Back to: Module: Helping with Instruction

## Introduction

Memorizing “the times tables” is something most of us remember doing. Many schools—perhaps most—still insist that students memorize these math facts. And most people seem to think it makes sense.

But the National Council of Teachers of Mathematics—the national association for math teachers—has standards that *don’t* insist on memorization! And neither do other state and national standards. Here’s the relevant standard (from the math teachers’ group):

In grades 3-5 all students should develop fluency in … multiplying whole numbers.

*Fluency* in this case means easy quickness. Those who teach math in grades 3-5 need to aim for fluency. Fluency is more complex than a timed test of dozens of math facts. It’s not so easy to judge. But fluency is much, much more powerful. Read on.

With fluency, every student *does not* need to know instantly that 12 x 12 is 144 or that 7 x 9 is 63! But students who don’t know these “by heart” should be able to *find the answer very quickly*. For instance, a student who is unsure of 12 x 12, should (for instance) be able to think fast as follows: 6 x 12 is 72 and twice that is 144, so 12 x 12 = 144. That’s an example of fluency. So fluency is reasoning instead of memorization. As it turns out, reasoning is a whole lot more powerful than memorization. Fluency is a strong first step toward making sense of math.

Some students have actual problems with memory that make it impossible for them to memorize facts. Other students—confronted with timed quizzes (for instance, completing 50 facts in 90 seconds)—experience panic that gets in the way of successful memorization. And they never get to fluency because their teachers and parents panic that “they don’t know their math facts.”

Regardless of the challenges, though, all students in grades 3-5 can improve their current level of fluency with basic facts. And so can many older students. Various scaffolds can assist the learning of multiplication facts to the point of fluency.

## Scaffolding Ideas

Here are some ideas for scaffolding fluency with basic multiplication facts:

- Perhaps some students need to understand repeated addition (3 + 3 + 3 = 9 is the same as 3 x 3 =9) and “skip counting” (for instance: 2, 4, 6, 8; 3, 6, 9, 12; 5, 10, 15, 20).
- It helps for students to know how multiplication and division are related—for instance, the way 2 x 3 = 6 and 6 / 2 = 3 are related. To get to that understanding they need lots of examples. It’s more about ideas than facts, though the ideas connect to the facts.
- Use one or more scaffolds that are recommended on the Internet or in other resources. Check with your instructional team for related plans and ideas. For lots of examples, type “YouTube skip counting” into your browser. You’ll get lots of skip-counting songs.
- The typical scaffold for nearly all students is (you guessed it): flash cards. Use them if helpful but plan a strategy. For instance, focus on a few facts that seem difficult; or reinforce the facts like 3 x 8, and 8 x 3; 4 x 7 and 7 x 4…and so forth).
- In addition to regular flashcards, some flash cards list the three related numbers: not just 2 x 4, but also 8—the “answer.” Some sets of cards give both the multiplication (2 x 4) and division (8 ÷ 4). Such cards allow a lot more variety in how they are used.
- Explain the pattern for 9 x 1, 9 x 2, 9 x 3, etc.: →
*first digit**0***9***1***8***2***7***3***6***4***5***5***4***6***3***7***2***8***1 ← second digit.**The pattern doesn’t directly build fluency, but appreciation of patterns is a key part of math! And the more students encounter displays and applications of the math facts, the more likely they are to recall them. - This written pattern for the nine times table has a version using the 10 fingers. The student will put both hands, fingers spread, on the table. The student will pick up one finger at a time, proceeding left to right. First, the left pinky is lifted. What remains are 9 fingers: 9 x 1 = 9. Second, the left ring finger comes up, and the pinky goes down. To the left of the left ring finger, is 1 finger (the pinky) and to the right are 8 fingers: 2 x 9 =18. The left middle finger comes up (all the others down) and we have 2 to its left, and 7 to its right: 3 x 9 =27…and so on.
*This is a whole lot easier to do than to read about! Probably someone you know can do it already and can show you.*

Because memorizing the times tables is so common a task, and because so many students have trouble with it, the web offers lots of scaffolding. You can work with your teacher (or team) to assemble a toolkit from these sources. A useful flash-card style drill appears at **https://www.mathsisfun.com/numbers/math-trainer-multiply.html**.

If suitable, work with the students in short sessions. At this site you can vary session length (from 10 seconds to 10 minutes) and you can vary the problem cutoff time (time allowed before the program treats non-response as an error) from 2 seconds to 8 seconds.