A few weeks ago, I mentioned in a column that "real" learning - that is, understanding - does not come from rote memorization. Almost immediately after that column appeared, I began to receive comments and questions from parents and teachers alike about the whole concept of memorization as a teaching and learning technique. Is it always a bad idea? Aren't there some things that children should commit to memory? And what's wrong with memorizing arithmetic combinations and spelling rules, anyway?

Good questions. In truth, there is nothing wrong with rote memorization ("rote" meaning mechanical repetition without much attention to comprehension). It is, in fact, an extremely valuable tool for certain purposes, but it is the nature of those purposes that makes all the difference.When I was in the eighth grade, everyone in my English class had to memorize the 22 helping verbs: is, are, was, were, am, be, been, do, does, did, can, could, shall, should, will, would, have been, has been, had been, may, might, and must. I can recite this list in my sleep today, but I have still not found a single, worthwhile use for it in the 33 years since I learned it. The purpose of this particular memory exercise is still unclear to me, and it is the type of classroom drill that parents have every right to question.

Now let's take another example. The purpose of using a multiplication table to memorize number combinations is at least twofold. First, such knowledge affords the child immediate information that can then be used to achieve some larger understanding. Oh sure, you could figure out WHY six groups of seven make one group of 42 on your way to answering 3897 x 416\ , but it isn't necessary and it gets in the way of understanding the problem at hand. Once you know the relationships between six, seven and 42, your memory should permit you to put those relationships to use quickly, accurately and with the confidence that those memorized facts are founded upon a genuine understanding.

Second, the multiplication table can help children discover those number relationships for themselves, and that type of discovery is what real learning is all about. I have known children who just never "saw" the sixes and sevens in 42 until they followed the table's columns that increased by sixes and rows that increased by sevens until they met at 42. Some children think of multiplication and division as being two distinct and separate operations until they have the almost-mystical realization that the multiplication table is actually a division table, too!

To help your children make a multiplication table of their own, take a sheet of sturdy paper or posterboard at least a foot square (13-by-13 inches is handy), and using a ruler, draw a grid with 13 equal spaces along each side. Leave the box at the upper left blank, and write the numbers 1-12 in the boxes across the top row and down the far left-hand column. Your child can then fill in the remaining squares so that the intersection of each row and column shows the product of the numbers written at the top and left.

Why use 1 through 12 instead of 1 through 10? Because 12s pop up so frequently (inches in a foot, months in a year, etc.) that memorizing these combinations can have frequent, practical applications.

Beyond the 12s lies another area of worthwhile memorization for older students and parents, too. Square numbers (8 x 8, 15 x 15, etc.) abound in daily life, and so committing to memory all the squares through 25 is a reasonable and purposeful exercise for all of us.