But Why Does It Work

Author: Susan Jo Russell
Publisher: Heinemann Educational Books
ISBN: 032508114X
Release Date: 2017
Genre: Education

If you ask students, "Why does that work?" do they know what you're asking and do you know what to listen for in their responses? Do you have images of what mathematical argument looks like in the elementary grades and how to help students learn to engage in this important practice? Do you have so much content to cover that finding time for this kind of work is difficult? But Why Does It Work? offers a simple, efficient teaching model focused on mathematical argument for developing the ability of students to justify their thinking and engage with the reasoning of others. Designed for individuals as well as study groups, this book includes access to classroom-ready instructional sequences, each built on a model supporting students in: noticing relationships across sets of problems, equations, or expressions articulating a claim about what they notice investigating their claim through representations such as manipulatives, diagrams, or story contexts using their representations to demonstrate why a claim must be true or not extending their thinking from one operation to another. Establishing a classroom culture where students gain confidence in their own mathematical voice and learn to value the contributions of their peers is a critical part of this work. The authors tell us, "If the idea underlying a student's reasoning is not made explicit, the opportunity for all students to engage in such thinking is lost." As students become a true community of mathematicians, they heighten each other's understanding by investigating questions, conjectures, and examples together. Enhanced with extensive video showing the instructional sequences in action-along with guiding focus questions and math investigations-But Why Does It Work? is a flexible approach that will help students confidently articulate and defend their reasoning, and share their deep thinking with others.

Proving in the Elementary Mathematics Classroom

Author: Andreas J. Stylianides
Publisher: Oxford University Press
ISBN: 9780198723066
Release Date: 2016-07-14
Genre:

Although proving is core to mathematics as a sense-making activity, it currently has a marginal place in elementary classrooms internationally. Blending research with practical perspectives, this book addresses what it would take to elevate the place of proving at elementary school. The book uses classroom episodes from two countries to examine different kinds of proving tasks and the proving activity they can generate in the elementary classroom. It examines further the role of teachers in mediating the relationship between proving tasks and proving activity, including major mathematical and pedagogical issues that arise for teachers as they implement each kind of proving task. In addition to its contribution to research knowledge, the book has important implications for teaching, curricular resources, and teacher education.

Teachers Professional Development and the Elementary Mathematics Classroom

Author: Sophia Cohen
Publisher: Routledge
ISBN: 9781135632267
Release Date: 2004-07-13
Genre: Education

This book illustrates the experiences of elementary school teachers across one year's time as they participated in a teacher development seminar focused on mathematics, and as a result changed their beliefs, their knowledge, and their practices. It explores these experiences as a means of understanding the learning that takes a teacher from a more traditional teaching practice to one that is focused on the ideas and understandings that students and teachers have of the subject matter. The work emerges from and reports on a unique data set from a two-year study of teacher learning that was funded by the Spencer and MacArthur foundations. The teachers, whose work is at the center of this study, were participants in the Developing Mathematical Ideas seminar (DMI), a mathematics teacher development seminar for elementary school teachers. This seminar is one example of intensive, domain-specific professional development. In this seminar teachers study elementary mathematics content to deepen their own understanding of it, they study the development among children of the ideas central to elementary mathematics, and they experience a teaching and learning environment consistent with the pedagogy envisioned by the National Council for Teachers of Mathematics' Principles and Standards for School Mathematics. The seminar is a nationally available teacher development curriculum, thus interested educators can gain access to the resources necessary to offer similar seminars in their own communities. Teachers' Professional Development and the Elementary Mathematics Classroom: Bringing Understandings to Light will be widely interesting to a broad audience, including mathematics teacher educators, teacher education researchers, policymakers, and classroom teachers. It will serve well as a text in a range of graduate courses dealing with teacher cognition/knowledge for teaching, mathematics methods, psychology of learning, and pedagogical theory.

Mathematical Models for Teaching

Author: Ann Kajander
Publisher: Canadian Scholars’ Press
ISBN: 9781551305561
Release Date: 2014-01-01
Genre: Mathematics

Students of mathematics learn best when taught by a teacher with a deep and conceptual understanding of the fundamentals of mathematics. In Mathematical Models for Teaching, Ann Kajander and Tom Boland argue that teachers must be equipped with a knowledge of mathematics for teaching, which is grounded in modelling, reasoning, and problem-based learning. A comprehensive exploration of models and concepts, this book promotes an understanding of the material that goes beyond memorization and recitation, which begins with effective teaching. This vital resource is divided into 15 chapters, each of which addresses a specific mathematical concept. Focusing on areas that have been identified as problematic for teachers and students, Mathematical Models for Teaching equips teachers with a different type of mathematical understanding--one that supports and encourages student development. Features: grounded in the most current research about teachers' learning contains cross-chapter connections that identify common ideas includes chapter concluding discussion questions that encourage critical thinking incorporates figures and diagrams that simplify and solidify important mathematical concepts offers further reading suggestions for instructors seeking additional information

Mathematics for Elementary School Teachers A Process Approach

Author: Mark A. Freitag
Publisher: Cengage Learning
ISBN: 9780618610082
Release Date: 2013-01-01
Genre: Mathematics

Freitag's MATHEMATICS FOR ELEMENTARY SCHOOL TEACHERS: A PROCESS APPROACH was developed using the five Content Standards from the NCTM Principles and Standards for School Mathematics, and the Common Core State Standards for Mathematics. Traditionally, books for pre-service elementary teachers have focused on problem solving. However, problem solving is not the only process through which mathematics is learned. It is also learned through mathematical reasoning, communication, representation, and connections. Recent trends in mathematics education now advocate implementing all five processes as a vital part of learning and doing mathematics. Consequently, you need to have concrete experiences with these processes that you will be required to teach. The goal of this book is to treat each of the processes equitably by using an approach in which the five processes serve as the central pedagogical theme. Most of the examples, exercises, and activities are designed to either model the processes or to directly engage you in working with them. As a result, you will not only come to understand the different processes, but also appreciate them as an integral to learning and doing mathematics. If this broader view can be instilled, you are more likely to give your students a more well-rounded and holistic view of mathematics once you enter the classroom. The content of the book is directly related to the mathematics that is taught in grades K - 8. The purpose is not to reteach elementary mathematics. Rather, the intent is to look at the content from a theoretical or generalized point of view, so that you can better understand the concepts and processes behind the mathematics you will teach. In short, the book focuses on the why behind the mathematics in addition to the how. Available with InfoTrac Student Collections http://gocengage.com/infotrac. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

Connecting Arithmetic to Algebra

Author: Susan Jo Russell
Publisher: Heinemann Educational Books
ISBN: 0325041911
Release Date: 2011
Genre: Education

"To truly engage in mathematics is to become curious and intrigued about regularities and patterns, then describe and explain them. A focus on the behavior of the operations allows students starting in the familiar territory of number and computation to progress to true engagement in the discipline of mathematics." -Susan Jo Russell, Deborah Schifter, and Virginia Bastable Algebra readiness: it's a topic of concern that seems to pervade every school district. How can we better prepare elementary students for algebra? More importantly, how can we help all children, not just those who excel in math, become ready for later instruction? The answer lies not in additional content, but in developing a way of thinking about the mathematics that underlies both arithmetic and algebra. Connecting Arithmetic to Algebra invites readers to learn about a crucial component of algebraic thinking: investigating the behavior of the operations. Nationally-known math educators Susan Jo Russell, Deborah Schifter, and Virginia Bastable and a group of collaborating teachers describe how elementary teachers can shape their instruction so that students learn to: *notice and describe consistencies across problems *articulate generalizations about the behavior of the operations *develop mathematical arguments based on representations to explain why such generalizations are or are not true. Through such work, students become familiar with properties and general rules that underlie computational strategies-including those that form the basis of strategies used in algebra-strengthening their understanding of grade-level content and at the same time preparing them for future studies. Each chapter is illustrated by lively episodes drawn from the classrooms of collaborating teachers in a wide range of settings. These provide examples of posing problems, engaging students in productive discussion, using representations to develop mathematical arguments, and supporting both students with a wide range of learning profiles. PLCs and book-study groups! Save $47.25 when you purchase 15 copies with the Book Study Bundle. Staff Developers: Available online, the Course Facilitator's Guide provides math leaders with tools and resources for implementing a Connecting Arithmetic to Algebra workshop or preservice course. For information on the PD course offered through Mount Holyoke College, download the flyer.

Teaching and Learning Proof Across the Grades

Author: Despina A. Stylianou
Publisher: Routledge
ISBN: 9781135856755
Release Date: 2010-09-23
Genre: Education

A Co-Publication of Routledge for the National Council of Teachers of Mathematics (NCTM) In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students’ understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.

Elementary and Middle School Mathematics

Author: John A. Van de Walle
Publisher: Allyn & Bacon
ISBN: 020538689X
Release Date: 2004
Genre: Education

New to the Fifth Edition *Inspired by one of the five process standards in the NCTM Principles and Standards', Content Connections is a new feature that highlights the connections between content areas in mathematics in various chapters. *NCTM icons appear in nearly every chapter and point explicitly to the alignment of this book with the NCTM Principles and Standards through excerpts and commentary. *Investigations in Number, Data, and Space and Connected Mathematics features spotlight examples of these two popular reform curricula for the K-5 and 6-8 grade levels, respectively. These excerpts occur in every chapter in Section 2, include an activity from the program along with a description of the surrounding content. The purpose is to acquaint students with materials reflecting the spirit of the NCTM Standards and the constructivist philosophy of the book. *Stop and Reflect features indicate places throughout the book to pause and become truly engaged with an idea. *Chapter 6, Planning in a Problem-Based Curriculum- includes a new step-by-step guide for planning student-centered lessons. A distinction between practice and drill provides a new perspective on the issue of drill. *

Elementary School Mathematics

Author: John A. Van de Walle
Publisher: Longman Publishing Group
ISBN: UCSC:32106018113511
Release Date: 1994
Genre: Mathematics

Elementary School Mathematics: Teaching Developmentally, Second Edition, provides detailed, comprehensive coverage of the teaching of mathematics in grades K-8. Firmly grounded in the NCTM Standards, the text develops a constructivist/developmental approach to learning mathematics that focuses on the nature of mathematical concepts and the manner in which children construct mathematical ideas. It aims to help teachers reconceptualize mathematics as a problem-solving endeavor not as a mastery of rules and procedures. An accessible synthesis of viewpoints and current research, this highly praised text includes topics ranging from the development of basic fact mastery to the important development of "number sense"--All the while encouraging teachers to consider the construction of mathematics from the child's perspective. In-depth treatment of curriculum content teaching, combined with hundreds of teaching activities aimed at conceptual development, makes Elementary School Mathematics a practical teacher's resource as well as a college text. Elementary School Mathematics: Teaching Developmentally, Second Edition, features: greater emphasis on the NCTM Curriculum Standards and the Professional Teaching Standards as the leading forces in mathematics education; a new chapter on mathematics as a "science of pattern and order"; a completely rewritten chapter on alternative assessment; expanded and improved problem-solving examples; enhanced coverage of grades 7 and 8; even more black-line masters with directions for making other useful materials; and an Instructor's Manual with instructional suggestions, activities for group homework, practicum ideas, and essay questions.

Mathematical Olympiad Challenges

Author: Titu Andreescu
Publisher: Springer Science & Business Media
ISBN: 0817646116
Release Date: 2008-12-04
Genre: Mathematics

Hundreds of beautiful, challenging, and instructive problems from algebra, geometry, trigonometry, combinatorics, and number theory Historical insights and asides are presented to stimulate further inquiry Emphasis is on creative solutions to open-ended problems Many examples, problems and solutions, with a user-friendly and accessible style Enhanced motivatio References

Teaching with Mathematical Argument

Author: Despina A. Stylianou
Publisher: Heinemann Educational Books
ISBN: 0325074526
Release Date: 2018
Genre: Education

Argumentation leads to deeper understanding Many students' interest in mathematics fades as they are asked to solve seemingly endless disconnected problems. Despina Stylianou and Maria Blanton show how building daily instruction around mathematical argumentation can enliven your classroom and re-engage your students. Teaching with Mathematical Argument explores how argumentation-discussing and debating a rich mathematical problem-provides all students a deeper understanding of mathematics. You'll find guidance for: understanding what argumentation is and building a classroom culture that supports it engaging every student in argumentation, not just "strong" or "high performing" students assessing your students' arguments and designing instruction responsive to their learning. The potential to transform student engagement At its core, argumentation helps students delve deeply into foundational mathematical concepts, enhancing their understanding and confidence along the way. Research shows that this type of instruction has the potential to transform student engagement and success in mathematics, and so Despina and Maria argue that mathematical arguments should have a far more central role in teaching and learning than most of us have ever considered. With ideas for structuring discussions and suggested tasks to try, this book will show you how to elevate argumentation in your instruction and harness its power for enhancing student learning.

The Effective Principal

Author: Barbara Scott Nelson
Publisher: Teachers College Press
ISBN: 0807746061
Release Date: 2005
Genre: Education

This volume examines how effective instructional leadership by principals and other school administrators is affected by their own knowledge and beliefs about learning, teaching, and subject matter. Using mathematics as a subject focus, the authors examine several specific aspects of instructional leadership, such as teacher supervision and classroom observation, curriculum selection, and student assessment. Nelson and Sassi provide detailed portraits of administrators at work, illuminating key decision-making situations and the actions they choose to take. This important volume looks at a new image of the school principal, one that is tied more closely to learning and teaching.The authors discuss and offer important implications for mathematics education, educational policy, and school improvement.