Strategies for Teaching Fractions: Using Error Analysis for Intervention and Assessment

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David B. Spangler

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    Preface

    Strategies for Teaching Fractions provides a practical intervention model with targeted hands-on materials for teachers to use when working with struggling learners. This strategy book is a comprehensive resource—providing the “what,” the “why,” and the “how”—to guide and support classroom work with fractions at whatever grades students may need intervention for these skills and concepts. As the second book in a series, Strategies for Teaching Fractions provides a natural continuation of the first book, Strategies for Teaching Whole Number Computation.

    There is probably universal agreement that student proficiency with fractions is essential. According to Foundations for Success: The Final Report of the National Mathematics Advisory Panel (2008), “A major goal for K–8 mathematics education should be proficiency with fractions, for such proficiency is foundational for algebra” (p. xvii).

    In 2010, the Council of Chief State School Officers and the National Governors Association released the Common Core State Standards for Mathematics (CCSS). These standards may be downloaded at http://www.corestandards.org/the-standards. The CCSS are organized into two related categories: (1) Standards for Mathematical Content (defining what students should understand and be able to do) and (2) Standards for Mathematical Practice (describing ways in which students should engage with mathematics based on processes and proficiencies). The CCSS place an emphasis on the teaching of fractions at Grades 3–6. The mathematical content and instructional activities in this book were written to closely align to these standards. Citations to the CCSS are frequently made where key standards are addressed. An overview of the Common Core State Standards for fractions is provided on the following page.

    It should be noted that the topics described in the CCSS carry an expectation that they be mastered by students no later than at the grade levels cited. However, it is assumed that students engage in appropriate prerequisite work prior to those grade levels and meaningful review and extensions subsequent to those grade levels.

    This book also addresses key concepts and skills related to fractions recommended by the National Council of Teachers of Mathematics (NCTM, 2000a, 2000b) in the Number and Operations Content Standard of their Principles and Standards for School Mathematics. At Grades 3–5, the NCTM expectations are for students to use models and benchmarks, to understand the various meanings of fractions, and to find equivalent fractions. At Grades 6–8, students are expected to compare and order fractions and to be fluent with all four operations with fractions to solve problems.

    In 2006, NCTM expanded upon those standards with its publication Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics: A Quest for Coherence. That document describes a new approach to curriculum development that focuses on areas of emphasis within each grade—recommending the most significant mathematical concepts and skills that should be taught at each grade level. Strategies for Teaching Fractions addresses the depth of coverage envisioned by the focal points with respect to number and operations for fractions for Grades 3–6. The focal points include the following: At Grade 3, students understand the meanings of fractions as parts of a whole, parts of a set, or points or distances on a number line. They also compare and order fractions using various strategies, including benchmarks. At Grade 4, students connect fractions and decimals. At Grade 5, students add and subtract fractions with like and unlike denominators. At Grade 6, students multiply and divide fractions. At all grades, students make estimates and solve problems.

    The pedagogy employed in this book is aligned with all five Process Standards outlined in the Principles and Standards for School Mathematics (NCTM, 2000a) as described below.

    • Problem solving: Problem-solving opportunities are embedded throughout the text.
    • Communication: Suggested questions that may be used directly with students are provided throughout the Intervention Activities; suggestions for student writing opportunities are also provided.
    • Reasoning and proof: Many of the suggested questions, along with the writing opportunities, require students to explain their reasoning and thinking.
    • Connections: This book makes connections to the study of probability, measurement, and geometry—thus providing a strong context for fractions. Because probability is often addressed in the final chapter of a textbook (and is often barely touched on in class), it is especially important to integrate work with it when studying fractions.
    • Representation: The concept development in the Intervention Activities utilizes manipulatives, number lines, fraction strips and circles, and other models and diagrams.

    Three types of knowledge crucial for teaching mathematics at any level are described at right. O'Donnell (2009) citing Hill et al. (2004) notes that teachers illustrate pedagogical content knowledge by “generating representations, interpreting student work, and analyzing student mistakes.” Strategies for Teaching Fractions addresses all three types of knowledge in a concise, user-friendly way. With a focus on how to apply specific pedagogy to specific content, this resource is especially strong in the area of pedagogical content knowledge. Because this book develops mainstream mathematics concepts in a way that is truly meaningful for students, teachers should find this book to be an effective supplement to any textbook program—from those on the traditional end of the continuum to those that are based on reform.

    Research Base for the Book

    Strategies for Teaching Fractions is informed by academic research conducted and analyzed during the past 35 or more years. In developing this book, the author provides a mix of this extensive research base with his personal experience of about 40 years in mathematics education. The author's experience includes teaching at various levels, including classroom (Grades 5–8), community college (in a remedial teaching lab), and university (methods courses).

    Although some errors that occur in students' work with fraction concepts and computation are due to carelessness or incorrect recall of number facts, many are due to misconceptions and the use of incorrect strategies. According to Pincus et al. (1975), “Too often, when teachers find errors in a child's work, they mark the example wrong, assume that the child did not master the basic facts, and prescribe further drill. Careful analysis of errors through observation and interviews with individual children is essential” (p. 581).

    A key premise of this book is that if teachers (1) analyze student work for error patterns (revealed through diagnostic tests, practice, activities, and student discourse/oral interviews) and (2) then provide timely, targeted, and meaningful intervention, student errors will decrease in frequency—while at the same time student understanding of concepts will increase. In particular, error analysis should enable a teacher to build on what students are doing correctly, while focusing on those areas for which the student needs additional work. By taking into account academic research on how students learn, the intervention strategies illustrated in this book should result in improved student performance and more positive student dispositions toward learning mathematics.

    How the Book Is Organized

    The book begins with a substantial section, “A Look at the Academic Research: Intervention in the Mathematics Classroom.” This academic research supports the goals and premises of the book, the pedagogical practices utilized in the Intervention Activities, and other aspects of intervention. These practices include accessing language, activating prior knowledge, scaffolding, using representations, using estimation and mental math, introducing alternative algorithms, differentiating instruction, participating in instructional games, using technology, and more.

    The research section concludes with a survey of key research findings related to the teaching of fraction concepts and computation. Included is an extensive discussion on the various meanings, or models, of fractions (addressed by NCTM at left). A hallmark of this book is the ongoing development of fraction concepts and computation through the use of a variety of fraction models.

    Next is a two-part section titled “Big Ideas in Fractions and Problem Solving.” The first part provides an overview of key understandings related to fraction concepts and fraction computation. These ideas include an overview of terminology, properties, and overall fraction number sense. The second part of this section is titled “Actions and Operations: Problem Structures for Addition, Subtraction, Multiplication, and Division.” Here, illustrative models based on fractions are provided for teachers to use with children to help them decide which operation to use to solve a given problem by thinking of actions that can be done with objects (or other representations) that relate to mathematical operations. By focusing on real-world fraction contexts for the operations, key connections are made between problem solving and computation. For illustrative models for each operation based on whole numbers, see Spangler (2010, pp. 17–23).

    According to Kilpatrick, Swafford, and Bradford (2001), “Studies in almost every domain of mathematics have demonstrated that problem solving provides an important context in which students can learn about number and other mathematical topics” (p. 420). The author suggests that the problem structures described in this section be used with students as they study each operation. Students should then be asked to write their own word problems based on these structures to enhance their understanding of the operation. Because problem solving is not the major focus of this book, teachers are encouraged to integrate additional resources for problem solving when they teach computation.

    Three main units then follow—one for fraction concepts, one for addition/subtraction of fractions and mixed numbers, and one for multiplication/division of fractions and mixed numbers. The units follow a predictable format. Each begins with a diagnostic test (in multiple-choice format), followed by an Item Analysis Table that keys student incorrect test responses to specific error patterns. (Each distractor on the tests is based on a specific error pattern.) A detailed section of error patterns with step-by-step Intervention Activities (the heart of the book) then follows. The Intervention Activities often include an alternative algorithm to go along with the traditional algorithm. It should be noted that the Intervention Activities (both for the traditional and for the alternative algorithms) may be used with students as part of the initial instruction—and are not just intended for use with students after they may have struggled with a concept.

    A hallmark of this book is its strong focus on teaching for understanding in developing the Intervention Activities for fraction concepts and for each operation. Guided questions (with suggested student responses) and hands-on experiences (including a guided discovery lesson to introduce each operation) are used to achieve student understanding. Following the Intervention Activities for each unit is a short section of practice (keyed to the Item Analysis Table). Supplemental material (blacklines) for estimation, instructional games, and follow-up activities is also included for each unit.

    Each main unit, along with the sections on academic research and “Big Ideas,” concludes with a set of Questions for Teacher Reflection. These open-ended questions are intended to provide springboards for discussion among teachers or preservice teachers who may be using this book in a professional development setting, workshop, or methods course.

    Intended Uses and Audience for the Book

    Strategies for Teaching Fractions is intended to serve a wide audience of educators, and there are a variety of ways to use the book.

    • Intervention program for fraction concepts and computation in pullout or full-classroom situations: The book may be used as a full-fledged intervention program for work with fractions. The book provides a comprehensive tool for quickly diagnosing and pinpointing trouble spots students are encountering—with specific Intervention Activities tailored to address the weaknesses identified.
    • Resource for elementary and middle grades classroom teachers: The book describes the types of errors struggling students frequently make with fraction concepts and computation. Teachers, especially those with limited experience teaching this content, should find this knowledge valuable as they teach these concepts.
    • Resource for special education teachers, including those who work with students in coteaching situations: These teachers—from the elementary level through high school—will find a wealth of material in this book to help them differentiate instruction for individual students or students working in small groups.
    • Resource for teachers and tutors working in developmental labs at the high school and community college levels: The book provides fresh approaches to help students who have been struggling for years to learn how to work with fractions.
    • Resource for instructional supervisors and curriculum coordinators: These educators should find this book to be an important tool to use as they work with their teachers.
    • Text that may be used with teachers and preservice teachers in mathematics methods courses, workshops, and professional development programs (including online programs): Educators at all grade levels should embrace this book as a key source of pedagogical content knowledge that they can directly use in their classrooms. The Questions for Teacher Reflection provide opportunities for teacher discussion and/or assignments in such settings.
    • Resource for educators interested in academic research on intervention: This book provides a handy compilation of important research findings of the past 35-plus years related to intervention for fraction concepts and computation.

    Acknowledgments

    I would like to thank Jessica Allan, Cassandra Seibel, Lisa Whitney, Cate Huisman, Jane Haenel, and Charline Wu of Corwin for their professional expertise on this project. I would also like to thank the many reviewers who provided much insight and direction in the development of the manuscript. Their contributions are greatly valued—and many of their suggestions were incorporated into the book.

    Additionally, Corwin gratefully acknowledges the following peer reviewers for their editorial insight and guidance:

    • Carol Amos
    • Math Teacher
    • Twinfield Union School
    • Plainfield, VT
    • Renee Bernhardt
    • Ed.S. Curriculum and Instruction
    • Cherokee County School District
    • Canton, GA
    • Marcia Carlson
    • Sixth Grade Math Teacher
    • Crestview Elementary School
    • Clive/West Des Moines, IA
    • Scott Currier
    • Math Teacher
    • Belmont High School
    • Belmont, NH
    • Susan D'Angelo
    • Teacher, Gifted Education
    • Pineview School for the Gifted
    • University of South Florida
    • Nokomis, FL
    • Jean Krsak
    • Learning Support Faculty
    • California State University, Fullerton
    • Imperial Beach, CA
    • Katharine Olson
    • Assistant Superintendent for Curriculum, Instruction, and Assessment
    • Northbrook School District 27
    • Northbrook, IL
    • Cathy Patterson
    • Assistant Principal
    • Evergreen Elementary
    • Diamond Bar, CA

    About the Author

    David B. Spangler has devoted his entire professional career of about 40 years to mathematics education. After graduating from Northern Illinois University with BS and MS degrees, he began his career as a mathematics teacher in middle school, where he taught in an individualized setting. Later he taught at Triton Community College, where he gained direct experience interacting with struggling students in a developmental math laboratory. He has also worked many years as a mathematics editor and editorial director for major publishers in the el-hi textbook industry.

    David is the author of Strategies for Teaching Whole Number Computation: Using Error Analysis for Intervention and Assessment (Corwin, 2010), Mathematics Explorations: Detective-style Activities for the Real World (Good Year Books, 2008, 2011), and Math for Real Kids (Good Year Books, 1997, 2005). He has also written numerous articles for journals, including Mathematics Teaching in the Middle School.

    Currently David teaches methods courses through National-Louis University and ActiveMath® Workshops, a professional development company he cofounded in 1994 (http://www.activemath.com). The courses, workshops, and in-district training he facilitates address special-needs students, intervention, and hands-on activities. He also provides staff development that specifically addresses the concepts in this book and in Strategies for Teaching Whole Number Computation. David has worked with literally thousands of students and teachers during his career.

    David is a frequent speaker at mathematics conferences of professional organizations, where he addresses teaching mathematics for meaning, teaching through real-world applications, and teaching through humor. He also delivers after-dinner talks on the subject of mathematical humor. The talk features mathematical blunders from the real world and suggestions for addressing mathematical illiteracy in our society.

    As a mathematics educator, David's goal is to teach mathematics for meaning rather than in a way that promotes rote memorization. This book was written to help teachers achieve that goal. Based on his extensive experience in mathematics education, David is uniquely qualified to write this book.

    David lives with his wife, Bonnie, in Northbrook, Illinois. They have three grown children, Ben, Jamie, and Joey. He would like to thank all of them for their love, support, and encouragement throughout his entire career.

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    Technology Resources Online

    The websites listed below provide links to suggested online interactive software programs (applets) that provide virtual manipulatives, games, activities, and tutorials that you may want to integrate into your lessons. For each website, the resources are organized according to the unit of this text that they support.

    AAA Math (This site provides a comprehensive set of interactive arithmetic lessons, with a focus on practice. Associated games are included.)

    http://www.aaastudy.com/fra.htm

    • Resources for Unit 1—Fraction Concepts

      Basic Fractions; Comparing Fractions; Converting Fractions; Divisibility Rules to Help Simplify Fractions; Relationships (relating fractions and decimals)

    • Resources for Unit 2—Addition/Subtraction of Fractions and Mixed Numbers Adding Fractions; Subtracting Fractions
    • Resources for Unit 3—Multiplication/Division of Fractions and Mixed Numbers Multiplying Fractions; Dividing Fractions

    http://Mr.Nussbaum.com (This site provides interactive games and other online resources for Grades K–8 in many disciplines.)

    http://www.mrnussbaum.com/mathgames.htm

    • Resources for Unit 1—Fraction Concepts

      Clara Fraction's Ice Cream Shop (game reinforcing converting improper fractions to mixed numbers); Death to Decimals (game reinforcing converting fractions to decimals)

    • Resources for Unit 2—Addition/Subtraction of Fractions and Mixed Numbers Tony Fraction's Pizza Shop (game reinforcing multiplying a fraction and a whole number)

    National Library of Virtual Manipulatives (This site, supported by the National Science Foundation, provides K–12 interactive, web-based virtual manipulatives and tutorials.)

    http://nlvm.usu.edu/en/nav/category_g_2_t_1.html

    • Resources for Unit 1—Fraction Concepts

      Factor Tree; Fraction Pieces; Fractions–Equivalent; Fractions–Naming; Fractions–Part of a Whole; Fractions–Comparing

    • Resources for Unit 2—Addition/Subtraction of Fractions and Mixed Numbers Fractions–Adding; Fractions–Rectangle Multiplication

    NCTM Illuminations (This site of the National Council of Teachers of Mathematics “illuminates” the teaching and learning of mathematics with online activities and games for each content strand of its Principles and Standards for School Mathematics.)

    http://illuminations.nctm.org/Activities.aspx?grade=2&grade=3&srchstr=fractions

    Shodor (Project Interactivate) (This site provides interactive activities that promote the use of modeling and simulation technologies. Each activity includes an instructor page that provides a list of the standards addressed, textbook alignment, and related resources.)

    http://www.shodor.org/interactivate/activities

    • Resources for Unit 1—Fraction Concepts

      Equivalent Fraction Finder; Equivalent Fractions Pointer; Fraction Four (game); Fraction Pointer (involves drawing visual models to represent fractions that are given on a number line); Fraction Sorter

    • Resources for Unit 3—Multiplication/Division of Fractions and Mixed Numbers Tortoise and Hare Race (activity based on Zeno's Paradox)

    TeacherLED Interactive Whiteboard Resources for Teachers (This site provides teaching and learning resources designed to be used with an interactive whiteboard.)

    http://www.teacherled.com/all-interactive-whiteboard-resources

    • Resources for Unit 1—Fraction Concepts

      Venn Factors; Venn Multiples (finding common factors or multiples using a Venn diagram); Lowest Common Multiple Carousel; Prime Factor Tree; Equivalent Fractions; Slide Puzzle Maths (puzzle game involving ordering fractions)

    The Math Forum: “MathTools” (MathTools is a digital library of K–12 online math resources. The site also includes a discussion forum.)

    http://mathforum.org/mathtools/sitemap2/m5

    • Resources for Unit 1—Fraction Concepts

      Writing Fractions; Equivalent Fractions; Comparing and Ordering; Simplifying Fractions; Fractions–Decimals

    • Resources for Unit 2—Addition/Subtraction of Fractions and Mixed Numbers Adding Fractions; Subtracting Fractions
    • Resources for Unit 3—Multiplication/Division of Fractions and Mixed Numbers Multiplying Fractions; Dividing Fractions

    Visual Fractions (This site provides online tutorials and games.)

    http://www.visualfractions.com

    • Resources for Unit 1—Fraction Concepts

      Identify Fractions; Rename Fractions; Compare Fractions; “Find Grampy,” “Find Grampy-Strict,” “Find Grammy” (games)

    • Resources for Unit 2—Addition/Subtraction of Fractions and Mixed Numbers Add Fractions; Subtract Fractions; “Platform Scales Addition,” “Fraction Scales Subtraction” (games)
    • Resources for Unit 3—Multiplication/Division of Fractions and Mixed Numbers Multiply Fractions; Divide Fractions

    CORWIN: A SAGE Company

    The Corwin logo—a raven striding across an open book—represents the union of courage and learning. Corwin is committed to improving education for all learners by publishing books and other professional development resources for those serving the field of PreK–12 education. By providing practical, hands-on materials, Corwin continues to carry out the promise of its motto: “Helping Educators Do Their Work Better.”

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