Those Times Tables

By Dr. Eugene Maier

I was in the stands at a middle-school basketball game. The mother of one of my grandson's teammates had noticed my "Math and The Mind's Eye" sweatshirt and, during a lull in the game, commented about it. We chatted briefly about math until the game resumed. In our conversation she mentioned that her son was having trouble in math. I gave her our web address and a couple of days later I got an e-mail from her elaborating on her son's difficulties.

Her son, a sixth-grader, she wrote, had never "mastered" the timed tests in elementary school. Recently he had scored in the seventies on a three-minute timed multiplication test and the teacher had announced that all those who didn't score 85% on the next one—in two weeks—would have to go to "homework club" until they did.

She told her son, in true Nike-town fashion, it was time to "just do it." So, they devoted Monday of a three-day weekend to the task and her son passed the tests. "It's great to be past that hurdle," she wrote. Meanwhile, her 4th grade daughter was struggling to up her score on a 5 minute, 100 problems test from 68% to the teacher-mandated 95%. Mom looked for the "mental stumbling blocks" getting in her daughter's way. Finding those—the nines—she "showed her some relationships" and daughter did fine.

Mom also recounted some of her own experiences. "With four kids," she wrote, "I just haven't devoted my life to this drilling." She was also reluctant to have her children experience what she went through in fifth grade to pass "those tests," the scores of which "were posted on the wall for everyone to know where every student stood at any point in time"; creating, she added with a bit of wryness, " another nurturing exposure to Math smile." She went on, "I've never been good at the pure 'memorization' game but succeeded with seeing or constructing relationships with material at hand as a way of latching it to the memory fibers of my brain."

I was reminded of my own family's stories about timed tests and multiplication facts. I remember a child in tears at the breakfast table because he got so nervous during timed tests on arithmetic facts he couldn't think, he said, even though he knew the answers.

I recall my youngest sister, after a game-playing session with her nephews, wishing she could do arithmetic as fast as they did and confessing that she had never learned to multiply. When I inquired what she meant, she told me that there were certain products she never could remember, like 8 x 7. She said she had to start with something she did remember, like 4 x 7 and count on by sevens until she got what she wanted. I told her she knew how to multiply; her system might take longer than her nephews but that was fine as long as it worked for her. Unfortunately, she finished school with the belief that she has very little math ability.

From what I gather, educators who insist that students have instant recall of the times tables believe that it's an essential basic skill, without which students will be hampered in their later mathematical development—especially when it comes to learning paper-and-pencil multiplication and division algorithms. But this confuses acquiring non-essential algorithmic skills with what is basic to all mathematics education: drawing out and developing the mathematical abilities that exist in every child, including the ability to devise one's own arithmetical procedures.

There is no question that being able to instantaneously recall arithmetical facts can be a convenience. However, one must weigh this convenience against the price paid in forcing rote memorization of facts and giving timed tests, especially when there are so many other ways of arriving at these facts, be it by counting, reading a chart, or punching a calculator. The principle costs are the negative emotions and beliefs that the struggle to memorize arouse, and the meaninglessness of rote learning.

As facts are stored in memory, so are the emotions evoked in acquiring them—and quite likely, from what is known about the brain, more vividly than the facts themselves. Also embedded in our psyche, are those erroneous beliefs about one's mathematical ability—such as my sister's—when too great an importance is placed on rapid recall of facts. Memorizing a page of multiplication facts is not something our brains are wired to do well, and for many of us requires considerable effort. It can be done in a verbatim fashion, like one memorizes a nonsense verse word by word. But, once in verbatim memory, it's also likely to carry about as much meaning as nonsense verse.

If one gives up the obsession with speed and the unwarranted emphasis on paper-and-pencil algorithms—if doing calculations quickly is the goal, use a calculator—and instead stress meaning and context, the multiplication facts will take care of themselves.

Most adults, not to mention school children, rattle off the sentence, "seven times eight equals fifty-six" with little attention to the meaning of the words they are saying. "Seven times eight" lends itself to several interpretations, depending on the model of multiplication one carries in one's head: some think of it as "seven eights," others as "seven taken eight times," still others as a 7 by 8 array. "Fifty-six" literally means "five tens and six." Saying "seven times eight is fifty-six" is simply saying, whatever one's interpretation of multiplication, that 7 x 8 can be rearranged to form 5 tens and 6.


multiplication model

Children shouldn't be expected to work on multiplication facts until they understand our numeration system is based on groupings of tens. (English-speaking students are at a disadvantage here. Other languages directly reflect the base ten nature of Arabic numerals. If, for example, our names for numbers were similar to the Chinese' names, instead of saying "twelve," we would say "one ten two"; "twenty-three" would be "two ten three"; "fifty-six" would be "five ten six." Studies have shown, at the age of four, the Chinese child, on the average, counts to 40, while the American counts to 15. When given some unit cubes and bars of ten and asked to use then to represent 25, Chinese children select two bars of ten and 5 units, while at the same age, most American children count out 25 units. Perhaps, when introducing counting to schoolchildren, we should use more literal names at first: "one ten and one, one ten and two, ...two tens, two tens and one, two tens and two" et cetera, and leave the standard "eleven, twelve, ...twenty, twenty-one," and so forth, till later.)

Once children understand the literal meaning of number names and how these names reflect the grouping-by-tens nature of our numeration system, using models based on their intuitive understanding of multiplication, they can develop their own times tables by converting products into groups of ten and recording the results. (Addition can be treated similarly, most folks can imagine in their minds how a stack of 8 blocks and one of 7 can be converted to stacks of 10 and 5 by moving 2 squares from the stack of seven to the stack of eight, that is, 8 + 7 is one ten and five, or 15).

In so doing, children will come to know the meaning behind the multiplication facts they are being asked to remember. They will have mental images, other than symbols, that convey these meanings. Games and other number activities in which number products are met in non-stressful settings will help them implant these facts in their memories. The recall may not always be instant, but, given the time to do so, they will have ways of recovering what's for the moment forgotten. And they won't be storing up strong emotions and negative messages that may not only block recall of a number fact but, as it has for my sister, lead to an aversion for all mathematical activity.