How Students Think When Doing Algebra

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.

First in the Arbor Algebra series. A writing-based, common sense, whimsical & engaging introduction to algebra for middle-grade math students.

A hilarious reeducation in mathematics-full of joy, jokes, and stick figures-that sheds light on the countless practical and wonderful ways that math structures and shapes our world. In Math With Bad Drawings, Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician. Truth and knowledge come in multiple forms: colorful drawings, encouraging jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Orlin shows us how to think like a mathematician by teaching us a brand-new game of tic-tac-toe, how to understand an economic crises by rolling a pair of dice, and the mathematical headache that ensues when attempting to build a spherical Death Star. Every discussion in the book is illustrated with Orlin's trademark "bad drawings," which convey his message and insights with perfect pitch and clarity. With 24 chapters covering topics from the electoral college to human genetics to the reasons not to trust statistics, Math with Bad Drawings is a life-changing book for the math-estranged and math-enamored alike.

This third edition of the perennial bestseller defines the recent changes in how the discipline is taught and introduces a new perspective on the discipline. New material in this third edition includes: A modernized section on trigonometry An introduction to mathematical modeling Instruction in use of the graphing calculator 2,000 solved problems 3,000 supplementary practice problems and more

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

This book is addressed to people with research interests in the nature of mathematical thinking at any level, topeople with an interest in "higher-order thinking skills" in any domain, and to all mathematics teachers. The focal point of the book is a framework for the analysis of complex problem-solving behavior. That framework is presented in Part One, which consists of Chapters 1 through 5. It describes four qualitatively different aspects of complex intellectual activity: cognitive resources, the body of facts and procedures at one's disposal; heuristics, "rules of thumb" for making progress in difficult situations; control, having to do with the efficiency with which individuals utilize the knowledge at their disposal; and belief systems, one's perspectives regarding the nature of a discipline and how one goes about working in it. Part Two of the book, consisting of Chapters 6 through 10, presents a series of empirical studies that flesh out the analytical framework. These studies document the ways that competent problem solvers make the most of the knowledge at their disposal. They include observations of students, indicating some typical roadblocks to success. Data taken from students before and after a series of intensive problem-solving courses document the kinds of learning that can result from carefully designed instruction. Finally, observations made in typical high school classrooms serve to indicate some of the sources of students' (often counterproductive) mathematical behavior.

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 identifies 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.

A New York Times–bestselling author looks at mathematics education in America—when it’s worthwhile, and when it’s not. Why do we inflict a full menu of mathematics—algebra, geometry, trigonometry, even calculus—on all young Americans, regardless of their interests or aptitudes? While Andrew Hacker has been a professor of mathematics himself, and extols the glories of the subject, he also questions some widely held assumptions in this thought-provoking and practical-minded book. Does advanced math really broaden our minds? Is mastery of azimuths and asymptotes needed for success in most jobs? Should the entire Common Core syllabus be required of every student? Hacker worries that our nation’s current frenzied emphasis on STEM is diverting attention from other pursuits and even subverting the spirit of the country. Here, he shows how mandating math for everyone prevents other talents from being developed and acts as an irrational barrier to graduation and careers. He proposes alternatives, including teaching facility with figures, quantitative reasoning, and understanding statistics. Expanding upon the author’s viral New York Times op-ed, The Math Myth is sure to spark a heated and needed national conversation—not just about mathematics but about the kind of people and society we want to be. “Hacker’s accessible arguments offer plenty to think about and should serve as a clarion call to students, parents, and educators who decry the one-size-fits-all approach to schooling.” —Publishers Weekly, starred review

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

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.