Tool for the Field #1

Introduction

Memorizing “the multiplication tables” (or “times tables”) is something almost all of us remember. Many schools—perhaps most—insist that students memorize these math facts. However, the National Council of Teachers of Mathematics—the national association for math teachers—has standards that don’t insist on memorization! And neither does the more recent Common Core. Here’s the relevant NCTM standard:

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. Assessing flawless memorization is easier, of course: either students get it, or they don’t. Fluency is more complex, not so easy to judge, and much more powerful. Read on.

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 sufficiently fluent that they can find the answer very quickly. For instance, a student who is unsure of 12 x 12, should (for instance) be able to say instantly that 6 x 12 is 72, and be able instantly to multiply 72 by 2 to get 144. That would be sufficient fluency.

Some students, of course, have memory problems that prevent them from memorizing facts, period. Other students—confronted with timed quizzes (for instance, completing 50 facts in 90 seconds)—enter a panicky mode that is counterproductive—it gets in the way of successful memorization.

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.

Scaffolding Ideas

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

  • Prior to working on fluency, perhaps a student needs 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). And it helps for them to know that 2 x 3 = 6 and that 6 / 2 = 3 (that is, to know lots of examples that show how multiplication and division are related).
  • Use one or more scaffolds that are recommended on the Internet or in other resources available to your instructional team. For an Australian example, see: http://www.education.vic.gov.au/school/teachers/teachingresources/ discipline/maths/continuum/pages/skipcount20.aspx). Check with your instructional team for related ideas and appropriate plans for particular students.
  • The typical scaffold for nearly all students is (you guessed it): flash cards. Use them if helpful, but plan a strategy (for instance, concentrate on a few facts that seem difficult; or reinforce the facts that are redundant—like 3 x 8, 8 x 3; 4 x 7; 7 x 4…and so forth).
  • In addition to regular flashcards, some flash cards list the three related numbers: not just 2 x 3, but also 6—the ‘answer’—and sometimes both the multiplication and the division symbol! The use of such cards is much more flexible than the simple form with a “problem” on one side and an “answer” on the other side.
  • Explain the pattern for 9 x 1, 9 x 2, 9 x 3, etc.:  → first digit 09 18 27 36 45 54 63 72 81 ← second digit. The pattern doesn’t directly build fluency, but appreciation of patterns is a key part of math! And the more students have a chance to observe displays and applications of the multiplication facts, the more likely they are to recall them.
  • If appropriate for the student’s “zone of proximal development,” the student can recreate this pattern 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 much easier to do than to read about!

Because “the times tables” are such a prominent feature of school learning, the web has a great many scaffolds available. You can work with your instructional team (or supervising teacher) to assemble quite a toolkit from which to scaffold fluency in basic multiplication facts. For instance, 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.