Both memorization and strategy matter in math fluency

If you’ve ever watched a child laboriously count on their fingers to solve 3 + 4, you know arithmetic fluency doesn’t come naturally. But the big question is: How do kids best learn math facts? Should they drill times tables until they’re automatic, or should they play games and explore number patterns to build “number sense”?

This debate has raged in education circles for decades. On one side, advocates of memorization insist that kids need quick recall of facts like 9 × 7 = 63. On the other side, reformers argue that drills create anxiety and shut down deeper thinking, urging instead for strategy-based teaching.

A report published in Psychological Science in the Public Interest by Nicole McNeil (University of Notre Dame), Nancy Jordan (University of Delaware), Alexandria Viegut (University of Wisconsin–Eau Claire), and Daniel Ansari (Western University, Canada) utilizes a new approach to answer this question. Instead of picking a winner, their article, What the Science of Learning Teaches Us About Arithmetic Fluency, shows that both camps are partly right and partly wrong.

The authors argue that memorization and strategy aren’t rivals at all. They’re partners. And when used together in the right way, they create the conditions kids need to become truly fluent in math.

What do we mean by “arithmetic fluency”?

Think of arithmetic fluency as being able to answer simple math problems, like 8 + 7 or 6 × 9, quickly, accurately, and almost without thinking. It’s not just memorization. It’s also the ability to flexibly use shortcuts and patterns: realizing, for instance, that 7 + 8 is just 7 + 7 + 1, or that if you know 6 × 7, then 7 × 7 is just one more group of seven.

McNeil and her colleagues stress that fluency is best described as “fluency with understanding.” It’s not about parroting answers, it’s about building a network of number relationships in long-term memory that frees up mental space for harder problems.

Some people argue that with calculators everywhere, why bother memorizing math facts at all? After all, your phone can spit out 8 × 7 faster than you can blink.

But the evidence says otherwise. Studies consistently show that students who can effortlessly recall basic math facts do better not just in math class, but in science and engineering too. One brain-imaging study found that high school students who relied on calculation for basic problems scored lower on the PSAT math section, while those who retrieved answers automatically scored higher.

In other words: fluency isn’t about bragging that you know your times tables. It’s about having the cognitive bandwidth to focus on deeper concepts like algebra, geometry, or physics, without getting bogged down in basic calculations.

And there are even more benefits. Early arithmetic skills predict long-term success in school and even socioeconomic outcomes decades later. One longitudinal study from the UK showed that math competence at age seven was a better predictor of adult income than reading ability. That’s how powerful arithmetic fluency can be.

So how do kids get from finger-counting to effortless recall? Cognitive science shows it’s a dance between working memory (the brain’s scratchpad) and long-term memory (the storehouse of knowledge).

At first, children solve problems explicitly and laboriously, counting every finger for 3 + 4. With practice, these steps get chunked into larger units, until eventually the answer “7” just pops into mind. That’s automaticity, the brain’s way of saving energy for more complex thinking.

But there’s a second, less obvious path: turning implicit knowledge into explicit understanding. A child may notice patterns without being able to explain them at first, then later articulate rules like “it doesn’t matter which order you add numbers.” These two processes, automation and reflection, work together to deepen both speed and understanding.

What works in practice?

The report goes beyond theory and offers concrete recommendations for teachers and parents. Here are some highlights. First, start early with number sense. Even toddlers show an intuitive grasp of “more” and “less.” Early play with counting, comparing, and grouping objects lays the groundwork for later fluency. Key milestones include understanding cardinality, that the last number you say when counting tells you how many objects are in the set. Second, mix memorization with meaning. Flashcards aren’t evil, but they work best when paired with strategy. For example, once a child can solve 5 + 5, they can build on it to solve 5 + 6. The goal is not rote recall in isolation, but recall supported by understanding. Third, use retrieval practice wisely. Quizzing students on math facts is one of the most effective ways to strengthen memory. But timing matters: it should come after accuracy is established, not before. That way, practice builds confidence instead of frustration. Four timed practice isn’t the enemy, if done right. The dreaded “timed test” has given speed practice a bad reputation. But the researchers argue that short bursts of time-limited retrieval (once accuracy is solid) can actually boost learning. The key is to distinguish between low-stakes practice and high-stakes testing. Finally, encourage reflection and discussion. Kids need time to compare strategies, talk through solutions, and reflect. This metacognitive work, thinking about thinking, cements flexible understanding.

So, the old tug-of-war between memorization and strategy is a false choice. Kids need both. Memorization provides efficiency. Strategies provide flexibility. Together, they fuel the kind of fluency that supports lifelong learning. The authors urge schools to integrate developmental cognitive science into teacher training, so educators understand not just what to teach, but why certain methods work. They also call for public awareness campaigns about early math milestones, so parents know that recognizing “how many” is just as important as learning to count to 20. Ultimately, arithmetic fluency is a hidden lever for opportunity. It shapes how easily children progress through school, how confidently they tackle advanced problem-solving, and even the career paths open to them later in life.

If you want to learn more, read the original article titled "What the Science of Learning Teaches Us About Arithmetic Fluency" on Psychological Science in the Public Interest at http://dx.doi.org/10.1177/15291006241287726.