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| conceptual understanding in math examples: Concept-Based Mathematics Jennifer T.H. Wathall, 2016-01-14 Give math students the connections between what they learn and how they do math—and suddenly math makes sense If your secondary-school students are fearful of or frustrated by math, it’s time for a new approach. When you teach concepts rather than rote processes, you show students math’s essential elegance, as well as its practicality—and help them discover their own natural mathematical abilities. This book is a road map to retooling how you teach math in a deep, clear, and meaningful way —through a conceptual lens—helping students achieve higher-order thinking skills. Jennifer Wathall shows you how to plan units, engage students, assess understanding, incorporate technology, and even guides you through an ideal concept-based classroom. Practical tools include: Examples from arithmetic to calculus Inquiry tasks, unit planners, templates, and activities Sample assessments with examples of student work Vignettes from international educators A dedicated companion website with additional resources, including a study guide, templates, exemplars, discussion questions, and other professional development activities. Everyone has the power to understand math. By extending Erickson and Lanning’s work on Concept-Based Curriculum and Instruction specifically to math, this book helps students achieve the deep understanding and skills called for by global standards and be prepared for the 21st century workplace. Jennifer Wathall’s book is one of the most forward thinking mathematics resources on the market. While highlighting the essential tenets of Concept-Based Curriculum design, her accessible explanations and clear examples show how to move students to deeper conceptual understandings. This book ignites the mathematical mind! — Lois A. Lanning, Author of Designing Concept-based Curriculum for English-Language Arts, K-12 Wathall is a master at covering all the bases here; this book is bursting with engaging assessment examples, discussion questions, research, and resources that apply specifically to mathematical topics. Any math teacher or coach would be hard-pressed to read it and not come away with scores of ideas, assessments, and lessons that she could use instantly in the classroom. As an IB Workshop Leader and instructional coach, I want this book handy on a nearby shelf for regular referral – it′s a boon to any educator who wants to bring math to life for students. — Alexis Wiggins, Instructional Coach, IB Workshop Leader and Consultant |
| conceptual understanding in math examples: Helping Children Learn Mathematics National Research Council, Division of Behavioral and Social Sciences and Education, Center for Education, Mathematics Learning Study Committee, 2002-07-31 Results from national and international assessments indicate that school children in the United States are not learning mathematics well enough. Many students cannot correctly apply computational algorithms to solve problems. Their understanding and use of decimals and fractions are especially weak. Indeed, helping all children succeed in mathematics is an imperative national goal. However, for our youth to succeed, we need to change how we're teaching this discipline. Helping Children Learn Mathematics provides comprehensive and reliable information that will guide efforts to improve school mathematics from pre-kindergarten through eighth grade. The authors explain the five strands of mathematical proficiency and discuss the major changes that need to be made in mathematics instruction, instructional materials, assessments, teacher education, and the broader educational system and answers some of the frequently asked questions when it comes to mathematics instruction. The book concludes by providing recommended actions for parents and caregivers, teachers, administrators, and policy makers, stressing the importance that everyone work together to ensure a mathematically literate society. |
| conceptual understanding in math examples: Conceptual and Procedural Knowledge James Hiebert, 2013-08-21 First Published in 1986. This book is intended for those people who are interested in how mathematics is learned. It is intended especially for those who are interested in the mental processes involved in becoming mathematically competent and the mental processes that inhibit such competency from developing. The volume opens with an overview of the issue and then traces the relationships between conceptual and procedural knowledge in mathematics from preschool days through the years of formal schooling. Mathematics educators and cognitive psychologists from a variety of perspectives contribute theoretical arguments and empirical data to illuminate the nature of the relationships and, in tum, the nature of mathematics learning. |
| conceptual understanding in math examples: Math on the Move Malke Rosenfeld, 2016-10-18 Kids love to move. But how do we harness all that kinetic energy effectively for math learning? In Math on the Move, Malke Rosenfeld shows how pairing math concepts and whole body movement creates opportunities for students to make sense of math in entirely new ways. Malke shares her experience creating dynamic learning environments by: exploring the use of the body as a thinking tool, highlighting mathematical ideas that are usefully explored with a moving body, providing a range of entry points for learning to facilitate a moving math classroom. ...--Publisher description. |
| conceptual understanding in math examples: Tools for Teaching Conceptual Understanding, Secondary Julie Stern, Krista Ferraro, Juliet Mohnkern, 2017-02-02 Students become experts and innovators through Concept-Based teaching Innovators don’t invent without a deep understanding of how the world works. With this foundation, they apply conceptual understanding to solve new problems. We want our students to not only retain ideas, but relate them to other things they encounter, using each new situation to add nuance and sophistication to their thinking. To do this, they need conceptual understanding. This book serves as a road map for Concept-Based teaching. Discover how to help students uncover conceptual relationships and transfer them to new situations. Specifically, teachers will learn: Strategies for introducing conceptual learning to students Four lesson frameworks to help students uncover conceptual relationships How to assess conceptual understanding, and How to differentiate concept-based instruction Look no further. For deep learning and innovative thinking, this book is the place to start. The authors tear down the false dichotomies of traditional vs innovative education and provide a practical toolkit for developing creativity and applying knowledge through Concept-Based learning. Every practitioner needs this book to juxtapose what worked well in the 20th Century with what is essential in the 21st Century and beyond. Michael McDowell, Superintendent Ross School District, Ross, CA While most good educators recognise the incredible value of teaching conceptually, it is challenging. The authors have created accessible, practical baby steps for every teacher to use. Dr. Vincent Chan, principal Fairview International School, Kuala Lumpur, Malaysia |
| conceptual understanding in math examples: Mathematical Mindsets Jo Boaler, 2015-10-12 Banish math anxiety and give students of all ages a clear roadmap to success Mathematical Mindsets provides practical strategies and activities to help teachers and parents show all children, even those who are convinced that they are bad at math, that they can enjoy and succeed in math. Jo Boaler—Stanford researcher, professor of math education, and expert on math learning—has studied why students don't like math and often fail in math classes. She's followed thousands of students through middle and high schools to study how they learn and to find the most effective ways to unleash the math potential in all students. There is a clear gap between what research has shown to work in teaching math and what happens in schools and at home. This book bridges that gap by turning research findings into practical activities and advice. Boaler translates Carol Dweck's concept of 'mindset' into math teaching and parenting strategies, showing how students can go from self-doubt to strong self-confidence, which is so important to math learning. Boaler reveals the steps that must be taken by schools and parents to improve math education for all. Mathematical Mindsets: Explains how the brain processes mathematics learning Reveals how to turn mistakes and struggles into valuable learning experiences Provides examples of rich mathematical activities to replace rote learning Explains ways to give students a positive math mindset Gives examples of how assessment and grading policies need to change to support real understanding Scores of students hate and fear math, so they end up leaving school without an understanding of basic mathematical concepts. Their evasion and departure hinders math-related pathways and STEM career opportunities. Research has shown very clear methods to change this phenomena, but the information has been confined to research journals—until now. Mathematical Mindsets provides a proven, practical roadmap to mathematics success for any student at any age. |
| conceptual understanding in math examples: Conceptual Mathematics F. William Lawvere, Stephen H. Schanuel, 2009-07-30 This truly elementary book on categories introduces retracts, graphs, and adjoints to students and scientists. |
| conceptual understanding in math examples: Concept-Based Curriculum and Instruction for the Thinking Classroom H. Lynn Erickson, 2007 This indispensable guide combines proven curriculum design with teaching methods that encourage students to learn concepts as well as content and skills for deep understanding across all subject areas. |
| conceptual understanding in math examples: Math, Better Explained Kalid Azad, 2015-12-04 Math, Better Explained is an intuitive guide to the math fundamentals. Learn math the way your teachers always wanted. |
| conceptual understanding in math examples: Open Middle Math Robert Kaplinsky, 2023-10-10 This book is an amazing resource for teachers who are struggling to help students develop both procedural fluency and conceptual understanding.. --Dr. Margaret (Peg) Smith, co-author of5 Practices for Orchestrating Productive Mathematical Discussions Robert Kaplinsky, the co-creator of Open Middle math problems, brings hisnew class of tasks designed to stimulate deeper thinking and lively discussion among middle and high school students in Open Middle Math: Problems That Unlock Student Thinking, Grades 6-12. The problems are characterized by a closed beginning,- meaning all students start with the same initial problem, and a closed end,- meaning there is only one correct or optimal answer. The key is that the middle is open- in the sense that there are multiple ways to approach and ultimately solve the problem. These tasks have proven enormously popular with teachers looking to assess and deepen student understanding, build student stamina, and energize their classrooms. Professional Learning Resource for Teachers: Open Middle Math is an indispensable resource for educators interested in teaching student-centered mathematics in middle and high schools consistent with the national and state standards. Sample Problems at Each Grade: The book demonstrates the Open Middle concept with sample problems ranging from dividing fractions at 6th grade to algebra, trigonometry, and calculus. Teaching Tips for Student-Centered Math Classrooms: Kaplinsky shares guidance on choosing problems, designing your own math problems, and teaching for multiple purposes, including formative assessment, identifying misconceptions, procedural fluency, and conceptual understanding. Adaptable and Accessible Math: The tasks can be solved using various strategies at different levels of sophistication, which means all students can access the problems and participate in the conversation. Open Middle Math will help math teachers transform the 6th -12th grade classroom into an environment focused on problem solving, student dialogue, and critical thinking. |
| conceptual understanding in math examples: Learning and Instruction National Research Council, Division of Behavioral and Social Sciences and Education, PANEL ON LEARNING AND INSTRUCTION., Strategic Education Research Partnership, 2003-12-04 The Strategic Education Research Partnership (SERP) is a bold, ambitious plan that proposes a revolutionary program of education research and development. Its purpose is to construct a powerful knowledge base, derived from both research and practice, that will support the efforts of teachers, school administrators, colleges of education, and policy officialsâ with the ultimate goal of significantly improving student learning. The proposals in this book have the potential to substantially improve the knowledge base that supports teaching and learning by pursuing answers to questions at the core of teaching practices. It calls for the linking of research and development, including instructional programs, assessment tools, teacher education programs, and materials. Best of all, the book provides a solid framework for a program of research and development that will be genuinely useful to classroom teachers. |
| conceptual understanding in math examples: Teaching Numeracy Margie Pearse, K. M. Walton, 2011-03-23 Transform mathematics learning from “doing” to “thinking” American students are losing ground in the global mathematical environment. What many of them lack is numeracy—the ability to think through the math and apply it outside of the classroom. Referencing the new common core and NCTM standards, the authors outline nine critical thinking habits that foster numeracy and show you how to: Monitor and repair students’ understanding Guide students to recognize patterns Encourage questioning for understanding Develop students’ mathematics vocabulary Included are several numeracy-rich lesson plans, complete with clear directions and student handouts. |
| conceptual understanding in math examples: Principles to Actions National Council of Teachers of Mathematics, 2014-02 This text offers guidance to teachers, mathematics coaches, administrators, parents, and policymakers. This book: provides a research-based description of eight essential mathematics teaching practices ; describes the conditions, structures, and policies that must support the teaching practices ; builds on NCTM's Principles and Standards for School Mathematics and supports implementation of the Common Core State Standards for Mathematics to attain much higher levels of mathematics achievement for all students ; identifies obstacles, unproductive and productive beliefs, and key actions that must be understood, acknowledged, and addressed by all stakeholders ; encourages teachers of mathematics to engage students in mathematical thinking, reasoning, and sense making to significantly strengthen teaching and learning. |
| conceptual understanding in math examples: Visible Learning for Mathematics, Grades K-12 John Hattie, Douglas Fisher, Nancy Frey, Linda M. Gojak, Sara Delano Moore, William Mellman, 2016-09-15 Selected as the Michigan Council of Teachers of Mathematics winter book club book! Rich tasks, collaborative work, number talks, problem-based learning, direct instruction...with so many possible approaches, how do we know which ones work the best? In Visible Learning for Mathematics, six acclaimed educators assert it’s not about which one—it’s about when—and show you how to design high-impact instruction so all students demonstrate more than a year’s worth of mathematics learning for a year spent in school. That’s a high bar, but with the amazing K-12 framework here, you choose the right approach at the right time, depending upon where learners are within three phases of learning: surface, deep, and transfer. This results in visible learning because the effect is tangible. The framework is forged out of current research in mathematics combined with John Hattie’s synthesis of more than 15 years of education research involving 300 million students. Chapter by chapter, and equipped with video clips, planning tools, rubrics, and templates, you get the inside track on which instructional strategies to use at each phase of the learning cycle: Surface learning phase: When—through carefully constructed experiences—students explore new concepts and make connections to procedural skills and vocabulary that give shape to developing conceptual understandings. Deep learning phase: When—through the solving of rich high-cognitive tasks and rigorous discussion—students make connections among conceptual ideas, form mathematical generalizations, and apply and practice procedural skills with fluency. Transfer phase: When students can independently think through more complex mathematics, and can plan, investigate, and elaborate as they apply what they know to new mathematical situations. To equip students for higher-level mathematics learning, we have to be clear about where students are, where they need to go, and what it looks like when they get there. Visible Learning for Math brings about powerful, precision teaching for K-12 through intentionally designed guided, collaborative, and independent learning. |
| conceptual understanding in math examples: Guided Math Workshop Laney Sammons, Donna Boucher, 2017-03-01 This must-have resource helps teachers successfully plan, organize, implement, and manage Guided Math Workshop. It provides practical strategies for structure and implementation to allow time for teachers to conduct small-group lessons and math conferences to target student needs. The tested resources and strategies for organization and management help to promote student independence and provide opportunities for ongoing practice of previously mastered concepts and skills. With sample workstations and mathematical tasks and problems for a variety of grade levels, this guide is sure to provide the information that teachers need to minimize preparation time and meet the needs of all students. |
| conceptual understanding in math examples: Concepts of Modern Mathematics Ian Stewart, 2012-05-23 In this charming volume, a noted English mathematician uses humor and anecdote to illuminate the concepts of groups, sets, subsets, topology, Boolean algebra, and other mathematical subjects. 200 illustrations. |
| conceptual understanding in math examples: Oxford Handbook of Numerical Cognition Roi Cohen Kadosh, Ann Dowker, 2015-07-30 How do we understand numbers? Do animals and babies have numerical abilities? Why do some people fail to grasp numbers, and how we can improve numerical understanding? Numbers are vital to so many areas of life: in science, economics, sports, education, and many aspects of everyday life from infancy onwards. Numerical cognition is a vibrant area that brings together scientists from different and diverse research areas (e.g., neuropsychology, cognitive psychology, developmental psychology, comparative psychology, anthropology, education, and neuroscience) using different methodological approaches (e.g., behavioral studies of healthy children and adults and of patients; electrophysiology and brain imaging studies in humans; single-cell neurophysiology in non-human primates, habituation studies in human infants and animals, and computer modeling). While the study of numerical cognition had been relatively neglected for a long time, during the last decade there has been an explosion of studies and new findings. This has resulted in an enormous advance in our understanding of the neural and cognitive mechanisms of numerical cognition. In addition, there has recently been increasing interest and concern about pupils' mathematical achievement in many countries, resulting in attempts to use research to guide mathematics instruction in schools, and to develop interventions for children with mathematical difficulties. This handbook brings together the different research areas that make up the field of numerical cognition in one comprehensive and authoritative volume. The chapters provide a broad and extensive review that is written in an accessible form for scholars and students, as well as educationalists, clinicians, and policy makers. The book covers the most important aspects of research on numerical cognition from the areas of development psychology, cognitive psychology, neuropsychology and rehabilitation, learning disabilities, human and animal cognition and neuroscience, computational modeling, education and individual differences, and philosophy. Containing more than 60 chapters by leading specialists in their fields, the Oxford Handbook of Numerical Cognition is a state-of-the-art review of the current literature. |
| conceptual understanding in math examples: How People Learn National Research Council, Division of Behavioral and Social Sciences and Education, Board on Behavioral, Cognitive, and Sensory Sciences, Committee on Developments in the Science of Learning with additional material from the Committee on Learning Research and Educational Practice, 2000-08-11 First released in the Spring of 1999, How People Learn has been expanded to show how the theories and insights from the original book can translate into actions and practice, now making a real connection between classroom activities and learning behavior. This edition includes far-reaching suggestions for research that could increase the impact that classroom teaching has on actual learning. Like the original edition, this book offers exciting new research about the mind and the brain that provides answers to a number of compelling questions. When do infants begin to learn? How do experts learn and how is this different from non-experts? What can teachers and schools do-with curricula, classroom settings, and teaching methodsâ€to help children learn most effectively? New evidence from many branches of science has significantly added to our understanding of what it means to know, from the neural processes that occur during learning to the influence of culture on what people see and absorb. How People Learn examines these findings and their implications for what we teach, how we teach it, and how we assess what our children learn. The book uses exemplary teaching to illustrate how approaches based on what we now know result in in-depth learning. This new knowledge calls into question concepts and practices firmly entrenched in our current education system. Topics include: How learning actually changes the physical structure of the brain. How existing knowledge affects what people notice and how they learn. What the thought processes of experts tell us about how to teach. The amazing learning potential of infants. The relationship of classroom learning and everyday settings of community and workplace. Learning needs and opportunities for teachers. A realistic look at the role of technology in education. |
| conceptual understanding in math examples: Mathematics Learning in Early Childhood National Research Council, Division of Behavioral and Social Sciences and Education, Center for Education, Committee on Early Childhood Mathematics, 2009-11-13 Early childhood mathematics is vitally important for young children's present and future educational success. Research demonstrates that virtually all young children have the capability to learn and become competent in mathematics. Furthermore, young children enjoy their early informal experiences with mathematics. Unfortunately, many children's potential in mathematics is not fully realized, especially those children who are economically disadvantaged. This is due, in part, to a lack of opportunities to learn mathematics in early childhood settings or through everyday experiences in the home and in their communities. Improvements in early childhood mathematics education can provide young children with the foundation for school success. Relying on a comprehensive review of the research, Mathematics Learning in Early Childhood lays out the critical areas that should be the focus of young children's early mathematics education, explores the extent to which they are currently being incorporated in early childhood settings, and identifies the changes needed to improve the quality of mathematics experiences for young children. This book serves as a call to action to improve the state of early childhood mathematics. It will be especially useful for policy makers and practitioners-those who work directly with children and their families in shaping the policies that affect the education of young children. |
| conceptual understanding in math examples: Introduction to Analysis Maxwell Rosenlicht, 2012-05-04 Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. 1968 edition. |
| conceptual understanding in math examples: Culturally Responsive Teaching Geneva Gay, 2010 The achievement of students of color continues to be disproportionately low at all levels of education. More than ever, Geneva Gay's foundational book on culturally responsive teaching is essential reading in addressing the needs of today's diverse student population. Combining insights from multicultural education theory and research with real-life classroom stories, Gay demonstrates that all students will perform better on multiple measures of achievement when teaching is filtered through their own cultural experiences. This bestselling text has been extensively revised to include expanded coverage of student ethnic groups: African and Latino Americans as well as Asian and Native Americans as well as new material on culturally diverse communication, addressing common myths about language diversity and the effects of English Plus instruction. |
| conceptual understanding in math examples: How Students Learn National Research Council, Division of Behavioral and Social Sciences and Education, Committee on How People Learn, A Targeted Report for Teachers, 2005-01-23 How do you get a fourth-grader excited about history? How do you even begin to persuade high school students that mathematical functions are relevant to their everyday lives? In this volume, practical questions that confront every classroom teacher are addressed using the latest exciting research on cognition, teaching, and learning. How Students Learn: History, Mathematics, and Science in the Classroom builds on the discoveries detailed in the bestselling How People Learn. Now, these findings are presented in a way that teachers can use immediately, to revitalize their work in the classroom for even greater effectiveness. Organized for utility, the book explores how the principles of learning can be applied in teaching history, science, and math topics at three levels: elementary, middle, and high school. Leading educators explain in detail how they developed successful curricula and teaching approaches, presenting strategies that serve as models for curriculum development and classroom instruction. Their recounting of personal teaching experiences lends strength and warmth to this volume. The book explores the importance of balancing students' knowledge of historical fact against their understanding of concepts, such as change and cause, and their skills in assessing historical accounts. It discusses how to build straightforward science experiments into true understanding of scientific principles. And it shows how to overcome the difficulties in teaching math to generate real insight and reasoning in math students. It also features illustrated suggestions for classroom activities. How Students Learn offers a highly useful blend of principle and practice. It will be important not only to teachers, administrators, curriculum designers, and teacher educators, but also to parents and the larger community concerned about children's education. |
| conceptual understanding in math examples: Vital Directions for Mathematics Education Research Keith R Leatham, 2014-07-08 This book provides a collection of chapters from prominent mathematics educators in which they each discuss vital issues in mathematics education and what they see as viable directions research in mathematics education could take to address these issues. All of these issues are related to learning and teaching mathematics. The book consists of nine chapters, seven from each of seven scholars who participated in an invited lecture series (Scholars in Mathematics Education) at Brigham Young University, and two chapters from two other scholars who are writing reaction papers that look across the first seven chapters. The recommendations take the form of broad, overarching principles and ideas that cut across the field. In this sense, this book differs from classical “research agenda projects,” which seek to outline specific research questions that the field should address around a central topic. |
| conceptual understanding in math examples: Mastering Math Manipulatives, Grades 4-8 Sara Delano Moore, Kimberly Rimbey, 2021-10-04 Put math manipulatives to work in your classroom and make teaching and learning math both meaningful and productive. Mastering Math Manipulatives includes everything you need to integrate math manipulatives—both concrete and virtual—into math learning. Each chapter of this richly illustrated, easy-to-use guide focuses on a different powerful tool, such as base ten blocks, fraction manipulatives, unit squares and cubes, Cuisenaire Rods, Algebra tiles and two-color counters, geometric strips and solids, geoboards, and others, and includes a set of activities that demonstrate the many ways teachers can leverage manipulatives to model and reinforce math concepts for all learners. It features: · Classroom strategies for introducing math manipulatives, including commercial, virtual, and hand-made manipulatives, into formal math instruction. · Step-by-step instructions for over 70 activities that work with any curriculum, including four-color photos, printable work mats, and demonstration videos. · Handy charts that sort activities by manipulative type, math topic, domains aligned with standards, and grade-level appropriateness. |
| conceptual understanding in math examples: Conceptual Model-Based Problem Solving Yan Ping Xin, 2013-02-11 Are you having trouble in finding Tier II intervention materials for elementary students who are struggling in math? Are you hungry for effective instructional strategies that will address students’ conceptual gap in additive and multiplicative math problem solving? Are you searching for a powerful and generalizable problem solving approach that will help those who are left behind in meeting the Common Core State Standards for Mathematics (CCSSM)? If so, this book is the answer for you. • The conceptual model-based problem solving (COMPS) program emphasizes mathematical modeling and algebraic representation of mathematical relations in equations, which are in line with the new Common Core. • “Through building most fundamental concepts pertinent to additive and multiplicative reasoning and making the connection between concrete and abstract modeling, students were prepared to go above and beyond concrete level of operation and be able to use mathematical models to solve more complex real-world problems. As the connection is made between the concrete model (or students’ existing knowledge scheme) and the symbolic mathematical algorithm, the abstract mathematical models are no longer “alien” to the students.” As Ms. Karen Combs, Director of Elementary Education of Lafayette School Corporation in Indiana, testified: “It really worked with our kids!” • “One hallmark of mathematical understanding is the ability to justify,... why a particular mathematical statement is true or where a mathematical rule comes from” (http://illustrativemathematics.org/standards). Through making connections between mathematical ideas, the COMPS program makes explicit the reasoning behind math, which has the potential to promote a powerful transfer of knowledge by applying the learned conception to solve other problems in new contexts. • Dr. Yan Ping Xin’s book contains essential tools for teachers to help students with learning disabilities or difficulties close the gap in mathematics word problem solving. I have witnessed many struggling students use these strategies to solve word problems and gain confidence as learners of mathematics. This book is a valuable resource for general and special education teachers of mathematics. - Casey Hord, PhD, University of Cincinnati |
| conceptual understanding in math examples: Daily Math Stretches: Building Conceptual Understanding Levels 3-5 Laney Sammons, Michelle Windham, 2011-02-01 Daily Math Stretches offers practice in algebraic thinking, geometry, measurement, and data for grades 3-5 to provide an early foundation for mastering mathematical learning. Written by Guided Math author Laney Sammons and with well-known, research-based approaches, this product provides step-by-step lessons, assessment information, and a snapshot of how to facilitate these math discussions in your classroom. Digital resources are also included for teacher guidance with management tips, classroom set-up tips, and interactive whiteboard files for each stretch. |
| conceptual understanding in math examples: 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. |
| conceptual understanding in math examples: Daily Math Stretches: Building Conceptual Understanding Levels 3-5 Sammons, Laney, 2017-03-01 Jumpstart your students’ minds with daily warm-ups that get them thinking mathematically and ready for instruction. Daily Math Stretches offers practice in algebraic thinking, geometry, measurement, and data for grades 3-5 to provide an early foundation for mastering mathematical learning. Written by Guided Math author Laney Sammons and with well-known, research-based approaches, this product provides step-by-step lessons, assessment information, and a snapshot of how to facilitate these math discussions in your classroom. Digital resources are also included for teacher guidance with management tips, classroom set-up tips, and interactive whiteboard files for each stretch. |
| conceptual understanding in math examples: Let's Play Math Denise Gaskins, 2012-09-04 |
| conceptual understanding in math examples: Reconsidering Conceptual Change: Issues in Theory and Practice Margarita Limón, L. Mason, 2007-05-08 This book is an important account of the state of the art of both theoretical and practical issues in the present-day research on conceptual change. Unique in its complete treatment of the questions that should be considered to further current understanding of knowledge construction and change, this book is useful for psychologists, cognitive scientists, educational researchers, curriculum developers, teachers and educators at all levels and in all disciplines. |
| conceptual understanding in math examples: Concept Mapping as an Assessment Tool for Conceptual Understanding in Mathematics Haiyue JIN, 2022-05-05 This book investigates the practicability and effectiveness of the concept map as a tool for assessing students’ conceptual understanding in mathematics. The author first introduces concept mapping and then employs it to investigate students’ conceptual understanding of four different mathematical topics. Alongside traditional scoring methods, she adopts Social Network Analysis, a new technique, to interpret student-constructed concept maps, which reveals fresh insights into the graphic features of the concept map and into how students connect mathematical concepts. By comparing two traditional school tests with the concept map, she examines its concurrent validity and discusses its strengths and drawbacks from the viewpoint of assessing conceptual understanding. With self-designed questionnaires, interviews, and open-ended writing tasks, she also investigates students and teachers’ attitudes toward concept mapping and describes the implications these findings may have for concept mapping’s use in school and for further research on the topic. Scholars and postgraduate students of mathematics education and teachers interested in concept mapping or assessing conceptual understanding in classroom settings will find this book an informative, inspiring, and overall valuable addition to their libraries. |
| conceptual understanding in math examples: Knowing and Teaching Elementary Mathematics Liping Ma, 2010-03-26 Studies of teachers in the U.S. often document insufficient subject matter knowledge in mathematics. Yet, these studies give few examples of the knowledge teachers need to support teaching, particularly the kind of teaching demanded by recent reforms in mathematics education. Knowing and Teaching Elementary Mathematics describes the nature and development of the knowledge that elementary teachers need to become accomplished mathematics teachers, and suggests why such knowledge seems more common in China than in the United States, despite the fact that Chinese teachers have less formal education than their U.S. counterparts. The anniversary edition of this bestselling volume includes the original studies that compare U.S and Chinese elementary school teachers’ mathematical understanding and offers a powerful framework for grasping the mathematical content necessary to understand and develop the thinking of school children. Highlighting notable changes in the field and the author’s work, this new edition includes an updated preface, introduction, and key journal articles that frame and contextualize this seminal work. |
| conceptual understanding in math examples: 5 Principles of the Modern Mathematics Classroom Gerald Aungst, 2015-10-09 Students pursue problems they’re curious about, not problems they’re told to solve. Creating a math classroom filled with confident problem solvers starts by introducing challenges discovered in the real world, not by presenting a sequence of prescribed problems, says Gerald Aungst. In this groundbreaking book, he offers a thoughtful approach for instilling a culture of learning in your classroom through five powerful, yet straightforward principles: Conjecture, Collaboration, Communication, Chaos, and Celebration. Aungst shows you how to Embrace collaboration and purposeful chaos to help students engage in productive struggle, using non-routine and unsolved problems Put each chapter’s principles into practice through a variety of strategies, activities, and by incorporating technology tools Introduce substantive, lasting cultural changes in your classroom through a manageable, gradual shift in processes and behaviors Five Principles of the Modern Mathematics Classroom offers new ideas for inspiring math students by building a more engaging and collaborative learning environment. Bravo! This book brings a conceptual framework for K-12 mathematics to life. As a parent and as the executive director of Edutopia, I commend Aungst for sharing his 5 principles. This is a perfect blend of inspiring and practical. Highly recommended! Cindy Johanson, Executive Director, Edutopia George Lucas Educational Foundation Aungst ignites the magic of mathematics by reminding us what makes mathematicians so passionate about their subject matter. Grounded in research, his work takes us on a journey into classrooms so that we may take away tips to put into practice today. Erin Klein, Teacher, Speaker, and Author of Redesigning Learning Spaces |
| conceptual understanding in math examples: Figuring Out Fluency in Mathematics Teaching and Learning, Grades K-8 Jennifer M. Bay-Williams, John J. SanGiovanni, 2021-03-02 Because fluency practice is not a worksheet. Fluency in mathematics is more than adeptly using basic facts or implementing algorithms. Real fluency involves reasoning and creativity, and it varies by the situation at hand. Figuring Out Fluency in Mathematics Teaching and Learning offers educators the inspiration to develop a deeper understanding of procedural fluency, along with a plethora of pragmatic tools for shifting classrooms toward a fluency approach. In a friendly and accessible style, this hands-on guide empowers educators to support students in acquiring the repertoire of reasoning strategies necessary to becoming versatile and nimble mathematical thinkers. It includes: Seven Significant Strategies to teach to students as they work toward procedural fluency. Activities, fluency routines, and games that encourage learning the efficiency, flexibility, and accuracy essential to real fluency. Reflection questions, connections to mathematical standards, and techniques for assessing all components of fluency. Suggestions for engaging families in understanding and supporting fluency. Fluency is more than a toolbox of strategies to choose from; it’s also a matter of equity and access for all learners. Give your students the knowledge and power to become confident mathematical thinkers. |
| conceptual understanding in math examples: What is a Mathematical Concept? Elizabeth de Freitas, Nathalie Sinclair, Alf Coles, 2017-06-22 Leading thinkers in mathematics, philosophy and education offer new insights into the fundamental question: what is a mathematical concept? |
| conceptual understanding in math examples: Getting Smart Tom Vander Ark, 2011-09-20 A comprehensive look at the promise and potential of online learning In our digital age, students have dramatically new learning needs and must be prepared for the idea economy of the future. In Getting Smart, well-known global education expert Tom Vander Ark examines the facets of educational innovation in the United States and abroad. Vander Ark makes a convincing case for a blend of online and onsite learning, shares inspiring stories of schools and programs that effectively offer personal digital learning opportunities, and discusses what we need to do to remake our schools into smart schools. Examines the innovation-driven world, discusses how to combine online and onsite learning, and reviews smart tools for learning Investigates the lives of learning professionals, outlines the new employment bargain, examines online universities and smart schools Makes the case for smart capital, advocates for policies that create better learning, studies smart cultures |
| conceptual understanding in math examples: The Knowledge Gap Natalie Wexler, 2020-08-04 The untold story of the root cause of America's education crisis--and the seemingly endless cycle of multigenerational poverty. It was only after years within the education reform movement that Natalie Wexler stumbled across a hidden explanation for our country's frustrating lack of progress when it comes to providing every child with a quality education. The problem wasn't one of the usual scapegoats: lazy teachers, shoddy facilities, lack of accountability. It was something no one was talking about: the elementary school curriculum's intense focus on decontextualized reading comprehension skills at the expense of actual knowledge. In the tradition of Dale Russakoff's The Prize and Dana Goldstein's The Teacher Wars, Wexler brings together history, research, and compelling characters to pull back the curtain on this fundamental flaw in our education system--one that fellow reformers, journalists, and policymakers have long overlooked, and of which the general public, including many parents, remains unaware. But The Knowledge Gap isn't just a story of what schools have gotten so wrong--it also follows innovative educators who are in the process of shedding their deeply ingrained habits, and describes the rewards that have come along: students who are not only excited to learn but are also acquiring the knowledge and vocabulary that will enable them to succeed. If we truly want to fix our education system and unlock the potential of our neediest children, we have no choice but to pay attention. |
| conceptual understanding in math examples: Number Sense Routines Jessica F. Shumway, 2011 Just as athletes stretch their muscles before every game and musicians play scales to keep their technique in tune, mathematical thinkers and problem solvers can benefit from daily warm-up exercises. Jessica Shumway has developed a series of routines designed to help young students internalize and deepen their facility with numbers. The daily use of these quick five-, ten-, or fifteen-minute experiences at the beginning of math class will help build students' number sense. Students with strong number sense understand numbers, ways to represent numbers, relationships among numbers, and number systems. They make reasonable estimates, compute fluently, use reasoning strategies (e.g., relate operations, such as addition and subtraction, to each other), and use visual models based on their number sense to solve problems. Students who never develop strong number sense will struggle with nearly all mathematical strands, from measurement and geometry to data and equations. In Number Sense Routines, Jessica shows that number sense can be taught to all students. Dozens of classroom examples -- including conversations among students engaging in number sense routines -- illustrate how the routines work, how children's number sense develops, and how to implement responsive routines. Additionally, teachers will gain a deeper understanding of the underlying math -- the big ideas, skills, and strategies children learn as they develop numerical literacy. |
| conceptual understanding in math examples: Math Expressions Karen C. Fuson, 2013 |
| conceptual understanding in math examples: Mathematics Explained for Primary Teachers Derek Haylock, Ralph Manning, 2014-06-19 Get access to an interactive eBook* when you buy the paperback! (Print paperback version only, ISBN 9781446285879) A Unique Blend of Digital and Print Learning Resources! 5 Star student reviews: “A must have for teachers-to-be, especially those who are a bit shaky on their maths knowledge!” “Not many maths books keep me fixated but this is one that is definitely worth the money.” “It is a book I will be using even when in the classroom.” Mathematics Explained for Primary Teachers develops your understanding of mathematical concepts and processes, and how children learn them, so you can confidently teach mathematics to primary children. Tried and tested, the fifth edition of Derek Haylock’s much loved textbook matches the 2014 curriculum requirements for England. Every chapter integrates children’s learning, classroom practice, and teacher’s own requirements for subject knowledge, making this the ideal text to guide you through your studies and beyond. More than just a book! The new edition is supported by FREE access to an interactive eBook and a companion website allowing you to use a wealth of teaching and learning resources. You can use the eBook to study where and when you want, and read, annotate and search the book on a tablet, laptop or PC. You can also visit study.sagepub.com/haylock5e to access: Videos by the author introduce core themes of each section and explain key mathematical processes. Links to the National Curriculum specify the statutory requirements for primary schools in England that relate to the mathematical content of each chapter. Learning and Teaching points highlight important issues you may face in the classroom and provide practical guidance for teaching. Self-assessment questions help check your understanding and provide immediate feedback to see how well you have done. Select SAGE journal articles to support literature reviews and wider reading. Lesson Plan Activities by Ralph Manning support content-focused chapters and contain creative mathematics tasks across the primary age range. A Student Workbook is also available to accompany this book, including over 700 practice problems to help you understand, apply and teach primary mathematics. Derek Haylock is an education consultant and writer with a background in mathematics teaching, teacher education and classroom-based research in mathematics education. Ralph Manning is an independent consultant in primary education. He has worked as a primary teacher and as a lecturer in primary teacher education for 18 years, following a career in IT. *interactivity only available through Vitalsource eBook |
Conceptual Understanding and Procedural Fluency in …
Both procedural fluency and conceptual understanding are necessary components of mathematical proficiency and mathematical literacy. In support of problem solving, teachers, …
MATHEMATICS TEACHING PRACTICE 6 - Kentucky
Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they …
CONCEPTUAL UNDERSTANDING, PROCEDURAL KNOWLEDGE …
Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples of concepts; use and interrelate models, …
WHAT IS CONCEPTUAL UNDERSTANDING? - Amazon Web …
Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples of concepts; use and interrelate models, …
Teaching Mathematics for Understanding - Pearson
• Conceptual understanding: Comprehension of math-ematical concepts, operations, and relations • Procedural fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and …
Teaching for Conceptual Understanding: Exciting Mathematics
Mathematicians make meaning in their world by setting up quantifiable and spatial relationships, by noticing patterns and transformations, by proving them as generalizations, and by searching …
Conceptual Understanding
Adding It Up describes conceptual understanding as “an integrated and functional grasp of mathematical ideas” (p. 118). Students should be able to understand the importance of …
Conceptual Understanding Framework - Newington …
Teacher example and task choice is important in creating a conceptual narrative. The examples and tasks teachers choose and the order in which they are sequenced can influence the …
Teaching Conceptual Understanding of Mathematics via a …
conceptual understanding of mathematics encourages students to be more independent and confident which evidences itself in students not shrinking back from challenging problems and …
Defining and measuring conceptual knowledge in mathematics
In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking lit-erature, and (2) how conceptual knowledge is defined, operationalized, and …
Promoting a Conceptual Understanding of Mathematics
Conceptual Understanding Promoting a This article provides an overview of the eight effective mathematics teaching practices fi rst described in NCTM’s Principles to Actions: Ensuring …
Rigor Conceptual Procedural Application - ASDN
Conceptual understanding: The Standards call for conceptual understanding of key concepts, such as place value and ratios. Teachers support students’ ability to access concepts from a …
UNIQUELY PRECISE: IMPORTANCE OF CONCEPTUAL …
The purpose of this paper is to (a) define mathematical language; (b) provide an example of a lack of precise conceptual understanding among pre-service math educators using the concept of …
Principles to Practice and the Subject Guide - IB MYP FOR …
conceptual understanding as they solve problems, analyse issues and evaluate decisions that can have an impact on themselves, their communities and the wider world. In the MYP, conceptual …
Conceptual Understanding and How to Assess It - AUT
Give examples of how equations can be useful. Comparative judgement. Abandon attempts to specify conceptual understanding in rubrics. Abandon attempts to anticipate every possible …
Conceptual and Procedural Knowledge of Mathematics: Does …
This study examined relations between children's conceptual understanding of mathematical equivalence and their procedures for solving equivalence problems (e.g., 3 + 4 + 5 = 3 + ).
Exploring Mathematics Learners’ Conceptual Understanding …
In this article, we explored learners’ conceptual understanding of coordinates and transformation geometry through concept mapping. A qualitative case study was employed in this study, and …
Teaching Mathematics for Understanding - Pearson
• Conceptual understanding: Comprehension of math-ematical concepts, operations, and relations • Procedural fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and …
Developing Strategic Competence by Teaching Using the …
conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and a productive disposition toward mathematics. In fact, the Common Core State Stan-dards for …
Conceptual Understanding and Procedural Fluency in …
Both procedural fluency and conceptual understanding are necessary components of mathematical proficiency and mathematical literacy. In support of problem solving, teachers, …
MATHEMATICS TEACHING PRACTICE 6 - Kentucky
Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they …
Promoting Student Conceptual Understanding of …
the most effective strategies for encouraging student conceptual understanding of mathematics. The future generation of teachers has the opportunity to teach mathematics in new ways that …
CONCEPTUAL UNDERSTANDING, PROCEDURAL …
Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples of concepts; use and interrelate models, …
WHAT IS CONCEPTUAL UNDERSTANDING? - Amazon Web …
Students demonstrate conceptual understanding in mathematics when they provide evidence that they can recognize, label, and generate examples of concepts; use and interrelate models, …
Teaching Mathematics for Understanding - Pearson
• Conceptual understanding: Comprehension of math-ematical concepts, operations, and relations • Procedural fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and …
Teaching for Conceptual Understanding: Exciting Mathematics
Mathematicians make meaning in their world by setting up quantifiable and spatial relationships, by noticing patterns and transformations, by proving them as generalizations, and by …
Conceptual Understanding
Adding It Up describes conceptual understanding as “an integrated and functional grasp of mathematical ideas” (p. 118). Students should be able to understand the importance of …
Conceptual Understanding Framework - Newington …
Teacher example and task choice is important in creating a conceptual narrative. The examples and tasks teachers choose and the order in which they are sequenced can influence the …
Teaching Conceptual Understanding of Mathematics via a …
conceptual understanding of mathematics encourages students to be more independent and confident which evidences itself in students not shrinking back from challenging problems and …
Defining and measuring conceptual knowledge in …
In the current work, we reviewed (1) how conceptual knowledge is defined in the mathematical thinking lit-erature, and (2) how conceptual knowledge is defined, operationalized, and …
Promoting a Conceptual Understanding of Mathematics
Conceptual Understanding Promoting a This article provides an overview of the eight effective mathematics teaching practices fi rst described in NCTM’s Principles to Actions: Ensuring …
Rigor Conceptual Procedural Application - ASDN
Conceptual understanding: The Standards call for conceptual understanding of key concepts, such as place value and ratios. Teachers support students’ ability to access concepts from a …
UNIQUELY PRECISE: IMPORTANCE OF CONCEPTUAL …
The purpose of this paper is to (a) define mathematical language; (b) provide an example of a lack of precise conceptual understanding among pre-service math educators using the concept of …
Principles to Practice and the Subject Guide - IB MYP FOR …
conceptual understanding as they solve problems, analyse issues and evaluate decisions that can have an impact on themselves, their communities and the wider world. In the MYP, conceptual …
Conceptual Understanding and How to Assess It - AUT
Give examples of how equations can be useful. Comparative judgement. Abandon attempts to specify conceptual understanding in rubrics. Abandon attempts to anticipate every possible …
Conceptual and Procedural Knowledge of Mathematics: Does …
This study examined relations between children's conceptual understanding of mathematical equivalence and their procedures for solving equivalence problems (e.g., 3 + 4 + 5 = 3 + ).
Exploring Mathematics Learners’ Conceptual …
In this article, we explored learners’ conceptual understanding of coordinates and transformation geometry through concept mapping. A qualitative case study was employed in this study, and …
Teaching Mathematics for Understanding - Pearson
• Conceptual understanding: Comprehension of math-ematical concepts, operations, and relations • Procedural fluency: Skill in carrying out procedures flexibly, accurately, efficiently, and …
Developing Strategic Competence by Teaching Using the …
conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and a productive disposition toward mathematics. In fact, the Common Core State Stan-dards for …
