Do your students suppose that 1/3 is greater than 1/2, since 3 is greater than 2? Do they believe that having "halves" means having two, and only two, congruent "pieces" of a whole? What tasks can you offer-what questions can you ask-to determine what your students know or don't know-and move them forward in their thinking? This book focuses on the specialised pedagogical content knowledge that you need to teach fractions effectively in grades 3-5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions-not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of fractions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning.

Do your students believe that division "doesn't make sense" if the divisor is greater than the dividend? Explore rich, researched-based strategies and tasks that show how students are reasoning about and making sense of mulitplication and division. This book focuses on the specialised pedagogical content knowledge that you need to teach multiplication and division effectively in grades 3-5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations - not only in their current work, but also in higher-level maths and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of multiplication and division. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. About the Series: You have essential understanding. It's time to put it into practise in your teaching. The Putting Essential Understanding into Practice Series moves NCTM's Essential Understanding Series into the classroom. The new series details and explores best practises for teaching the essential ideas that students must grasp about fundamental topics in mathematics - topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life and questions for reader reflection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, understanding into practise, instructional strategies and assessment that pedagogical content knowledge entails. Maximise the potential of student-centred learning and teaching by putting essential understanding into practise.

Using strengths-based approaches to support development in mathematics It's time to re-imagine what's possible and celebrate the brilliance multilingual learners bring to today's classrooms. Innovative teaching strategies can position these learners as leaders in mathematics. Yet, as the number of multilingual learners in North American schools grows, many teachers have not had opportunities to gain the competencies required to teach these learners effectively, especially in disciplines such as mathematics. Multilingual learners-historically called English Language Learners-are expected to interpret the meaning of problems, analyze, make conjectures, evaluate their progress, and discuss and understand their own approaches and the approaches of their peers in mathematics classrooms. Thus, language plays a vital role in mathematics learning, and demonstrating these competencies in a second (or third) language is a challenging endeavor. Based on best practices and the authors' years of research, this guide offers practical approaches that equip grades K-8 teachers to draw on the strengths of multilingual learners, partner with their families, and position these learners for success. Readers will find: * A focus on multilingual students as leaders * A strength-based approach that draws on students' life experiences and cultural backgrounds * An emphasis on maintaining high expectations for learners' capacity for mastering rigorous content * Strategies for representing concepts in different formats * Stop and Think questions throughout and reflection questions at the end of each chapter * Try It! Implementation activities, student work examples, and classroom transcripts With case studies and activities that provide a solid foundation for teachers' growth and exploration, this groundbreaking book will help teachers and teacher educators engage in meaningful, humanized mathematics instruction.

Do your students have “concept images” that limit their ideas of shapes to specific examples, oriented in particular ways? Do they confuse the size of an angle with the length of the rays in a drawing of an angle? When solving problems, do they mix up the concepts of perimeter—the distance around a shape, measured in linear units—and area—the region within the shape, measured in square units? What tasks can you offer—what questions can you ask—to determine what they know or don’t know—and move them forward in their thinking? This book focuses on the specialized pedagogical content knowledge that you need to teach geometry and measurement effectively in grades 3–5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with geometry and measurement—not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of geometry and measurement. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. Putting Essential Understanding of Geometry and Measurement into Practice in Grades 3–5 is the eighth title in NCTM’s highly useful and very readable Putting Essential Understanding into Practice Series, edited by Barbara J. Dougherty. Each volume in the series builds on the companion volume in the earlier Essential Understanding Series to show teachers how to implement their understanding of mathematics in the classroom.