how problems can be the vehicle for learning mathematics itself, and presents a comprehensive pedagogy based on classroom discourse, assessment, critical and creative thinking.

How can you broaden student thinking and help them develop their independence and confidence as problem solvers? Real-life problems are a remarkable tool to stretch student thinking and help them develop a deeper understanding of mathematics and its role in everyday life. Rather than using textbook exercises, the book argues that solving real-world problems promotes flexibility and encourages students to adjust and grow their thinking. It inspires them to consider alternatives and apply math in authentic contexts. You will find practical ways to engage students in critical thinking, develop their independence, and make connections with the world.

'Geometry is often given less time in the teaching timetable than other aspects of mathematics. This book encourages practitioners to think about and raise its profile, indeed achieving what its title suggest' - Primary Practice `This creative, innovative and fascinating book/CD package is one you "MUST BUY". All prospective, new and experienced teachers of mathematics can use it to transform their teaching. All readers can use it to reignite their fascination with mathematics' - Professor Sylvia Johnson, Sheffield Hallam University 'This book exudes activity and interactivity. Moreover, it provides challenge in the context of a significant pedagogy, one that is not just present but actually made explicit. It is undoubtedly a book to learn geometry with, but also one to learn to think more deeply about geometry, about its nature and essence, and also about its teaching and learning' - David Pimm Developing Thinking in Geometry enables teachers and their support staff to experience and teach geometric thinking. As well as discussing key teaching principles, the book and accompanying interactive CD include many activities that encourage readers to extend their own learning, and consequently their teaching practices. The book is constructed around the following key themes: - invariance; - language and points of view; - reasoning using invariance; - visualising and representing. These themes draw on teaching principles developed by the team at the Open University's Centre for Mathematics Education which has a 20-year track record of innovative approaches to teaching and learning geometry. This is a 'must have' text for all primary mathematics specialists, secondary and Further Education mathematics teachers and their support staff. Anyone who wishes to create an understanding and enthusiasm for geometry based upon firm research and effective practice, will enjoy this book. This is the course reader for the Open University Course ME627 Developing Geometric Thinking

To define better techniques of mathematics education, this book combines a knowledge of cognitive science with mathematics curriculum theory and research. The concept of the human reasoning process has been changed fundamentally by cognitive science in the last two decades. The role of memory retrieval, domain-specific and domain-general skills, analogy, and mental models is better understood now than previously. The authors believe that cognitive science provides the most accurate account thus far of the actual processes that people use in mathematics and offers the best potential for genuine increases in efficiency. As such, they suggest that a cognitive science approach enables constructivist ideas to be analyzed and further developed in the search for greater understanding of children's mathematical learning. Not simply an application of cognitive science, however, this book provides a new perspective on mathematics education by examining the nature of mathematical concepts and processes, how and why they are taught, why certain approaches appear more effective than others, and how children might be assisted to become more mathematically powerful. The authors use recent theories of analogy and knowledge representation -- combined with research on teaching practice -- to find ways of helping children form links and correspondences between different concepts, so as to overcome problems associated with fragmented knowledge. In so doing, they have capitalized on new insights into the values and limitations of using concrete teaching aids which can be analyzed in terms of analogy theory. In addition to addressing the role of understanding, the authors have analyzed skill acquisition models in terms of their implications for the development of mathematical competence. They place strong emphasis on the development of students' mathematical reasoning and problem solving skills to promote flexible use of knowledge. The book further demonstrates how children have a number of general problem solving skills at their disposal which they can apply independently to the solution of novel problems, resulting in the enhancement of their mathematical knowledge.

Stanford mathematician and NPR Math Guy Keith Devlin explains why, fun aside, video games are the ideal medium to teach middle-school math. Aimed primarily at teachers and education researchers, but also of interest to game developers who want to produce videogames for mathematics education, Mathematics Education for a New Era: Video Games as a Medium for Learning describes exactly what is involved in designing and producing successful math educational videogames that foster the innovative mathematical thinking skills necessary for success in a global economy. Read the author's monthly MAA column Devlin's Angle

First Published in 1991. Routledge is an imprint of Taylor & Francis, an informa company.

Fundamental Constructs in Mathematics Education is a unique sourcebook which has been crafted from a collection of classic tasks, extracts and texts that have been quoted repeatedly in mathematics education literature. Linked together by the editors'' narrative, the book provides a fascinating examination of key constructs in mathematics education. The book is divided into two parts. The first part examines ''thinking about the learner'' and includes the following constructs: constructivisms, activity theory and didactics. Beginning with a chapter dedicated to the classic tasks used by researchers to ''probe'' learners'' understanding, readers are encouraged to try these theories themselves with learners and be knowledgeable when they encounter them in other writing. The second part focuses on ''thinking and teaching'' and includes issues of getting started, keeping going and bringing to a conclusion. Bringing together writing from Balacheff, Brousseau, Bruner, Cobb, Comfrey, Freudenthal, Greeno, Marton, Piaget, Schon, Vygotsky and many others, this unique examination of constructs in mathematics education will be a valuable resource for anyone reading literature related to learning mathematics be they a teacher, adviser or a student on a masters or PhD course.

Mathematics and Science education have both grown in fertile directions in different geographic regions. Yet, the mainstream discourse in international handbooks does not lend voice to developments in cognition, curriculum, teacher development, assessment, policy and implementation of mathematics and science in many countries. Paradoxically, in spite of advances in information technology and the “flat earth” syndrome, old distinctions and biases between different groups of researcher’s persist. In addition limited accessibility to conferences and journals also contribute to this problem. The International Sourcebooks in Mathematics and Science Education focus on under-represented regions of the world and provides a platform for researchers to showcase their research and development in areas within mathematics and science education. The First Sourcebook on Asian Research in Mathematics Education: China, Korea, Singapore, Japan, Malaysia and India provides the first synthesized treatment of mathematics education that has both developed and is now prominently emerging in the Asian and South Asian world. The book is organized in sections coordinated by leaders in mathematics education in these countries and editorial teams for each country affiliated with them. The purpose of unique sourcebook is to both consolidate and survey the established body of research in these countries with findings that have influenced ongoing research agendas and informed practices in Europe, North America (and other countries) in addition to serving as a platform to showcase existing research that has shaped teacher education, curricula and policy in these Asian countries. The book will serve as a standard reference for mathematics education researchers, policy makers, practitioners and students both in and outside Asia, and complement the Nordic and NCTM perspectives.

Written by an experienced teacher and teacher educator with widespread experience of teaching mathematics in the UK and internationally, Understanding and Teaching Primary Mathematics combines pedagogy and subject knowledge to build confidence and equip you with all the skills and know-how you need to successfully teach mathematics to children of any age. This fourth edition has been fully updated to reflect the latest research developments and initiatives in the field, including a brand-new chapter on ‘Mastery and mathematics’ and ‘The Singapore approach’ which reflects the current international interest in these approaches to learning and teaching mathematics. Extra features also include helpful callouts to the book’s revised and updated companion website, which offers a shared site with a range of resources relevant to both this book and its companion volume, Teaching for Mathematical Understanding. Stimulating, accessible and well-illustrated, with comprehensive coverage of subject knowledge and pedagogy, Understanding and Teaching Primary Mathematics is an essential purchase for trainee and practising teachers alike.

John Mason has been a prominent figure in the research field of mathematics education for several decades. His principal focus has been thinking about mathematical problems, supporting those who wish to foster and sustain their own thinking and the thinking of others.

Becoming a Successful Teacher of Maths is a practical guide for newly qualified teachers of secondary mathematics. It develops the essential core knowledge, skills and understanding demanded by the new DfEE requirements for courses of initial teacher training. It is based on research findings relating to the organisation and management of maths classrooms, teaching approaches, assessment and the common misconceptions which often hinder pupils' progress in key areas of the National Curriculum. Theoretical principles are exemplified through case-study material. Suggestions for school-based activities are made. While being a practical 'how to' guide for beginning teachers, it also offers critical insights for more experienced teachers reflecting on their practice.

The book has been written by an international group of very activeresearchers and scholars who have a passion for the study of Chinesemathematics education. It aims to provide readers with a comprehensiveand updated picture of the teaching and learning of mathematicsinvolving Chinese students from various perspectives, including theways in which Chinese students learn mathematics in classrooms, schools and homes, the influence of the cultural and socialenvironment on Chinese students'' mathematics learning, and thestrengths and weaknesses of the ways in which Chinese learnmathematics

'This is an outstanding book: it should be high on the list of any primary school teacher's set of references and a required text for pre-service teachers.' Australian Primary Mathematics Classroom In our technology-rich world, numeracy is just as important as the smartphone in your pocket. Students need to develop mathematical ways of seeing the world and strong problem-solving skills, and those foundations are taught in the primary school classroom. Teaching Mathematics in Primary Schools covers the mathematical content taught in primary and middle years, always emphasising how students can connect what they learn in mathematics with other curriculum areas and with the world beyond the classroom. The authors draw on the latest international research to show how teachers can develop a rich repertoire of classroom teaching techniques, and effective planning, assessment and reporting methods. They outline approaches to creating supportive learning environments for all students, and to building their knowledge and confidence in using mathematics. This third edition has been updated throughout and includes a new chapter on numeracy. Evidence-based uses of digital technologies to support learning and teaching are included in every chapter. With practical strategies that can be implemented in the classroom, this book is an invaluable resource for pre-service and early career primary and middle years mathematics teachers.

This edited volume is written in memoriam of Professor Emeritus Ubiratan D’Ambrosio (1932 – 2021), who was a well-known Brazilian mathematics educator and historian of mathematics. This book explores the diverse facets of D’Ambrosio’s work as well as his legacy and the later adaptation of his ideas around the globe. It starts with a preface written by his son, Alexandre D' Ambrosio, who shares his personal experiences growing up with this father and his love for discovery. The book is then divided into four sections: Past and Future: Ubi’s Way of Seeing Education in the Present Roots of Ethnomathematics Ethnomathematics in Action Trends in Ethnomathematics It features diverse points of view and experiences that explore mathematics and culture from researchers in the Americas, Africa, Europe and South Asia. Chapters range from personal explorations of D’Ambrosio’s impact to broader views of his research and work. This book forms part of the growing understanding of Ubiratan D’Ambrosio’s life, research, and the legacy he has left for millions of researchers, students and teachers worldwide. This book is appealing to anyone involved in mathematics education research as well as those interested in the history and future of mathematics education.