Having spent a year as a Mathematics Response to Intervention Teacher in a low-performing school, I can attest to the difficulty many elementary school students experience when learning, or attempting to learn, the basic multiplication facts. This difficulty arises usually arises from one common, but ineffective, practice – rote memorization.

An argument I have often heard made by elementary school teachers who employ rote memorization in their classroom is, “learning the multiplication facts is like getting better at sports, practice makes perfect.” I introduce these teachers to Vince Lombardi, the legendary football-coach and wordsmith. Vince Lombardi famously said, “Practice doesn’t make perfect, perfect practice makes perfect.”

Rote memorization of the basic multiplication facts is practice. But, it is necessarily imperfect practice. Imagine a student sitting across from a teacher or parent equipped with a set of multiplication flash cards. The student’s interlocutor presents to the student a card which reads *6 x 7*. The student’s job is to somehow find the product of *6* and *7* in her mind and produce it. The product of *6* and *7* is quite likely nowhere to be found in the poor student’s mind. Otherwise, the adult would not be asking the student to produce the product. The student is reduced to guessing the correct product.

Suppose, however, the student had some method of finding the product of *6* and *7* even though that fact was not part of her memory – some method of finding the product reliably, every time she was asked to find it. Each time the student found the product *42*, the fact that *6 x 7 = 42* would be reinforced in her mind. Eventually, the fact *6 x 7 = 42* would become part of the student’s memory, and the student will have learned this multiplication fact.

The method I used as an RTI Teacher relied on ‘skip-counting’ – beginning with some whole number, state that number and then every multiple of that number. For example, skip-counting by *3* looks like;

*3*, *6*, *9*, *12*, *15*, *18*, *21*, *24*, *27*, *30*, *33*, *36*, *39*, *42*, *45*, …

Then, the product of *3* and *7*, *3 x 7*, is found by skip-counting by three until the *7*th multiple of *3*;

*3*, *6*, *9*, *12*, *15*, *18*, *21*

Each time the student voices a multiple of *3*, she counts the multiple. I preferred to teach my students to keep count by pressing a finger onto their desk with each multiple counted. I call this the student’s ‘bookkeeping’ method.

Some of the skip-count patterns are more easily learned than others, and when teaching this strategy to my students, I employed skip-count patterns for *2*, *3*, *5*, and *10*. Once a student masters the *2* skip-count pattern, and then the *3* skip-count pattern, I would teach the ‘double skip-count’ heuristic.

The *2*, *5*, and *10* skip-count patterns come quite naturally to most elementary school age students. It is necessary, however, to teach the *2* skip-count pattern to *40* rather than *20* (*20 = 2 x 10*). This facilitates using the *2* skip-count pattern to effect the *4* skip-count pattern.

The *4* skip-count pattern is nothing more than the 2 skip-count pattern with each second multiple voiced. It look like;

*2*, **4**, *6*, **8**, *10*, **12**, *14*, **16**, *18*, **20**, *22*, **24**, *26*, **28**, *30*, **32**, *34*, **36**, *38*, **40**

Those multiples that are in bold-face above are the multiples of *4*. They are every second multiple of *2*. It is quite easy to teach a student to employ their skip-count bookkeeping method with each second multiple of *2* rather than each multiple of *2*.

The *6* skip-count pattern is the *3* double skip-count pattern, similar to the relationship between *2* and *4* skip-count patterns. It is employed in a manner analogous to the *4* skip-count pattern, but using the *3* skip-count pattern as the underlying pattern. Just as it is necessary to teach the *2* skip-count pattern to *40* rather than *20*, it is necessary to teach the *3* skip-count pattern to *60* rather than *30*. (*6 x 10 = 60*)

In my teaching practice, I teach the *2*, *3*, *4*, *5*, *6*, and *10* skip-count patterns. (In fact, many students master the *2*, *5*, and *10* skip-count patterns with very little instruction, and the *4* skip-count pattern with only slightly more instruction.) The *3*, and consequently the *6*, skip-count pattern often takes quite a bit of effort on the part of students to master.

While there are certainly skip-count patterns for *6*, *7*, and *9*, their difficulty to master make them less than useful. For the *9* times facts, I employ a different heuristic (the topic of a later post), and I do not teach any method for finding the *6* and *7* times facts. It is unnecessary.

Multiplication of the whole numbers is commutative. That is, for any whole numbers *n* and *m*, *n x m = m x n*. It makes no difference in which order the factors *n* and *m* are written. The consequence of this property of multiplication of the whole numbers is that any time *7* or *8* are exactly one of the factors of a multiplication fact, the fact can be commuted, leaving as the result the possibility of employing some skip-count pattern to find the product. (This does not apply to *7 x 9*, or *8 x 9*.) Then, only the products *7 x 7*, and *7 x 8*, *8 x 7* have no heuristic to assist in finding the product. Essentially, a student need only ‘memorize’ two multiplication facts in order to master the basic multiplication facts from *0 x 0* to *10 x 10*.

Euro (nee Mark)