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Uncovering Student Thinking in Mathematics, Grades 6-12

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Table of Contents

Foreword by Johnny W. Lott Preface Acknowledgments About the Authors 1. Mathematics Assessment Probes Introduction What Types of Understandings and Misunderstandings Does a Mathematics Assessment Probe Uncover? How Were the Mathematics Assessment Probes Developed? What Is the Structure of a Mathematics Assessment Probe? What Additional Information Is Provided With Each Mathematics Assessment Probe? What Mathematics Assessment Probes Are Included in the Book? 2. Instructional Implications Differentiating Instruction Assessing Points of Entry Analyzing Trends in Student Thinking Giving Student Interviews Promoting Student-to-Student Dialogue Developing Vocabulary Allowing for Individual Think Time Improving Students' Process Skills Assessing Effectiveness of Instructional Activities Moving Beyond the Individual Classroom Summary 3. Number and Operations Assessment Probes Probe 1: What Is the Value of the Digit? Teachers' Notes: What Is the Value of the Digit? Probe 1a: What Is the Value of the Digit? Variation Probe 2: What Is the Meaning of 2/3? Teachers' Notes: What Is the Meaning of 2/3? Probe 2a: What Is the Meaning of 2/3? Variation Probe 3: Is It Equivalent? Teachers' Notes: Is It Equivalent? Probe 3a: Is It Equivalent? Variation Probe 4: What's Your Estimate? Teachers' Notes: What's Your Estimate? Probe 5: Is It an Estimate? Teachers' Notes: Is It an Estimate? Probe 6: Is It Simplified? Teachers' Notes: Is It Simplified? Probe 6a: Is It Simplified? Algebraic Variation Probe 7: Where Is a Million? Teachers' Notes: Where Is a Million? Probe 8: How Low Can You Go? Teachers' Notes: How Low Can You Go? Probe 8a: How Low Can You Go? Variation Probe 9: What's Your Addition Strategy? Teachers' Notes: What's Your Addition Strategy? Probe 9a: What's Your Addition Strategy? Decimals Variation Probe 9b: What's Your Addition Strategy? Fractions Variation Probe 10: What's Your Subtraction Strategy? Teachers' Notes: What's Your Subtraction Strategy? Probe 10a: What's Your Subtraction Strategy? Decimals Variation Probe 10b: What's Your Subtraction Strategy? Fractions Variation Probe 11: What's Your Multiplication Strategy? Teachers' Notes: What's Your Multiplication Strategy? Probe 11a: What's Your Multiplication Strategy? Decimals Variation Probe 12: What's Your Division Strategy? Teachers' Notes: What's Your Division Strategy? Probe 12a: What's Your Division Strategy? Decimals Variation 4. Geometry, Measurement, and Data Assessment Probes Probe 1: What's the Measure? Teachers' Notes: What's the Measure? Probe 1a: What's the Measure? Variation Probe 2: Are Area and Perimeter Related? Teachers' Notes: Are Area and Perimeter Related? Probe 3: What's the Area? Teachers' Notes: What's the Area? Probe 4: What's the Capacity? Teachers' Notes: What's the Capacity? Probe 5: Is It Transformed? Teachers' Notes: Is It Transformed? Probe 6: Are They Similar? Teachers' Notes: Are They Similar? Probe 7: What Do You Mean? Teachers' Notes: What Do You Mean? Probe 7a: What Do You Mean? Variation Probe 8: Name of the Graph? Teachers' Notes: Name of the Graph? Probe 9: Graph Construction? Teachers' Notes: Graph Construction? 5. Algebra and Data Assessment Probes Probe 1: Equal or Not Equal? Teachers' Notes: Equal or Not Equal? Probe 2: Is It the Same as a+b? Teachers' Notes: Is It the Same as a+b? Probe 3: M & N's? Teachers' Notes: M & N's? Probe 4: What's the Substitute? Teachers' Notes: What's the Substitute? Probe 5: Is It True? Teachers' Notes: Is It True? Probe 6: Solving Equations? Teachers' Notes: Solving Equations? Probe 7: Correct Representation of the Inequality? Teachers' Notes: Correct Representation of the Inequality? Probe 8: Is It a Variable? Teachers' Notes: Is It a Variable? Probe 9: Binomial Expansion? Teachers' Notes: Binomial Expansion? Probe 10: Is It Quadratic? Teachers' Notes: Is It Quadratic? Resource A: Note Template for QUEST Cycle References Index

About the Author

Cheryl Rose Tobey is a senior mathematics associate at Education Development Center (EDC) in Massachusetts. She is the project director for Formative Assessment in the Mathematics Classroom: Engaging Teachers and Students (FACETS) and a mathematics specialist for Differentiated Professional Development: Building Mathematics Knowledge for Teaching Struggling Students (DPD); both projects are funded by the National Science Foundation (NSF). She also serves as a director of development for an Institute for Educational Science (IES) project, Eliciting Mathematics Misconceptions (EM2). Her work is primarily in the areas of formative assessment and professional development. Prior to joining EDC, Tobey was the senior program director for mathematics at the Maine Mathematics and Science Alliance (MMSA), where she served as the co-principal investigator of the mathematics section of the NSF-funded Curriculum Topic Study, and principal investigator and project director of two Title IIa state Mathematics and Science Partnership projects. Prior to working on these projects, Tobey was the co-principal investigator and project director for MMSA's NSF-funded Local Systemic Change Initiative, Broadening Educational Access to Mathematics in Maine (BEAMM), and she was a fellow in Cohort 4 of the National Academy for Science and Mathematics Education Leadership. She is the coauthor of six published Corwin books, including seven books in the Uncovering Student Thinking series (2007, 2009, 2011, 2013, 2014), two Mathematics Curriculum Topic Study resources (2006, 2012), and Mathematics Formative Assessment: 75 Practical Strategies for Linking Assessment, Instruction and Learning (2011). Before joining MMSA in 2001 to begin working with teachers, Tobey was a high school and middle school mathematics educator for ten years. She received her BS in secondary mathematics education from the University of Maine at Farmington and her MEd from City University in Seattle. She currently lives in Maine with her husband and blended family of five children. Carolyn B. Arline is a secondary mathematics educator, currently teaching high school students in Maine. Carolyn also works as a teacher leader in the areas of mathematics professional development, learning communities, assessment, systematic school reform, standards-based teaching, learning and grading, student-centered classrooms, and technology. She has previously worked as a mathematics specialist at the Maine Mathematics and Science Alliance (MMSA) and continues her work with them as a consultant. Carolyn is a fellow of the second cohort group of the Governor's Academy for Science and Mathematics Educators and serves as a mentor teacher with the current cohort. She participated as a mathematics mentor in the NSF-funded Northern New England Co-Mentoring Network (NNECN) and continues her role as a mentor teacher. She serves as a board member of the Association of Teachers of Mathematics in Maine (ATOMIM) and on local curriculum committees. Carolyn received her B.S. in secondary mathematics education from the University of Maine.


"One of the most effective tools for teachers to gather valuable formative assessment data. This resource provides an easily accessible, yet powerful way for teachers to elicit and analyze student work that truly reflects student thinking. The probes are targeted to specific concept areas that are easily connected to daily lesson plans and curriculum materials." -- Michelle Parks, Educational Consultant "Using probes with my students has helped me plan and modify my instruction. The range of student explanations elicit rich mathematical discourse." -- Johnette R. Winfrey, Math Specialist "A tool for middle school and high school mathematics teachers that combines the effective use of research, formative assessment, and ready-to-go student tasks. This resource belongs int he hands of any mathematics educator who seeks to balance procedural knowledge, conceptual understanding, and skills for each student." -- Christine Downing, Mathematics School Improvement Coach and Educational Consultant "An efficient teaching tool to uncover and correct misconceptions students have across all mathematics strands. This book provides numerous opportunities for increased student achievement that are both powerful and exciting for educators." -- Jennifer Petersen, K-12 Math Curriculum Facilitator "The book is timely. The probes will be beneficial to teachers and very useful to districts developing local assessments." -- Debra Scarpelli, Mathematics Teacher "The authors contribute to the field by answering the question: How do I help my struggling students who hold a variety of misconceptions or have limited prior knowledge of the content?" -- Lesa M. Covington Clarkson, Assistant Professor "Cheryl Rose and Carolyn Arline provide a valuable resource to teachers who are interested in determining how their students think and answer questions in mathematics classes." -- From the Foreword by Johnny W. Lott

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