How to Teach Multiplication Table Factors through Nine

One of the most irritating phenomena that I confront in teaching math to elementary, middle school, and high school students enrolled in the New Salem Educational Initiative is the failure of teachers in the Minneapolis Public Schools to have imparted knowledge of basic multiplication table factors, those including the numbers 0 through 9. In my observation, this unfortunate circumstance results from two causes: 1) education professors in general, and math education professors in particular, who see learning basic multiplication factors as rote memorization of the kind that they abhor; and 2) classroom teachers who typically suffer from having matriculated at university departments of education and are all too ready to retreat into lazy pedagogy that exalts the calculator.

All of this is a great shame, because full memorization of multiplication factors through 9 is necessary efficiently to find common denominators, reduce fractions, calculate decimals and percentages, solve proportions, find ratios and probability, solve algebraic equations, and perform a variety of mathematical operations in geometry, trigonometry, and calculus. And the shame is further intensified when one discovers that young children love to learn multiplication table factors when properly taught.

So consider this approach:

I always start with "0" and "1" as multiplied by the factors through 9, even when students have long since learned that "0" times any number is "0," and "1" times any number is that latter number. I rarely go through the whole group of factors through 9, but I do give students a chance to do one or two of each of these, upon which I say, "Wow, look at that--- you already know 20 of these cards. We just have to look at the ones involving 2 through 9 now."

So then I move on to the "2's," which tend to go very fast. I don't have a problem with students using their fingers to count by "2's," although after all factors "0" through "9" have been memorized or gained quick responses, I do begin to insist that students move so fast as to make counting on the fingers impossible. By this time, the "2's" have in any case typically been memorized.

I then skip to the "5's," which involve the counting exercise that comes easily to most students, so that I can then say, "See there, we don't have that far to go now. You already have mastered the table with factors using '0,' '1,' '2,' and '5.'" The student's confidence builds and what might have once seemed a daunting task now seems very manageable.

Then I back up to the "3's" and "4's." These are not quite so easily countable as the "2's" and "5's," but most students do not have much trouble first counting by
"3's" and then performing the corresponding multiplication tasks. And "4's" are still low enough that this holds true for this set, as well.

I have by this time also introduced my students to the pneumonic potential of rhyme by pointing out that "5 times 5 is 25" (vocal stress on the 5) and "6 times 4 is 24" (vocal stress on the 4). Particularly in this age of hip-hop, students love this sort of word play.

After the "3's" and "4's," I skip ahead to the "9's," at which point I teach them my version of the "9 trick." To my astonishment, few students are taught this easy pneumonic device in today's classrooms. I get "0" and "1" as multiplied by "9" out of the way and then focus on the trick, which is simply that the first digit in the product is the number sequentially just before the number by which "9" is multiplied, then the second number is that first digit subtracted from "9." So in
6 times 9, the first digit is "5" (just before "6"), and the second digit is "4"
("5"subtracted from "9") to yield "54." Similarly, the trick reveals the first digit in the product of 8 and 9 to be the number right before "8" and the second digit to result from subtracting that "7" from "9" to get "2" and thus yields a product of "72." The trick works throughout the numbers multiplied by "9" from "0" through "9."

So all of this usually goes quite quickly, allowing us to focus on the three numbers in the "0" through "9" sequence that tend to come most slowly for students: "6,"
"7," and "8." But by this time, there really are just a few individual sets of factors that the student has not seen in earlier sets of factors, because she or he has already seen these numbers multiplied by "0," "1," "2," "3," "4," "5," and "9." And the "6's" come rather quickly after I point out that in addition to the rhyming value of "6" times "4" is "24," that same sort of rhyming value exists in "6" times "6" is "36."

Students can then settle in to master the remaining "7's" and "8's," focusing especially on the most difficult five pairings for most students: "6" times
"7"; "7" times "7"; "7" times "8"; "6" times "8"; and "8" times "8." But "6 times 8 is 48" features another rhyme, and many students remember that "7" times
"8" is "56" after I tell them that this is the hardest one in the deck; interestingly, this observation (based on my experience in teaching multiplication over the years) helps students remember the correct product when they are presented with "7" times
"8."

I have found that this method incrementally builds student mastery and confidence, enabling young people to master the multiplication table with factors "0" through
"9" quickly. All of my Grade 2 students master this fundamental skill and knowledge set, and I have a few Grade 1 students who have already mastered the multiplcation table through "9" by the end of that academic year.

My students to a person love to learn multiplication in the way described above, and in the knowledge of this kind of response deserve to have this skill and knowledge set imparted to them as early in their K-12 years as possible.

An Overview of the Successful Approach of the New Salem Educational Initiative

In previous series on "The Importance of No Child Left Behind" and "Teaching the Child from a Poor and Dysfunctional Family" I have emphasized the importance of public education in a democracy, systemic failings of our current public education system upon which No Child Left Behind has shone a bright light, and important tenets for teaching children whom we need to reach if we are to transform education into the agent for genuine democracy that it should be.

In this article, as I resume posts of greater regularity (if not quite the torrid pace that I was setting for awhile), I offer in brief form the key features of the New Salem Educational Initiative that make it an agent for highly effective education and democratic transformation. I will in the course of time (although not necessarily immediately following upon this article) discuss all of these features more fully.

The New Salem Educational Initiative proceeds on the basis of tenets that I have previously emphasized as defining an "excellent education":

1) an excellent teacher;

2) a strong liberal arts curriculum.

The great bulk of students enrolled in the New Salem Educational Initiative participate in the small-group format for which I am the only teacher. I cordially invite anyone reading this article to come and see me in action as I teach my students in groups of one to five at New Salem Missionary Baptist Church at 2519 Lyndale Avenue North in Minneapolis. I have 17 groups whom I serve after school and on weekends, seven days a week. The students range from Grade K students to Grade
12. The emphasis is on getting students who have been functioning well below level of school enrollment up to grade level in math and reading as quickly as possible. When this is accomplished, I put my students on a college preparatory track in which they engage with challenging material from across the liberal arts curriculum with vocabulary pertinent to a variety of content areas, including fiction and poetry; history; economics; natural science; and the fine arts.

In the course of a 40-year career I have developed highly effective approaches to teaching the mathematical sequence of the K-8 years, including basic operations, fractions, decimals, percents, proportions, probability, ratios; and the Algebra I-Geometry-Algebra II-Trigonometry-Calculus sequence of the 9-12 years. Explicit vocabulary acquisition and close critical questioning regarding content undergird my approach to teaching students to be effective readers across a breadth of subject area material.

I love my students and they know it. I love teaching and they observe it in every session. I know exactly what I want my students to accomplish each and every session, and this is manifested in the logically sequenced progress that my students can see from week to week. We approach what we do with joy. We embrace education as the stuff of life. We spend virtually all of our time in straight-ahead approaches that waste no time. The fun that we have comes from the joy of acquiring new knowledge and skill sets, and from the happy and jocular manner that I bring to the task at hand, not from time-wasting and trivial games.

Willie Mays once said, "I am the best baseball player that I have ever seen."

And I would say, "I am the best teacher that I have ever seen."

There is no substitute for effective pedagogy. There is no surrogate for academic training to the level of a Ph.D. I am thinking all the time, every day, about how to impart the skill and knowledge that I bring to the art of teaching so that the model that I have developed can be generally implemented.

In the delivery of an excellent education, an excellent teacher and a strong liberal arts curriculum are paramount.

More briefly on this particular day, the New Salem Educational Initiative also thrives on the following programmatic features:

>>> transportation that I provide personally, going wherever I need to go, even for those students whose residences shift often and may include a multiplicity of possiblities (stayng variously with mom, dad, grandma, grandpa or close family friends on any given week);

>>> detailed knowledge of family situations and very close relationships that I develop with family members in every one of my students' households.

>>> genuine concern for the students and their families, with offers to connect them with resources needed by families of the inner city as they strive to meet the many challenges of their lives.


These are the major features through which students in the New Salem Educational Initiative rise to grade level performance and embark on a college preparatory track. The first features of excellent teacher and strong liberal arts curriculum are necessary for an excellent education. The other programmatic features--- flexible transportation; knowledge of family situations and close human relationships with the members of those families; and genuine concern combined with effective action in helping people meet their life challenges--- are those that recognize the additional responsibility of any educational system that seeks to meet the needs of all of its students and thus advance education as the agent of democratic transformation that it should be.