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| teaching fractions and ratios for understanding: Teaching Fractions and Ratios for Understanding Susan J. Lamon, 1999 A text that crosses traditional boundaries to include many of the elements that are integrated in the teaching/learning enterprise: mathematics content, teacher understanding, student thinking, teaching methods, instructional activities, and assessment. |
| teaching fractions and ratios for understanding: MORE! Teaching Fractions and Ratios for Understanding Susan J. Lamon, 2013-06-17 More is not an answer key but a resource that provides the scaffolding for the groundbreaking approach to fraction and ratio instruction presented in its companion text, Teaching Fractions and Ratios. Keeping the focus on the reasoning needed to properly understand and teach rational numbers, More shows teachers how to engage in powerful ways of thinking so that they can, in turn, enhance the mathematical education of their students. Like its companion text, More has been heavily expanded and reorganized, including even more student work, templates for key manipulatives, and an emphasis on applications to everyday life. Based on the content chapters in Teaching Fractions and Ratios, each chapter includes: In-depth Discussions of selected problems and their solutions. Supplementary Activities and a collection of challenging problems involving fractions. Praxis Preparation Questions geared to the content of each chapter. |
| teaching fractions and ratios for understanding: More Teaching Fractions and Ratios for Understanding Susan J. Lamon, 2011 |
| teaching fractions and ratios for understanding: Developing Essential Understanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics in Grades 6-8 Joanne Elizabeth Lobato, Amy B. Ellis, Randall Inners Charles, 2010 A series for teaching mathematics.--P. [1] of cover. |
| teaching fractions and ratios for understanding: Putting Essential Understanding of Fractions Into Practice in Grades 3-5 Kathryn Bouchard Chval, John Lannin, Dusty Jones, 2013 Do your students suppose that 1/3 is greater than 1/2, since 3 is greater than 2? Do they believe that having “halves” means having two, and only two, congruent “pieces” of a whole? What tasks can you offer—what questions can you ask—to determine what your students know or don’t know—and move them forward in their thinking? This book focuses on the specialised pedagogical content knowledge that you need to teach fractions effectively in grades 3–5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions—not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of fractions. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. |
| teaching fractions and ratios for understanding: Fractions, Decimals, Ratios, and Percents , 1994 |
| teaching fractions and ratios for understanding: Unpacking Fractions Monica Neagoy, 2017-03-10 For years, the teaching and learning of fractions has been associated with rote memorization. But this mechanical approach to instruction—which strips students of an ability to reason or make sense of math—has resulted in a failure of understanding. Author Monica Neagoy, drawing on decades of research studies, evidence from teacher practice, and 25 years of experience working around the world with teachers, students, and parents, addresses seven big ideas in the teaching and learning of fractions in grades 2–6. Each idea is supported by a vignette from a real classroom, common misconceptions, a thorough unpacking of productive mathematical thinking, and several multistep and thought-provoking problems for teachers to explore. She offers three fundamental reasons why it’s imperative for us to take a closer look at how we teach fractions: 1. Fractions play a key role in students’ feelings about mathematics. 2. Fractions are fundamental to school math and daily life. 3. Fractions are foundational to success in algebra. While a solid grounding in algebra is necessary for a STEM career, the worthy goal of “algebra for all” will not be possible until “fractions for all” is a reality. Unpacking Fractions provides teachers with concrete strategies for achieving that reality—in short, helping all students gain the knowledge they need to feel at ease with fractions. |
| teaching fractions and ratios for understanding: A Focus on Ratios and Proportions Marjorie M. Petit, Robert E. Laird, Matthew F. Wyneken, Frances R. Huntoon, Mary D. Abele-Austin, Jean D. Sequeira, 2020-05-17 This resource offers a groundbreaking effort to make mathematics education research on ratios and proportions readily accessible and understandable to preservice and in-service teachers of grades 6 to 8. Using extensive annotated samples of student work and based on research gathered in the Ongoing Assessment Project (OGAP), A Focus on Ratios and Proportions teaches readers how students develop understanding and fluency involving ratio and proportion concepts. Special features include: A close focus on student work, including 150+ annotated pieces of student work, to help teachers improve their ability to recognize, assess and monitor their students’ errors and misconceptions, as well as their developing conceptual understanding. A focus on the OGAP Ratios and Proportions Progression, based on research conducted with hundreds of teachers and thousands of pieces of student work. Sections on how Common Core State Standards for Math (CCSSM) are supported by math education research. Student work samples and vignettes to illuminate the research, as well as end of chapter Looking Back questions and Instructional Links, which allow teachers to analyze evidence of student thinking and strategies and consider instructional responses. An accompanying eResource, available online, offers an answer key as well as extensive explanation of the Looking Back questions. Like A Focus on Multiplication and Division and A Focus on Fractions, this book is designed to bridge the gap between what math education researchers know and what teachers need to know in order to better understand evidence in student work and make effective instructional decisions. |
| teaching fractions and ratios for understanding: Fractions in Disguise Edward Einhorn, 2014-03-25 Equal parts action and humor add up to a wholly entertaining introduction to simplifying fractions, in this one-of-a-kind math picture book story. When a valuable fraction goes missing, George Cornelius Factor (a.k.a. GCF) vows to track it down. Knowing that the villainous Dr. Brok likes to disguise his ill-gotten fractions, G.C.F. invents a Reducer—half ray gun, half calculator— that strips away the disguise, reducing the fraction to its lowest common denominator and revealing its true form. With the Reducer in hand, George seeks out Dr. Brok in hopes of retrieving the missing fraction. David Clark’s illustrations are packed with humorous details as well as clearly defined fractions and their corresponding reduction equations. |
| teaching fractions and ratios for understanding: Let's Play Math Denise Gaskins, 2012-09-04 |
| teaching fractions and ratios for understanding: Real-Life Math Tom Campbell, 1998 Covers key middle school and high school topics in the context of everyday life scenarios. Teaches fractions, equivalent fractions, finding patterns, ratio, solving proportions, rates, and unit pricing. |
| teaching fractions and ratios for understanding: Teaching Fractions and Ratios for Understanding Susan J. Lamon, 2020-03-23 Written in a user-friendly, conversational style, the fourth edition of this groundbreaking text helps pre-service and in-service mathematics teachers build the comfort and confidence they need to begin talking to children about fractions and ratios, distilling complex ideas and translating research into usable ideas for the classroom. For two decades, Teaching Fractions and Ratios for Understanding has pushed readers beyond the limits of their current understanding of fractions and rational numbers, challenging them to refine and explain their thinking without falling back on rules and procedures they have relied on throughout their lives. All of the material offered in the book has been used with students, and is presented so that readers can see the brilliance of their insights as well as the issues that challenge their understanding. Each chapter includes children’s strategies and samples of student work for teacher analysis, as well as activities for practicing each thinking strategy, designed to be solved without rules or algorithms, using reasoning alone. The fourth edition of this popular text has been updated throughout and includes new examples of student work, updated artwork, and more. As with previous editions, an equally valuable component of this text is the companion book MORE! Teaching Fractions and Ratios for Understanding (2012), a supplement that is not merely an answer key but a resource that provides the scaffolding for the groundbreaking approach to fraction and ratio instruction explored here. MORE! includes in-depth discussions of selected problems in the main text, supplementary activities, Praxis preparation questions, more student work, and templates for key manipulatives. |
| teaching fractions and ratios for understanding: A Focus on Fractions Marjorie M. Petit, Robert E. Laird, Edwin L. Marsden, Caroline B. Ebby, 2015-07-24 A Focus on Fractions is a groundbreaking effort to make the mathematics education research on how students develop their understanding of fraction concepts readily accessible and understandable to pre- and in-service K– 8 mathematics educators. Using extensive annotated samples of student work, as well as vignettes characteristic of classroom teachers’ experiences, this book equips educators with the knowledge and tools to reveal students’ thinking so that they can modify their teaching and improve student learning of fraction concepts. A Focus on Fractions 2nd edition includes sections on the Common Core State Standards for Mathematics and the Ongoing Assessment Project (OGAP) Fraction Framework integrated into each chapter as well as a new chapter on the OGAP Fraction Progression and how it can be used for formative assessment purposes. This updated edition assists teachers in translating research findings into their classroom practice by conveying detailed information about how students develop fraction understandings. Additional images and examples serve to flesh out and supplement the newly-introduced concepts in this updated and expanded edition. Special Features: Looking Back Questions at the end of each chapter provide teachers the opportunity to analyze student thinking and consider instructional strategies for their own students. Instructional Links help teachers relate concepts from the chapter to their own instructional materials and programs. Big Ideas frame the chapters and provide a platform for meaningful exploration of the teaching of fractions. Answer Key posted online offers extensive explanations of in-chapter questions. New sections devoted to the CCSSM and OGAP Fraction Progression are woven throughout the book as well as a new stand alone chapter on the OGAP Fraction Progression. The OGAP Fraction Framework is an all-new eResource, now available as a free download from the book’s website: www.routledge.com/9781138816442. |
| teaching fractions and ratios for understanding: Strategies for Teaching Fractions David B. Spangler, 2011-08-17 Analyzing student work for error patterns (such as on a diagnostic test) and then providing timely, specific, and meaningful intervention will decrease the probability that the errors will be repeated in the future. Taking into account academic research on how students learn, Spangler provides a practical intervention model and specific hands-on materials for teachers to use when working with the many students who struggle with fractions. The step-by-step intervention strategies illustrated in this book will result in improved student performance and improved student dispositions towards mathematics. It will contain the following elements: reproducibles for fully functional diagnostic tests practice pages for exercises keyed to the diagnostic tests and error patterns, and pages to practice alternative algorithms and estimation teacher resources for hands-on activities, game sheets and pieces, and more--Provided by publisher. |
| teaching fractions and ratios for understanding: Putting Essential Understanding of Ratios and Proportions Into Practice in Grades 6-8 Travis Olson, Melfried Olson, Hannah Slovin, 2015 Focuses on the specialized pedagogical content knowledge that you need to teach ratios and proportions effectively in grades 6-8. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with ratios and proportions. |
| teaching fractions and ratios for understanding: More: In-depth Discussion of the Reasoning Activities in Teaching Fractions and Ratios for Understanding Susan J Lamon, 2005-10 First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company. |
| teaching fractions and ratios for understanding: Developing Essential Understanding of Rational Numbers for Teaching Mathematics in Grades 3/5 Carne Barnett-Clarke, 2010 What is the relationship between fractions and rational numbers? Can you explain why the product of two fractions between 0 and 1 is less than either factor? How are rational numbers related to irrational numbers, which your students will study in later grades? How much do you know… and how much do you need to know? Helping your upper elementary school students develop a robust understanding of rational numbers requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about rational numbers. It is organised around four big ideas, supported by multiple smaller, interconnected ideas-essential understandings. Taking you beyond a simple introduction to rational numbers, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls and dispel misconceptions. You will also learn to develop appropriate tasks, techniques and tools for assessing students’ understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently. |
| teaching fractions and ratios for understanding: Fostering Geometric Thinking Mark Driscoll, Rachel Wing Dimatteo, Michael Egan, Johannah Nikula, 2017-05-12 The Trends in International Mathematics and Science Study has shown that American schools have consistently helped students understand algebraic and statistical concepts, leading to high achievement internationally. Now it's time to do the same for geometry. Mark Driscoll, author of the powerful and popular Fostering Algebraic Thinking, takes up the challenge and leads you to new, research-based ways to improve how your students conceptualize and apply geometric ideas. With Fostering Geometric Thinking any math teacher can discover essential, practical ideas for helping students cultivate geometric habits of mind that lead to success in this crucial mathematical subject. The book focuses on rigorous, problem-based teaching that encourages students to deepen their thinking in three key geometric strands: geometric properties geometric transformations measurement of geometric objects. Fostering Geometric Thinking shows you how the interplay of these strands helps students devise multiple solutions and develop a broader sense of geometric principles. It's loaded with helpful resources, including: engaging problems to use in your classroom examples of student solutions to these problems transcripts of classroom interactions online resources featuring in-the-field footage of students working through open-ended problems highlighted in the book. Geometry is a vital component of mathematical understanding, and it's time that it received the same attention that algebra and statistics do. With engaging problems and straightforward suggestions that can help students deepen, recognize, and describe their thinking, Fostering Geometric Thinking is the resource you need to ensure that when it comes to geometry, your students know all the angles. |
| teaching fractions and ratios for understanding: Fractions Janette Maree Bobis, 2011 |
| teaching fractions and ratios for understanding: The Cambridge Handbook of Cognition and Education John Dunlosky, Katherine A. Rawson, 2019-02-07 This Handbook reviews a wealth of research in cognitive and educational psychology that investigates how to enhance learning and instruction to aid students struggling to learn and to advise teachers on how best to support student learning. The Handbook includes features that inform readers about how to improve instruction and student achievement based on scientific evidence across different domains, including science, mathematics, reading and writing. Each chapter supplies a description of the learning goal, a balanced presentation of the current evidence about the efficacy of various approaches to obtaining that learning goal, and a discussion of important future directions for research in this area. It is the ideal resource for researchers continuing their study of this field or for those only now beginning to explore how to improve student achievement. |
| teaching fractions and ratios for understanding: Teaching Maths to Pupils with Different Learning Styles Tandi Clausen-May, 2005-09-28 'Tackles an area of the curriculum many teachers lack confidence in.' 'Lots of good starting points'. 'Spans a lot of material and is strong on diverse learning styles.' 'Clear explanation and good visual layour, very innovative in approach.' - Judging Panel for NASEN/TES Book Award 'The book is rich in lively teaching suggestions and in insights into the impact of different forms of explanation' - Debate '[C]arries us away from narrow views of ability and special needs and into the consideration of difference. The author takes us through lively discussions of many aspects of mathematics learning. Each section offers learning and teaching ideas involving visual and kinaesthetic approaches. The book is a compendium of sound ideas rather than a collection of startlingly new approaches. But throughout it has the great strength of being exceptionally clear in its arguments, descriptions and drawings. The design is generally helpful with plenty of illustrations, as befits the book's message. There are handy pages of photocopiable resources. This is a lively and often passionate account of ways of ensuring that multi-sensory approaches infect mathematics learning. As the author says, pictures in the mind can help all pupils. We might add, They help all teachers too' - TES Extra for Special Needs 'If you have found pupils struggling to understand some aspects of mathematics at any age then this book is for you. It is a very readable book that would interest all those who work in classrooms, whether as a teacher or support worker with all ages and abilities, for those who work with older pupils as it gives possible approaches to use with those for whom basic skills are weak or have difficulty in understanding some of the concepts required of GCSE examinations' - Alison Parish, Second in Mathematics Department, Stowmarket High School, Suffolk Read the full review as posted on the Association of Teachers of Mathematics website! 'It is a highly practical book. One strength is the way that it develops a topic from the very basics through to the harder concepts. There are a large number of activities that are 'ready to run' but these really are just a starting point for teachers to begin thinking about teaching topics in a different way, and from these teachers will be able to develop their own approach. Although this book is focusing on pupils who are visual and kinaesthetic learners, the great majority of learners adopt a mixture of learning styles, so this approach will benefit the entire class. Worth a read!' - Maths Coordinator's File 'This excellent and very informative teaching resource is about teaching mathematics to pupils who have learning differences. [It] is very practical and easy to read. A really nice feature is the inclusion of photocopiable resource sheets allowing readers to try out easily the ideas suggested in the book. This resource is highly recommended and will be very suitable for maths teachers in primary and secondary schools, SENCOs and teaching assistants' - British Journal of Special Education 'This book is about making mathematics visible and tangible -- not something that just lies flat on the page. Dipping into it will provide instantly usable suggestions across a variety of topics at different levels: from early number concepts through to fractions and ratios, algebra, aspects of geometry (including angles and circles), and data handling. When you get a chance to read it more thoroughly you will find arguments for using these approaches, consideration of some of the pitfalls to avoid, and inspiration to develop different ways of helping students to achieve deep and connected understandings. For any teacher who wants to provide students with opportunities for visual and kinaesthetic learning in mathematics' - The Australian Association of Mathematics Teachers Inc. 'A very good book, offering teachers, SENCOs and teaching assistants guidelines, strategies and practical activities to access the thought processes of pupils with different learning styles. It has an easy-to-read format giving suggestions, rather than dictat, on the use of models to think and is a unique document for those who have input into the furthering of the teaching and learning of mathematics' - Mathematics in School How can you make maths exciting and meaningful for all your pupils? Some pupils find even basic concepts in mathematics difficult to grasp and it can be a challenge to make lessons accessible to all. This book offers practising teachers a range of approaches to making maths clear for struggling students. It looks at the different ways in which maths can be taught so that pupils with different learning styles can be stimulated. Maths is visible and tangible - not something that just lies flat on the page. Included are: - ideas to be used in lessons - suggestions for exciting, visual ways to teach basic concepts - lots of practical advice and guidance. The book shows teachers how to unlock mathematics for all their learners, and it encourages the use of a variety of methods to teach the subject. It provides a valuable resource for maths teachers in both primary and secondary schools, for SENCOs and teaching assistants, and for those delivering initial teacher training or inservice courses. Tandi Clausen-May is an educational researcher responsible for the development of a range of mathematics curriculum and assessment materials. She delivers popular workshops on teaching mathematics around the United Kingdom. She also writes regular articles on mathematics teaching for educational journals and newspapers. |
| teaching fractions and ratios for understanding: Pictorial Mathematics Guillermo Mendieta, 2006-02 Pictorial Mathematics is a comprehensive and engaging resource for teaching and learning second through algebra level mathematics. It uses multiple representations and effective visuals to help learners with a wide variety of learning styles to develop a strong conceptual understanding of each concept. Pictorial Mathematics provides the perfect bridge between the abstract and the concrete. Its 400 pages are packed with invaluable tools to help teachers, parents and the learner develop meaning, connections and a deeper conceptual understanding of key mathematical concepts. Inside, you?ll find such resources as:More than 1,000 engaging visual exercises, Powerful and engaging models for the development of conceptual understanding of place value, fractions, ratios, geometry, the four operations and algebra concepts, Energizing tasks -- for small groups, large classrooms, or individuals alike, Master guides to create personalized pictorial problems, Tools to differentiate instruction, A complete set of 38 pictorial templates: from printable manipulatives to graphing paper with suggested activities for these. For previews of the book go to www.pictorialmath.com. |
| teaching fractions and ratios for understanding: Ratio and Proportion David Ben-Chaim, Yaffa Keret, Bat-Sheva Ilany, 2012-07-30 Ratio and Proportion—Research and Teaching in Mathematics Teachers’ Education offers its readers an intellectual adventure where they can acquire invaluable tools to turn teaching ratio and proportion to professionals and school children into an enjoyable experience. Based on in-depth research, it presents a deep, comprehensive view of the topic, focusing on both the mathematical and psychological-didactical aspects of teaching it. The unique teaching model incorporates both theoretical and practical knowledge, allowing instructors to custom-design teacher courses according to their speci?c needs. The book reports on hands-on experience in the college classes plus teachers’ experience in the actual classroom setting. An important feature is the extensive variety of interesting, meaningful authentic activities. While these activities are on a level that will engage pre- and in-service mathematics teachers in training, most can also be utilized in upper elementary and middle school classes. Accompanying the majority of these activities are detailed remarks, explanations, and solutions, along with creative ideas on how to conduct and expand the learning adventure. While primarily written for educators of mathematics teachers, this book can be an invaluable source of information for mathematics teachers of elementary and middle school classes, pre-service teachers, and mathematics education researchers. |
| teaching fractions and ratios for understanding: Number Talks Sherry Parrish, Ann Dominick, 2016 This resource was created in response to the requests of teachers--those who want to implement number talks but are unsure of how to begin, and those with experience who want more guidance in crafting purposeful problems.--Page 4 de la couverture. |
| teaching fractions and ratios for understanding: Word Problems with Fractions Paul R. Robbins, Sharon K. Hauge, 1999 Prepares all students for standardized testing Builds essential critical-thinking and problem-solving skills Provides real-life situations for meaningful connections to science, computer science, math history, and other topics See other Word Problems titles |
| teaching fractions and ratios for understanding: Integrating Research on Teaching and Learning Mathematics Elizabeth Fennema, Thomas P. Carpenter, Susan J. Lamon, 1991-01-01 During the last decade there were significant advances in the study of students' learning and problem solving in mathematics, and in the study of classroom instruction. Because these two research programs usually have been conducted individually, it is generally agreed now that there is an increasing need for an integrated research program. This book represents initial discussions and development of a unified paradigm for studying teaching in mathematics that builds upon both cognitive as well as instructional research. |
| teaching fractions and ratios for understanding: Understanding Numbers in Elementary School Mathematics Hongxi Wu, 2011 This is a textbook for pre-service elementary school teachers and for current teachers who are taking professional development courses. By emphasizing the precision of mathematics, the exposition achieves a logical and coherent account of school mathematics at the appropriate level for the readership. Wu provides a comprehensive treatment of all the standard topics about numbers in the school mathematics curriculum: whole numbers, fractions, and rational numbers. Assuming no previous knowledge of mathematics, the presentation develops the basic facts about numbers from the beginning and thoroughly covers the subject matter for grades K through 7. Every single assertion is established in the context of elementary school mathematics in a manner that is completely consistent with the basic requirements of mathematics. While it is a textbook for pre-service elementary teachers, it is also a reference book that school teachers can refer to for explanations of well-known but hitherto unexplained facts. For example, the sometimes-puzzling concepts of percent, ratio, and rate are each given a treatment that is down to earth and devoid of mysticism. The fact that a negative times a negative is a positive is explained in a leisurely and comprehensible fashion. |
| teaching fractions and ratios for understanding: The Teaching of Fractions Edward Wildeman, 1923 |
| teaching fractions and ratios for understanding: More Teaching Fractions and Ratios for Understanding Susan J. Lamon, 2014-04-04 This book is a supplement to the text Teaching Fractions and Ratios for Understanding. It is not merely an answer key, but a resource that includes in-depth discussions of the problems in the text; develops and extends discussion of the issues, teaching problems, and other considerations raised in the chapters; and contains additional problems--with and without solutions--that instructors may find helpful for assessment purposes. In the second edition, MORE has been reorganized so that supplementary activities immediately follow the discussions of the activities from the text. Solutions to these supplemental activities have been removed to facilitate the use of MORE in courses. Nevertheless, because the number of activities in each chapter of the text has increased and complete discussions of all of them are provided in MORE, there is ample opportunity to learn reasoning methods. |
| teaching fractions and ratios for understanding: Glencoe Math, Course 1, Student Edition McGraw-Hill Education, 2014-06-06 The Glencoe Math Student Edition is an interactive text that engages students and assist with learning and organization. It personalizes the learning experience for every student. The write-in text, 3-hole punched, perfed pages allow students to organize while they are learning. |
| teaching fractions and ratios for understanding: Saxon Math 6/5 Wrialey, Saxon Publishers, 2004-09 |
| teaching fractions and ratios for understanding: Developing Essential Understanding of Geometry for Teaching Mathematics in Grades 6-8 Nathalie Sinclair, David Pimm, Melanie Skelin, 2012-01 Why are there so many formulas for area and volume, and why do some of them look alike? Why does one quadrilateral have no special name while another has several, like square, rectangle, rhombus, and parallelogram—and why are all these names useful? How much do you know … and how much do you need to know? Helping your students develop a robust understanding of geometry requires that you understand this mathematics deeply. But what does that mean? This book focuses on essential knowledge for teachers about geometry. It is organized around four big ideas, supported by multiple smaller, interconnected ideas—essential understandings. Taking you beyond a simple introduction to geometry, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students—and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing students’ understanding of the topic. |
| teaching fractions and ratios for understanding: Extending Children's Mathematics Susan B. Empson, Linda Levi, 2011 With the collaboration of a number of dedicated teachers and their students, Susan Empson and Linda Levi have produced a volume that is faithful to the basic principles of CGI while at the same time covering new ground with insight and innovation. -Thomas P. Carpenter This highly anticipated follow-up volume to the landmark Children's Mathematics: Cognitively Guided Instruction addresses the urgent need to help teachers understand and teach fraction concepts. Fractions remain one of the key stumbling blocks in math education, and here Empson and Levi lay a foundation for understanding fractions and decimals in ways that build conceptual learning. They show how the same kinds of intuitive knowledge and sense making that provides the basis for children's learning of whole number arithmetic can be extended to fractions and decimals. Just as they did in Children's Mathematics and Thinking Mathematically, Empson and Levi provide important insights into children's thinking and alternative approaches to solving problems. Three themes appear throughout the book: building meaning for fractions and decimals through discussing and solving word problems the progression of children's strategies for solving fraction word problems and equations from direct modeling through relational thinking designing instruction that capitalizes on students' relational thinking strategies to integrate algebra into teaching and learning fractions. With illuminating examples of student work, classroom vignettes, Teacher Commentaries from the field, sample problems and instructional guides provided in each chapter, you'll have all the tools you need to teach fractions and decimals with understanding and confidence. |
| teaching fractions and ratios for understanding: N-Gen Math 7 Bundle - 20 Kirk Weiler, 2021-10 |
| teaching fractions and ratios for understanding: Elementary and Middle School Mathematics: Pearson New International Edition John A. Van de Walle, Karen Karp, Jennifer M. Bay-Williams, 2013-07-29 For Elementary Mathematics Methods or Middle School Mathematics Methods Covers preK-8 Written by leaders in the field, this best-selling book will guide teachers as they help all PreK-8 learners make sense of math by supporting their own mathematical understanding and cultivating effective planning and instruction. Elementary and Middle School Mathematics: Teaching Developmentally provides an unparalleled depth of ideas and discussion to help teachers develop a real understanding of the mathematics they will teach and the most effective methods of teaching the various mathematics topics. This text reflects the NCTM and Common Core State Standards and the benefits of problem-based mathematics instruction. |
| teaching fractions and ratios for understanding: N-Gen Math 6: Bundle-20 Kirk Weiler, 2021-10 |
| teaching fractions and ratios for understanding: Acquisition of Mathematics Concepts and Processes Richard A. Lesh, Marsha Landau, 1983 |
| teaching fractions and ratios for understanding: Principles and Standards for School Mathematics , 2000 This easy-to-read summary is an excellent tool for introducing others to the messages contained in Principles and Standards. |
| teaching fractions and ratios for understanding: Making Sense of Fractions Randy Palisoc, 2014-08-06 Frustrated with fractions? They're easier than you think. Fractions have been given a bad rap. Understanding fractions is essential in math, yet too many students struggle to make sense of them. Even if students have performed well in math early in elementary school, they seem to hit a brick wall when it comes the abstract concepts involved with fractions. This is how Making Sense of Fractions helps. This unit challenges the long-held assumption that fractions are difficult because they're abstract. Fractions are actually more concrete and more logical than most people realize. Making Sense of Fractions helps students make these realizations, and it has a different philosophy from past approaches. This unit explains fractions in a more concrete way that makes logical sense to students (and to adults), making fractions easier to learn than before. Essential concepts include: * Simplifying Fractions * Adding Fractions * Subtracting Fractions * Multiplying Fractions * Dividing Fractions * Improper Fractions and Mixed Numbers * GCF (Greatest Common Factor) and LCM (Lowest Common Multiple) |
| teaching fractions and ratios for understanding: Ratio, Proportion, and Percent Instructional Fair, 2001-01-01 Your students will develop a greater understanding of the math concepts required for mastery of the new NCTM Standards. Easy-to-follow instructions, fun-to-solve puzzles and riddles, and many self-checking activities make these books a hit in any middle school math class. |
TEACHING FRACTIONS AND RATIOS FOR UNDERSTANDING
TEACHING FRACTIONS AND RATIOS FOR UNDERSTANDING Essential Content Knowledge and Instructional Stategies for Teachers SUSAN J. LAMON Fourth Edition
Teaching Fractions And Ratios For Understanding
Teaching Fractions and Ratios for Understanding (2012), a supplement that is not merely an answer key but a resource that provides the scaffolding for the groundbreaking approach to …
TEACHING FRACTIONS AND RATIOS FOR UNDERSTANDING
This resource book accompanies Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Teaching Strategies for Teachers. It was originally intended as a …
Teaching Fractions for Understanding: Addressing Interrelated …
The concept of fractions is perceived as one of the most difficult areas in school mathematics to learn and teach. The most frequently mentioned factors contributing to the complexity is …
Fractions: Teaching for Understanding - Australian Association of ...
A strong “sense of fractions” is fundamental to developing conceptual understanding of rational numbers, and includes flexibility with visual representations of fractions, a sense of the …
Teaching Fractions and Ratios for Understanding: Essential …
Teaching Fractions and Ratios for Understanding: Essential Content Knowledge and Instructional Strategies for Teachers, 2nd edition, by Susan J. Lamon, pushes readers to teach for …
Understanding fractions: interpretations and representations
When teaching fractions, we need to take into account that fractions can be interpreted in several different ways (Kieran, 1976, 1993). The interpretations are part-whole, ratio, operator, …
Fractions: Teaching for Understanding - albert.aamt.edu.au
Very often the teaching sequence in the mathematics curriculum proceeds as: fractions, decimals, percents, ratios, in that order, with very little attempt at integration. What we describe here is a …
Teaching interrelated concepts of fraction for understanding and ...
Fractions are perceived as one of the most difficult areas in school mathematics to learn and teach. The most frequently mentioned factors contributing to the complexity are fractions …
Fractions: Teaching for Understanding - Australian Association of ...
understanding of fractions and the wider domain of rational numbers can be obtained, because fractions form an important part of middle years mathematics curriculum, underpinning the …
A systematic review on teaching fraction for understanding …
pertaining to ‘teaching fractions’ or ‘understanding fractions’ and ‘representation’, this study utilized PRISMA’s procedure in analysing previously published articles. This review reveals …
Teaching Fractions and Ratios for Understanding
Teaching Fractions and Ratios for Understanding (2012), a supplement that is not merely an answer key but a resource that provides the scaffolding for the groundbreaking approach to …
Lesson plan Ratios and fractions - The Education and Training …
In this lesson, ratios and fractions are presented together, with a single diagram that represents both the ratio and the fraction. The diagrams provide insight into mathematical structure (Key …
FRACTIONS
FRACTIONS? IN SUMMARY • Seeing fractions as a set of interrelated concepts encompassing both dividing and measuring may help students to understand them • Considering fractions as …
Part 2 - Percentages, fractions and decimals
All teachers can support the foundations for proportional thinking by providing targeted teaching that deepens students’ conceptual understanding. This includes problem solving and …
An ASCD Study Guide for Unpacking Fractions: Classroom-Tested ...
This ASCD Study Guide is meant to enhance your understanding of the concepts and practical ideas presented in Unpacking Fractions: Classroom-Tested Strategies to Build Students' …
TEACHERS’ UNDERSTANDING OF RATIOS AND THEIR …
fractions and ratios. We used two instruments to collect data from 11 teachers and relied on Knowledge in Pieces as a lens for considering understandings teachers have and how …
Understanding the Challenges and Strategies in Teaching Fractions ...
17 Sep 2024 · Teaching fractions effectively at the primary school level is a persistent challenge faced by mathematics educators worldwide. This study explores the specific difficulties …
Understanding Fractions Begins with Literacy - ed
In this article, we outline strategies for helping students advance their conceptual knowledge of fractions by utilizing two sound literacy practices, vocabulary development (Bay-Williams & …
Systematic Interventions for Teaching Ratios - JSTOR
We chose three essential understandings related to ratios from Developing Un-derstanding of Ratios, Proportions, and Proportional Reasoning for Teaching Mathematics in Grades 6–8 …
Mastery Professional Development Materials - NCETM
Several important considerations are key to students gaining a secure and deep understanding of the mathematics within this theme, namely: • that any two numbers can be connected and compared using a multiplier • that fractions, ratios and percentages are all different ways of expressing multiplicative relationships and operations
The Number Line Model for Conceptual Understanding of Fractions
modeling to encourage conceptual understanding of topics. Teachers have been making adjustments to their curriculum in preparation to implement the new standards. The purpose of this paper is to provide a better understanding of what research shows when teaching fractions on a number line. This concept was
Understanding Fractions Begins with Literacy - ed
Understanding Fractions Begins with Literacy Teri Bingham, Ed.D. West Texas A&M University Regina Chanel Rodriguez, Ph.D. West Texas A&M University Abstract Transitioning from teaching whole number operations to teaching fraction operations can prove difficult, even for the seasoned mathematics teacher.
Teachers’ views of practical work in the teaching of fractions: a …
Teaching fractions 601 learners to understand and answer. Other approaches were the number line (one respondent) and physical objects that learners could handle (three respondents). Table 1 The role of practical work in teaching fractions Statement Strongly agree Agree Disagree Strongly disagree Practical work has a place in the teaching of ...
15 Chapter Developing Fraction Concepts - Pearson
Equivalent fractions are ways of describing the same amount by using different-sized fractional parts. Fractions can be compared by reasoning about the relative size of the fractions. Estimation and reasoning are important in teaching understanding of fractions. 15 Chapter M15_VAND8930_09_SE_C15.indd 339 08/12/14 11:54 AM
TEACHING - Brick Math
numerator and denominator as they proceed toward ratios in later content. Brick Math journal: After students build their models, have them draw the mod- ... depending on their level of understanding fractions. ... SHIRLEY DISSELER | TEACHING FRACTIONS USING LEGO® BRICKS 13 SUGGESTED BRICKS Size Number 1x1 4 1x2 6-8 1x4 4-6 2x2 4-6 2x4 9-12 ...
Part 2: Proportional thinking with fractions, decimals and …
See teaching considerations for multiplicative thinking. Multiplicative thinking and proportional reasoning are complex. Students should be supported to ... • the link between fractions and ratios builds an understanding when simplifying ratios, for example, 2:8 could be simplified to 1:4 because I know 2 eighths is the same as 1 over 4 ...
Teaching Fractions And Ratios For Understanding (2024)
Teaching Fractions And Ratios For Understanding Teaching Fractions and Ratios for Understanding Susan J. Lamon,2020-03-23 Written in a user friendly conversational style the fourth edition of this groundbreaking text helps pre service and in service mathematics teachers build the comfort and confidence they need to begin talking to children ...
Decimals and Percentages, Ratio and Proportion - Brockenhurst C …
The principal focus of mathematics teaching in upper key stage 2 is to ensure that pupils extend their understanding of the number system and place value to include larger integers. This should develop the connections that pupils make between multiplication and division with fractions, decimals, percentages and ratio.
Paying Attention to Fractions - K-12 - Fractions Learning Pathways
Why Is Understanding Fractions Important? “No area of elementary school mathematics is as mathematically rich, cognitively complicated, and difficult to teach as fractions, ratios, and proportionality. These ideas all express mathe-matical relationships: fractions and ratios are ‘relational’ numbers. They are the first place in
Queensland Students Understanding of Fractions: Evidence from …
used to represent fractions (Hiebert 1988 cited in Saxe, Taylor, McIntosh and Gearhart 2005). Although students are introduced to fractions in primary school, the use of this knowledge transcends the classroom . In everyday life there are many situations that require fractional understanding and representation (multiplicative and proportional
An ASCD Study Guide for Unpacking Fractions: Classroom-Tested ...
2. Teaching equivalent fractions—regrettably—often comes down to teaching the equivalent fractions algorithm: a⁄ b= (n × a)⁄ (n × b). Reflect on how you might teach the concept of equivalence first, before introducing the algorithm. Take a simple example, such as "⅔ is equivalent to 4⁄ 6." Rather than showing or telling students ...
Fractions: Teaching for Understanding - albert.aamt.edu.au
Understanding fraction equivalence Ni (2001) contends that in many mathematics classrooms, understanding of fraction equivalence is considered mastery of the rule, “multiply or divide the numerator and denominator of a fraction by the same number” (p. 413). Research shows that it is much more (e.g., Lamon, 2005). Understanding
Fractions: Teaching for Understanding - Australian Association …
contexts and concepts involving fractions, decimals, percents, ratios, similarity, scaling, probability, and linear relationships. Very often the teaching sequence in the mathematics curriculum proceeds as: fractions, decimals, percents, …
DEVELOPING AND TESTING A SCALE FOR MEASURING STUDENTS’ UNDERSTANDING …
notion of fractions as ratios. Those included expressing the relative size of two quantities using a fraction (tasks 9-10), identifying fractions as ratios (task 25) and comparing ratios, based either on quantitative (13-15) or qualitative information (30-31) (the latter five tasks were related to the “orange juice problem”).
Learning to Think Mathematically with the Ratio Table
that might be used to further their ability and interest in teaching with ratio tables. The book contains ideas about how the ratio table can be used to multiply, to divide, to combine equivalent ratios, to model equivalent fractions, and to structure informal mental math strategies, among other things. Each chapter has a blend of teaching ideas,
Primary teachers preferred fraction models and manipulatives for ...
When teaching fractions, teachers make instructional decisions about if, when, and how to use the many di erent types of fraction models and manipulatives. In this study, we sought
RUNNING HEAD: Improving Teachers’ Fraction Understanding 1 Can Teaching ...
8 Jul 2019 · The purpose of this causal-comparative study was to gain insight into whether teaching fractions improves teachers’ understanding of fractions. University master’s students (n = 25) conducted tutoring focused on fraction magnitude using number line representations across 39 sessions (3 per week) at grades 3-5.
Developing Essential Understanding - National Council of …
A Series for Teaching Mathematics Developing Essential Understanding of Developing Essential Understanding of Ratios, Proportions & Proportional Reasoning • Grades 6–8 NCTM 13482 Developing Essential Understanding How do you refute the claim that all ratios are fractions? How are ratios related to fractions?
Five Key Ideas to Teach Fractions and Decimals with Understanding …
Strother,'S.,'Brendefur,'J.'L.,'Thiede,'K.,'&'Appletion,'S.'(2016).'Five'Key'Ideas'to'Teach'Fractions'and'Decimals'with'Understanding.)Advances)in)
Conceptual Understanding of Fractions and Decimals for Fourth …
to add and subtract fractions with unlike denominators. Sixth grade students should be able to interpret and compute quotients of fractions. In order to create this level of understanding, students need rich, rigorous tasks promoting a deep insight into fractions and decimals (NGACCSSO, 2010). Purpose of the Study
Using Digitally Enhanced Tangible Materials for Teaching Fractions ...
• Ratios. Fractions as ratios express the relationship between two quantities, their relative magnitude. This is useful for size comparisons. In order for students to understand this interpretation of fractions, they should be able to realize the correlation between the two quantities and how these quantities are able to change simultaneously.
Teaching Fractions in the Intermediate Phase: A Neuroscience …
Fractions are often problematic for learners to fully grasp. Teachers must therefore determine various approaches for teaching fractions to reinforce understanding. More so, teachers need to determine strategies that cater for a diverse learner population and promote meaningful learning. In the quest to find a possible solution
Fractions: Teaching for Understanding
Understanding the size of fractions . In a task designed to get at students’ understanding of the “size” of fractions, we used the . Construct a Sum. task (Behr, Wachsmuth, & Post, 1985). The student is asked to place number cards in the boxes to make fractions so that when you add them, the answer is as close to one as + + F
Five Favorite Techniques for Teaching About Fractions 2.1
Wrap Up – Connecting Fractions with Whole Numbers • Concept of Units – the denominator is the unit based on some part of the ”ones” unit. • Fractions as Numbers – fit on a number line just like whole numbers and can be used to count parts of objects. • Equivalent Fractions – Big Idea that numbers can be
Low-Performing Students and Teaching Fractions for Understanding…
Teaching Fractions for Understanding: An Interactional Analysis Susan B. Empson, University of Texas at Austin This article presents an analysis of two low-performing students’ experiences in a ...
Developing a Theoretical Framework for Examining Student Understanding …
Teaching Fractions and Ratios for Understanding (2008a) and its companion text More (2008). This text was chosen in large part because it served the dual purpose of providing research into many of the areas that cause issues with learning fractions as …
Number Sense and Numeration, Fractions Gr. 6_7
Unit Plan Grade 6/7 – Number Sense and Numeration Fractions, Ratios and Percents –Term 3 Grade 6 OEs and SEs: OEs: • read, represent, compare, and order whole numbers to 1 000 000, decimal numbers to ... understanding that ratios are comparisons. Understand equivalent fractions. Day Problem/Checkpoint Intent 8 Warm-up: A lion can run 550 ...
Teaching fractions; Educational practices series; Vol.:22; 2011
teaching strategies, but all are focused on improving students’ conceptual understanding of fractions. We define conceptual knowledge of fractions as knowledge of what fractions mean, for example their magnitudes and relations to physical quantities, an understanding of why arithmetic procedures with fractions are
Core concept 3.1: Understanding multiplicative relationships
• NCETM Secondary Professional Development materials: 3.1 Understanding multiplicative relationships, pages 23–27 • NCETM: Using mathematical representations at KS3: Double number line and ratio tables. 3.1.3 Understand that fractions are an example of a multiplicative relationship and apply this understanding to a range of contexts
Using paper folding, fraction walls, and number lines - ed
whole–part relations, ratios, quotients, measures, and operators (Kieren, 1980; 1988). Kieren suggested that children have to develop the appropriate ... teaching fractions for understanding is being used in many Victorian primary and secondary schools: state, catholic, and independent.
Fractions: Teaching for Understanding - Nationellt centrum för ...
powerful insights into children’s understanding of key mathematical ideas, and their preferred strategies in solving problems. Combining formative assessment with an understanding of the “big ideas” of fractions informs the classroom teacher’s planning, enabling a clear focus for the teaching of fractions in the middle years.
Relating Decimals, Fractions, and Ratios for Deeper Understanding…
fractions to make more but smaller unit fractions. The fraction stays the same size overall. The fraction looks bigger because there are more partsand you do not see that these parts are smaller unless you make a drawing. You can also make equivalent fractions by groupingthe unit fractions and dividingsymbolically.
On the Teaching and Learning of Fractions through a Conceptual ...
understanding of fractions is due to the reproductive teaching of mathematics through the mechanical repetition of various rules that a textbook provides.
Teachers’ views of practical work in the teaching of fractions: a …
Teaching fractions 601 learners to understand and answer. Other approaches were the number line (one respondent) and physical objects that learners could handle (three respondents). Table 1 The role of practical work in teaching fractions Statement Strongly agree Agree Disagree Strongly disagree Practical work has a place in the teaching of ...
Neurocognitive Architectures and the Nonsymbolic Foundations of ...
Understanding fractions is critically important for the development of mathematical ... capacities of the human cognitive architecture for nonsymbolic ratios, and teaching methods that
3 Focusing on Ratio and Rate - National Council of Teachers of …
Focus in Grade 6: Teaching with Curriculum Focal Points 46 ... Using fractions to build an understanding of ratios A ratio is a multiplicative comparison of two numbers. As students begin to use what they know about fractions to understand ratios, they will learn, for example, if there are 4 children and 2 adults in a family, the ratio that com
Fractions: Teaching for Understanding - archive.aamt.edu.au
research literature on key concepts of fractions and how these develop, and yet this research has had little impact on state and national curriculum documents and even less impact on classroom practice. Much of the research on fractions is contained within broader research programs in relation to the rational number system.
FRACTIONS, RATIOS AND PROPORTIONAL REASONING - Homi …
FRACTIONS, RATIOS AND ... Students’ difficulties in understanding multiplicative relationships ... MULTIPLE APPROACHES TO TEACHING FRACTIONS Sharing approaches of learning fractions Measure approaches of learning fractions. PEDAGOGY FOR SHARING IN FRACTIONS!In sharing chapatismaking different equal parts (e.g. 1/10)
Pre-Service Primary School Teachers' Mathematical Knowledge for
Teaching Fractions through Problem Solving ... understanding of other number types and algebraic operations in later years of children‟s school experience. ... The study of fractions, ratios ...
A guide to the use of practical tasks and manipulatives in the teaching …
in the teaching of fractions and decimals with children aged 3 to 11 Rose Griffiths, Jenni Back and Sue Gifford ... whole and can access ratios from an early age, so there is no need to defer ratio ... sub-length of a unit length’ so leading to understanding fractions on a number line and decimal fractions in measurement contexts (1984:281). ...
Using the Number Line to Promote Understanding of Fractions …
introducing fractions and teaching students a variety of fractions concepts, as it is ―an effective way to develop students’ understanding of fractions as numbers with magnitudes…‖ (Siegler et al., 2010, p. 20). Number lines can be used when teaching students both the part-whole and measurement interpretation of fractions.
DRAWING ON A THEORETICAL MODEL TO STUDY STUDENTS
UNDERSTANDINGS OF FRACTIONS ABSTRACT. Teaching and learning fractions has traditionally been one of the most prob ... offer a detailed description and analysis of students' understanding of fractions, necessary both for the teaching of fractions and the develop ... whereas ratios are con ceptualized as the comparison of two quantities of the ...
DRAWING ON A THEORETICAL MODEL TO STUDY STUDENTS
offer a detailed description and analysis of students’ understanding of fractions, necessary both for the teaching of fractions and the develop-ment of relevant tests, such the one used in the present study. Yet with-out disregarding the possible limitations of more comprehensive studies,
Fractions, Fractions Everywhere - charlotteteachers.org
Teaching Standards: See Appendix 1 for teaching standards addressed in this unit. Synopsis: Realizing that larger number of my students struggled with the various ways to interpret fractions or unit rates of items, I decided to create a unit focused on fractions with specific attention to ratios, proportional reasoning and unit rates.
13th International Congress on Mathematical Education Hamburg, …
the teaching of fractions, understanding of different meanings should be addressed to help students ... relevant numbers, or are ratios of two integral numbers. They argued that, as for children ...
