Chris Frawley, M.Ed.

Mr. Toledo, a fifth grade teacher, presents a lesson on solving multistep math problems involving multiplication. As small groups of students begin to work together to solve five practice problems, he walks around the room listening to their explanations. Mr. Toledo notices that several of the students are slower to complete the problems because they don’t know their multiplication facts with accuracy and automaticity. As a result, some of the groups have incorrect answers to their problems. He knows it’s time for a more intensive intervention for these students to learn their basic math facts but isn’t sure which intervention to use.

Students who are fluent with their facts are more likely to complete math problems more quickly and will have more opportunities to respond.

Mr. Toledo’s experience is not uncommon. Many students struggle to recall their basic addition, subtraction, multiplication, and division facts with accuracy and automaticity. Mr. Toledo observed, as many teachers do, that his students’ inability to recall the basic facts affects their performance on assignments and tests. How can he and other teachers assist students with developing math fact fluency? This article provides suggestions for interventions to develop math fact fluency.

#### What is math fact fluency?

Math fact fluency is the ability to accurately and quickly recall basic addition, subtraction, multiplication, and division facts (Burns, 2005; McCallum, Skinner, & Turner, 2006; Poncy, Skinner, & Jaspers, 2006). Math fluency is often calculated by determining a student’s digits correct per minute for a specific set of facts (e.g., addition, division). Students who possess fluency can recall facts with automaticity, which means they typically think no longer than two seconds before responding with the correct answer. Burns (2005) explains that automaticity is evident when a student solves a problem faster through recall than performing a mental algorithm. Students with math fact fluency have received instruction in the basic math facts and understand the concepts of addition, subtraction, multiplication and division. They possess strategies to determine the answers to math problems (e.g., counting strategies, number line) and willingly use them when necessary. However, the use of these strategies can slow a student and affect the accuracy of his work.

Students who recall their basic facts accurately and quickly have greater cognitive resources available to learn more complex tasks or concepts (McCallum, et al., 2006; Poncy, et al., 2006). For example, if Sarah knows her division facts with proficiency, then she has more cognitive resources available to learn and acquire new skills to complete advanced problems such as 3,496 ÷ 437 = 8. Additionally, students who are fluent with their facts are more likely to complete math problems more quickly and will have more opportunities to respond because it takes less cognitive effort to complete the math tasks. As McCallum, et al. (2006) and Poncy, et al. (2006) point out, having more opportunities to respond can increase students’ skill acquisition, fluency, and generalization. This, in turn, can influence whether students choose to engage in their assigned math work. The ability to quickly recall math facts lessens students’ anxiety and increases their confidence to engage in more challenging math tasks. Engagement in these challenging tasks is likely to lead to increased skill development. For example, Sarah is proficient with her multiplication facts and her class has learned how to multiply three-digit numbers with regrouping. Sarah works diligently to complete each of the ten assigned practice problems by using what she learned in class. Since she is fluent with her multiplication facts, Sarah can concentrate on how to solve the problem without having to stop to calculate each fact.

#### How can teachers assist students develop math fact fluency?

There are many interventions available to help students develop math fact fluency. For example, teachers can use flashcards, songs, raps, games, worksheet drills, or timed tests to assist students with accuracy and recall of basic math facts. Since the components of each intervention vary, it’s difficult to determine which components are the most effective. To investigate this question, Codding, Burns, and Lukito (2011) conducted a meta-analysis to analyze the components of math fact fluency interventions. The meta-analysis synthesized seventeen single-case design studies that included 55 elementary students (24 boys and 25 girls, average age 10). The participants were identified as struggling math learners, with 54% identified as having a disability (i.e., learning disability, cognitive disability, behavioral disorder). Codding et al. found that math interventions that contain the components of practice with modeling and drill produce the largest treatment effects. Among the interventions that have these components, Taped-Problems, Cover, Copy, and Compare, and Incremental Rehearsal are math fact fluency interventions that can be implemented easily with students.

#### What is practice with modeling?

Practice with modeling occurs when a student’s basic math fact practice incorporates self-modeling or teacher-directed modeling (Codding, et al., 2011). The addition of auditory and/or visual models with practice provides students additional repetitions that can lead to increased retention and recall of math facts. Taped-Problems and Cover, Copy and Compare are two interventions that include the practice with modeling component. Both interventions can be implemented with an individual student or with a small group of students.

Developed by McCallum, Skinner and Hutchins (2004), the Taped-Problems intervention requires students to respond to recorded basic math facts. Students write answers to a numbered list of math problems (e.g., 6 + 8 = ___) as they listen to an audio recording of math facts (see Figure 1 for a more detailed explanation). The interval between when the problem is spoken and the answer is given can vary from a short period of time (e.g., 1 second) to a longer period of time (e.g., 5 seconds). This variance allows students to respond independently and use the audio recording for immediate feedback and to produce quicker, more accurate responses. The Taped Problems intervention provides students practice writing the facts accurately while listening to each problem and the correct answer as they respond to the model that is presented visually on paper.

#### Figure 1

**Taped-Problems Intervention**

Materials:

- Audio recordings of sets of basic math facts problems (e.g., a set of 15 addition facts). At least three sets should be recorded. Note: The teacher determines the interval between the problem and the given answer. This interval remains consistent throughout the recording.
- Student answer sheet for the set of math facts on the audio recording
- Head phones (if needed)

Procedure:

Instruct the student to listen to the audio recording of a set of basic math facts. As each problem is spoken, the student is to “try to answer faster than the recording” by writing the answer for each problem before he hears it on the audio recording. If the answer is given before he can write it down, then the student writes the correct answer that he hears. If the student writes an incorrect response, he writes a slash mark through his answer and then writes the correct answer is written next to it. The student is to follow along with the recording and not move ahead.

The procedure is repeated multiple times with the recordings of the other sets of basic math facts problems.

The time interval can be very short (e.g., 1 second) in the beginning to help the student develop accuracy and to provide practice with modeling. Then, the intervals can be increased to help the student become more independent in his responses, and, finally decreased to promote more automaticity.

(McCallum, Skinner, Turn, & Saecker, 2006; Poncy, Skinner, & Jaspers, 2007)

Cover, Copy, and Compare is another intervention that uses practice with modeling. Originally designed to assist students to increase their spelling accuracy, it was adapted to help students develop math fact proficiency (Poncy, et al., 2006). The procedures for Cover, Copy, and Compare require the student to record answers to a list of math problems that are written on the left side of a paper. Then, the student covers and rewrites the problem and answer on the right side of the paper. He compares the second answer to the first answer and makes needed corrections. For an extended description of the intervention, see Figure 2. As with the Taped-Problems intervention, Cover, Copy, and Compare provides students with immediate feedback as they compare the answers for the math problems in the left column to the problems in the right column. Additionally, it prevents students from practicing inaccurate answers and reinforces their correct responses to math facts. The multiple practice opportunities reinforce the accuracy of the students’ answers and result in increased proficiency.

#### Figure 2

**Cover, Copy and Compare**

Materials:

Sheet of basic math facts, problems and their answers are written on the left side of the paper (e.g., 5 x 6 = 30)

Procedure:

- Give the student the sheet of math problems.
- Teach the student to study the problem and answer that is on the left side of the paper.
- Tell the student to cover the problem and answer on the left side of the page.
- The student writes the problem on the right side of the paper.
- The student uncovers the problem and answer on the left side of the page and compares it to what he wrote on the right side of the page.
- If the problem and answer are written correctly on the right side of the page, then the student moves to the next problem.
- If the answer is incorrect, then the student is to rewrite the correct response.

Variations to Cover, Copy and Compare:

- If the answer is incorrect, the student writes the problem and correct answer more than once.
- The student can respond verbally to the problems instead of writing them.

(Poncy, Skinner, & Jaspers, 2007)

#### What is drill?

Drill rehearsal interventions have been determined to be successful with assisting students with developing math fact fluency. These interventions increase the number of practice trials on individual skill items (e.g., math facts) and lead to increased retention (Burns, 2005; Codding, et al., 2011). Incremental Rehearsal develops automaticity through multiple opportunities to respond by using a specific set of items. Students are provided a set of math fact flash cards to review (e.g., a set of 10). The set of flash cards contains 90% facts that are already known to the student and 10% facts that are unknown to the student. The student follows a specific procedure for reviewing the flash cards until the unknown fact(s) become known. Then, the procedure is repeated for the next set of unknowns (see Figure 3 for more information). Burns (2005) explains that interspersing known and unknown math facts creates an appropriate level of challenge for the student by not giving him more items (facts) than he can learn at one time. The key component of the Incremental Rehearsal intervention is the multiple opportunities for the student to practice and respond to individual math facts. As a result, students become more accurate and respond more quickly.

#### Figure 3

**Incremental Rehearsal**

Materials:

Blank cards to make flashcards

Procedure:

- Assess the student (e.g., use pre-made flash cards, a timed drill) to determine facts he knows with automaticity.
- From the list of facts that student knows with automaticity (knowns), select nine. Write each fact on a blank flashcard.
- From the facts the student does not know (unknowns), select ten facts. Write each fact on a blank flashcard.
- Take the nine knowns and one card from the unknown stack of fact cards.
- Present the first unknown fact flashcard and ask the student to attempt to answer it aloud. If he is incorrect, provide the correct answer.
- Present the first known fact flashcard and ask the child to answer it aloud.
- Present the original unknown from Step 5 and again ask the child to attempt to answer it aloud.
- If the child answers it correctly, it now becomes known. Start the procedure again at Step 4 using a different unknown.
- If the child answers it incorrectly, then continue the process by presenting two knowns to the student before asking him to attempt to answer the unknown again. The next step would be to present three knowns with the unknown, the four knowns with the unknown, etc.
Example flashcard sequence:

First unknown, known

First unknown, known, known

First unknown, known, known, known

First unknown, known, known, known, known

First unknown, known, known, known, known, known

First unknown, known, known, known, known, known, known

First unknown, known, known, known, known, known, known, known

First unknown, known, known, known, known, known, known, known, known

First unknown, known, known, known, known, known, known, known, known, known

- Repeat until all unknowns become knowns.

(East Carolina University, 2011)

Teachers like Mr. Toledo want their students to be successful. If students do not know their basic math facts with accuracy and automaticity, they will continue to struggle to learn new math concepts and to solve more complex problems. Codding, et al. (2011) determined that the math fact fluency intervention components of practice with modeling and drill result in increased student performance in learning math facts. Taped-Problems, Cover, Copy and Compare and Incremental Rehearsal can be used with students who need intensive support with developing math fact fluency. Each intervention is easy to teach and students can use it with independence after it has been learned.

**References**

Burns, M. K. (2005). Using incremental rehearsal to increase fluency of single-digit multiplication facts with children identified as learning disabled in mathematics computation. Education and Treatment of Children, 28, 237-249.

Codding, R.S., Burns, M.K., & Lukito, G. (2011). Meta-analysis of mathematic basic-fact fluency interventions: A component analysis. Learning Disabilities Research & Practice, 26, 36-47.

East Carolina University. (2011). Common reason for academic failure: They have not spent enough time doing it. Intervention name: Incremental rehearsal. Retrieved from http://teacherpages.nhcs.net/schools/parsley/karlysokolowski/Documents/Sight%20Word%20Activities/Incremental%20Rehearsal.pdf

McCallum, E., Skinner, C.H., & Hutchins, H. (2004). The taped-problems intervention: Increasing division fact fluency using a low-tech self-managed time-delay intervention. Journal of Applied School Psychology, 20(2), 129-147.

McCallum, E., Skinner, C. H., Turner, H., & Saecker, L. (2006). The taped-problems intervention: Increasing multiplication fact fluency using a low-tech, classwide, time-delay intervention. School Psychology Review, 35, 419-434.

Poncy, B. C., Skinner, C. H., & Jaspers, K. E. (2006). Evaluating and comparing interventions designed to enhance math fact accuracy and fluency: Cover, copy, and compare versus taped problems. Journal of Behavioral Education, 16, 27-37.