The job of teaching math is not the same as it was when I went to school. Shortly after I graduated elementary school back in the 1960s the new math was invented, and in the new millennium we have "New-New Math" (which some have renamed "Know No Math").
By far, the most significant innovation to come along in elementary school math is the widespread use of calculators. Some people, in fact, have advocated eliminating the teaching of arithmetic altogether. They claim that calculators have made knowing how to do arithmetic an archaic skill. You wouldn't walk thirty miles if you could drive, or drive 4,000 miles if you could fly. Why add numbers on paper, if you can use a calculator?
Harm or Help?
Adding machines have been around almost forever, especially if you count abacuses. Scientific calculators have been around since just after I was an undergrad, thirty plus years ago, when those nifty devices replaced the slide-rules on engineering students' belts. They have had decades to prove their worth as educational tools.
What are the results?
For those who learn how to do the arithmetic first, and then learn to use a calculator, we have the testimony of Susan Richman in the Pennsylvania Homeschoolers Winter 2006-2007 newsletter:
"This past fall in our Pennsylvania Homeschoolers Testing Service, we welcomed kids in 5th grade and up to use calculators for the first time - this is allowed for the problem solving section of the Terra Nova test that we administer. Basically there was little difference in how the students did, even though about half the students did bring calculators with them... The College Board and ACT tests have both allowed calculator use for a number of years... Scores have remained pretty constant, even with calculator use."
As with many skills, proficiency in mathematics depends more on the ability of the person holding the tool than it does on the tool.
For those who don't learn the arithmetic, you get high-school students who, in actual truth, can't multiply six times four without a calculator. This doesn't bother the publishers of elementary school math textbooks. Everyday Mathematics: Teacher's Reference Manual Grades 4-6, published by SRA McGraw Hill says,
"The authors of Everyday Mathematics do not believe it is worth students' time and effort to fully develop highly efficient paper-and-pencil algorithms [step-by-step procedures] for all possible whole number, fraction, and decimal division problems. Mastery of the intricacies of such algorithms is a huge endeavor, one that experience tells us is doomed to failure for many students. It is simply counter-productive to invest many hours of precious class time on such algorithms. The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator."
Textbook series like this claim that calculators are an absolute necessity.
Even textbooks that teach "algorithms" for doing arithmetic aren't necessarily trustworthy. It depends on which ones they teach. Beware of textbook series that use methods for long multiplication such as: "cluster problems," "partial products method" (one textbook says, "If this method becomes too cumbersome, use a calculator"), or "lattice method" (elegant, but time-consuming). For division, beware of "cluster problems" or "partial quotients" methods. For a complete explanation of why these other methods confuse kids see the 15-1/2-minute video "Math Education: An Inconvenient Truth" at www.youtube.com/watch?v=Tr1qee-bTZI.
My conclusion: Calculators should never be used to teach elementary school mathematics. The purpose of elementary school math is to teach the arithmetic skills essential for high-school mathematics. Kids will never learn these skills if their math is all in their calculators and not in their heads.
I'll take this one step further: Elementary-school kids shouldn't be allowed to use calculators at all, at least not in school. A child prodigy who does the family accounting and taxes can use a calculator for that, but no problem in an elementary school math program should require the use of a calculator.
Calculating on Paper
What is the traditional way to teach children to multiply two numbers? First, teach the multiplication tables from zero to nine. A paper math drill product such as Calculadder, or a software drill product such as Math Drill Express or Barnum Software's Quarter Mile can help. See other Reader Award winners in the Math Drill category on page 52. Or just make your own flashcards or worksheets.
Second, teach them to multiply a multi-digit number times a single-digit number. Multiply one digit at a time from right to left. If the product is one digit long, write it down. If it is two digits long, write down the second digit and carry the first one. When you multiply the next digit, add the carry to the result, then write down the second digit and carry the first again. the result looks something like this:
Third, learn to do step two without writing the carries. You won't be able to write carries once you get to step four, because you will be multiplying by the top number more than once.
Fourth, multiply a multi-digit number by another multi-digit number. You multiply the top number by each digit of the bottom number, making a "partial product," lining it up so the last digit of the partial product is under the digit being multiplied. Then add the partial products to get the answer. The result looks like this:
And that's all there is to multiplication!
Kids of this age don't need to understand why this works. In fact, most of them get confused when adults try to explain it. However, a child who is proficient at long multiplication using the standard method I just demonstrated will have no trouble at all understanding how place value works with partial products, once he or she is old enough to do algebra.
Teaching long division is done just the same way. First you teach "the division tables," for example, how many times does 7 go into 42? Then you teach your student to divide one digit into two with a remainder, e.g. 53 ÷ 8 = 6 r5. Finally you teach long division:
That's all there is to it... without a calculator.