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Mathematics Explainer

What Is Math ‘Fact Fluency,’ and How Does It Develop?

By Stephen Sawchuk — May 01, 2023 6 min read
Conceptual Illustration of various math symbols floating above a grid and waves.
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A key part—though surely not the only part—of early-grades math is ensuring students get the basic arithmetic functions down and, beyond that, making sure they’re able to swiftly and automatically recall addition number combinations and the times tables.

Those skills help kids advance when multidigit arithmetic, fractions, and long division enter the picture—allowing them to focus on more complex problem-solving instead of simple computation.

Nevertheless, a surprising number of misconceptions about fact fluency abound. One common debate—which probably stretches all the way back to the introduction of the abacus, let alone calculators, computers, and Google—concerns whether kids really need to know the multiplication tables when those facts are readily at hand elsewhere. (Hint: They do.)

In a recent Education Week survey of about 300 math educators, most agreed that it’s “essential” for students to have fact fluency in order to work on higher-order, conceptual math problems. But more than a quarter said that it was “helpful, but not essential.”

EdWeek interviewed researchers and reviewed dozens of studies for this explainer. Read on—or jump straight to the bibliography at the end.

Why does math fact fluency matter?

The basic reason why math fact fluency matters, cognitive scientists say, is that it frees up brainpower or working memory to do more complex mathematical work—like figuring out how to structure a multistep word problem, model a solution, or puzzle out systems of equations. It’s harder for students to do those things when they’re simply trying to work through basic arithmetic.

Also, being able to automatically recall math facts seems to be especially important for multiplication: Students have fewer rapid backup strategies to lean on in multiplication if they haven’t stored the times tables in their long-term memory.

For single-digit addition, kids can get pretty fast using strategies like “counting on” from the highest number being added (i.e., 6+5 is counted as “7, 8, 9, 10, 11.”) This becomes impractical in multiplication.

“When you don’t know 6x8, and you’re doing an algebra problem with multiplication, you have to take the time and attention to add 8 and 8 and 8 and 8 and 8 and 8,” said Robert Siegler, a professor of psychology and education at Teachers College, Columbia University. “And, ultimately, you can’t regenerate these forever, as the math gets more complicated.”

Research finds that fluency in these facts is linked with progress in later grades; multiplication in particular is linked to success with fractions, a common tripping-up point for many young students.

How does math fact fluency develop?

There’s a generally accepted understanding that students’ knowledge of the addition number combinations begins by counting up all the digits in a problem and evolves to the more conceptual understanding that whole numbers can be “decomposed,” or broken down or recombined, in a variety of ways. All these stages appear to help students secure the answer to number combinations in their long-term memory.

Many students learn their addition facts without explicit instruction in these strategies, but those with math learning disabilities will need more intensive help. A variety of approaches based in helping kids hone those strategies are promising (see the next heading for details).

In general, there’s much less research on how math fact fluency develops in multiplication. The limited available evidence suggests that there are fewer universal strategies that transfer.

Students generally find 0 and 1 multiplication facts easy because they’re effectively rules-based: Any number times 0 is 0, and any number times 1 is itself. Students tend to skip-count the five’s time tables (5, 10, 15, 20, etc.) And students can use a decomposition strategy for nine’s by remembering their 10’s facts and quickly subtracting 9 from each. Other combinations seem to be somewhat harder to learn and harder for students to store in their long-term memory for instant retrieval.

That is partly why timed exercises are so associated with multiplication: It’s a way of assessing whether students are truly recalling from memory or whether they’re still using the backup strategies.

What are the best ways to develop fact fluency?

Frustratingly, there isn’t a clear answer here. Most of the research derives from interventions for students with math difficulties, and those studies use a mix of approaches, so it’s hard to identify which element makes the most difference.

In one study on addition facts, for example, researchers compared four different approaches: one was traditional drill work, in which students were shown simple addition and subtraction facts in a computer program and asked to restate them, with a visual aid embedded. In the others, students received explicit teaching beforehand in counting-up or decomposition strategies, in some cases with explicit help practicing and with work on word problems mixed in. All those approaches seemed to be helpful and were linked with improved fluency.

There is far less direct research on multiplication strategies and how they contribute to fact fluency. One study found in comparing two different approaches—one a traditional approach focused on memorization, and a second that integrated some strategies, including number-line work—both seemed to be helpful.

Are timed exercises bad for kids? Do they cause math anxiety?

Timed exercises are often used as a way to measure whether students have committed math facts to memory. They seem to be especially popular in multiplication, which is generally harder for students to master.

They’ve also gotten a bad rap over the years, as some educators question whether they could cause or exacerbate students’ math anxiety. But there’s no conclusive evidence on that point—mostly because there are no empirical studies that directly try to measure the question.

See Also

Illustration of a child in motion jumping easily across number block formations  and equations.
J.R. Bee for Education Week

Math anxiety shows up even in young students and can interfere with working memory. Anecdotally, both students and teachers recount feeling stressed when taking timed tests, but it’s less clear that the tests themselves trigger math anxiety or inaccuracy. One study found that removing a timed element improved accuracy on a basic arithmetic test, though another found no difference in accuracy for students with high anxiety on timed vs. untimed exercises.

The difficulty of the math—and whether it’s being graded—also seems to affect matters. A meta-analysis on math anxiety found that it did not interfere with students’ accuracy on simple math problems as much as when problems were more cognitively demanding or related to getting a grade. It also found that these links were higher at the secondary level, rather than at the elementary level.

Some educators point out that if done badly, timed exercises can exacerbate disparities among students.

“In too many classrooms, those ‘mad minute’ type things are creating a dichotomy,” warns Dylan Kane, a 7th grade math teacher in the Lake County district in Leadville, Colo. “The kids who know most of the facts and are developing that automaticity and long-term memory of the facts. [But] the ones who don’t know them are deriving them on their fingers and skip-counting. And if you derive it once and don’t achieve it from memory, you’re not developing it very well.”

A better approach, some teachers say, may be to individualize timed exercises so students are motivated to improve their own time, rather than being compared with one another.

Researchers interviewed: Robert Siegler, Columbia University; Nicole McNeil, University of Notre Dame; Daniel Ansari, University of Western Ontario.

Special thanks to Michael Pershan, whose bibliographies helped focus the questions for this explainer.

There is a vast amount of research on early arithmetic, particularly addition, but less research on how this knowledge should inform teaching, curriculum, and some of the specific problems of practice, including timed testing. Here are some critical studies that helped inform this explainer.

Ashcraft, M. H., and Kirk, E. P. (2001). “The relationships among working memory, math anxiety, and performance.” Journal of Experimental Psychology: General. 130, 224–237.
Ashcraft, M. H., & Krause, J. A. (2007). “Working memory, math performance, and math anxiety.” Psychonomic Bulletin & Review, 14, p. 243-248.
Fuchs, Lynn S., Sarah R. Powell, Pamela M. Seethaler, Douglas Fuchs, Carol L. Hamlett, Paul T. Cirino, and Jack M. Fletcher. (2010). “A Framework for Remediating Number Combination Deficits.” Exceptional Children. 76(2): p. 135–165.
Fuchs, L.S., Newman-Gonchar, R., Schumacher, R., Dougherty, B., Bucka, N., Karp, K.S., Woodward, J., Clarke, B., Jordan, N. C., Gersten, R., Jayanthi, M., Keating, B., and Morgan, S. (2021). “Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades.” (WWC 2021006). Washington, DC: National Center for Education Evaluation and Regional Assistance (NCEE), Institute of Education Sciences, U.S. Department of Education. Retrieved from http://whatworks.ed.gov/.
Fuchs, Lynn. S., Sarah R. Powell, Pamela M. Seethaler, Paul T. Cirino, Jack M. Fletcher, Douglas Fuchs, Carol L. Hamlett, and Rebecca O. Zumeta. “Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial.” (2009.) Journal of Educational Psychology, 101(3): 561–576.
Grays, Sharnita D., Katrina N. Rhymer, and Melissa D. Swartzmiller. “c.” (2017). Behavioral Education 26, p. 188-200.
Namkung, Jessica M., Peng Peng, and Xin Li. “The Relation Between Mathematics Anxiety and Mathematics Performance Among School-Aged Students: A Meta-Analysis.” (2019) Review of Educational Research June 2019, Vol. 89, No. 3, p. 459–496
National Mathematics Advisory Panel. “Foundations for Success: The Final Report of the National Mathematics Advisory Panel.” (2008). U.S. Department of Education: Washington, D.C.
National Mathematics Advisory Panel. “Reports of the Task Groups and Subcommittees.” (2008). U.S. Department of Education: Washington, D.C.
Pershan, Michael. “Questions and Answers About Addition Fact Research.” (2022) Retrieved from https://pershmail.substack.com/p/questions-and-answers-about-addition
Pershan, Micahel. “Questions and Answers about Multiplication fact Research.” (2023). Retrieved from https://pershmail.substack.com/p/questions-and-answers-about-multiplication
Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362.
Sherin, Bruce and Karen Fuson. (2005) “Multiplication Strategies and the Appropriation of Computational Resources.” Journal for Research in Mathematics Education. 36(4) p. 347-395
Van der Ven, S. H. G., Straatemeier, M., Jansen, B. R. J., Klinkenberg, S., & van der Maas, H. L. J. (2015). “Learning multiplication: An integrated analysis of the multiplication ability of primary school children and the difficulty of single digit and multidigit multiplication problems.” Learning and Individual Differences, 43, 48-62.
Woodward, John. Developing Automaticity with Multiplication Facts: Integrating Strategy Production With Timed Practice Drills. (2006). Learning Disability Quarterly 29, p. 269-283.

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