Argumentation is, to quote Alan Schoenfeld, the heart and soul of mathematics (Schoenfeld 2009). It has distinguished mathematics from other disciplines since the time of Euclid. However, the practice of building mathematical arguments, including informal justifications, is not always at the center of mathematics instruction, particularly in K–8 grades. With this book, we hope to help you incorporate argumentation into your own teaching.
Argumentation -the activity of building mathematical arguments- is central to mathematical thinking. Indeed, argumentation is a practice, not a skill or a set of facts to be memorized but a way of thinking about mathematics. It is a habit of mind that should characterize students' attempts to solve mathematical problems, and as such, it takes time and deliberate effort to develop. Mathematically proficient students use argumentation (along with other practices) as a way to approach mathematics and as a lens for a deeper understanding of mathematics. Teachers of mathematically proficient students provide opportunities for students to practice argumentation and advance their understanding of this practice.
Mathematics at its core is a science of sense-making. Argumentation is a process that helps us make sense of the relationships and generalizations we notice in mathematical quantities and objects. Argumentation is also the process through which mathematical claims are vetted; mathematicians build arguments by explaining and justifying solutions to their peers. The process is completed when the community accepts or rejects the argument. In this sense, mathematics evolves by virtue of a conversation among those who solve problems. Hence, argumentation is a pivotal tool that guides mathematical discourse and helps us navigate the terrain of mathematics.
In this book, Despina Stylianou and Maria Blanton navigate argumentation as it pertains to school mathematics, particularly in the formative years of middle school. There is no better time to explore these ideas; new standards across the nation identify “constructing viable arguments” as part of our curricula and assessments. This book aims to support you in expanding your understanding of argumentation and in providing resources to make argumentation a hallmark of everyday instruction for all students.
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Despina A. Stylianou is a Professor of Mathematics Education at The City College of New York. She is also the Director of Mathematics in the City, a center for K–8 professional development focused on mathematics education. Her research interests lie in the area of mathematical cognition; her work explores the mathematical skills, sensibilities, and habits of mind, and actions that are critical to doing, learning, and using mathematics proficiently.
Maria Blanton is a Senior Scientist at TERC in Cambridge, Massachusetts. Prior to joining TERC, she was a Professor of Mathematics Education in the STEM Education Department at the University of Massachusetts, Dartmouth. Her work focuses on understanding issues of teacher and student learning associated with algebraic thinking in the elementary grades.