A Research-Based Approach to Math Fact Fluency (That Also Promotes A Love of Mathematics!)

Leveraging Learning Science in Math Instruction

PreK-12 math teachers often rely upon worksheets and timed drills to develop math fact fluency in students. While research on the effectiveness of worksheets and timed drills varies, some research indicates that such approaches to developing fact fluency — particularly drilling for speed prematurely — can negatively affect students’ feelings toward math.

While drill, both timed and untimed, can be appropriate in certain circumstances, other approaches can foster more positive attitudes towards mathematics while also developing math fact fluency. Learning science research can help us identify those strategies and integrate them into math instruction. By harnessing what we know about how learning happens, we can design learning experiences that are not only effective in promoting positive outcomes, but also enjoyable for students.

First, it’s important to understand the definition and characteristics of math fact fluency. How do we know if a student is truly fluent in math facts? Gina Kling, mathematics education researcher and author, points us to two core aspects of fluency:

  • Fluency requires students to notice relationships and use strategies. It’s about the speed at which a student produces an answer, but it’s also about the methods they used to produce that answer and the flexibility they have in their thinking.
  • However, fluency should not require rote memorization. Instead, students should either have a fact meaningfully memorized or be able to produce that fact through a highly efficient, automatically executed, strategy. A student has mastered a math fact if they can produce an answer within 3 seconds, through either recall or a highly efficient strategy application (Kling & Bay-Williams 2015).

According to Kling, fluency is developed when students have the opportunity to deliberately and explicitly move through three developmental phases by building reasoning strategies. In general, children begin solving math facts through counting (Phase 1), progress to using reasoning strategies to derive unknown facts (Phase 2), and finally, develop mastery with their facts (Phase 3). If students simply memorize math facts as rote facts (in other words, skip Phase 2), they might fail to develop important conceptual understandings, which puts them at a disadvantage when attempting to engage in more advanced math work (Kling & Bay-Williams 2015). “Fluency allows us to retrieve and use math facts with a certain automaticity, having internalized them from prior experience,” according to Learning Scientist Claire Cook. “It’s not about memorization per se — just as a master chef doesn’t go about selecting the right ingredients in the right amounts because he’s memorized recipes, but rather because he knows what he’s doing at that level without thinking about it too hard or too explicitly.”

Premature drill for speed and other practices that rely heavily on rote memorization should be replaced or supplemented with practices that promote reasoning — and, in many cases, replace math anxiety with math positivity. Learning science research has enabled us to identify a variety of practices that encourage students to develop conceptual understandings, reasoning skills, and fact fluency — all in a math-positive learning environment. We have deliberately integrated such practices into Everyday Mathematics, a program built on decades of research about how learning happens. Here are a few of the research-based practices Everyday Mathematics employs to develop fact fluency and foster math positivity:

Interleaving

We know that developing fact fluency requires practice. But what kind of practice is most effective? While timed worksheets with many similar facts can sometimes foster math anxiety, particularly when they are used too early in the learning process or when they are tied to high-stakes outcomes, certain kinds of worksheets do have their place in math instruction — particularly when they employ a technique called interleaving. Interleaving essentially involves practicing skills in a rotating or variable order rather than starting with one skill and mastering it completely before moving on to another. For example, a math facts worksheet or activity might include both addition and subtraction problems, including some set within story contexts. Or a facts game might include practice with all four operations (addition, subtraction, multiplication, and division). Research shows that this kind of practice is more effective for mastering skills in a variety of domains (sports, even music), including math. For more on interleaving and the research mentioned above, see:

Games

Returning to the phases of fact fluency development: it’s critical that students do not skip phases that give them the opportunity to develop reasoning skills. Explicitly teaching reasoning strategies doesn’t mean simply teaching a specific strategy and then asking students to use it to solve a problem — in that case, students may use the strategy in a rote way, without understanding why it works. Instead, explicitly teaching reasoning should involve encouraging students to develop and share efficient strategies they can use and allowing them to practice. Games provide students with the opportunity to practice these reasoning strategies and move through the developmental stages of mastering math facts at their own pace (Bay-Williams & Kling 2014). And because games are fun, students play them willingly and even on their own, thus getting much more practice than they would from worksheets.

Formative Assessment

Finally, consider how formative assessment strategies contribute to math positivity and support instructional iteration. Timed tests can limit an educator’s ability to understand what strategies a student uses to solve a problem. To conduct formative assessment in a way that advances math fluency without promoting math anxiety, Gina Kling recommends a few practices:

  • Interviews, where teachers have open discussions with students about the strategies they use to solve a problem and utilize tools to chart student progress
  • Observations, where teachers might watch students employing strategies during math game play and record their observations on structured tools
  • Journaling, where students have the opportunity to create a written record of their reasoning

According to Kling, quizzes can have their place in a math-positive, productive formative assessment strategy — as long as they focus on efficiency and strategy use (Kling & Bay-Williams 2014).

For a deeper understanding of the way learning science serves as a foundational research basis for Everyday Mathematics, see:

References

Bay-Williams, Jennifer M., and Gina Kling. “Enriching Addition and Subtraction Fact Mastery through Games.” Teaching Children Mathematics, vol. 21, no. 4, 2014, p. 238., doi:10.5951/teacchilmath.21.4.0238.

Bay-Williams, Jennifer M., and Gina Kling. “Math Fact Fluency: 60+ Games and Assessment Tools to Support Learning and Retention.” ASCD and NCTM: 2019.

Kling, Gina, and Jennifer M. Bay-Williams. “Assessing Basic Fact Fluency.” Teaching Children Mathematics, vol. 20, no. 8, Apr. 2014, p. 488., doi:10.5951/teacchilmath.20.8.0488.

Kling, Gina, and Jennifer M. Bay-Williams. “Three Steps to Mastering Multiplication Facts.” Teaching Children Mathematics, vol. 21, no. 9, 2015, p. 548., doi:10.5951/teacchilmath.21.9.0548.

Pan, Steven C. “The Interleaving Effect: Mixing It Up Boosts Learning.” Scientific American, Springer Nature America, Inc., 4 Aug. 2015, www.scientificamerican.com/article/the-interleaving-effect-mixing-it-up-boosts-learning/.