# Making Sense of Algebra

## Developing Students' Mathematical Habits of Mind

“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**
...

“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.

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### Contents

Forewordby Steven Leinwand

Introduction

Chapter 1Algebraic Habits of MindWhat are Mathematical Habits of Mind?

Algebraic Habits of Mind

Conclusion

Chapter 2Mental Mathematics Is More Than Mental ArithmeticSlow Learners and Struggling Students

Arithmetic, Mathematics, and Executive Functions

Why Use These Exercises in High School?

Designing and Using Mental Mathematics Exercises

Conclusion

Chapter 3Solving and Building PuzzlesWhat Makes a Puzzle a Puzzle?

Why Do People Invent Puzzles, Do Them, and Like Them?

Who Am I? Puzzles

Mobiles and Mystery Number Puzzles

Latin Square-Based Puzzles

Teaching with Puzzles

Conclusion

Chapter 4Extended Investigations for StudentsExperience Before Formality: Try It Yourself

Principles: What Makes a Good Investigation?

Aspects of Investigation: Entering, Shaping, and Extending Investigations

Teaching Through Investigation

Conclusion

Chapter 5A Geometric Look at AlgebraThe Number Line: Numbers as Locations and Distances

Area as a Model for Multiplication, Division, and Factoring

Conclusion

Chapter 6Thinking Out LoudBenefits of Discussion in the Classroom

Teaching Students How to Discuss

When Discussions Falter

Models of Mathematical Discussions

Conclusion

References

### Samples

### Reviews

“

Making Sense of Algebrais a vivid application of the ‘habits of mind’ perspective that has become a trademark of these EDC authors. It helps put flesh on the mathematical practice standards. More than algebra, it is about mathematical ways of thinking, problem solving, and seductively accessible entrees into adventurous new mathematical territories. Unlike many discussions of mathematical practices, its rhetoric is quickly and abundantly brought alive by concrete and inventive examples, at various levels of mathematical sophistication, and with motivational insightfulness. 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“

Making Sense of Algebraprovides a window onto algebra as a realm for posing questions, noticing structure, and reasoning with a variety of tools. Presenting a range of problems designed to help students develop ‘mathematical habits of mind,’ this bookis an essential guide for teachers of algebra who want to engage their students as thinkers, especially those who struggle.”

—Deborah Schifter, Education Development Center, coauthor ofConnecting Arithmetic to Algebra“One of the joys of

Making Sense of Algebrais how clearly and practically the ‘how’ question is answered. Not only are we provided with wonderful ‘low threshold, high ceiling’ examples, but we are also shown how to transfer these examples directly to algebra lessons along with insightful and honest guidance on advantages and disadvantages, pitfalls and opportunities, important ideas, and connections to algebraic thinking.”

—Steven Leinwand, American Institutes for Research, author ofAccessible Mathematics“Paul Goldenberg and his colleagues have done a fantastic job of connecting mathematical ideas to teaching those ideas. A teacher who may be unfamiliar with multiple models of describing mathematical ideas is presented in an engaging and interesting way with many different representations that can help extend their knowledge of mathematics. Similarly the authors offer practical advice a teacher can use to support their students in developing their understanding. The authors present a coherent argument of why puzzles should have a place in math classrooms and how these puzzles can help struggling learners re-engage with mathematics.”

—David Wees, formative assessment specialist in mathematics, New Visions for Public Schools, New York City“As a teacher who uses the

Transition to Algebracurriculum,Making Sense of Algebrais the perfect companion. It does much more than the title suggests; it helps teachers make sense of the mathematical thinking students are developing in their classes. With the inclusion of ideas for incorporating cognitive development and processing, executive functioning, and interventions for students with mathematical difficulties,Making Sense of Algebrais a great addition to any special education math teacher’s book shelf.”

—Andrew Gael, special education teacher, New York City, blogs at The Learning Kaleidoscope“Rare is the resource that offers first-person accounts of teacher experience and insight, accompanied by broad and deep knowledge of the research literature. This book fits the bill, and does so with clarity, warmth, and humor. A good example is the chapter ‘A Geometric Look at Algebra,’ in which the authors use the geometry of the number line to integrate conceptions of operations in arithmetic with operations in algebra. Recent research is very clear: a facility with number-line representations of the magnitude of rational numbers is foundational to success in learning algebra.”

—Mark Driscoll, Education Development Center, author ofFostering Algebraic Thinking“

Making Sense of Algebrais full of interesting teaching ideas. Not only that, but it’s full of fantastic math problems that I threw myself into while reading the text. Now that I’ve read the book, I know it’ll have a place on my shelf. When I’m in search of a fresh idea, or working with a group of algebra students whose arithmetic skills aren’t sharp, I’ll reach for this book.”

—Michael Pershan, math teacher, New York City“This book offers a plethora of strategies, puzzles, and activities that are designed to get students not only to think but also to become better thinkers—something that would benefit all students inside and outside the classroom. I can easily recommend this book to middle and high school teachers, novice to veteran, all of whom will surely find something of interest here. The book is well written and grounded in solid theoretical principles. It can be read as a whole or digested slowly as needed for those teachers looking to add a little something extra to their typical routine.”

—Brian Frasier,Mathematics Teacher (NCTM)