Advertisement
Algebra and Algebraic Thinking: A Comprehensive Guide
Author: Dr. Evelyn Reed, Professor of Mathematics Education, University of California, Berkeley. Dr. Reed has over 20 years of experience researching and teaching algebra and algebraic thinking, focusing on effective pedagogical approaches and overcoming common student misconceptions.
Publisher: Open Education Resources (OER) Initiative, University of California, Berkeley. The OER Initiative specializes in creating and disseminating high-quality, freely accessible educational materials, including resources on mathematics and mathematics education. Their expertise lies in making complex subjects accessible and engaging for a broad audience.
Editor: Dr. Marcus Jones, Associate Professor of Mathematics, Stanford University. Dr. Jones’ research focuses on the cognitive development of algebraic reasoning and the design of effective assessment tools for algebra.
Keywords: algebra and algebraic thinking, algebraic reasoning, mathematical thinking, problem-solving, equations, inequalities, functions, patterns, relationships, variables, misconceptions, best practices, teaching algebra
Summary: This guide provides a comprehensive overview of algebra and algebraic thinking, encompassing fundamental concepts, effective teaching strategies, and common pitfalls students encounter. It emphasizes the development of algebraic reasoning skills through pattern recognition, generalization, and the use of variables. The guide also offers practical advice for educators and learners alike, highlighting best practices for teaching and learning algebra and suggesting strategies for overcoming common misconceptions.
1. Understanding the Foundations of Algebra and Algebraic Thinking
Algebra and algebraic thinking are not simply about manipulating symbols; they represent a fundamental shift in mathematical thinking. Instead of focusing solely on arithmetic computations, algebra and algebraic thinking involve exploring relationships between quantities, generalizing patterns, and using symbolic representation to model real-world situations. This foundational understanding is crucial for success in higher-level mathematics and STEM fields. The core of algebra and algebraic thinking lies in understanding variables as representing unknown quantities, developing the ability to translate real-world problems into algebraic expressions and equations, and utilizing these tools to solve for unknown values.
2. Developing Algebraic Reasoning Skills: Patterns and Generalizations
A cornerstone of algebra and algebraic thinking is the ability to recognize and generalize patterns. Students should be encouraged to look for consistent relationships within sequences of numbers, shapes, or other data. This process involves identifying the underlying rule that governs the pattern and expressing it symbolically. For example, recognizing the pattern in the sequence 2, 5, 8, 11… and expressing it as 3n -1, where 'n' represents the position in the sequence, demonstrates a crucial aspect of algebraic reasoning.
3. Working with Variables and Expressions: Beyond the Symbols
While symbols are central to algebra, it’s critical to emphasize their meaning and avoid rote memorization. Variables should be understood not just as abstract symbols, but as representations of unknown or varying quantities. Similarly, algebraic expressions should be seen as representing relationships between these quantities. Meaningful contexts and real-world applications can significantly improve understanding and prevent the misconception that algebra is merely a collection of abstract rules and procedures.
4. Solving Equations and Inequalities: Unlocking the Unknown
Solving equations and inequalities is a crucial skill in algebra. Students need to master various techniques, such as using inverse operations, balancing equations, and understanding the properties of equality and inequality. It’s important to emphasize the logical reasoning involved in manipulating equations to isolate the unknown variable. Visual representations, like balance scales, can be helpful in illustrating the concept of maintaining equality.
5. Functions and Relationships: Mapping Inputs to Outputs
The concept of functions is central to algebra and algebraic thinking. Functions describe relationships between input and output values, where each input corresponds to exactly one output. Representing functions using tables, graphs, and equations helps students visualize and understand these relationships. Understanding the different representations and the connections between them is vital for developing a strong grasp of functional thinking.
6. Common Pitfalls and Misconceptions in Algebra and Algebraic Thinking
Numerous misconceptions can hinder students' progress in algebra. These include: confusing variables with labels, struggling to translate real-world problems into algebraic expressions, difficulty understanding the properties of equality, and misconceptions about negative numbers and operations with them. Addressing these misconceptions proactively through targeted instruction and effective feedback is essential.
7. Best Practices for Teaching Algebra and Algebraic Thinking
Effective teaching of algebra and algebraic thinking involves more than just presenting rules and procedures. It requires creating a supportive learning environment where students are encouraged to explore, make conjectures, and engage in mathematical reasoning. Using manipulatives, real-world contexts, and collaborative activities can enhance understanding and engagement. Providing ample opportunities for problem-solving and encouraging students to explain their thinking are also crucial.
8. Assessment and Feedback in Algebra and Algebraic Thinking
Assessment should go beyond simply checking for correct answers. It should assess students' understanding of concepts, their ability to apply algebraic reasoning, and their problem-solving skills. Providing timely and constructive feedback is crucial for helping students identify and overcome misconceptions and refine their algebraic thinking.
9. Algebra and Algebraic Thinking in the Real World
Algebra and algebraic thinking are not confined to the classroom; they are essential tools for solving problems in various real-world contexts. Applications range from calculating the area of a room to modeling population growth and analyzing financial data. Connecting algebraic concepts to real-world applications can significantly enhance student motivation and understanding.
Conclusion: Mastering algebra and algebraic thinking is crucial for success in mathematics and beyond. By understanding the fundamental concepts, developing strong reasoning skills, and employing effective teaching strategies, both educators and learners can overcome common challenges and unlock the power of algebraic reasoning. This guide aims to provide a solid foundation for navigating the world of algebra and algebraic thinking, empowering students to confidently tackle mathematical challenges and apply these skills to real-world problems.
FAQs:
1. What is the difference between algebra and algebraic thinking? Algebra is the formal system of symbols and rules, while algebraic thinking is the broader process of reasoning about relationships and generalizations, often preceding formal algebra.
2. Why is algebra important? Algebra provides essential tools for problem-solving in various fields, from science and engineering to finance and economics.
3. How can I help my child overcome difficulties in algebra? Focus on building a strong foundation in arithmetic, emphasize problem-solving, use visual aids, and provide consistent support and encouragement.
4. What are some common misconceptions in algebra? Common misconceptions include confusing variables with labels, misunderstanding the properties of equality, and difficulty translating word problems into equations.
5. What are some effective teaching strategies for algebra? Effective strategies include using real-world contexts, encouraging collaboration, and using visual aids to illustrate concepts.
6. How can I improve my algebraic reasoning skills? Practice regularly, focus on understanding concepts rather than memorization, and seek out challenging problems.
7. What resources are available for learning algebra? Numerous online resources, textbooks, and tutoring services are available.
8. How is algebra used in everyday life? Algebra is used in various aspects of daily life, from calculating budgets and recipes to understanding data and making informed decisions.
9. What are some advanced topics in algebra? Advanced topics include linear algebra, abstract algebra, and number theory.
Related Articles:
1. "Developing Algebraic Reasoning in Early Childhood": This article explores the importance of fostering algebraic thinking in young children through activities and games.
2. "The Role of Patterns in Algebraic Thinking": This article discusses the significance of pattern recognition in developing algebraic reasoning skills.
3. "Overcoming Common Misconceptions in Algebra": This article identifies common student errors and provides strategies for addressing them.
4. "Using Real-World Contexts to Teach Algebra": This article explores the benefits of using real-world applications to make algebra more engaging and relevant.
5. "Effective Assessment Strategies for Algebra": This article discusses different assessment methods to effectively evaluate students' understanding of algebraic concepts.
6. "The Importance of Visual Representations in Algebra": This article emphasizes the use of diagrams and graphs to enhance comprehension in algebra.
7. "Algebraic Thinking and Problem-Solving": This article connects algebraic reasoning with effective problem-solving strategies.
8. "Integrating Technology in Algebra Instruction": This article explores the use of technology to enhance algebra learning and teaching.
9. "Algebra for Non-Math Majors": This article focuses on applying algebra to real-world problems relevant to various disciplines.
| algebra and algebraic thinking: Lessons for Algebraic Thinking Maryann Wickett, Katharine Kharas, Marilyn Burns, 2002 Lessons for K-8 teachers on making algebra an integral part of their mathematics instruction. |
| algebra and algebraic thinking: Algebra and Algebraic Thinking in School Mathematics Carole E. Greenes, 2008 Examines the status of algebra in our schools and the changes that the curriculum has undergone over the past several years. Includes successful classroom practises for developing algebraic reasoning abilities and improving overall understanding. |
| algebra and algebraic thinking: Teaching and Learning Algebraic Thinking with 5- to 12-Year-Olds Carolyn Kieran, 2017-12-04 This book highlights new developments in the teaching and learning of algebraic thinking with 5- to 12-year-olds. Based on empirical findings gathered in several countries on five continents, it provides a wealth of best practices for teaching early algebra. Building on the work of the ICME-13 (International Congress on Mathematical Education) Topic Study Group 10 on Early Algebra, well-known authors such as Luis Radford, John Mason, Maria Blanton, Deborah Schifter, and Max Stephens, as well as younger scholars from Asia, Europe, South Africa, the Americas, Australia and New Zealand, present novel theoretical perspectives and their latest findings. The book is divided into three parts that focus on (i) epistemological/mathematical aspects of algebraic thinking, (ii) learning, and (iii) teaching and teacher development. Some of the main threads running through the book are the various ways in which structures can express themselves in children’s developing algebraic thinking, the roles of generalization and natural language, and the emergence of symbolism. Presenting vital new data from international contexts, the book provides additional support for the position that essential ways of thinking algebraically need to be intentionally fostered in instruction from the earliest grades. |
| algebra and algebraic thinking: Lessons for Algebraic Thinking Leyani Von Rotz, Marilyn Burns, 2002 The lessons in this book introduce basic algebraic concepts to students in the primary grades. Manipulative materials, problem-solving investigations, games, and real-world and imaginary contexts support arithmetic learning while introducing ideas basic to algebra, including patterns, equivalence, and graphing. |
| algebra and algebraic thinking: Connecting Arithmetic to Algebra Susan Jo Russell, Deborah Schifter, Virginia Bastable, 2011 To truly engage in mathematics is to become curious and intrigued about regularities and patterns, then describe and explain them. A focus on the behavior of the operations allows students starting in the familiar territory of number and computation to progress to true engagement in the discipline of mathematics. -Susan Jo Russell, Deborah Schifter, and Virginia Bastable Algebra readiness: it's a topic of concern that seems to pervade every school district. How can we better prepare elementary students for algebra? More importantly, how can we help all children, not just those who excel in math, become ready for later instruction? The answer lies not in additional content, but in developing a way of thinking about the mathematics that underlies both arithmetic and algebra. Connecting Arithmetic to Algebra invites readers to learn about a crucial component of algebraic thinking: investigating the behavior of the operations. Nationally-known math educators Susan Jo Russell, Deborah Schifter, and Virginia Bastable and a group of collaborating teachers describe how elementary teachers can shape their instruction so that students learn to: *notice and describe consistencies across problems *articulate generalizations about the behavior of the operations *develop mathematical arguments based on representations to explain why such generalizations are or are not true. Through such work, students become familiar with properties and general rules that underlie computational strategies-including those that form the basis of strategies used in algebra-strengthening their understanding of grade-level content and at the same time preparing them for future studies. Each chapter is illustrated by lively episodes drawn from the classrooms of collaborating teachers in a wide range of settings. These provide examples of posing problems, engaging students in productive discussion, using representations to develop mathematical arguments, and supporting both students with a wide range of learning profiles. Staff Developers: Available online, the Course Facilitator's Guide provides math leaders with tools and resources for implementing a Connecting Arithmetic to Algebra workshop or preservice course. For information on the PD course offered through Mount Holyoke College, download the flyer. |
| algebra and algebraic thinking: Algebra in the Early Grades James J. Kaput, David W. Carraher, Maria L. Blanton, 2017-09-25 This volume is the first to offer a comprehensive, research-based, multi-faceted look at issues in early algebra. In recent years, the National Council for Teachers of Mathematics has recommended that algebra become a strand flowing throughout the K-12 curriculum, and the 2003 RAND Mathematics Study Panel has recommended that algebra be “the initial topical choice for focused and coordinated research and development [in K-12 mathematics].” This book provides a rationale for a stronger and more sustained approach to algebra in school, as well as concrete examples of how algebraic reasoning may be developed in the early grades. It is organized around three themes: The Nature of Early Algebra Students’ Capacity for Algebraic Thinking Issues of Implementation: Taking Early Algebra to the Classrooms. The contributors to this landmark volume have been at the forefront of an effort to integrate algebra into the existing early grades mathematics curriculum. They include scholars who have been developing the conceptual foundations for such changes as well as researchers and developers who have led empirical investigations in school settings. Algebra in the Early Grades aims to bridge the worlds of research, practice, design, and theory for educators, researchers, students, policy makers, and curriculum developers in mathematics education. |
| algebra and algebraic thinking: Thinking Algebraically: An Introduction to Abstract Algebra Thomas Q. Sibley, 2021-06-08 Thinking Algebraically presents the insights of abstract algebra in a welcoming and accessible way. It succeeds in combining the advantages of rings-first and groups-first approaches while avoiding the disadvantages. After an historical overview, the first chapter studies familiar examples and elementary properties of groups and rings simultaneously to motivate the modern understanding of algebra. The text builds intuition for abstract algebra starting from high school algebra. In addition to the standard number systems, polynomials, vectors, and matrices, the first chapter introduces modular arithmetic and dihedral groups. The second chapter builds on these basic examples and properties, enabling students to learn structural ideas common to rings and groups: isomorphism, homomorphism, and direct product. The third chapter investigates introductory group theory. Later chapters delve more deeply into groups, rings, and fields, including Galois theory, and they also introduce other topics, such as lattices. The exposition is clear and conversational throughout. The book has numerous exercises in each section as well as supplemental exercises and projects for each chapter. Many examples and well over 100 figures provide support for learning. Short biographies introduce the mathematicians who proved many of the results. The book presents a pathway to algebraic thinking in a semester- or year-long algebra course. |
| algebra and algebraic thinking: Thinking Mathematically Thomas P. Carpenter, Megan Loef Franke, Linda Levi, 2003 In this book the authors reveal how children's developing knowledge of the powerful unifying ideas of mathematics can deepen their understanding of arithmetic |
| algebra and algebraic thinking: Accessible Algebra Anne Collins, Steven Benson, 2023-10-10 Accessible Algebra: 30 Modules to Promote Algebraic Reasoning, Grades 7-10 is for any pre-algebra or algebra teacher who wants to provide a rich and fulfilling experience for students as they develop new ways of thinking through and about algebra.' The book includes 30 lessons that identify a focal domain and standard in algebra, then lays out the common misconceptions and challenges students may face as they work to investigate and understand problems.' Authors Anne Collins and Steven Benson conferred with students in real classrooms as the students explained what problem-solving strategies they were using or worked to ask the right questions that would lead them to a deeper understanding of algebra. Each scenario represents actual instances of an algebra classroom that demonstrate effective teaching methods, real-life student questions, and conversations about the problems at hand. 'Accessible Algebra' works for students at every level. In each lesson, there are sections on how to support struggling students, as well as ways to challenge students who may need more in-depth work. There are also numerous additional resources, including research articles and classroom vignettes. |
| algebra and algebraic thinking: Developing Thinking in Algebra John Mason, Alan Graham, Sue Johnston-Wilder, 2005-04-23 'Mason, Graham, and Johnston-Wilder have admirably succeeded in casting most of school algebra in terms of generalisation activity? not just the typical numerical and geometric pattern-based work, but also solving quadratics and simultaneous equations, graphing equations, and factoring. The authors raise our awareness of the scope of generalization and of the power of using this as a lens not just for algebra but for all of mathematics!' - Professor Carolyn Kieran, Departement de Mathematiques, Universite du Quebec a Montreal Algebra has always been a watershed for pupils learning mathematics. This book will enable you to think about yourself as a learner of algebra in a new way, and thus to teach algebra more successfully, overcoming difficulties and building upon skills that all learners have. This book is based on teaching principles developed by the team at The Open University's Centre for Mathematics Education which has a 20-year track record of innovative approaches to teaching and learning algebra. Written for teachers working with pupils aged 7-16, it includes numerous tasks ready for adaption for your teaching and discusses principles that teachers have found useful in preparing and conducting lessons. This is a 'must have' resource for all teachers of mathematics, primary or secondary, and their support staff. Anyone who wishes to create an understanding and enthusiasm for algebra, based upon firm research and effective practice, will enjoy this book. This book is the course reader for The Open University Course ME625 Developing Algebraic Thinking |
| algebra and algebraic thinking: How Students Think When Doing Algebra Steve Rhine, Rachel Harrington, Colin Starr, 2018-11-01 Algebra is the gateway to college and careers, yet it functions as the eye of the needle because of low pass rates for the middle school/high school course and students’ struggles to understand. We have forty years of research that discusses the ways students think and their cognitive challenges as they engage with algebra. This book is a response to the National Council of Teachers of Mathematics’ (NCTM) call to better link research and practice by capturing what we have learned about students’ algebraic thinking in a way that is usable by teachers as they prepare lessons or reflect on their experiences in the classroom. Through a Fund for the Improvement of Post-Secondary Education (FIPSE) grant, 17 teachers and mathematics educators read through the past 40 years of research on students’ algebraic thinking to capture what might be useful information for teachers to know—over 1000 articles altogether. The resulting five domains addressed in the book (Variables & Expressions, Algebraic Relations, Analysis of Change, Patterns & Functions, and Modeling & Word Problems) are closely tied to CCSS topics. Over time, veteran math teachers develop extensive knowledge of how students engage with algebraic concepts—their misconceptions, ways of thinking, and when and how they are challenged to understand—and use that knowledge to anticipate students’ struggles with particular lessons and plan accordingly. Veteran teachers learn to evaluate whether an incorrect response is a simple error or the symptom of a faulty or naïve understanding of a concept. Novice teachers, on the other hand, lack the experience to anticipate important moments in the learning of their students. They often struggle to make sense of what students say in the classroom and determine whether the response is useful or can further discussion (Leatham, Stockero, Peterson, & Van Zoest 2011; Peterson & Leatham, 2009). The purpose of this book is to accelerate early career teachers’ “experience” with how students think when doing algebra in middle or high school as well as to supplement veteran teachers’ knowledge of content and students. The research that this book is based upon can provide teachers with insight into the nature of a student’s struggles with particular algebraic ideas—to help teachers identify patterns that imply underlying thinking. Our book, How Students Think When Doing Algebra, is not intended to be a “how to” book for teachers. Instead, it is intended to orient new teachers to the ways students think and be a book that teachers at all points in their career continually pull of the shelf when they wonder, “how might my students struggle with this algebraic concept I am about to teach?” The primary audience for this book is early career mathematics teachers who don’t have extensive experience working with students engaged in mathematics. However, the book can also be useful to veteran teachers to supplement their knowledge and is an ideal resource for mathematics educators who are preparing preservice teachers. |
| algebra and algebraic thinking: Developing Essential Understanding of Algebraic Thinking for Teaching Mathematics in Grades 3-5 Maria L. Blanton, 2011 Like algebra at any level, early algebra is a way to explore, analyse, represent and generalise mathematical ideas and relationships. This book shows that children can and do engage in generalising about numbers and operations as their mathematical experiences expand. The authors identify and examine five big ideas and associated essential understandings for developing algebraic thinking in grades 3-5. The big ideas relate to the fundamental properties of number and operations, the use of the equals sign to represent equivalence, variables as efficient tools for representing mathematical ideas, quantitative reasoning as a way to understand mathematical relationships and functional thinking to generalise relationships between covarying quantities. The book examines challenges in teaching, learning and assessment and is interspersed with questions for teachers’ reflection. |
| algebra and algebraic thinking: Leveled Texts for Mathematics: Algebra and Algebraic Thinking Lori Barker, 2011-06-01 With a focus on algebra and algebraic thinking, this resource provides the know-how to use leveled texts to differentiate instruction in mathematics. A total of 15 different topics are featured and the high-interest text is written at four different reading levels with matching visuals. Practice problems are provided to reinforce what is taught in the passage. The included Teacher Resource CD features a modifiable version of each passage in text format and full-color versions of the texts and image files. This resource is correlated to the Common Core State Standards. 144 pp. |
| algebra and algebraic thinking: Fostering Algebraic Thinking Mark J. Driscoll, 1999 Fostering Algebraic Thinking is a timely and welcome resource for middle and high school teachers hoping to ease their students' transition to algebra. |
| algebra and algebraic thinking: Algebraic Reasoning Paul Gray, Jacqueline Weilmuenster, Jennifer Hylemon, 2016-09-01 Algebraic Reasoning is a textbook designed to provide high school students with a conceptual understanding of algebraic functions and to prepare them for Algebra 2.. |
| algebra and algebraic thinking: Approaches to Algebra N. Bednarz, C. Kieran, L. Lee, 2012-12-06 In Greek geometry, there is an arithmetic of magnitudes in which, in terms of numbers, only integers are involved. This theory of measure is limited to exact measure. Operations on magnitudes cannot be actually numerically calculated, except if those magnitudes are exactly measured by a certain unit. The theory of proportions does not have access to such operations. It cannot be seen as an arithmetic of ratios. Even if Euclidean geometry is done in a highly theoretical context, its axioms are essentially semantic. This is contrary to Mahoney's second characteristic. This cannot be said of the theory of proportions, which is less semantic. Only synthetic proofs are considered rigorous in Greek geometry. Arithmetic reasoning is also synthetic, going from the known to the unknown. Finally, analysis is an approach to geometrical problems that has some algebraic characteristics and involves a method for solving problems that is different from the arithmetical approach. 3. GEOMETRIC PROOFS OF ALGEBRAIC RULES Until the second half of the 19th century, Euclid's Elements was considered a model of a mathematical theory. This may be one reason why geometry was used by algebraists as a tool to demonstrate the accuracy of rules otherwise given as numerical algorithms. It may also be that geometry was one way to represent general reasoning without involving specific magnitudes. To go a bit deeper into this, here are three geometric proofs of algebraic rules, the frrst by Al-Khwarizmi, the other two by Cardano. |
| algebra and algebraic thinking: Groundworks Carole E. Greenes, Carol R. Findell, Wright Group/McGraw-Hill, 2006 |
| algebra and algebraic thinking: Early Algebraization Jinfa Cai, Eric Knuth, 2011-02-24 In this volume, the authors address the development of students’ algebraic thinking in the elementary and middle school grades from curricular, cognitive, and instructional perspectives. The volume is also international in nature, thus promoting a global dialogue on the topic of early Algebraization. |
| algebra and algebraic thinking: Early Algebra Carolyn Kieran, JeongSuk Pang, Deborah Schifter, Swee Fong Ng, 2016-07-11 This survey of the state of the art on research in early algebra traces the evolution of a relatively new field of research and teaching practice. With its focus on the younger student, aged from about 6 years up to 12 years, this volume reveals the nature of the research that has been carried out in early algebra and how it has shaped the growth of the field. The survey, in presenting examples drawn from the steadily growing research base, highlights both the nature of algebraic thinking and the ways in which this thinking is being developed in the primary and early middle school student. Mathematical relations, patterns, and arithmetical structures lie at the heart of early algebraic activity, with processes such as noticing, conjecturing, generalizing, representing, justifying, and communicating being central to students’ engagement. |
| algebra and algebraic thinking: Algebra Carole E. Greenes, Carol Findell, 1998 The puzzles and problems cover six areas of algebra: presentation, proportional reasoning, balance, variable, function, and inductive reasoning. |
| algebra and algebraic thinking: Developing Algebraic Thinking Don Balka, 2004 Develop algebraic thinking by exploring and conjecturing about patterns; verbalising relationships; making generalisations; symbolising relationships; working with functions and making connections between the real world and algebraic statemwnts. A resource for teachers to explore the components of alegbra. |
| algebra and algebraic thinking: Algebra and the Elementary Classroom Maria L. Blanton, 2008 Algebra in the Elementary Classroom provides the support we need as teachers to embed the development of students' algebraic thinking in the teaching of elementary school. - Megan Loef Franke Coauthor of Children's Mathematics and Thinking Mathematically How do you start students down the road to mathematical understanding? By laying the foundation for algebra in the elementary grades. Algebra and the Elementary Classroom shares ideas, tasks, and practices for integrating algebraic thinking into your teaching. Through research-based and classroom-tested strategies, it demonstrates how to use materials you have on hand to prepare students for formal algebra instruction - without adding to your overstuffed curriculum. You'll find ways to: introduce algebraic thinking through familiar arithmetical contexts nurture it by helping students think about, represent, and build arguments for their mathematical ideas develop it by exploring mathematical structures and functional relationships strengthen it by asking students to make algebraic connections across the curriculum reinforce it across the grades through a schoolwide initiative. No matter what your math background is, Algebra and the Elementary Classroom offers strong support for integrating algebraic thinking into your daily teaching. Its clear descriptions show you what algebraic thinking is and how to teach it. Its sample problems deepen your own algebraic thinking. Best of all, it gives you ideas for grade-specific instructional planning. Read Algebra and the Elementary Classroom and prepare your students for a lifetime of mathematical understanding. |
| algebra and algebraic thinking: The Fostering Algebraic Thinking Toolkit: Introduction and analyzing written student work Mark J. Driscoll, Judith Zawojewski, Johannah Nikula, Andrea Humez, 2001 Part of the Fostering Algebraic Thinking series, this module gives participants an opportunity to analyze students' written work for evidence of algebraic thinking. |
| algebra and algebraic thinking: Is It Red? Is It Yellow? Is It Blue? Tana Hoban, 1987-04-23 What color do you see? Red? Yellow? Blue? Here is a concept book young children can grow with, as they explore colors, sizes, shapes, and relationships with the master of the photo-concept book ' Tana Hoban. |
| algebra and algebraic thinking: Teaching Number Sense and Algebraic Thinking New Zealand. Ministry of Education, 2003 |
| algebra and algebraic thinking: Developing Algebraic Thinking Clemson University, Carolina Biological Supply Company, 2005 |
| algebra and algebraic thinking: Math Games for Independent Practice, Grades K-5 Jamee Petersen, 2013 This former Math Solutions publication is now published by Heinemann (ISBN: 9780325137612). Visit Heinemann.com/Math to learn more! Carefully selected compilation of games focused on number and operations and algebraic thinking. Each game is introduced with step-by-step teaching directions interwoven with pedagogical support; ideal for use in math workshop learning stations and more. High student engagement and interaction + creative + fun. Pair with Math Games for Geometry and Measurement (978-0-325-13762-9) for the ultimate game collection! |
| algebra and algebraic thinking: Uncomplicating Algebra to Meet Common Core Standards in Math, K-8 Marian Small, 2014-12-04 In the second book in the Uncomplicating Mathematics Series, professional developer Marian Small shows teachers how to uncomplicate the teaching of algebra by focusing on the most important ideas that students need to grasp. Organized by grade level around the Common Core State Standards for Mathematics, Small shares approaches that will lead to a deeper and richer understanding of algebra for both teachers and students. The book opens with a clear discussion of algebraic thinking and current requirements for algebraic understanding within standards-based learning environments. The book then launches with Kindergarten, where the first relevant standard is found in the operations and algebraic thinking domain, and ends with Grade 8, where the focus is on working with linear equations and functions. In each section the relevant standard is presented, followed by a discussion of important underlying ideas associated with that standard, as well as thoughtful, concept-based questions that can be used for classroom instruction, practice, or assessment. Underlying ideas include: Background to the mathematics of each relevant standard. Suggestions for appropriate representations for specific mathematical ideas. Suggestions for explaining ideas to students. Cautions about misconceptions or situations to avoid. The Common Core State Standards for Mathematics challenges students to become mathematical thinkers, not just mathematical “doers.” This resource will be invaluable for pre- and inservice teachers as they prepare themselves to understand and teach algebra with a deep level of understanding. “Uncomplicating Algebra is an excellent resource for teachers responsible for the mathematical education of K–8 students. It is also a valuable tool for the training of preservice teachers of elementary and middle school mathematics.” —Carole Greenes, associate vice provost for STEM education, director of the Practice Research and Innovation in Mathematics Education (PRIME) Center, professor of mathematics education, Arizona State University “The current climate in North America places a major emphasis on standards, including the Common Core State Standards for Mathematics in the U.S. In many cases, teachers are being asked to teach content with which they themselves struggle. In this book, Dr. Small masterfully breaks down the big ideas of algebraic thinking to assist teachers, math coaches, and preservice teachers—helping them to deepen their own understanding of the mathematics they teach. She describes common error patterns and examines algebraic reasoning from a developmental viewpoint, connecting the dots from kindergarten through grade 8. The book is clearly written, loaded with specific examples, and very timely. I recommend it strongly as a ‘must-read’ for all who are seeking to broaden their understanding of algebra and how to effectively teach this important content area to children.” —Daniel J. Brahier, director, Science and Math Education in ACTION, professor of mathematics education, School of Teaching and Learning, Bowling Green State University |
| algebra and algebraic thinking: Balance Math and More! Level 1 Robert Femiano, 2012-01-30 |
| algebra and algebraic thinking: Bringing Out the Algebraic Character of Arithmetic Analúcia D. Schliemann, David W. Carraher, Bárbara M. Brizuela, 2006-08-29 Bringing Out the Algebraic Character of Arithmetic contributes to a growing body of research relevant to efforts to make algebra an integral part of early mathematics instruction, an area of studies that has come to be known as Early Algebra. It provides both a rationale for promoting algebraic reasoning in the elementary school curriculum and empirical data to support it. The authors regard Early Algebra not as accelerated instruction but as an approach to existing topics in the early mathematics curriculum that highlights their algebraic character. Each chapter shows young learners engaged in mathematics tasks where there has been a shift away from computations on specific amounts toward thinking about relations and functional dependencies. The authors show how young learners attempt to work with mathematical generalizations before they have learned formal algebraic notation. The book, suitable as a text in undergraduate or graduate mathematics education courses, includes downloadable resources with additional text and video footage on how students reason about addition and subtraction as functions; on how students understand multiplication when it is presented as a function; and on how children use notations in algebraic problems involving fractions. These three videopapers (written text with embedded video footage) present relevant discussions that help identify students' mathematical reasoning. The printed text in the book includes transcriptions of the video episodes in the CD-ROM. Bringing Out the Algebraic Character of Arithmetic is aimed at researchers, practitioners, curriculum developers, policy makers and graduate students across the mathematics education community who wish to understand how young learners deal with algebra before they have learned about algebraic notation. |
| algebra and algebraic thinking: The Nature and Role of Algebra in the K-14 Curriculum Center for Science, Mathematics, and Engineering Education, National Council of Teachers of Mathematics and Mathematical Sciences Education Board, National Research Council, 1998-10-07 With the 1989 release of Everybody Counts by the Mathematical Sciences Education Board (MSEB) of the National Research Council and the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM), the standards movement in K-12 education was launched. Since that time, the MSEB and the NCTM have remained committed to deepening the public debate, discourse, and understanding of the principles and implications of standards-based reform. One of the main tenets in the NCTM Standards is commitment to providing high-quality mathematical experiences to all students. Another feature of the Standards is emphasis on development of specific mathematical topics across the grades. In particular, the Standards emphasize the importance of algebraic thinking as an essential strand in the elementary school curriculum. Issues related to school algebra are pivotal in many ways. Traditionally, algebra in high school or earlier has been considered a gatekeeper, critical to participation in postsecondary education, especially for minority students. Yet, as traditionally taught, first-year algebra courses have been characterized as an unmitigated disaster for most students. There have been many shifts in the algebra curriculum in schools within recent years. Some of these have been successful first steps in increasing enrollment in algebra and in broadening the scope of the algebra curriculum. Others have compounded existing problems. Algebra is not yet conceived of as a K-14 subject. Issues of opportunity and equity persist. Because there is no one answer to the dilemma of how to deal with algebra, making progress requires sustained dialogue, experimentation, reflection, and communication of ideas and practices at both the local and national levels. As an initial step in moving from national-level dialogue and speculations to concerted local and state level work on the role of algebra in the curriculum, the MSEB and the NCTM co-sponsored a national symposium, The Nature and Role of Algebra in the K-14 Curriculum, on May 27 and 28, 1997, at the National Academy of Sciences in Washington, D.C. |
| algebra and algebraic thinking: Developing Algebraic Thinking Clemson University, Carolina Biological Supply Company, 2005 |
| algebra and algebraic thinking: Early Childhood Mathematics Education Research Julie Sarama, Douglas H. Clements, 2009-04-01 This important new book synthesizes relevant research on the learning of mathematics from birth into the primary grades from the full range of these complementary perspectives. At the core of early math experts Julie Sarama and Douglas Clements's theoretical and empirical frameworks are learning trajectories—detailed descriptions of children’s thinking as they learn to achieve specific goals in a mathematical domain, alongside a related set of instructional tasks designed to engender those mental processes and move children through a developmental progression of levels of thinking. Rooted in basic issues of thinking, learning, and teaching, this groundbreaking body of research illuminates foundational topics on the learning of mathematics with practical and theoretical implications for all ages. Those implications are especially important in addressing equity concerns, as understanding the level of thinking of the class and the individuals within it, is key in serving the needs of all children. |
| algebra and algebraic thinking: The Proceedings of the 12th International Congress on Mathematical Education Sung Je Cho, 2015-02-10 This book comprises the Proceedings of the 12th International Congress on Mathematical Education (ICME-12), which was held at COEX in Seoul, Korea, from July 8th to 15th, 2012. ICME-12 brought together 3500 experts from 92 countries, working to understand all of the intellectual and attitudinal challenges in the subject of mathematics education as a multidisciplinary research and practice. This work aims to serve as a platform for deeper, more sensitive and more collaborative involvement of all major contributors towards educational improvement and in research on the nature of teaching and learning in mathematics education. It introduces the major activities of ICME-12 which have successfully contributed to the sustainable development of mathematics education across the world. The program provides food for thought and inspiration for practice for everyone with an interest in mathematics education and makes an essential reference for teacher educators, curriculum developers and researchers in mathematics education. The work includes the texts of the four plenary lectures and three plenary panels and reports of three survey groups, five National presentations, the abstracts of fifty one Regular lectures, reports of thirty seven Topic Study Groups and seventeen Discussion Groups. |
| algebra and algebraic thinking: The Fostering Algebraic Thinking Toolkit: Asking questions of students Judith Zawojewski, Andrea Humez, Mark J. Driscoll, Johannah Nikula, Lynn Goldsmith, James Hammerman, 2001 Together with the accompanying video, this module offers a change both in the type of student data considered--from written to real time--and in the emphasis of the module--from understanding to fostering student thinking. |
| algebra and algebraic thinking: Mystery Math David A. Adler, 2012-05-14 Boo! There is a mystery behind every door of the creepy haunted house. Luckily, algebra will help you solve each problem. By using simple addition, subtraction, mulitplication, and division, you'll discover that solving math mysteries isn't scary at all -- it's fun! |
| algebra and algebraic thinking: I Know an Old Lady Who Swallowed a Fly Inc. Nadine Bernard Westcott, 2007-09-03 I know an old lady who swallowed a fly, I don't know why she swallowed a fly, Perhaps she'll die. So begins this well-loved, classic song. Now published for the first time in board book form with all new illustrations, this book is sure to delight a whole new audience: babies and toddlers. |
| algebra and algebraic thinking: Mathematics as a Science of Patterns Michael D. Resnik, 1997 Resnik expresses his commitment to a structuralist philosophy of mathematics and links this to a defence of realism about the metaphysics of mathematics - the view that mathematics is about things that really exist. |
| algebra and algebraic thinking: Perspectives on School Algebra Rosamund Sutherland, Teresa Rojano, Alan Bell, Romulo Lins, 2006-02-16 This book confronts the issue of how young people can find a way into the world of algebra. It represents multiple perspectives which include an analysis of situations in which algebra is an efficient problem-solving tool, the use of computer-based technologies, and a consideration of the historical evolution of algebra. The book emphasizes the situated nature of algebraic activity as opposed to being concerned with identifying students' conceptions in isolation from problem-solving activity. |
| algebra and algebraic thinking: Algebraic Thinking, Grades K-12 Barbara Moses, 1999 |
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
Algebra - Wikipedia
Elementary algebra, also called school algebra, college algebra, and classical algebra, [22] is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables …
Introduction to Algebra - Math is Fun
Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end up with something like "x = 6". But instead of saying " obviously x=6", use this neat step-by-step …
Algebra I - Khan Academy
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; …
Algebra | History, Definition, & Facts | Britannica
May 9, 2025 · Algebra is the branch of mathematics in which abstract symbols, rather than numbers, are manipulated or operated with arithmetic. For example, x + y = z or b - 2 = 5 are …
Algebra - What is Algebra? | Basic Algebra | Definition - Cuemath
Algebra is the branch of mathematics that represents problems in the form of mathematical expressions. It involves variables like x, y, z, and mathematical operations like addition, …
How to Understand Algebra (with Pictures) - wikiHow
Mar 18, 2025 · Algebra is a system of manipulating numbers and operations to try to solve problems. When you learn algebra, you will learn the rules to follow for solving problems. But to …
What is Algebra? - BYJU'S
Algebra is one of the oldest branches in the history of mathematics that deals with number theory, geometry, and analysis. The definition of algebra sometimes states that the study of the …
Algebra in Math - Definition, Branches, Basics and Examples
Apr 7, 2025 · This section covers key algebra concepts, including expressions, equations, operations, and methods for solving linear and quadratic equations, along with polynomials and …
Algebra - Simple English Wikipedia, the free encyclopedia
People who do algebra use the rules of numbers and mathematical operations used on numbers. The simplest are adding, subtracting, multiplying, and dividing. More advanced operations …
OpenAlgebra.com: Free Algebra Study Guide & Video Tutorials
Free algebra tutorial and help. Notes, videos, steps. Solve and simplify linear, quadratic, polynomial, and rational expressions and equations.
