A Look into Fact Fluency with Nancy Anderson

Multiplication Fact Fluency Cards are an easy yet effective system to master multiplication and division facts to 100. Students develop fluency with one set of facts at a time, using facts they now know to solve unknown related facts. Each card provides a visual model, an array of dots, to aid in learning the multiplication fact. Students apply their knowledge of each multiplication fact to the related multiplication "turn-around" fact, cutting the number of facts to be learned in half! Answers can be found on the back of each card along with other related multiplication and division equations.

We get an inside look into the development and creation of these cards with author Nancy Anderson.

Nancy Anderson, Ed.D. | Math teacher, author, consultant | @NAndersonMath | www.nancyandersonmath.com

What is your background in education?

I have been working in the field of mathematics education for about 25 years. I began teaching math at the elementary level and enjoyed that so much that I decided to go back to college and take more mathematics courses so that I could become a middle school math teacher. I taught at the middle school level for several years – and loved it – before going back to school once more to get my doctorate in curriculum and teaching with a focus on mathematics education. Afterward, I became a K-8 mathematics curriculum coordinator and did some work at the college level too. I am currently teaching high school, so I can officially say I’ve taught math to students from every grade kindergarten through grade 12.

Throughout my career, I have also been fortunate enough to hold other roles within the field of mathematics education but outside of the classroom as a curriculum specialist, professional developer, math coach, and school administrator. But the classroom is definitely “my happy place” and the place where I feel most at home and most fulfilled.

Why are you so passionate about multiplication fluency?

As a 25-year-plus math educator, who has had different roles, in different school settings and with many different age groups of students, math facts have been a constant presence in my work. They were commonplace in my work as an elementary math specialist and K-8 curriculum coordinator – and an area where very regularly colleagues would seek out my support. In my work as an interventionist, multiplication fact fluency was a very common reason why students were selected to work with me. In my work as a middle school teacher, I’m amazed by how often multiplication facts appear in grade-level skills and topics like the topic of finding unit rates or solving equations, or multiplying integers. Even in my own work as a high school classroom teacher, multiplication fact fluency comes up there too – often laden with a lot of emotions from students who tell me how embarrassed they are that they don’t know their multiplication facts. And this really bothers me because they’re such skilled and capable students! They should not feel this way and they don’t have to feel this way because multiplication fact fluency is an area where we as math educators know a lot about how to help students succeed. No, there are no quick fixes but we do know what works and what resources are available to help students learn their facts. Fact fluency is a solvable problem in the field of mathematics education and if these multiplication fact fluency cards can play some part in eradicating the shame and embarrassment that so many students feel when it comes to their facts, I would feel very grateful for the chance to be a part of this solution.

What is your definition of fluency?

I define multiplication fact fluency in terms of three characteristics:

1. Quick recall – Students are fluent with their facts when they can state or produce a particular product (equal to a multiplication expression up to 10 × 10) without actively using a mental or written strategy. Although no set time defines “quick recall,” three seconds is often considered the upper bound. In other words, if a student can say or produce a product in three seconds or less, they likely either “just know” the fact or can call up the strategy quickly enough so that it becomes part and parcel of the fact itself. When a student has quick recall, they can call up a particular product when it’s needed so that the facts become helpful tools in their arithmetic and problem solving tool kits.
2. Retention – Students are fluent with their facts if they remember them long after a particular unit of instruction or practice session.
3. Application – Students are fluent with their facts if they can apply them in the context of more complex problems and when they recognize a particular fact is needed in a novel or unfamiliar context or problem scenario. For example, students are fluent with their multiplication facts if they can apply their facts to solve multiplicative compare and area problems just as readily as they can solve more well known “equal groups” problems.

How did you develop the Multiplication Fact Fluency Cards?

Across the course of my entire career as a math educator, multiplication fact fluency has had a constant presence in my work. As a classroom teacher, I encountered many students in the upper elementary grades and beyond who were capable and confident math thinkers but had trouble recalling some or a lot of multiplication facts. For my students with documented learning disabilities, deficits in fact fluency seemed to always be an unfinished learning goal. As a math specialist, informal conversations with colleagues would almost always drift toward the topic of multiplication facts and what to do about the many students who were not yet fluent. And as a math coordinator, I had to soothe the anxieties of many school principals after they encountered a group of students who didn’t yet know their facts.

For a long while, my response to concerns about students’ multiplication fact fluency was pretty standard. I told people, “Don’t worry about it.” I assured my colleagues that as long as students continued to have opportunities to solve problems, develop their own strategies, and understand the various meanings of multiplication, fluency with facts would come organically. I also reminded them that there was not a linear relationship with fact fluency and more advanced mathematics skills such as multi-digit multiplication and operations with fractions. Students could successfully engage with these content areas even if they did not have every fact through 10 × 10 committed to memory.

And although these reassurances would quell my colleagues’ concerns for a bit, they often would seek me out again when they did not see the growth in fluency that I assured them would take place. As the school year marched on, they grew increasingly worried about their students’ fluency and, understandably so, frustrated with me for my “pollyannaish” or laissez faire attitude. My colleagues told me that they were doing all of the things I said would help such as relying conceptually rich curriculum materials, giving students time to develop their own strategies, and teaching students the “why” as well as the “how” when it came to multiplication. And yet, they were not seeing these practices lead to fact fluency. Instead, what they saw was students who were not yet fluent with their multiplication facts losing ground to those who were. For example, class discussions about partial product and compensation strategies for multi-digit multiplication were entirely different experiences for students with fact fluency (e.g., knowing 3 × 8 = 24, 6 × 4 = 24) than for those without. And when students got to fractions, those with quick recall of their multiplication facts formed important generalizations about representation and equivalence that remained obscured to those without. I could no longer deny that fact fluency was creating divergent pathways for students and told my colleagues I was ready to help.

I asked my colleagues to send me a list of students who needed support with their multiplication facts and worked with these students one-on-one on a regular basis. I spent the first few sessions asking students questions about which facts they “just knew,” which ones they thought were hard to remember, and listening as they worked to derive unknown facts. These data points not only helped me understand the reasons these students were struggling to develop quick recall of their facts but also helped me develop a set of strategies for filling in their gaps. This resource, ‘Multiplication Fact Fluency,” is the culmination of that work.

How do students use the card sets?

Each fluency card shows one multiplication fact as a numeric expression and geometric array. Students develop fact fluency by working with a small group (or set) of facts. There are five sets in the resource and each set contains no more than 11 cards. In addition, each fact in a new set is connected to a fact that students learned in a previous set (e.g., a fact they already “just know”). With this approach, students develop fact fluency by continually building on what they already know. The product for each fact is written on the back of each card so that students can affirm or correct their answers in real time and without assistance from the teacher.

Students should practice only one set of cards at a time. They can use the cards independently or with a partner. They should look at the expression at the top of the front of the card. If they “just know” the product, they can say it, turn it over to check, and move on to the next card. If they don’t know the product, they should use the array to help them find it. Again, they should turn the card over to check their answer. They should practice the cards in the set for about five minutes two to three times per week. This type of practice is essential in order for students to move from using the strategy to come up with the product to “just knowing” or having quick recall of the product. For example, consider the flashcard for 4 × 7 which shows students the expression 4 × 7 and its array divided into two halves to prompt students to find the product by recalling the product of 2 × 7 and doubling it. Each time the student practices the card, they will enact the strategy (e.g., thinking about 2 × 7, calling up the product 14, and adding 14 + 14 = 28), until the strategy becomes so familiar to the student that it either becomes automatic or activated so quickly (e.g., “4 × 7 … 14 … 28”) that the student says “I just knew it,” when asked how they got their answer.

This resource helps students achieve multiplication fact fluency with fluency cards that promote efficient number sense strategies and instructional routines that empower students to transition from strategy enactment to quick recall.

Why are the problems on the back of the cards so critical?

Multiplication Fact Fluency cards utilize the commutative property of multiplication to cut students’ fluency workload in half. The arrays can be used to show students that when we reverse the number of rows and size of rows, the total number of dots in the array does not change. The first card on the back of each card lists this related “turnaround fact.” Once students develop quick recall of most of the cards in a set, they can incorporate these facts in their practice too.

The other equations on the backs of the cards are the related missing factor multiplication equations and division equations. And these facts are tremendously important for helping students develop the last facet of fluency – application. Multiplication fact fluency – and in particular applying the facts to solution strategies – is essential for performing a wide range of other strategies and skills within arithmetic and other strands including geometry, measurement, probability and eventually algebra. And in many of these skill areas, students need to be able to navigate between “the fact” and the related missing factor and division equations. Consider, for instance, rewriting 56/8 as a whole number. To complete this process, a student could think about the expression as a division – 56 divided by 8. Then, to find the answer, they could turn this division into a missing factor multiplication equation (8 times what is 56). In order to answer that question, it would be helpful to have quick recall of the multiplication fact 8 x 7 = 56. So in just this one example, we can see that fact fluency (i.e., quick recall of 8 x 7) is definitely a key component but so too is connecting this fact to division and missing factor multiplication. And that’s what the backs of the cards aim to do – help students develop quick recall of their multiplication facts and connect each facts to their related division and missing factor equations so that they can move flexibility between these different equations as they use them in their work with division, fractions, other forms of rational numbers, and eventually algebra, and beyond.

Check out Multiplication Fact Fluency Cards today!

Nancy is an experienced classroom teacher, curriculum specialist, author, and professional developer. Her publications include "Talk Moves: A Teacher's Guide for Using Classroom Discussions in Math Class," "Good Questions for Teaching High School Math," "Good Questions for Math Teaching, Grades 5-8," and "What’s Right About Wrong Answers: Learning from Math Mistakes, Grades 4-5." Nancy earned her undergraduate degree, Master of Education, and Doctor of Education from Boston University.