Ways to Think About Mathematics

Engaging in mathematical investigations yourself will help you be a better mathematics teacher!

National, state, and local curriculum requirements and recommendations increasingly emphasize active student involvement in exploratory investigations. Teachers now face new demands when they field unexpected student discoveries, determine when some open-ended activity has played out its usefulness or is just on the verge of paying off in a major way, and judge which student-initiated directions are likely to lead to development of important mathematical ideas and which are dead ends. These pedagogical judgments depend heavily on deep mathematical knowledge.

Ways to Think About Mathematics uses immersion experiences in algebra, geometry, and statistics to help mathematics teachers improve their knowledge and understanding of mathematical concepts. By experiencing open-ended problems, making and checking conjectures, and evaluating problem solving strategies, every math teacher can become better prepared to deal with day-to-day classroom decisions.

Funded by the National Science Foundation and successfully field-tested in a wide variety of professional development and preservice settings, the materials in this book integrate mathematical thinking, effective teaching practices, and explicit connections to exemplary curricula. Because it is aligned with the principles and standards of the National Council of Teachers of Mathematics but is not fixed to any single curriculum, the materials in this book can be aligned with any state or district standards.

Emphasizing all components of mathematical investigation, this accessible volume focuses on:

• Mathematical ways of thinking
• Discovering connections between mathematical ideas
• Posing and analyzing problems
• Making, modifying, testing, and proving conjectures
• Linking higher level mathematics to the content of grade 6-12 classrooms

Ways to Think About Mathematics gives teachers the opportunity to grow as learners and teachers of mathematics by including problems for teachers to explore as well as selected "Problems for the Classroom" for the classes they teach.

Steven R. Benson is an Associate Professor of Mathematics at Lesley University in Cambridge, Massachusetts, where he teaches a variety of mathematics content courses for traditional undergraduate students and in-service mathematics teachers. Before joining the Lesley faculty, Dr. Benson was a Research Scientist at Education Development Center, Inc., where he was involved in a wide variety of projects, most of which involved the development of curricula for mathematics students and teachers. He has also facilitated preservice and in-service teacher professional development workshops across the U.S. and internationally (including serving as consultant to the Ministry of Education in Azerbaijan), directed a research project investigating the genesis and development of mathematical talent in Mathematical Olympians, and edited the problem calendar section of the Mathematics Teacher journal published by the National Council of Teachers of Mathematics. Prior to joining the EDC staff in June 2000, he held mathematics faculty positions at St. Olaf College, Santa Clara University, University of New Hampshire, and University of Wisconsin-Oshkosh, and is currently a co-director of the Master of Science for Teachers program at the University of New Hampshire. He received his PhD from the University of Illinois, working under the direction of Leon McCulloh in algebraic number theory.

Susan Addington is a professor of Mathematics at California State University,San Bernardino. Though her doctoral work was in arithmetic algebraic geometry (a field that includes almost every mathematical area except statistics),she now spends most of her time thinking about math education. Her educational interests include teacher preparation (elementary through college), ethnomathematics, and the use of technology to teach traditionally difficult ideas. Susan is married to David Dennis, a math educator, historian of mathematics, jazz saxophonist, and virtuoso gardener.

Nina P.Arshavsky is a Math Coach at Boston Public Schools, providing professional development to high school math teachers. She received her doctoral degree in educational psychology from the University of Wisconsin-Madison and taught mathematics in middle and high schools in both Russia and the US. Dr. Arshavsky worked on a variety of research, professional development, and curriculum development projects in the Center for Mathematics Education at EDC and is a co-author of Impact Mathematics: Algebra and More for the Middle Grades, a comprehensive 6-8 mathematics curriculum. She has published and presented numerous research papers, and has taught college courses in cognitive development and human development in infancy and early childhood.

Al Cuoco is Senior Scientist and Director of the Center for Mathematics Education at Education Development Center. Before coming to EDC, he taught high school mathematics for 24 years to a wide range of students in the Woburn, Massachusetts public schools. A student of Ralph Greenberg, Dr. Cuoco received his Ph.D. in mathematics from Brandeis; his mathematical interests and publications have been in algebraic number theory. His favorite publication is his 1991 article in the American Mathematical Monthly, described by his wife as “an attempt to explain a number system no one understands with a picture no one can see.”

E. Paul Goldenberg is a Senior Scientist and the Principal Investigator of three recent NSF-funded projects: Connecting with Mathematics, materials for teacher professional development programs; Problems with a Point,a Web-accessible, searchable database of orchestrated problem sets; and Math Workshop: A Comprehensive Elementary Curriculum for Skill, Mathematical Ability, and Real Thinking, a project that will develop materials for a K-5 comprehensive mathematics curriculum that promotes rather than requires professional development. He was previously PI on the Connected Geometry curriculum development project, and of a 3-year,NSF-funded research project into the nature of learning with geometry software. He has over 35 years of experience in elementary, secondary, and post-secondary teaching, teacher enhancement, and education research. He is widely published and has conducted workshops and seminars on a variety of topics in mathematics education.