Anyone who wishes to be mathematically proficient needs to “just know” simple arithmetic — 4 + 9 = 13, for example, or 3 × 7 = 21. Students who do not learn such facts tend to have difficulties learning more advanced mathematics.

Anyone who wishes to be mathematically proficient needs to “just know” simple arithmetic — 4 + 9 = 13, for example, or 3 × 7 = 21. Students who do not learn such facts tend to have difficulties learning more advanced mathematics.

The reason for this is not that children cannot solve, for example, a multidigit arithmetic problem without knowing these basic facts, but rather that it is far more difficult to do so. When students achieve “automaticity” easy access to these facts—they free up mental resources that they can use to think about other aspects of the math, such as place value, complexities of renaming, and, of course, the problem they were solving in the first place. In general, they can give more attention to more complex matters. Without automaticity, students have a hard time learning multidigit calculations—both written and mental. They also face a roadblock in learning multiplication and division and then in understanding and later calculating with fractions.

Many teachers assume that students must “just memorize” their basic facts by rote. To them, that implies timed practices, worksheets, and flash cards. If you have to memorize facts, you need to practice each one separately, again and again, quickly. They often believe, “That’s how we learned them, and that worked well.”

The problem is, worldwide research shows that the way most people in the United States think about arithmetic facts and children’s learning of them, and the language they use, may harm more than help. To see why, we have to step back and rethink each of these assumptions.

There are three primary misconceptions embedded in these assumptions:

## Misconception 1: Arithmetic Facts Are Disconnected Items

To many people, the term fact often means a piece of disconnected information.But almost nothing in mathematics is disconnected. Knowing an arithmetic fact well—that is, fluently—means far more than knowing a simple, isolated piece of information.

## Misconception 2: Learning = Memorization

Research suggests that producing basic facts is not just a simple process of “rotely memorizing” each fact and then mentally “looking each up” in our memories. Retrieval of arithmetic facts is part of a complex process.

## Misconception 3: Students Learn Through Rote Practice

Memorization without understanding, drill without developing concepts and strategies, are not effective ways to teach or learn arithmetic facts, much less the edifice that is mathematics.

An important goal of early mathematics is students’ flexible, fluent, and accurate knowledge of arithmetic facts. Learning these facts is not about rote memorization. Seeing and using patterns, and building relationships, can free children’s cognitive resources to be used in other tasks. Number fact instruction that focuses on encouraging children to look for patterns and relations can generalize to problem-solving situations and can free attention and effort for other tasks.

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**Linda Ruiz Davenport** is the Director of K–12 Mathematics for Boston Public Schools and supports mathematics teaching and learning district-wide.

Follow her on Twitter @LindaD_BPSMath

**Connie S. Henry** is an Assistant Director of K–12 Mathematics for Boston Public Schools. She has taught and coached math for many years.

Follow her on Twitter **@ConnieS_Henry**

**Douglas H. Clements **is the Kennedy Endowed Chair in Early Childhood Learning and Distinguished University Professor at the University of Denver.

Follow him on Twitter @DHClements

**Julie Sarama **is the Kennedy Endowed Chair in Innovative Technologies and Distinguished University Professor at the University of Denver.** **

Follow her on Twitter @JulieSarama