Table of Contents

Jossey-Bass Teacher

Title Page




The Author

About SEDL

About This Book

Summary of Content

Approach and Philosophy


Julian's Story

Rationale and Purpose

Who Benefits from this Book?

Chapter One: The Problem with Math Is English (and a Few Other Things)

Why Language and Symbolism?

What We are Teaching

Turning the Tide: A Sampling of Approaches

Mathematics is About Relationships

Connecting the Pieces and Looking Ahead

Chapter Two: Why a Language Focus in Mathematics?

The Convergence of Mathematics and English: More than Just Vocabulary

Problems Based on the English Language

A Number of Problems with number

Chapter Three: Language and Symbolism in Traditional Instruction

Shortcomings of Traditional Instruction

More Language and Symbolism Issues: Adding Fuel to the Fire

Tell Me Again Why the Language Focus in Math?

Chapter Four: So What Does Conceptual Understanding Look Like?

It Starts with Definitions

Making Connections in Math: Beyond Connecting Dots

The Interpretation and Translation of Math


Chapter Five: The Order of Operations: A Convention or a Symptom of What Ails Us?

The Roots of the Rules

The Natural Order: A Mathematical Perspective

Conclusion: A Conceptual Understanding of the Order of Operations

Chapter Six: Using Multiplication as a Critical Knowledge Base

Understanding Key Definitions and Connections

Interpreting Multiplication

Using the Power of the Distributive Property

Feeling Neglected: The Units in Multiplication

Conclusion: Small Details, Huge Impact

Chapter Seven: Fractions: The “F Word” in Mathematics

Defining Fractions: Like Herding Cats

The Fraction Kingdom

Interpreting Fractions


Chapter Eight: Operations with Fractions

Adding and Subtracting Fractions

Multiplying Fractions

Dividing Fractions


Chapter Nine: Unlocking the Power of Symbolism and Visual Representation


Visual Representation

The Power of Interpretation: Three Perspectives of Trapezoids


Chapter Ten: Language-Focused Conceptual Instruction

Language Focus: Beyond the Definitions

The Secrets to Solving Word Problems

Suggested Instructional Strategies


Chapter Eleven: Mathematics: It's All About Relationships!

Language and Symbolism: Vehicles for Relationship Recognition

Relationships and Fractions

Proportional Reasoning

Relationships: Important Considerations

Relationships: Making Powerful Connections


Chapter Twelve: The Perfect Non-Storm: Understanding the Problem and Changing the System

A Systemic Issue

Math Makeover




Jossey-Bass Teacher

Jossey-Bass Teacher provides educators with practical knowledge and tools to create a positive and lifelong impact on student learning. We offer classroom-tested and research-based teaching resources for a variety of grade levels and subject areas. Whether you are an aspiring, new, or veteran teacher, we want to help you make every teaching day your best.

From ready-to-use classroom activities to the latest teaching framework, our value-packed books provide insightful, practical, and comprehensive materials on the topics that matter most to K–12 teachers. We hope to become your trusted source for the best ideas from the most experienced and respected experts in the field.

Title Page


This publication would not have been possible without the support, patience and encouragement of family, friends, and workplace colleagues. Sincere appreciation goes to Dr. Victoria Dimock, chief program officer at SEDL, for providing the opportunity and SEDL support necessary for completion of this extensive task. In addition, her support, commitment, and guidance were essential throughout the process. Special recognition is deserved by Joni Wackwitz, SEDL communications specialist, who was instrumental in the editing process. Her thorough review and thoughtful insights resulted in clear communication of the intended messages as well as smooth transitions and connections among topics.

My colleagues at SEDL are the most professional and knowledgeable educators I have ever had the honor of working with, and their support was instrumental in this endeavor as well. Special appreciation is deserved by Dr. Stephen Marble, former program manager and supervisor, for bringing me on board at SEDL and providing the opportunity to grow as a mathematics educator through the design and delivery of professional development for teachers. His guidance, reflective questions, and continued encouragement helped to bring out dormant mathematics knowledge within me that I did not realize existed.

The list of acknowledgments would be inadequate without the inclusion of the educators that helped to shape my future and my life. I was fortunate to have had caring and expert teachers during my 12 years in the Karnes City (Texas) Independent School District. It was their example and dedication that influenced my choice of education as a career. Recognition also goes to the faculty of the College of Education at Texas A&M University for their expertise and guidance during my teacher preparation and certification. Special thanks also to fellow teachers and staff at McAllen High School (McAllen, Texas) where I began my career, as well as fellow faculty at Moody High School (Corpus Christi, Texas) where I spent the majority of my time in the mathematics classroom. In particular, much appreciation to Mr Franciso Moreida, Moody High School mathematics department chair, who helped me grow as a teacher during my 11 years there.

I would be remiss if I did not acknowledge the students that I had the honor of teaching during my years in the mathematics classroom. Quite honestly, I often learned as much from them as they learned from me. Their questions, thinking, and approaches to mathematics expanded my horizons and raised my level of expertise. And their growth and appreciation helped energize me to do more and try harder.

Above all, I give thanks to the Lord Almighty for providing me with the insights, the experiences, the support, and opportunities, as well as the depth of thinking and the perseverance to reach this milestone.



To my wonderful wife, Yolanda, for her understanding and unwavering behind-the-scenes support.


To my immediate family members for their encouragement and patience.


To my SEDL colleagues for their support and guidance during the entire research and writing process.


To my departed siblings whose memories served as the inspiration to fulfill their unrealized potential through my work.


And most of all, to the memory of my parents, Carlos and Luisa Molina, who believed in me and unselfishly sacrificed so that I could avail myself of the educational opportunity they were never afforded.

The Author

Concepcion Molina, EdD, grew up in a small south Texas town. Although born a Texan, he began school knowing very little English because Spanish was the primary language spoken at home. His entire elementary through high school education took place in that small town. He did well and was the first in his family to attend college. But first, he served four years in the U.S. Air Force as an accounting specialist.

After being honorably discharged, he attended Texas A&M University–College Station and graduated with honors with a Bachelor of Science degree in Educational Curriculum and Instruction with teaching certifications in secondary mathematics and Spanish. After graduation, his initial experience in the field of education was as a mathematics teacher at McAllen High School in the Texas Rio Grande valley. He remained for three years and taught Fundamentals of Mathematics, Pre-Algebra, Algebra I, and Geometry.

The next education stop was as a college admissions counselor at Texas A&M University—College Station. In that five-year span, Molina counseled visitors to campus on admissions, housing, and financial aid, represented the university at college nights and career fairs throughout Texas, and helped administer a scholarship program for academically gifted black and Hispanic students. The mathematics classroom beckoned, which led to teaching high school mathematics in Corpus Christi, Texas, for the next 11 years. The classes he taught included Algebra I, Algebra II, Geometry, Trigonometry, and Pre-Calculus. While teaching he earned a Master of Science degree in Educational Administration from Corpus Christi State University.

In 1998 he took on a new challenge by moving to Austin, Texas, to join the Southwest Educational Development Laboratory, a nonprofit educational agency that is now known as SEDL. As a program specialist he used scientifically based research to guide the design, piloting, and delivery of mathematics professional development training to clients in the five-state Southwest Consortium for the Improvement of Mathematics and Science Teaching (SCIMAST) region. In addition, he assisted and led in the planning, design, and delivery of the Consortium's regional forums and state meetings, as well as collaborating with other educational organizations and disseminating information that supported and promoted SCIMAST work. During this stage of his career, he finished the doctoral work that he had begun while still in Corpus Christi. In 2004, he earned an EdD in Educational Leadership from Texas A&M University–Corpus Christi.

When the SCIMAST program ended, Molina became part of SEDL's Texas Comprehensive Center and Southeast Comprehensive Center. As a staff member of both centers, he assisted and supported the state departments of education and their intermediate agencies in their efforts to implement No Child Left Behind (NCLB). Tasks and projects included the planning, design, and delivery of Comprehensive Center regional forums and meetings, assisting states with revision of state content standards, as well as designing and delivering professional development training requested by state agency staff on such topics as mathematics content, high school dropout prevention and high school reform models, systemic school reform, and mathematics instruction for EL students (English learners). This variety of experience and research has led to expertise in mathematics instruction, instructional leadership, teacher quality, professional development, and systemic reform.

About SEDL

SEDL ( is a nonprofit education research, development, and dissemination organization based in Austin, Texas. Throughout our 45-year existence, two central ideas have guided our work. The first is one of purpose—serving the educational needs of children in poverty. We believe that a quality education is an essential mechanism for freeing both individuals and society from the ravages and inequities of poverty. The second idea is one of means—bridging the gap between research and practice. We believe a responsive, effective educational system must be grounded in a strong research base that is tightly linked to practice.

SEDL is committed to sustainable research- and experience-based solutions. We work at national, regional, state, and local levels to develop and study approaches to strengthen educational policy and practice. We also provide professional development, technical assistance, and information services tailored to the specific needs of our diverse constituencies, which include educational practitioners, policymakers, families, and other researchers.

Our current efforts address five program areas: improving school performance, strengthening teaching and learning in core content areas, integrating technology into teaching and learning, involving family and community in student learning, and connecting disability research to practice. Work in these areas concentrates on pre-K–16 education and on underserved students, particularly those living in poverty and English language learners. Among our current projects are rigorous scale-up research studies of mathematics and reading interventions, and two technical assistance centers that work intensively with state education agencies to help boost achievement among low-performing schools.

About This Book

The goal of this book is to raise readers' mathematics expertise while simultaneously explaining the critical role of language and symbolism in understanding mathematics conceptually. The main body of the text focuses on fundamental mathematical concepts that fall primarily in the algebra and number and operation strands of the mathematics content standards. At the same time, the text explores the relationships between and connections among key mathematics topics to illustrate how a basic understanding of more complex concepts can be developed while teaching fundamental ideas.

Summary of Content

The journey begins with an investigation of issues related to language and symbolism in mathematics and mathematics instruction. Chapter One provides a sampling of the problems in both the language and math arenas. Chapter Two builds on that start with an in-depth look at why language is so problematic, and Chapter Three extends the conversation with a focus on language-based hurdles that permeate traditional instruction. Chapter Four begins the transition to a mathematics focus by examining the meaning and makeup of conceptual understanding in the math classroom. Next, Chapter Five uses the order of operations to illustrate how language-focused conceptual instruction leads to a deeper understanding than procedure-based traditional instruction. Chapter Six then spotlights the concept of multiplication, which transitions the focus from math instruction to math content.

The remaining chapters continue to explore how to help students develop a conceptual understanding of mathematics. Because fractions are such an issue for students—and even some teachers—two chapters are devoted to the topic. Chapter Seven examines the concept of fractions, with a deliberate slant to the language and symbolism involved; and Chapter Eight provides a deep look at the computation of fractions. Chapter Nine has a content focus, with the objective of merging the deep understanding of both concepts and their associated language and symbolism in a way that each bolsters and supports the other. Chapters Ten and Eleven apply all the preceding information by examining what conceptual math instruction looks like in action. Chapter Ten investigates language-focused instructional strategies, such as how fundamental concepts can be used to connect to more advanced topics, and Chapter Eleven focuses on how relationships in mathematics can serve as a powerful strategy for deepening understanding. Finally, Chapter Twelve recommends changes in the U.S. mathematics education system that would enable teachers to implement more effectively many of the ideas in this book.

Approach and Philosophy

All teachers want their students to do well and to learn as much as possible. In their quest to boost learning, teachers often seek out new lessons and activities to use in their classrooms. Providing such lessons or activities is not the purpose of this book, although teachers are encouraged to use the models and strategies provided with their students. In some ways, creativity is a function of expertise. Rather than just provide activities, the philosophy of this book is to help teachers understand mathematics at a conceptual level so that they can develop their own activities to deepen students' learning. To maximize the book's utility and value, readers should experience the mathematics, answering questions or solving problems as they appear in the reading. The intent of these questions and problems is to take readers' content expertise to a higher level as well as to prompt readers to reflect on their personal knowledge of fundamental mathematics.

Through the process of answering questions and solving problems, readers will enhance their level of pedagogical content knowledge. Part of that learning will occur as readers identify the shortcomings in traditional instruction. Another part of that learning will occur as readers work through purposeful problems and activities to explore conceptual-level mathematics from a number of perspectives. The central theme of this text is that with a conceptual understanding of mathematics, teachers are empowered to do the following:


Mathematics education in the United States has been the focus of a great deal of scrutiny. U.S students continue to perform poorly on state, national, and international math exams and continue to fall behind their global peers in mathematics performance. Although some progress has been made to address these issues, more work is needed. For years, the primary focus of mathematics reform has been on improving classroom instructional practices, and rightfully so. However, the best teaching practices in the world and the most advanced educational technology are of little use if the math content and the communication of that content are not of equal quality. This book focuses on these two often overlooked issues in mathematics education: (1) what we teach and (2) how we communicate it.

Julian's Story

Julian sits in his first-grade math class. He listens attentively because he enjoys the class and wants to please the teacher. Today is his first exposure to the topic of length, which the teacher refers to as a duration of time. Julian watches closely as the teacher moves the hands of a clock from one position to another and asks questions such as, “How long did it take to drive to grandmother's house?” The lesson is interesting, and Julian feels he understands. But a few weeks later, the teacher says the more common meaning of length is as a measure of distance, such as the length of a table. Then, later in the year, during an English lesson, Julian learns that length can also refer to a piece of some item, such as a length of rope. He is surprised and slightly puzzled by all these different meanings of the word. Things get even more complicated in his math education when Julian learns that if a length is vertical, it is called height, even though the units of measurement remain the same. He scratches his head when he is told his height is 37 inches, but his legs are 16 inches long. Some time later, Julian learns in basic geometry that the sides of a rectangle are called the length and the width. He is given an assignment where he must find the length of the width of a rectangle given the length of the length. Sometimes, Julian gets a little confused.

But mathematics remains Julian's favorite subject. He goes to college and gets certified to be a high school math teacher. To obtain this credential, he has to take calculus and a number of other rigorous mathematics courses. On his first day as a teacher, he walks into his freshman math class feeling confident and well prepared. While reviewing some basics with the class, one of the students asks why we invert and multiply when we divide by a fraction. Julian is at a loss, saddened to realize he does not know despite his level of education. The only answer he has is “That's the rule.” The experience eats at him all day. That evening, Julian reflects on his knowledge of mathematics. He is teaching algebra as well as freshman math. Nervously, he wonders, “Do I know enough to teach these classes?” He looks ahead at the content of the algebra textbook. Slowly, he starts to panic at the thought of all the different questions students might ask. “What if someone asks why something to the zero power is 1?” Julian worries. He has no idea of the answer. As a student, he blindly accepted and memorized the various rules and procedures presented—just as his teachers told him to do. The panic turns to fear as he wonders what his response will be if a student asks what a logarithm is, rather than how to convert an equation from exponential to logarithmic form. Shaken and dejected, Julian realizes he does not know mathematics at the level needed to teach it well.

Rationale and Purpose

Julian's story is my story. I began grade school knowing very little English. At the time, no bilingual or dual language programs were available, nor were teachers aware of all the EL (English Learner) strategies we have today. With regard to the challenge of learning English and mathematics at the same time, quite simply, I lived it. The inherent issues related to learning a language and academic content simultaneously led me to develop an acute awareness and appreciation of the problems and nuances of language, particularly in mathematics. Many of these problems go unnoticed by math educators. Addressing the neglected issue of language in mathematics and math instruction was a driving force in why I decided to write this book.

In addition to language, I wanted to focus this book on the mathematics content being taught in our education system. The start of my teaching career was a rude awakening regarding the depth of mathematics taught in the U.S. education system. I was a product of that system, and although I had made good grades and was certified to teach math at the secondary level, my initial classroom experiences clearly revealed the lack of depth of my mathematics knowledge. The true test of knowledge is revealed when one must teach it. In my first few years as a teacher, I struggled, but began deepening my knowledge of mathematics, learning more about what concepts were and why things work the way they do. And it was a focus on language that was the bridge in the transition from my being a good student of rules and procedures to a teacher of deep, conceptual mathematics.

In time, I left the classroom and began designing and delivering professional development in mathematics. This work focused primarily on middle schools, but included elementary schools as well. In my professional development sessions, my goal was to improve teachers' math content expertise by simultaneously using and modeling effective instructional practices grounded in research. During this time, I began to really examine the content of mathematics. The combination of teaching advanced math to high school students and fundamental math to elementary and middle school teachers enabled me to see surprising vertical connections across topics and concepts that previously had eluded me. In addition, during this time in my career, I did a doctoral dissertation that focused on the content knowledge of middle school math teachers. The volumes of research on this topic combined with my personal experiences as a student, teacher, and trainer transformed my perspective about what we are teaching and how we are communicating it.

As a compilation of hundreds of “aha” moments that I experienced both as a learner and a teacher, this book aims to improve the math content expertise of the reader. Teaching is a complex endeavor, and subject-matter expertise is only one of the many interrelated components involved. However, a teacher obviously cannot teach what he or she does not know. My dissertation research revealed that many elementary math teachers know the facts and procedures they teach, but have a weak understanding of the conceptual basis behind them—just as I did when I began teaching (Kilpatrick et al., 2001; Molina, 2004). Moreover, the educational system sends teachers the message that if they get good grades in college and get certified to teach, they know the content they need. The result is a teaching force with deficiencies in content expertise—and in many cases, neither teachers nor district and campus leaders even realize the deficiencies exist. As a result, educational reform in mathematics tends to focus on how instructors teach and overlooks what they teach.

This book uses a unique language-focused perspective to bring to light the deeper content knowledge that math teachers need as well as issues with how teachers communicate that content. The role of language and symbolism in learning and understanding mathematics is generally slighted in the U.S. education system, including K–12 education, teacher preparation and professional development programs, and state standards. Thus, the intersection of these issues leads to a specific focus on the role of language and symbolism in understanding mathematics conceptually.

The term language as used in this book goes far beyond the simplistic idea of vocabulary. Because language and symbolism are an integral part of teaching any mathematics concept, these two topics not only constitute separate chapters but are integrated throughout the text. The National Council of Teachers of Mathematics (NCTM) standards emphasize the importance of communication in mathematics, particularly in students' explanations and justifications of their mathematical thinking. However, in this publication, mathematical language and symbolism are viewed from a content perspective, where each aspect and nuance of mathematical language and symbolism is considered an integral part of the content to be learned. When I was a high school teacher, a sign above my classroom door read, “Se Habla Algebra,” meaning “I speak algebra.” More broadly interpreted, it states that the language of mathematics is spoken and understood here, which sums up the approach and philosophy of this publication.

Who Benefits from this Book?

The ultimate goal of any educational publication is to improve student learning. To this end, this publication is geared primarily toward K–12 teachers who provide classroom instruction in mathematics, although the bulk of the content focuses on middle school mathematics. The language-focused conceptual mathematics presented in this text should be a new and refreshing approach to novice as well as veteran teachers. Readers who work in the classroom will increase their content knowledge while learning how to address language-based problems in mathematics, which in turn will improve instruction and student learning.

Improving the mathematics content knowledge of current and future teachers is a problem of scale, however. For this reason, a secondary audience for this publication consists of leaders in mathematics education. This group includes, but is not limited to, campus mathematics coaches and specialists, district mathematics coordinators, and state directors of mathematics. For example, this publication should prove valuable to educators who provide technical assistance and professional development to math teachers at the elementary and middle school levels. These leaders can accelerate change and affect a larger audience. Likewise, campus administrators can benefit from this book by seeing the type of conceptual mathematics that students should be learning. With this knowledge, administrators can ensure that teachers know and provide instruction in mathematics at a conceptual level. In addition, university staff in colleges of education can utilize this resource to ensure they produce a future teaching force of content experts.

Chapter One

The Problem with Math Is English (and a Few Other Things)

Many people do not consider English as playing a significant role in math, except in word problems. My hope in the forthcoming pages is to change that perspective. A well-known proverb says that to truly understand another's perspective, you must walk a mile in that person's shoes. Not everyone has experienced the struggle of learning both academic content and a new language at the same time. True, this double burden makes learning mathematics much more of a challenge. However, the phrase “the problem with math is English” applies to all students, not just those whose native language is not English. Language struggles are embedded in mathematics, which in many ways is its own language. These problems often occur at the critical juncture of math instruction and content. Two major issues in mathematics education that result from this merger are often overlooked: (1) the language and symbolism of mathematics, which in turn greatly influence (2) the mathematics itself—the content that we teach—and by association, how we communicate that content. The following scenarios introduce some key concepts related to these issues, which this book will explore in-depth.

Why Language and Symbolism?

Imagine you are a middle school student taking the state's required progress exam in mathematics. As you begin the test, you feel confident about your answers to the first few items. But then you read Item 5: “Find the arglif of a nopkam if the betdosyn is 12.” Try as you will, the problem has you stumped. You finally give up, make a guess, and move on to the next problem. Unfortunately, you come across 14 other test items that confuse you in similar ways. Once again, the best you can do is guess at the answers. Later, you find out you did not pass the exam primarily because of those 15 items.

When most people see or hear the term mathematics, the initial thoughts that come to mind are numbers, computation, rules, and procedures. But the root cause of the student's inability to solve Item 5 is not because of a lack of knowledge of computation, rules, or procedures. If the strange terms arglif, nopkam, and betdosyn meant “area,” “circle,” and “diameter,” respectively, Item 5 would become the following: “Find the area of a circle if the diameter is 12.” However, if a student does not know the meanings of area, circle, and diameter, the terms might as well be arglif, nopkam, and betdosyn because they still hold no meaning.

This scenario is a rudimentary example of the key role that language plays in the understanding of mathematics. The student's difficulty is not in reading English, but in understanding the language of mathematics, and it makes no difference whether the student is fluent in English or not. A language problem still exists. Although perhaps not obvious, language is as critical in mathematics as in any other discipline. Moreover, the role of language in mathematics entails far more than vocabulary or definitions, encompassing a broad landscape of language-based issues, which are explored in this book.

Beyond Words: The Symbolism of Mathematics

Box 1.1
Solve the two tasks below:
a. n = 1 + 3 + 5 + 7
b. images

For many people, the first problem in Box 1.1 is child's play, whereas the second poses a serious challenge. The interesting paradox about these two problems is that although the second seems far more difficult, they are, in fact, the same problem. They are just presented differently. The first problem seems simple because most people can easily interpret what the numerals and symbols are telling them to do. The second problem, however, will literally be Greek to many people because they have no idea what those symbols mean. The problem becomes much simpler once the symbols are explained. The symbol ∑ is a summation symbol. The n =1 in the subscript means that 1 is the first value of n in the expression 2n – 1, and the superscript 4 indicates the last value to be used in that expression. Thus, the task is to determine the value of 2n – 1 when n is 1, 2, 3, and 4, and then to find the sum of those values. These steps result in the expression 1 + 3 + 5 + 7. Simple!

These two problems illustrate the key role of symbolism and visual representation in mathematics. The interpretation and subsequent understanding of mathematics concepts are heavily dependent not only on language but also on the symbols that are an inherent component of the discipline. These two problems also illustrate a scenario seen in far too many classrooms when concepts or ideas in mathematics that are actually quite simple are presented in a way that is far more complex, much to the detriment of student learning. In other words, there are too many instances in mathematics instruction where a simple concept or idea is somehow prodded and molded, either by the math education system or teachers, and unveiled to students as something that appears to be far more complicated than it really is.

The Language of Mathematics Instruction

Box 1.2
You have been teaching the challenging concept and skill of division by a proper fraction. You write the problem images on the board. You then state, “Class, how many times does images go into 10?”

Refer to the question asked in Box 1.2. If teachers tend to teach the same way they were taught, it follows they will tend to teach using the same language. Over the years, numerous elementary and middle school teachers have presented the expression 10 ÷ images as the question, “How many times does images go into 10?” As students themselves, these teachers accepted this interpretation of the symbols and moved on regardless of how much sense the interpretation made—or did not make—to them. Once they became teachers, they used the same language, thus passing on the torch to their students. Quite honestly, what can a student create to model a context where images goes into 10? The question as posed really makes no sense. The language of instruction in mathematics often makes the conceptual meaning almost impossible to grasp, but students survive by blindly following procedures that enable them to get the correct answers that result in good grades. As educators, however, we must reflect and ask ourselves what level of mathematics are students actually learning, and is that depth of knowledge acceptable?

What We are Teaching

Box 1.3
Next week's unit of instruction will focus on the multiplication of mixed numbers. You need to ensure that students have the prerequisite skills and knowledge to learn this new topic. What are these prerequisite skills and knowledge? Make a list.

Your generated list from Box 1.3 would likely include many prerequisite skills or concepts that focus on how to multiply mixed numbers. Think of your experiences as a math student and, if applicable, as a math teacher. Much of the instruction and learning in math classrooms is focused on how-tos. In the United States, we value good old American know-how. To learn mathematics at a deep conceptual level, however, know-how is not enough. Just as important, if not more so, is good old American know-what and know-why. In other words, students need to understand what a basic concept is and why that concept works as it does. In the scenario in Box 1.3, students need first to understand what multiplication is conceptually, then use that knowledge to understand why the process works as it does. Knowing only how to multiply, at best, results in basic memorization of rules and procedures.

Not surprisingly, many adults in the United States perceive mathematics simply as a conglomeration of facts, rules, computations, and procedures. After all, that is the type of mathematics they were taught. The K–12 math education system often focuses on arithmetic and efficiency (or algorithms), but mathematics is far more than that. If teachers only know, and subsequently teach, arithmetic and algorithms rather than a conceptual understanding of mathematics, then that limited knowledge will be the baton passed on to future generations.

What Is Missing: The Need for Definitions

An interesting paradox in mathematics is that one can know how to do something without understanding what the concept or process truly is. For example, students can know how to multiply without understanding what multiplication is conceptually. I successfully navigated numerous math courses knowing how to use π in formulas while having no clue as to what π meant as a concept. Quite literally, the what is missing from math instruction. Since a definition tells us what something is, it makes sense to emphasize definitions as a core element of instruction.

Box 1.4
Answer the following:
a. Define equation (noun).
b. Define graph (noun).

Defining key mathematical terms helps students build their understanding of important concepts. Students should be able to provide precise yet simple definitions of basic terms, such as those in Box 1.4. For teachers, these types of definitions paint a clear picture of students' depth of understanding. Incorrect student responses can reveal misunderstandings and gaps in knowledge. In addition, patterns that emerge from students' definitions of basic concepts can provide clues about the effectiveness of instruction, the curriculum, and even the larger mathematics education system.

Patterns of Misunderstanding

As a young high school math teacher, I often made assumptions regarding students' content knowledge, especially their mastery of fundamental concepts in earlier grades. I adopted the strategy of having students define basic math concepts not only to build understanding but also to expose the areas where students' knowledge was weak. The approach revealed some interesting patterns over the years. For example, high school students in higher-level math classes most commonly define an equation, one of the most basic concepts in mathematics, as “when you solve for x.” This definition is a clear misconception of what an equation is, but the root cause was not so evident. After much reflection and analysis, I realized the origin lay in the state's mathematics content standards. In the state standards in effect at that time, the term equation did not appear until the sixth grade. Moreover, the context for this first appearance focused on learning to solve simple linear equations. This initial focus had likely contributed to students' misconception of an equation being “when you solve for x.”

The concept of an equation is usually defined as a mathematical sentence that states that one quantity is the same as another quantity. In other words, the quantity expressed on the left side of the equal sign is the same as the quantity expressed on the right side, regardless of how those quantities are represented. For some reason, the U.S. educational system often waits to formally include the idea of equations in state mathematics standards until fifth or sixth grade. However, do students not begin their experiences with this concept early in elementary school? Is 2 + 1 = 3 not an equation? There is no requirement that an equation must involve variables! Yet even many adults struggle with the correct definition and gravitate back to the notion that an equation must have “letters” and that a solution must be found.

Earlier in the chapter, I discussed how the mathematics education system sometimes takes something that is relatively simple and manages to make it inordinately complex. An equation is actually an extremely simple concept that can be ingrained very early in a child's education. Young children love to learn what they think are complex or sophisticated words. So why do we not simultaneously teach that 3 + 4 = 7 is an equation when students learn basic arithmetic? This early introduction would help solidify students' fundamental understanding of an equation and simultaneously reduce the misconception that an equation must somehow involve a variable and be solved.

Similarly, when I asked high school students to define the term graph as a noun, their responses revealed another interesting pattern. What emerged was the common idea of a graph as the grid itself or the x and y axes. An investigation of the possible root causes of this off-target perspective of a graph pointed to both instruction and curriculum materials as the culprits. How many times have math teachers instructed students to plot the points or to draw the curve “on the graph”? How many textbooks ask students to illustrate their responses “on the graph”? With this phrase repeated ad nauseam, is it any wonder that students begin to define a graph as the grid itself? These patterns reinforce the importance and impact of language, both written and oral, on students' perceptions and understanding of fundamental mathematical concepts.

Turning the Tide: A Sampling of Approaches

Mathematics instruction is an extremely complex enterprise with multiple interrelated factors. However, logically, the foundation for instruction should be the mathematics itself—the concepts and big ideas—not skills and algorithms, although they do play an important role. This conceptual foundation, in turn, necessitates a paradigm shift from the how-to of mathematics to the what and the why. The result is language-focused instruction based on conceptual understanding. The following examples provide a sampling of approaches used in this type of instruction and an overview of the chapters to come.

Multiple Perspectives

Box 1.5
You have taught your students the multiplication of mixed numbers using the standard algorithm: Convert to improper fractions, multiply, then simplify and convert back to a mixed number if applicable. You are tutoring Michael because he still does not get it. Using images as an example, think about how you would go about helping Michael gain a conceptual understanding of mixed-number multiplication using other perspectives or approaches. Write down an explanation of how you would help Michael.

Review the example in Box 1.5. If you had difficulty thinking of different approaches to help Michael understand how to multiply mixed numbers, you are definitely not alone. As mentioned, we tend to teach not only how we were taught but also what we were taught, with multiple implications. Teachers can teach only what they know and only to the depth of their own understanding. If teachers' only experience with the multiplication of mixed numbers is the standard algorithm and their college training, or subsequent professional development has not taken them beyond that, then that one approach is what they will teach.

Out of training and habit, teachers can easily begin to view a concept or approach a problem from just one perspective. We know students learn in different ways; thus, we should teach mathematical concepts and skills using multiple approaches and perspectives that relate to how individual students learn. So what can we do for Michael in the above scenario? We might try teaching multiplication of mixed numbers from a numeracy perspective using the definition of multiplication. Or we might try a geometry-measurement perspective using an area model. We could also try an algebraic perspective through the use of the distributive property. All these approaches and the power of the connections among them can help students develop conceptual understanding. This instructional scenario will be revisited in Chapter Eight to illustrate how to use multiple perspectives to help students understand the concept of multiplying mixed numbers.

So What Was the Question?

Box 1.6
You have been teaching the challenging concept and skill of division by a proper fraction. You write the problem images on the board. You then state, “Class, how many times does images go into 10?” The question is followed by silence. Finally, one student volunteers and states, “I sort of understood back when we had stuff like how many times does 3 go into 12, but no matter how hard I try, I can't picture in my head how many times images can go into 10. What does that mean?” How do you respond?

The scenario in Box 1.6 is an extension of the one in Box 1.2. Again, the focus is on the tendency for teachers to use the same language used to teach them. Suppose the students are older, and a high school teacher poses this same problem to review basic computation. A few students give the incorrect solution of 5, while others give the correct solution of 20 but with trepidation and uncertainty. The teacher then challenges the students to state the expression 10 ÷ images as a question without using phrases such as “goes into” or “divided by.” It should come as no surprise that the high school students are stumped despite their maturity and experience.

The scenarios in Boxes 1.2 and 1.6 reveal how teachers' language choices, often unconsciously influenced by how they were taught, can confuse students and even erode the understanding of the mathematics involved. Most teachers would be at a loss to explain what it means for images to “go into” 10 or to describe division without using this phrase. And without a deeper understanding of division, the high school teacher above would not be able to pose that challenge to students as an instructional strategy.

This dilemma illustrates two additional problems in K–12 mathematics instruction. First, instruction tends to focus on finding answers without any focus on students' understanding of the question. When students truly understand the question being asked, they can answer it correctly with full confidence. But often, students will present a correct answer hesitantly, revealing a lack of real understanding of the task. Second, despite being in high school, many students, even high-achievers in advanced math classes such as Algebra II, still do not truly understand basic concepts such as division. True, students can easily recite facts such as 42 divided by 6 is 7, but they do not understand division conceptually. There is a huge difference between using procedures or memorizing facts and truly understanding concepts and why procedures work. And the language used can contribute to this lack of understanding.

Mathematics is About Relationships

Box 1.7
Solve the following:
The state champion in football is determined by a single-elimination playoff system: If you lose, you are out. How many games would the state need to hold if 37 teams were involved in the playoffs?

Refer to the task in Box 1.7. One could solve this problem by laboriously drawing playoff brackets to determine the number of necessary games. There is, however, a much simpler approach to the solution. The key is to look for the relationships