The Fact Tactics™ Fluency Program

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Acknowledgments

The idea for the Fact Tactics™ Fluency Program began when my children were in elementary school, and now my oldest is a teacher! Thank you, Alex and Jessica, for sharing your strategies for determining products of basic facts and helping me make sense of how focusing on reasoning with facts can lead to powerful understandings in mathematics. And thank you to my husband, Marc, for engaging in all the family mathematics discussions throughout the years and believing in my vision for this book.

Taking an idea and making it accessible to others can be a long and arduous process. My dear friends and co-leaders of DNA Math, Ed Nolan and Thomasenia Adams, provided invaluable guidance, insight, motivation, and immeasurable reflections along that path.

A key component of a program like this is its feasibility. Thank you, Melissa Logsdon and all the amazing students, teachers, and administrators at Cheatham Park Elementary School, for piloting the program and giving me valuable feedback.

You would think that this should be where the journey ends. I’ve got a worthwhile and usable program. However, if it is not communicated clearly and nobody knows about it, how much good will it do? That’s where Douglas Rife comes into the picture. I shared a draft of this book with him, and he readily encouraged me to move forward. Douglas—thank you for continuing to see the vision of my ideas and for assigning Christine Hood as my production editor to take the Fact Tactics Fluency Program to publication. Thank you, also, to Shik Love, Kelly Rockhill, Rian Anderson, Lauren Ware, and so many other amazing people at Solution Tree for supporting me in spreading the message.

Finally, thank you in advance to each and every teacher who chooses to take their students on the path toward Fact Tactics fluency. Your students’ potential to make sense of strategies for basic multiplication facts and use those strategies to develop fact fluency are the driving force behind the Fact Tactics Fluency Program.

Taylor Bronowicz

Math Teacher

Sparkman Middle School

Toney, Alabama

Erin Fedina

Supervisor of Mathematics, Science, and Gifted Education

Howell Township Public Schools

Howell, New Jersey

Nanci Smith Jones

Consultant and Associate Professor of Mathematics and Education, retired

Peoria, Arizona

Rea Smith Math Facilitator

Rogers Public Schools

Rogers, Arkansas

Gwendolyn Zimmermann Director

Adlai E. Stevenson High School

Lincolnshire, Illinois

About the Author

Juli K. Dixon, PhD, is a professor of mathematics education at the University of Central Florida (UCF) in Orlando. Prior to joining the faculty at UCF, Dr. Dixon was a secondary mathematics educator at the University of Nevada, Las Vegas, and a public school mathematics teacher in urban school settings at the elementary, middle, and secondary levels. Dr. Dixon is focused on improving teachers’ knowledge of teaching mathematics so they can support their students in communicating and justifying mathematical ideas.

Dr. Dixon is a prolific writer who has authored and coauthored books, textbooks, chapters, and articles. She is also a lead author on the Making Sense of Mathematics for Teaching professional development book and video series as well as for the K–12 mathematics textbooks and personalized instruction programs Go Math!, Into Math, Into AGA, and Waggle with Houghton Mifflin Harcourt. Especially important to Dr. Dixon is the need to teach each and every student. She often shares her personal story of supporting her own children with special needs to learn mathematics in an inclusive setting. Dr. Dixon published A Stroke of Luck: A Girl’s Second Chance at Life with her daughter, Jessica Dixon. A sought-after speaker, Dr. Dixon has delivered keynotes and professional development throughout North America.

Dr. Dixon received a bachelor’s degree in mathematics and education from the State University of New York at Potsdam, a master’s degree in mathematics education from Syracuse University, and a doctorate in curriculum and instruction with an emphasis in mathematics education from the University of Florida. Dr. Dixon is a leader in DNA Mathematics.

To learn more about Dr. Dixon’s work supporting children with special needs, visit www.astrokeofluck.net. To learn more about Dr. Dixon’s work supporting teachers, follow her @thestrokeofluck on Twitter.

To book Juli K. Dixon for professional development, contact pd@SolutionTree.com.

Introduction

It is reasonable to say that all teachers want students to become fluent with their basic facts for multiplication. Basic facts for multiplication refers to problems with one-digit factors, such as 3 × 4 or 9 × 9. What is less clear is what the term fluency means, and when and how students should develop fluency with basic multiplication facts.

The seminal work by the National Research Council (NRC, 2001) Adding It Up: Helping Children Learn Mathematics defines procedural fluency as “skill in carrying out procedures flexibly, accurately, efficiently, and appropriately” (p. 5). According to this definition, for students to be considered fluent with basic multiplication facts, they would need to be able to determine the results of multiplying one-digit factors like 6 × 7 quickly and accurately

Fluency does not mean students would just need to know the facts without thinking. In this case, students could determine the product of 6 × 7 efficiently by thinking 3 × 7 is 21 and 21 + 21 is 42. Here, the student appropriately determines the product of 6 × 7 through the flexible application of the doubling strategy, in which they multiply by half of one of the factors and then double that product. This sort of fluency is based on teaching mathematics for understanding. Is this the same fluency you think about when referring to basic multiplication facts? For many parents, teachers, and administrators, the answer is no. Often, they think of fluency as “just knowing.” However, that is not fluency, but rather, automaticity.

In Adding It Up (NRC, 2001), the authors describe automaticity as the ability to execute procedures without conscious thought and indicate that this should be a goal for fluency with basic multiplication facts. So how do students develop procedural fluency that leads to automaticity with basic multiplication facts? When should automaticity be the focus of instruction? What should we do when students struggle with developing automaticity? What we, as teachers, can all likely agree on is that saying the facts loudly and more often hasn’t been a productive strategy for students who struggle.

Some strategies for developing automaticity rely on the use of flash cards, timed tests, or computer programs that serve as both automated flash cards and timed tests. These often work well for students who already know their basic multiplication facts but less so for those who struggle. They also serve just one purpose: helping students memorize. Teachers spend so much time helping students memorize their facts that it seems like this time should result in something beyond just fact recall.

In Principles to Actions: Ensuring Mathematical Success for All , the National Council of Teachers of Mathematics (NCTM, 2014a) encourages teachers to “build procedural fluency from conceptual understanding” (p. 10). When this occurs, students become critical consumers of the mathematics they encounter. They make it their own. Further, in Catalyzing Change in Early Childhood and Elementary Mathematics, DeAnn Huinker (NCTM, 2020) suggests that students should develop a deep understanding of mathematics as one of four key recommendations for a successful path with mathematics. The teacher’s role in this process is to “build a strong foundation of deep mathematical understanding, emphasize reasoning and sense making, and ensure the highestquality mathematics education for each and every child” (NCTM, 2020, p. 9).

When students have a deep understanding of mathematics, they are more likely to determine whether responses to tasks are reasonable. In mathematics classrooms where the goal is to develop deep understanding of mathematics, it is likely that students will explain and justify their mathematical thinking and make sense of the thinking of others (NCTM, 2020). According to Thomas Carpenter, Megan Loef Franke, and Linda Levi (2003):

Students who learn to articulate and justify their own mathematical ideas, reason through their own and others’ mathematical explanations, and provide a rationale for their answers develop a deep understanding that is critical to their future success in mathematics and related fields. (p. 6)

With respect to basic multiplication facts, this means students can reason about the facts rather than just recall them. This perspective is supported by research from the Program for International Student Assessment (PISA). Jo Boaler and Pablo Zoido (2016) analyzed data from PISA that includes both students’ knowledge of and approach to mathematics. Their analysis indicates that there are three ways that students seem to grasp new mathematical ideas: (1) memorizing, (2) making connections, and (3) self-monitoring by evaluating what they know and focusing on concepts they have yet to learn. According to the researchers’ analysis, the students who learned through memorizing were the lowest achievers in mathematics among the three groups. Additionally, according to Jo Boaler, Cathy Williams, and Amanda Confer (2015), “Mathematics facts are important but the memorization of math facts through times table repetition, practice and timed testing is unnecessary and damaging” (p. 1).

So, what would a productive program focused on developing automaticity with basic multiplication facts by growing reasoning skills related to multiplication look like? The Fact Tactics™ Fluency Program!

The Fact Tactics Fluency Program

The Fact Tactics Fluency Program provides an alternative to the drill-and-kill (or rote memorization) process educators commonly use to teach basic multiplication facts in so many classrooms. This process can lead to mathematics anxiety, which many people experience throughout their lives.

Figure

illustrates the typical cycle that leads to this avoidable outcome.

Push to learn basic facts

Encounter barriers to quick recall of basic facts due to stress and mathematics anxiety

Focus on memorization through drill and kill

Develop mathematics anxiety in students because of drill and kill and timed tests

Use timed tests for assessment

I.1

FIGURE I.1: Typical cycle for basic facts instruction.

Instead of this typical cycle, the Fact Tactics Fluency Program is based on connecting what students learn as they make sense of multiplication by emphasizing and supporting procedural fluency to develop automaticity. According to NCTM’s (2023) position statement on procedural fluency, “Basic facts should be taught using number relationships and reasoning strategies, not memorization” (p. 2). Through this program, students hone their reasoning skills to foster a deep understanding of mathematics.

This book will guide you through a process that offers continued support, linking prior knowledge and learned facts to future learning—a pathway to fact fluency, as shown in figure I.2.

Focus on developing fact fluency through building reasoning skills.

Introduce facts in an order that supports students connecting new ideas to previous ideas and limits the number of basic facts they need to memorize.

Practice facts with an emphasis on sense making and use formative assessment to guide support.

Provide challenge for students who need more.

Achieve automaticity.

Educators can seamlessly integrate the Fact Tactics Fluency Program into any existing mathematics program for students in grades 3–6. This program is intended for classroom and resource teachers, interventionists, coaches, administrators, tutors, and parents to support learners in making sense of and developing automaticity with basic multiplication facts.

FIGURE I.2: Fact Tactics fluency pathway.

About This Book

This book is organized into three chapters. The first chapter, “Making Sense of Fact Strategies for Multiplication,” illustrates the difference between additive reasoning and multiplicative reasoning, explores strategies for deriving basic facts, and highlights the properties of operations used to develop multiplicative reasoning. The properties of operations discussed are the commutative property of multiplication, the associative property of multiplication, and the distributive property of multiplication over addition.

The second chapter, “Promoting Grounded Fluency With the Fact Tactics Fluency Program,” describes the components of the Fact Tactics Fluency Program and how they work together to develop and promote automaticity with basic multiplication facts following a specific sequence. The six tactics that make up the program include the following.

1. The grounding tactic

2. The linking tactic

3. The strategic repetition tactic

4. The review tactic

5. The assessment tactic

6. The extension tactic

The grounding and linking tactics facilitate making sense of multiplication and building on that by deriving products from known facts. The strategic repetition and review tactics support students incorporating a spiral review (practicing previously encountered facts over time) of facts in meaningful ways while connecting to sense making for multiplication. The assessment tactic supports data collection in a formative assessment process, and the extension tactic provides students who have mastered the basic multiplication facts with more challenging tasks.

The third chapter, “Engaging in the Fact Tactics Fluency Program,” provides an overview of the program’s twenty-week plan for building reasoning skills for multiplication to develop automaticity with basic facts. The plan for each week includes the focus fact for the week; a sample Fact Tactics web, which is a graphic that links multiple strategies to the focus fact or facts and includes multiplication strategies with explanations for some of the strategies; and an extension task for students who need more challenge. The Fact Tactics cards and assessments for each week are included in the appendices.

Quick recall of basic multiplication facts is frequently a gatekeeper to many mathematics topics. For students who struggle to memorize their facts, there are often limited opportunities to explore other options beyond drill and kill to learn them. It is my hope that the Fact Tactics Fluency Program will transcend recall of basic facts and increase student access to mathematics proficiency, providing a gateway to increased success with mathematics for each and every learner.

Making Sense of Fact Strategies for Multiplication

Building reasoning skills for multiplication is part of mathematical conceptual understanding and requires attention to strategies rather than just solutions. This emphasis on how students solve problems shifts focus from the answer to the process used to determine the answer. The mathematical process of applying the structure of multiplicative reasoning sets up students for success in algebra. In 2008, the National Mathematics Advisory Panel (NMAP) released a report that determined the best way to prepare students for algebra. It reports that by the end of grade 5 or 6:

Children should have a robust sense of number. . . . It must clearly include a grasp of the meaning of the basic operations of addition, subtraction, multiplication, and division. It must also include use of the commutative, associative, and distributive properties; computational facility; and the knowledge of how to apply the operations to problem solving. (p. 17)

The authors of the report go on to say that “conceptual understanding of mathematical operations, fluent execution of procedures, and fast access to number combinations together support effective and efficient problem solving” (NMAP, 2008, p. 17).

The multiplicative reasoning developed through a focus on strategies supports foundational topics such as multiplying and factoring polynomials. Consider 7 × 8. If a

student applies multiplicative reasoning to determine the product of 7 × 8 by thinking 7 × 5 and add 7 × 3, the student is applying the distributive property of multiplication over addition in that 7 × 8 can be represented as 7 × (5 + 3), and the 7 is then distributed across the 5 and the 3. This leads to success with simplifying a problem such as 7(x + 3) in later grades, and it must be preceded by a deep understanding of multiplication.

Connecting Multiplication and Addition

Prior to beginning the Fact Tactics Fluency Program, students must first establish an understanding of multiplication as a way of determining the total number of objects in a collection of groups of objects. You might approach this conceptual development using contexts such as baskets of apples, plates of cookies, and so on. Students should see that they can model word problems before they look for more efficient ways to compute with the expressions that model them. For example, a student might solve the following problem using drawings: Alex has 4 boxes of cupcakes. Each box has 6 cupcakes. How many cupcakes does she have? The drawings would likely show four rectangles with six small circles in each rectangle. The student could determine the total number of cupcakes by counting all the small circles. Another student might determine the total using repeated addition such as the following: 6 + 6 = 12, 12 + 6 = 18, and 18 + 6 = 24. Yet another student might skip count and think, 6, 12, 18, 24, while holding up four fingers one at a time to keep track of the number of boxes. Eventually, students should come to model this problem with the expression 4 × 6 and think of the problem as four groups of six.

At this point, you can introduce students to strategies based on multiplication rather than counting by ones, repeated addition, or skip counting. If a student knows the twos facts, they can use the fact 2 × 6 to help find the product of 4 × 6 without needing to add 6 + 6 first. The student might say, “I know that 2 × 6 = 12 and 12 + 12 = 24, so 4 × 6 = 24.” By knowing the twos facts, the student has someplace to start, so they can find the new fact by using multiplication facts already known. This is called using derived facts.

Making the Most of Memorization and the Multiplication Table

The Fact Tactics Fluency Program builds a progression of facts that provides a foundation for students so they can use multiplicative reasoning with derived facts as a strategy in developing fluency. For this reason, there are a few facts that students need to know prior to starting the program. Before beginning the program, students should develop quick recall with their twos facts (the doubles) and their fives facts. Also, they should

know the product of 3 × 3 from recall. If students struggle with this, they can think, 2 × 3 = 6, add 3 is 9 to develop this fact. This fact is not included in the Fact Tactics progression because the strategies for determining the product of 3 × 3 are so limited, and the program is based on using strategy choices to determine the products of basic facts.

Once students know the twos facts, the fives facts, and the product of 3 × 3, they can use those facts to determine the twenty remaining basic multiplication facts. There are only twenty facts remaining because students will apply the commutative property of multiplication (which states that the product of two factors is the same regardless of the order of the factors) to learn each combination of facts that have the same factors using the same strategy, such as 4 × 6 and 6 × 4. These facts are called partner facts in the Fact Tactics Fluency Program.

Figure 1.1 provides a multiplication table with the twenty Fact Tactics facts highlighted in dark gray squares and the partner facts highlighted in light gray squares. You will build on these facts throughout this program. The ones, twos, and fives facts, and the product of 3 × 3, are in white squares because they do not require memorization. The ones facts just require an understanding that multiplying any number by 1 results in the original number.

Students build on a small set of known facts to develop fluency and, ultimately, automaticity, with all the facts in the multiplication table.

FIGURE 1.1: Multiplication table identifying Fact Tactics facts.

Making Sense of Mathematics for Basic Facts Instruction

Students likely developed the twos facts in second grade and the fives facts are often learned by skip counting. Skip counting is based on counting strategies rather than derived facts. While it is useful to skip count to develop fluency with twos and fives facts, this program does not promote the use of skip counting for other facts. Using skip counting to learn twos and fives facts is important so students have a foundation on which to start their exploration of derived facts. Using skip counting beyond that promotes memorization for the sake of memorization rather than as a building block for using properties of operations, such as the distributive property of multiplication over addition, to multiply.

While students can certainly use skip counting to determine the product of two singledigit factors, it is not strategic in ways consistent with the goals of this program. Skip counting does not qualify as a Fact Tactics strategy, and neither does using tally marks. Drawing groups of objects and counting the total number of objects in the groups, drawing and counting tally marks, and skip counting can all be classified as counting strategies and are based in additive reasoning.

Fact Tactics fluency refers to using multiplicative reasoning to develop quick recall of basic multiplication facts. Consider the fact 4 × 6. Thinking 6 + 6 + 6 + 6 = 24 relies solely on additive reasoning. Thinking 6 , 12, 18, 24 is also based on additive reasoning. Determining the product of 4 × 6 by thinking 2 × 6 is 12 and double 12 is 24 is based on multiplicative reasoning. It is still considered multiplicative reasoning if a student thinks 2 × 6 add 2 × 6, so 12 add 12 is 24, even though addition is used in determining the final product. This strategy is based on use of the distributive property of multiplication over addition in that 4 × 6 is broken down to (2 + 2) × 6 = (2 × 6) + (2 × 6), so multiplication is still central to the computation. The strategy is classified as a Fact Tactics strategy. If a student provides a strategy based on additive reasoning, acknowledge that this strategy will lead to the product, but then challenge them to come up with a strategy that uses multiplication or support the student to make sense of a multiplicative strategy that a peer provides.

The way you make sense of basic multiplication facts for teaching is somewhat different from how students should make sense of them; think of this as mathematics knowledge for teaching basic multiplication facts. This knowledge should influence how you discuss fact strategies with students. It helps you know where to linger as well as what questions to ask to ensure that the focus is on Fact Tactics strategies rather than counting strategies. For example, if a student uses drawings or tally marks to determine the product of 4 × 6, you may accept the response but then don’t ask other students to explain the thinking represented in the strategy. In other words, accept the solution but don’t linger on the process.

However, if another student says, “I know that 3 × 6 is 18, and I used that to determine that 4 × 6 is 24,” it might make sense to linger here and ask a classmate who used counting strategies to unpack how this student could have used 3 × 6 to determine the product of 4 × 6. The first strategy used counting, while the second strategy used the distributive property of multiplication over addition by thinking of 4 × 6 as (3 + 1) × 6 or 3 × 6 + 6. It’s important that the time spent on developing fluency with basic multiplication facts also supports students making sense of and using properties of operations.

Let’s explore this with another example. Consider the fact 7 × 8. Figure 1.2 provides a Fact Tactics web for this fact, along with its partner fact, 8 × 7. I describe Fact Tactics webs in detail in chapter 2 (page 13). Here I use the web to highlight several strategies for determining the product of 7 × 8.

If a student shares the strategy 4 × 7 and double to determine the fact 7 × 8, the teacher should know that the student is applying the commutative property of multiplication to use the partner fact 8 × 7 to know 7 × 8. So how is the student getting from 8 × 7 to 4 × 7 and double? The student is using the associative property of multiplication (which states that the product of three or more factors is the same, regardless of how they are grouped) by replacing the factor 8 with 2 × 4 so that the fact is represented as (2 × 4) × 7 and then 2 × (4 × 7). This translates to finding 4 × 7 first, because it is in parentheses, and then multiplying that product by 2 or finding 2 × 28. Wow, there is a

FIGURE 1.2: Fact Tactics web for 7 × 8.

lot of mathematics behind coming up with the Fact Tactics strategy 4 × 7 and double! Is it important for students to make sense of all those steps and properties? Ultimately, you want students to name and be able to justify the use of these properties of operations, but that is not the learning goal for the Fact Tactics Fluency Program. For Fact Tactics fluency, the goal is to use the properties of operations, along with facts that students know, to derive the focus fact. It is enough for a student to say, “The product of 7 × 8 is 56, because I know that 4 × 7 is 28, and I can double 28 to get 56. So, 7 × 8 (and 8 × 7) is 56.”

Additionally, it is not necessary for all students to use this strategy. You should not encourage students who struggle to double 28 using mental computation to use this strategy. The goal of this program is for students to make sense of one another’s strategies and then choose and use the strategy that supports their personal strengths.

It is helpful for you, as the teacher, to become comfortable with the different strategies and know the properties of operations involved in deriving facts using this program so you can help students make sense of the strategies. For this reason, you should plan for instruction by anticipating the strategies students might suggest each week. Chapter 3 (page 27) provides sample Fact Tactics webs for each focus fact as well as a discussion of the mathematics behind the strategies. This discussion is for you more than it is for students. The knowledge you build about the strategies will help you facilitate rich discussions as students share their thinking. Your understanding of mathematics for teaching basic multiplication facts will translate to your precise use of vocabulary and applicable properties as you support classroom discourse and sense making around fact fluency.

Each week, you will generate a class Fact Tactics web for the focus fact for that week. It is crucial that you generate the fact strategies as a class through discourse with students. Students might share strategies that are not provided with the sample web. If the strategy works and the product derives from other facts rather than solely by repeated addition or counting strategies, then you should include it in the web. The goal is for students to have input and choice with the strategies they use to develop Fact Tactics fluency. However, there are also suggestions you can use in chapter 3 for when students don’t provide the strategies you seek.

Conclusion

In this chapter, you explored the difference between additive and multiplicative reasoning as you made sense of the foundations for the Fact Tactics Fluency Program. The next chapter introduces the sequence of the Fact Tactics facts and the six tactics you will use to implement the program. Through the six tactics, you will support students in making sense of multiplication strategies, using facts they know to derive other facts, practicing new facts and facts previously explored, engaging in self-monitoring, and developing productive perseverance.

How can students achieve a deep understanding of mathematics, one that allows them to go beyond rote memorization to explain and justify their thinking? In The Fact Tactics™ Fluency Program: Building Reasoning Skills for Multiplication in Grades 3–6 , author Juli K. Dixon introduces an innovative twenty-week program that teachers can seamlessly integrate into existing mathematics instruction. The program is built on six tactics that offer an alternative to drill-and-kill instruction, connecting what students learn as they make sense of multiplication by emphasizing and supporting procedural fluency to develop automaticity.

READERS WILL:

• Understand the six tactics involved in engaging with the Fact Tactics Fluency Program

• Discover how to seamlessly integrate the twenty-week program into mathematics instruction

• Learn how to help students transcend rote memorization and access the tools to explain and justify their thinking

• Explore the process, resources, and assessments to implement the program

• Learn how to support a deep understanding of mathematics beyond the program

“Memorizing multiplication facts can be a source of anxiety, frustration, and eventual disaffection from mathematics. In The Fact Tactics Fluency Program, author Juli K. Dixon moves beyond rote memorization by positioning fluency with mathematics facts as a byproduct of thinking, reasoning, and number experiences. Students use the strategies in this program to achieve fluency through active sense making and the co‑construction of ideas that build networks of connections between what is known and not known, familiar and unfamiliar, and fluency and reason.”

Peter Liljedahl, Professor, Faculty of Education, Simon Fraser University, Vancouver, Canada

“Juli Dixon’s book, The Fact Tactics Fluency Program, provides practical, ready to implement strategies to help students grow their multiplication fact fluency. The program, grounded in decades of research, offers six tactics for teachers to introduce, reinforce, and assess students’ development of mathematics reasoning skills and automaticity. If you want to move away from repetitive, memorization based strategies toward reasoning and sense making to help students learn multiplication, then this book is for you!”

Zandra de Araujo, Chief Equity Officer & Math Principal, Lastinger Center for Learning, University of Florida

ISBN 978-1-958590-21-8

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