Common Mistakes in Teaching Elementary Math—And How to Avoid Them (Paperback) book cover

Common Mistakes in Teaching Elementary Math—And How to Avoid Them

By Fuchang Liu

© 2017 – Routledge

158 pages | 86 B/W Illus.

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About the Book

Learn the most effective ways to teach elementary math, no matter how much experience you have with the subject. In this book, Fuchang Liu takes you through many common mistakes in math instruction and explains the misunderstandings behind them. He points out practices that should be avoided, helping you to adjust your lessons so that all students can achieve success.

You’ll discover how to…

-          Increase your confidence with core math principles and reasoning

-          Set your students on the path toward eventually developing more complex math skills

-          Improve student achievement by approaching problems in logical yet creative ways

-          Overcome common challenges faced by students and teachers

-          Teach problem solving for different learning styles

Every chapter reconsiders well-established ways of teaching all areas of elementary math, from addition and subtraction to statistics and graphs. Helpful examples and tips are scattered throughout the book, offering revisions to the way these topics are often presented in the classroom. Also included are group study ideas for principals and instructional coaches so your school or district can work on the book together. With this practical guide, you’ll be ready to help students truly develop their math understanding.


Children’s misunderstandings about mathematics ideas stand in their ways to learn mathematics well and effectively in elementary schools. Common Mistakes in Teaching Elementary Math and How to Avoid Them, is the first book, I know, that directly explains teachers’ own misunderstandings about mathematics ideas, how they reproduce them among their students, and more importantly, what they can do to address these issues in their mathematics teaching. Dr. Liu accomplished all the above successfully by engaging readers in experiencing what teachers do and how students react in elementary classrooms using a series of stories. A great book for mathematics teachers, teacher educators, and policy makers.

----Jian Wang, Helen DeVitt Jones Chair in Teacher Education, Texas Tech University

This book appeals to pre-service and elementary teachers who are not math content knowledge experts. Fuchang Liu offers a "quick-fix" for those elementary teachers who are aware that they are struggling as math teachers, and are seeking help.

--- Elizabeth Simpson, Math Coach, Lamphere School District, MI

Table of Contents

Meet the Author


Chapter 1 Counting

Counting Shouldn't Start at 0

What Are Tally Marks for?

It May Not Be Fast—The Purpose of Skip-Counting

Our Number System Is Base-Ten, Not Base-Twelve

Chapter 2 Number Properties

A Red Marble Isn't More Than a Blue One

Playing Cards and Dice Aren't Ideal Things to Make Multidigit Numbers With

It's Odd Not to Consider 0 as Even

14 and 37 Don't Belong in the Same List

Chapter 3 Addition

4 + 4 Isn't Simply 8 Bars Put Together

Make Pictorial Representations More Than a "Literal" Translation

Don’t Hop From Square One

You Can't Add Apples and Oranges Together

Chapter 4 Subtraction

Don't Make the Kittens Disappear

Don't Ever Say "Subtract the Smaller Number From the Larger One"

"Neither a Borrower nor a Lender Be"—Why We Shouldn't Borrow

Can We Subtract a Larger Number From a Smaller One?

10 + 5 – 8 = 15 – 8: No Way to Make It Simpler?

Chapter 5 Multiplication

The Formidable 169-Cell Multiplication Table

"You Must Put a Zero in the Ones Place"

Can You Move Over One Place Value?

Line Multiplication: Why It Won't Work

Chapter 6 Division

The Bigger Number Doesn't Always Go Inside

What's 0 ¸ 0?

Division Isn't Always Repeated Subtraction

"Dad, Mom, Sister, Brother, Rover": Where Is Dad?

Divide Doesn't Always Yield a Smaller Number

Chapter 7 The Order of Operations

Aunt Sally Is Evil – The Order of Operations

The Order "M/D Before A/S" Isn't Haphazard

Are Negative and Subtract Really Different?

Chapter 8 Algebra

An Equal Sign Means Equal

What Does Adding Up Numbers Landed Have to Do With Finding Factors?

Timelines Aren't Good Candidates for Teaching Negative Numbers

The Worst Example in Teaching Exponents

Chapter 9 Geometry: Bits and Pieces

Don't Count the Diagonals on a Grid

"All 3-D Shapes Have an Extra Third Dimension of Height"

Don't Use "Vertical" to Find the Horizontal Value

The Two Sides of a Symmetrical Figure Aren't Exactly the Same

Don't Use Tiles to Figure Out the Perimeter

Chapter 10 Geometry: Common Geometric Figures

Length Doesn't Necessarily Mean Longer

A Rectangle's Orientation Doesn't Matter Either

"A Rectangle Has Two Longer Sides and Two Shorter Sides"

What's Wrong With Saying "Triangles, Rectangles, Squares, and Hexagons"?

Base Doesn't Necessarily Mean "Side at Bottom"

Three Sides Don't Necessarily Make a Triangle

How Many Sides Does a Circle Have?

Chapter 11 Time-Telling

"A Quarter in Time Means 15"

Does 1 on Analog Clocks Mean 5 Minutes?

"Why Does the Time on My Analog Clock Look Weird?"

Which Hand Pointing to 12 Makes 12 O'clock?

Why We Shouldn't Jump Around Between 8:00 and 9:00

Chapter 12 Fractions

A Condition for Using Fractions: Equivalent Parts

"Bottom Number" and "Top Number" Aren't Nearly Sufficient

for Defining Denominators and Numerators

What Does 1/2x Mean?

A Fraction Doesn't Address "How Many"

Denominators Should Be Substantially Different for Easy Comparison

Percentage Shouldn't Be Juxtaposed With Fractions and Decimals

Chapter 13 Decimals

Changing the Value Scheme of Base-10 Blocks Not a Good Idea

When Is It Appropriate to Read a Decimal as a Fraction?

25.0 and 25 Aren't Exactly the Same

Keep a Few More Decimal Places

When Rounding at Intermediate Steps

Chapter 14 Simple Statistics and Graphs

Were the Children in Your Class Born, on Average, on the 12.8th?

"The Mean Is the Average"

Where in the World Does the Mode Ever Get Used?

Do All Your Children Have an Equal Chance of Winning?

Use Different Types of Data for a Beginning Lesson on Graphs

Chapter 15 Measurement

Why Do We Need Nonstandard Measurement?

What Is Standard Measurement?

What Are Centimeters Used to Measure?

Leave Out Units of Measure at Intermediate Steps

How Do You Say the Word for "1000 Meters"?

Fractions Don't Belong With Metric Measurements

Chapter 16 Computational Estimation

Is Computational Estimation an Educated Guess?

Are All Estimates Good Ones?

How Practical Is an Estimate If It Takes Four Minutes?

Be Extremely Careful When Rounding Single-Digit Numbers

There Is Often More Than One Way to Round a Number

Chapter 17 Odds and Ends

Appendix: Suggestions for Using Common Mistakes in Teaching Elementary Math with Study Groups

About the Author

Fuchang Liu

Wichita, Kansas, United States

Learn more about Fuchang Liu >>

Fuchang Liu is Associate Professor of Mathematics Education at Wichita State University. Previously, he taught high school math in Louisiana and Texas.

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Download the Appendix.