 Volume VIII, Issue III Heather Clayton, the author of Making the Standards Come Alive!, is the principal of Mendon Center Elementary School in Pittsford Central School District, New York. She is also a co-author of Creating a Culture for Learning published by Just ASK.

Math Fact Fluency

For years, a focus of mathematics instruction has been around the learning of basic facts, which are the single digit combinations for all four operations-addition, subtraction, multiplication, and division. Without knowledge of the basic facts, the future study of fractions and decimals, money, measurement, and geometry can be extremely difficult. However, what is the best way to learn and practice math facts? What about students who are already in middle school and still don’t have automaticity? Which practices will finally help those math facts “stick?”

We know that the ability to use facts to solve problems is an expectation of the math standards. In order for students to be able to solve increasingly complex problems, they need to have a strong foundation with basic math facts. Math fact fluency begins in Kindergarten and for some students, extends well into middle school.

When students enter middle school with a knowledge of their math facts, they are free to focus on rigorous concepts and complex problems. However before a student can retain and apply math facts, there needs to be a foundation of understanding. Without meaning and understanding behind each operation, practicing math facts in isolation will not lead to long term retention. However when students combine conceptual understanding and strategies with memorization, a solid foundation has been built for the retention of math facts.

The standards call for fluency with math facts. Being fluent in mathematics is more than just rote memorization of number combinations. According to the Common Core Standards for Mathematics (CCSM), procedural fluency is defined as a “skill in carrying out procedures flexibly, accurately, efficiently and appropriately.” This aligns with Arthur Baroody’s definition that basic fact fluency is “the efficient, appropriate, and flexible application of single-digit calculation skills.” In order to assess fluency, attention needs to be given to flexibility, accuracy, efficiency and the use of appropriate strategies.

What is Fluency?

Flexibility: When students are able to choose, understand, and explain different methods for solving problems.

Accuracy: When students consider the meaning of an operation, re-read their work carefully, and check the reasonableness of their answer. When a student is accurate, their answers are mathematically correct.

Efficiency: When students produce accurate answers efficiently by using strategic thinking to carry out computation, without being limited by too many steps. A student gets to their answer relatively quickly.

Use of appropriate reasoning strategies: When students are fluent, they select and apply the appropriate strategies when solving problems. Students use number relationships and known facts as a basis for the strategies they choose to figure out unknown facts. If a student is struggling to learn specific facts, they should go back to the use of strategies and not rely on rote memorization.

Phases of Fluency

In order to gain fluency with basic math facts, students move through three developmental phases as described by Arthur Baroody. Without moving through these phases and without the strong development of reasoning strategies, students aren’t achieving fluency.

• Phase 1: Counting – Students use objects or verbal counting to figure out an answer. For example, for 3 + 8 a student will start with 8 and count on 9, 10, 11).
• Phase 2: Reasoning Strategies – Students use facts they know to determine facts they don’t know. For example, if a student knows that 5 + 5 = 10, then they know that 6 + 5 is one more, 11.
Reasoning strategies help students to solve fact problems when they have forgotten the answer. They use strategic thinking to regenerate the answer rather than relying on counting or simply guessing.
• Phase 3: Mastery – Students produce answers accurately and quickly, without counting or applying any reasoning strategies. These are answers the students “just knows” automatically.
Why not just have students memorize? When students are asked to move on to memorizing facts before they have developed number sense and strategies, they become limited in their ability to correctly apply their facts and recognize when they have made errors. When developmental stages are skipped, students have far too many facts to learn in isolation and rather than gaining flexibility with their thinking, students likely rely on counting strategies.
Traditional practices such as flash cards, timed tests, and drills skip the second phase of reasoning strategies. Without strategic thinking, students may not retain their facts in the long run. This is when we see students who are in grades 4 and beyond and still don’t know their math facts.

Teaching Reasoning Strategies

Reasoning strategies are important to teach for each of the four operations.

Addition is the joining of two or more sets. A common strategy is counting on, such as finding 5 + 2 by saying “5, 6, 7”. Another strategy is using the commutative property, or “turn around facts.” For instance, if you know 6 + 7 then you also know 7 + 6. Yet another way, is children using what they already know to figure out unknown problems. They might begin with doubles (if you know 5 + 5 is 10, then you know 5 + 6 is 11), or combining simpler problems (when solving 9 + 7, if you know 10 + 7 is 17, then 9 + 7 is one less).

Subtraction
Subtraction can be viewed as “taking away,” the comparing of two numbers, the distance between two numbers on a number line, or a missing addend problem such as 4 + ___ = 11.
When helping a student to find strategies for answering subtraction problems, it is helpful to relate subtraction to addition.

Multiplication
Multiplication typically begins at the start of third grade and is the joining of equal sets. Arrays, or area grids, are often used to illustrate the concept of multiplication, along with repeated addition of equal sized groups.
In multiplication, students should start with facts they know and then build from there. For instance, students may begin with x 2’s, and double every number. Then students can easily learn 4’s by knowing that they are just the 2’s doubled. The 5’s are familiar to students, as they think about familiar counting sequences (skip counting). Yet another strategy is breaking down multiplication problems into easier problems. For example, children might break down 6 x 5 into 2 x 5 and 2 x 5 and 2 x 5.

Division
Division is repeated subtraction and equal sharing/partitioning. Stories and manipulatives are very helpful to children as they learn their division facts. For instance, Jamie had a bag of 18 pieces of candy. She gave one child 6 pieces, another child 6 pieces, and another child 6 pieces, and is now out of candy. She has subtracted 6 from her bag of candy 3 different times. Or think about Mark who shared a tray of 12 cookies with 4 friends. Also helping your student to see that 12 divided by 3 is the same as 3 x __ = 12 gets them to use the multiplication facts they know to solve division problems.

Resources for Practicing Reasoning Strategies

Math games are an engaging way to support students’ learning about numbers and operations. When played consistently, math games can support students’ development of computational fluency and strategic thinking.

Number talks typically last between 5 and 15 minutes and engage the whole class in strategy discussion. During a number talk, a teacher presents a meaningfully crafted computation problem. Students then solve the problem mentally and share their strategies with the class. While students are sharing their strategies, teachers are looking for accuracy, efficiency, and flexibility with their thinking.

Alycia Zimmerman shares resources and tips related to number talks at www.scholastic.com/teachers/blog-posts/alycia-zimmerman/number-talks-grow-mathematical-minds/

Assessing Math Fact Fluency and Reasoning Strategies

If done well, the assessment of basic math fact fluency can provide a teacher with invaluable information about what a student knows, doesn’t know, as well as any misconceptions the student may possess. These assessments do not need to be lengthy or complicated; it’s actually the use of a few well-constructed questions that can yield an abundance of information about student progress and next steps for instruction.

According to the article “Assessing Basic Fact Fluency” by Gina Kling and Jennifer Bay-Williams, there are three important ways to glean important data about students understanding.

• Interviews
• Engaging in a brief discussion with a student provides a window into their thinking and reasoning. Also, as students talk through how they are solving problems, there is an increased likelihood that they begin to check the reasonableness of their answers and self- correct when it is necessary. Listed below are some sample questions that can be used when interviewing individual students:
• Write a problem on an index card. Ask the student the answer to the question, as well as how they found the answer and if they know of another way to solve the problem.
• Ask students how they can use the answers to facts they know to solve facts they don’t know. For example, as questions like “How can 5 + 5 help you to solve 7 + 5?” or “How can 3 x 3 help you to solve 4 x 3?”
• Observations
Math class is full of opportunities to observe children at work. Pull up alongside a student as they are playing a math game, completing a partner task, or completing independent work and notice what they are doing. Notice the answers they are getting along with how they are arriving at those answers. Notice the facts students “just know” versus those they need to use reasoning strategies to solve. Are the strategies they are using efficient? Are they using a variety of strategies or just one? Do they self-correct if an answer is wrong?
• Writing
Asking students to write about how they are solving problems provides an important window into their thinking. Provide students with simple prompts they can respond to in a journal and track their responses over time. Ask students questions like:

• Bobby used the fact 8 X 5 to help him solve 8 x 7. How did he do that? Use pictures, words, and/or numbers to explain.
• A friend is having trouble solving 6 + 7. How could you help them?
• Raoul explained to a friend that 8 + 9 = 16. Is he correct? Why or why not?

Why Not Just Use Timed Tests?

There is a growing body of research that indicates using timed tests can actually be detrimental to a student’s confidence in the area of mathematics. For even the strongest of math students, timing them with their facts can be anxiety provoking and restrict their ability to think flexibly. Furthermore, when timing fact tests, the opportunity to see how students think and reason is missed.

One suggestion King and Williams give is to use “fluency quizzes” instead of timed tests. On a fluency quiz, students are only assessed on their foundational facts, or those facts that can be used to derive other facts using a strategy. The foundational facts in addition are one more than (6 + 1, 8 + 1), two more than (6 + 2, 8 + 2), combinations that make ten (3 + 7, 6 + 4), and doubles (4 + 4, 9 + 9). The foundational facts for multiplication include x 2, x 5, and x 10. When students master their addition and multiplication facts they can use them to fluently solve subtraction and division facts.

On the fluency quiz, in addition to asking the students to solve the foundational fact, ask them if it was a fact they knew already or if they used a strategy to solve it. Students could even choose just one problem they solved and write about how they solved it.

Three BIG IDEAS on Math Fact Fluency

• Fact instruction is about strategy instruction.
• For fact retention and understanding, students need to move through all phases of fluency and NOT go from counting to memorization.
• Fluency is not acquired without strategies, and mastery is not acquired without fluency.

Resources and References

Baroody, Arthur. “Why children have difficulties mastering the basic number combinations and how to help them.” Teaching Children Mathematics, August 2006.

Kling, Gina and Jennifer Bay-Williams. “Assessing basic fact fluency.” Teaching Children Mathematics, 2014.

O’Connell Susan and John SanGiovanni. Mastering the Basic Math Facts in Addition and Subtraction. Portsmouth NH: Heinemann, 2011.

_________________. Mastering the Basic Math Facts in Multiplication and Division. Portsmouth: Heinemann, 2011.

Parrish, Sherry. Number Talks: Helping Children Build Mental Math and Computation Strategies. Sausalito, CA: Math Solutions, 2010.

Van De Walle, John, Karen Karp, and Jennifer Bay-Williams. Elementary and Middle School Mathematics: Teaching Developmentally, Tenth Edition. New York: Pearson, 2018. Permission is granted for reprinting and distribution of this newsletter for non-commercial use only. Please include the following citation on all copies: