Teaching With Mathematical Argument

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.

This research-based book brings tough Standards for Mathematical Practice 3 standards for mathematical argumentation and critical reasoning alive - all within a thoroughly explained four-part model that covers generating cases, conjecturing, justifying, and concluding.

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.

"Chris is a math teacher and debate coach, and this book is about how he uses the tools of debate to get his students arguing in math class. Students are highly engaged and learn mathematics more deeply when using his techniques. The book is short and clear and will help readers learn how to incorporate debate in an incremental, totally not overwhelming way because Chris has broken the work down into a lovely step-by-step sequence. It's a super practical and readable manual, written by a working teacher for working teachers"--

For too many students, mathematics consists of facts in a vacuum, to be memorized because the instructor says so, and to be forgotten when the course of study is completed. In this all-too-common scenario, young learners often miss the chance to develop skills—specifically, reasoning skills—that can serve them for a lifetime. The elegant pages of Teaching Mathematical Reasoning in Secondary School Classrooms propose a more positive solution by presenting a reasoning- and discussion-based approach to teaching mathematics, emphasizing the connections between ideas, or why math works. The teachers whose work forms the basis of the book create a powerful record of methods, interactions, and decisions (including dealing with challenges and impasses) involving this elusive topic. And because this approach shifts the locus of authority from the instructor to mathematics itself, students gain a system of knowledge that they can apply not only to discrete tasks relating to numbers, but also to the larger world of people and the humanities. A sampling of the topics covered: Whole-class discussion methods for teaching mathematics reasoning. Learning mathematical reasoning through tasks. Teaching mathematics using the five strands. Classroom strategies for promoting mathematical reasoning. Maximizing student contributions in the classroom. Overcoming student resistance to mathematical conversations. Teaching Mathematical Reasoning in Secondary School Classrooms makes a wealth of cutting-edge strategies available to mathematics teachers and teacher educators. This book is an invaluable resource for researchers in mathematics and curriculum reform and of great interest to teacher educators and teachers.

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

Mathematics as the Science of Patterns: Making the Invisible Visible to Students through Teaching introduces the reader to a collection of thoughtful, research-based works by authors that represent current thinking about mathematics, mathematics education, and the preparation of mathematics teachers. Each chapter focuses on mathematics teaching and the preparation of teachers who will enter classrooms to instruct the next generation of students in mathematics. The value of patterns to the teaching and learning of mathematics is well understood, both in terms of research and application. When we involve or appeal to pattern in teaching mathematics, it is usually because we are trying to help students to extract greater meaning, or enjoyment, or both, from the experience of learning environments within which they are occupied, and perhaps also to facilitate remembering. As a general skill it is thought that the ability to discern a pattern is a precursor to the ability to generalize and abstract, a skill essential in the early years of learning and beyond. Research indicates that the larger problem in teaching mathematics does not lie primarily with students; rather it is with the teachers themselves. In order to make changes for students there first needs to be a process of change for teachers. Understanding the place of patterns in learning mathematics is a predicate to understanding how to teach mathematics and how to use pedagogical reasoning necessary in teaching mathematics. Importantly, the lack of distinction created by the pedagogical use of patterns is not immediately problematic to the student or the teacher. The deep-seated cognitive patterns that both teachers and students bring to the classroom require change. Chapter 1 opens the book with a focus on mathematics as the science of patterns and the importance of patterns in mathematical problem solving, providing the reader with an introduction. The authors of Chapter 2 revisit the work of Po lya and the development and implementation of problem solving in mathematics. In Chapter 3, the authors present an argument for core pedagogical content knowledge in mathematics teacher preparation. The authors of Chapter 4 focus on preservice teachers’ patterns of conception as related to understanding number and operation. In Chapter 5 the authors examine the role of visual representation in exploring proportional reasoning, denoting the importance of helping learners make their thinking visible. The authors of Chapter 6 examine patterns and relationships, and the importance of each in assisting students’ learning and development in mathematical understanding. The authors of Chapter 7 examine the use of worked examples as a scalable practice, with emphasis on the importance of worked examples in teaching fraction magnitude and computation is discussed. In Chapter 8, the authors expand on the zone of proximal development to investigate the potential of Zankov’s Lesson in terms of students analyzing numerical equalities. The authors of Chapter 9 focus on high leverage mathematical practices in elementary pre-service teacher preparation, drawing into specific relief the APEX cycle to develop deep thinking. In Chapter 10, the author focuses on number talks and the engagement of students in mathematical reasoning, which provides opportunities for students to be sensemakers of mathematics. Chapter 11 presents an epilogue, focusing on the importance of recognizing the special nature of mathematics knowledge for teaching.