How Students Think When Doing Algebra
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How Students Think When Doing Algebra
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Author : Steve Rhine
language : en
Publisher: IAP
Release Date : 2018-11-01
How Students Think When Doing Algebra written by Steve Rhine and has been published by IAP this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-11-01 with Education categories.
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.
Understanding College Algebra
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Author : Eric Kuennen
language : en
Publisher:
Release Date : 2018-08-03
Understanding College Algebra written by Eric Kuennen and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-08-03 with categories.
College Algebra actually makes sense and is something that you can figure out and understand why it works. This text focuses on a conceptual understanding of the big ideas in algebraic thinking, engaging the student in authentic problem solving and exploring the logical reasoning that underlies the various techniques and procedures in college algebra. An Inquiry-based Approach. Each section starts with a Class Activity to engage students in actually doing mathematics. Doing math is not just calculating or following a procedure. Doing math is figuring things out: investigating, making and testing conjectures, making arguments, and communicating your reasoning to others. The class activities are designed to highlight big algebraic ideas and spark a discussion of algebraic habits of mind, as well as students' alternate conceptions that lead to common algebra mistakes. Students are asked to analyze solutions, explore representations, explain why valid methods for simplifying expressions or solving equations work, and explain why invalid methods do not work. This book is intended to be read. Often math textbooks do not end up being read, but instead are used merely as a reference for their step-by-step procedures. Each section of this text has a "Read and Study" section that discusses the mathematics raised by the Class Activity and focuses on the mathematical reasoning and proof needed to nurture longer-lasting understanding of the content. This is meant to be read slowly and carefully, with pencil in hand. We pose questions that you should think about and answer before reading on. When we do work out an example, we do so to discuss the big ideas and illustrate our reasoning, not with the intention of providing you with a model to copy. Exercises vs. Problems. In the homework, we distinguish between "exercises" and "problems." Exercises are more routine, intended to give you more practice thinking about the big concepts. In contrast, the "problems" are intended to be problematic, to take time to explore, develop and make connections, and often to extend your reasoning to develop new ideas. We do not include "answers" to these homework exercises and problems. Why struggle and persevere to figure something out and understand it when you can just look it up? Mathematics is not about getting the right answer; it's about figuring things out. It's about logical reasoning and being able to justify that what you claim is true. This doesn't mean that you are on your own. We will do our best in the Read and Study sections to discuss the big ideas, offer explanations, and show you some good examples of problem solving and making mathematical arguments. This text addresses the topics of a standard course in College Algebra, with the following sections: 1. Introduction to Algebra 2. Algebra as Generalized Arithmetic 3. Graphs of Equations and Relations 5. Algebraic Expressions and the Order of Operations 6. Defining and Interpreting Symbols 7. The Laws of Algebra 8. The Distributive Law 9. Justifying Properties 10. Cancellation Facts and Myths 11. Exponents 12. Roots of Numbers 13. Irrational and Imaginary Numbers 14. Testing and Justifying Simplifications 15. Types of Equations 16. Properties of Equality and Solving Equations 17. Techniques for Solving Equations 18. Inverse Functions 19. Finding Inverse Function Formulas 20. The Distance Formula 21. Finding Equations for Graphs 22. Function Forms 23. Area Models for Quadratic Expressions 24. Quadratic Forms 25. Transformations of Functions 26. The Fundamental Theorem of Algebra 27. Rational Functions 28. Exponential and Logarithm Functions 29. The Natural Exponent Base 30. Solving Equations Review
Where S The Math
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Author : Stephanie Lessman
language : en
Publisher:
Release Date : 2004
Where S The Math written by Stephanie Lessman and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with categories.
Understanding College Algebra
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Author : Eric W Kuennen
language : en
Publisher: Independently Published
Release Date : 2021-06-21
Understanding College Algebra written by Eric W Kuennen and has been published by Independently Published this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-06-21 with categories.
College Algebra actually makes sense and is something that you can figure out and understand why it works. This text focuses on a conceptual understanding of the big ideas in algebraic thinking, engaging the student in authentic problem solving and exploring the logical reasoning that underlies the various techniques and procedures in college algebra. An Inquiry-based Approach. Each section starts with a Class Activity to engage students in actually doing mathematics. Doing math is not just calculating or following a procedure. Doing math is figuring things out: investigating, making and testing conjectures, making arguments, and communicating your reasoning to others. The class activities are designed to highlight big algebraic ideas and spark a discussion of algebraic habits of mind, as well as students' alternate conceptions that lead to common algebra mistakes. Students are asked to analyze solutions, explore representations, explain why valid methods for simplifying expressions or solving equations work, and explain why invalid methods do not work. This book is intended to be read. Often math textbooks do not end up being read, but instead are used merely as a reference for their step-by-step procedures. Each section of this text has a "Read and Study" section that discusses the mathematics raised by the Class Activity and focuses on the mathematical reasoning and proof needed to nurture longer-lasting understanding of the content. This is meant to be read slowly and carefully, with pencil in hand. We pose questions that you should think about and answer before reading on. When we do work out an example, we do so to discuss the big ideas and illustrate our reasoning, not with the intention of providing you with a model to copy. Exercises vs. Problems. In the homework, we distinguish between "exercises" and "problems." Exercises are more routine, intended to give you more practice thinking about the big concepts. In contrast, the "problems" are intended to be problematic, to take time to explore, develop and make connections, and often to extend your reasoning to develop new ideas. We do not include "answers" to these homework exercises and problems. Why struggle and persevere to figure something out and understand it when you can just look it up? Mathematics is not about getting the right answer; it's about figuring things out. It's about logical reasoning and being able to justify that what you claim is true. This doesn't mean that you are on your own. We will do our best in the Read and Study sections to discuss the big ideas, offer explanations, and show you some good examples of problem solving and making mathematical arguments. This text addresses the topics of a standard course in College Algebra, with the following sections: 1.Features of Algebraic Thinking 2. Algebraic Symbols 3. Sequences of Operations 4. Properties of Operations 5. The Distributive Law 6. Additive and Multiplicative Inverses 7. Using Inverses 8. Exponents 9. Roots of Numbers 10. Irrational and Imaginary Numbers 11. Testing and Justifying Simplifications 12. Types of Equations 13. Properties of Equality and Solving Equations 14. Techniques for Solving Equations 15. The Distance Formula 16. Finding Equations for Graphs 17. Ellipses 18. Function Defintions 19. Functional Thinking 20. Function Forms 21. Linear Function Forms 22. Quadratic Expressions 23. Quadratic Functions 24. Transformations of Functions 25. Polynomials 26. Rational Functions 27. Exponential and Logarithm Functions 28. The Natural Exponent Base 29. Inverse Functions 30. Finding Inverse Function Formulas 31. Solving Equations Review
Making Sense Of Algebra
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Author : Ernest Paul Goldenberg
language : en
Publisher: Heinemann Educational Books
Release Date : 2015
Making Sense Of Algebra written by Ernest Paul Goldenberg and has been published by Heinemann Educational Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with Education categories.
"This book has much to offer teachers of middle and high school algebra who wish to implement the Common Core Standards for all of their students." -Hyman Bass, Samuel Eilenberg Distinguished University Professor of Mathematics & Mathematics Education, University of Michigan "One of the joys of Making Sense of Algebra is how clearly and practically the 'how' question is answered." -Steven Leinwand, American Institutes for Research, author of Accessible Mathematics "Paul Goldenberg and his colleagues have done a fantastic job of connecting mathematical ideas to teaching those ideas." -David Wees, New Visions for Public Schools, New York City Every teacher wants to help students make sense of mathematics; but what if you could guide your students to expect mathematics to make sense? What if you could help them develop a deep understanding of the reasons behind its facts and methods? In Making Sense of Algebra, the common misconception that algebra is simply a collection of rules to know and follow is debunked by delving into how we think about mathematics. This "habits of mind" approach is concerned not just with the results of mathematical thinking, but with how mathematically proficient students do that thinking. Making Sense of Algebra addresses developing this type of thinking in your students through: using well-chosen puzzles and investigations to promote perseverance and a willingness to explore seeking structure and looking for patterns that mathematicians anticipate finding-and using this to draw conclusions cultivating an approach to authentic problems that are rarely as tidy as what is found in textbooks allowing students to generate, validate, and critique their own and others' ideas without relying on an outside authority. Through teaching tips, classroom vignettes, and detailed examples, Making Sense of Algebra shows how to focus your instruction on building these key habits of mind, while inviting students to experience the clarity and meaning of mathematics-perhaps for the first time. Discover more math resources at Heinemann.com/Math
Practical Algebra
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Author : Bobson Wong
language : en
Publisher: John Wiley & Sons
Release Date : 2022-04-14
Practical Algebra written by Bobson Wong and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-14 with Mathematics categories.
The most practical, complete, and accessible guide for understanding algebra If you want to make sense of algebra, check out Practical Algebra: A Self-Teaching Guide. Written by two experienced classroom teachers, this Third Edition is completely revised to align with the Common Core Algebra I math standards used in many states. You’ll get an overview of solving linear and quadratic equations, using ratios and proportions, decoding word problems, graphing and interpreting functions, modeling the real world with statistics, and other concepts found in today’s algebra courses. This book also contains a brief review of pre-algebra topics, including arithmetic and fractions. It has concrete strategies that help diverse students to succeed, such as: over 500 images and tables that illustrate important concepts over 200 model examples with complete solutions almost 1,500 exercises with answers so you can monitor your progress Practical Algebra emphasizes making connections to what you already know and what you’ll learn in the future. You’ll learn to see algebra as a logical and consistent system of ideas and see how it connects to other mathematical topics. This book makes math more accessible by treating it as a language. It has tips for pronouncing and using mathematical notation, a glossary of commonly used terms in algebra, and a glossary of symbols. Along the way, you’ll discover how different cultures around the world over thousands of years developed many of the mathematical ideas we use today. Since students nowadays can use a variety of tools to handle complex modeling tasks, this book contains technology tips that apply no matter what device you’re using. It also describes strategies for avoiding common mistakes that students make. By working through Practical Algebra, you’ll learn straightforward techniques for solving problems, and understand why these techniques work so you’ll retain what you’ve learned. You (or your students) will come away with better scores on algebra tests and a greater confidence in your ability to do math.
Classroom Ready Rich Algebra Tasks Grades 6 12
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Author : Barbara J. Dougherty
language : en
Publisher: Corwin Press
Release Date : 2023-03-15
Classroom Ready Rich Algebra Tasks Grades 6 12 written by Barbara J. Dougherty and has been published by Corwin Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-03-15 with Education categories.
Stop algebra from being a mathematical gatekeeper. With rich math tasks, all students can succeed. Every teacher strives to make instruction effective and interesting, yet traditional methods of teaching algebra are not working for many students! That’s a problem. But the answer isn’t to supplement the curriculum with random tasks. Classroom Ready-Rich Math Tasks for Grades 6-12 equips you with a cohesive solution--50+ mathematical tasks that are rich, research-based, standards-aligned, and classroom-tested. The tasks: Are organized into learning progressions that help all students make the leap from arithmetic to algebra Offer students interesting mathematics problems to think about and solve so math is investigative, interactive, and engaging Provide opportunities for you to connect new content to prior knowledge or focus on an underdeveloped concept Engage students in conceptual understanding, procedural practice, and problem solving through critical thinking and application Come with downloadable planning tools, student resource pages, and extension questions Include additional support for students who may be struggling Every learner deserves opportunities to engage in meaningful, rigorous mathematics. And every teacher can develop mathematical thinking and reasoning abilities in students. Part of the bestselling series spanning elementary and middle school, Classroom-Ready Rich Algebra Tasks, Grades 6-12 is a powerful add-on to any core mathematics program at your school.
Best Strategies For Pre Algebra With Basic Algebra
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Author : Alco Mathematic Tutorial Series
language : en
Publisher:
Release Date : 2016-04-21
Best Strategies For Pre Algebra With Basic Algebra written by Alco Mathematic Tutorial Series and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-04-21 with categories.
This textbook is designed around the philosophy that students do better in math and science if they have a real perception of the fundamental concepts of mathematics. The content within this textbook first places an emphasis on developing a strategy for solving math problems which will mentally lead the problem solver towards a solution. Secondly, this text encourages the problem solver to visualize images such as charts, graphs, sketches or models to support that strategy. These steps encourage a student to think their way through each problem so they will understand the concepts rather than to make an attempt to memorize a way to get an answer. Successful math students understand the concepts while other students attempt to memorize and duplicate. This text encourages students to develop an understanding approach to problem solving as they solve more than 1100 progressively challenging problems. "Best Strategies for Pre-Algebra with Basic Algebra" is written in a concise and sequential manner that will promote student interest and efficiency. This text demonstrates the necessary fundamentals taught in Pre-Algebra and it also includes two additional chapters of Algebra, which emphasizes the use of variables, as well as, graphing, writing, and solving linear equations. The content of this book was written to fulfill the needs of any middle school or high school Pre-Algebra course. Any student of higher level mathematics could also use this text to refresh their memory on the fundamentals of Pre-Algebra. The dimensions of this 6" by 9" textbook makes it light and easy to carry. Special important facts, concepts, or diagrams are emphasized in color. The concise descriptions of why and how problems are simplified will keep students interested. This text should fulfill the requirements of any Pre-Algebra class or could be used to supplement any school's current math program. Also "Best Strategies for Pre-Algebra with Basic Algebra" would be a great text for home schooling.
Algebra And The Elementary Classroom
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Author : Maria L. Blanton
language : en
Publisher: Heinemann Educational Books
Release Date : 2008
Algebra And The Elementary Classroom written by Maria L. Blanton and has been published by Heinemann Educational Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008 with Education categories.
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.
Lectures On Linear Algebra
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Author : I. M. Gelfand
language : en
Publisher: Courier Corporation
Release Date : 1989-01-01
Lectures On Linear Algebra written by I. M. Gelfand and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 1989-01-01 with Mathematics categories.
Prominent Russian mathematician's concise, well-written exposition considers n-dimensional spaces, linear and bilinear forms, linear transformations, canonical form of an arbitrary linear transformation, and an introduction to tensors. While not designed as an introductory text, the book's well-chosen topics, brevity of presentation, and the author's reputation will recommend it to all students, teachers, and mathematicians working in this sector.