Readers who follow Family Learning will come to recognize that almost every column is aimed at a particular date or an occasion celebrated during the week that column is published. My purpose in doing this is to provide some "excuse" for parents to bring up a particular topic and use it as a vehicle for learning in the home. Some learning topics, though, just don't lend themselves readily to this sort of introduction, and mathematics is one of those.
I was reading the other day about a German mathematician named Johann Karl Friedrich Gauss (pronounced GOWSS), who was a mathematical genius even as a child. From the time he was 3 years old, he could solve complicated arithmetic problems without even the aid of pencil or paper.When he was 6, he was asked to add up all the numbers from 1 through 100. He paused for a few seconds and then announced the correct answer: 5,050. How did he do this? And, I wondered, how long would it take me to work out the same calculation? Well, the fascinating part of this story, at least for me, is that the important question is not "How long?" but "How?".
Gauss thought about the problem in a completely different way than most of us would. Instead of barging ahead with calculating a hundred different sums, he saw that the problem could be solved in one step.
In those few seconds of deliberation, Gauss saw that if all the numbers 1 through 100 were arranged in a row, the first and last numbers would add up to 101 (1 + 100\ 101). So, too, would the second number and the second from the last number add up to 101 (2 + 99\ 101); and so would the pairs formed by the third, fourth, fifth and so on, numbers from each end of the row. There would, then, be 50 such pairs of numbers (the last pair would be 50 + 51), each pair adding up to 101. So the sum of all the numbers 1 through 100 is really just 50 groups of 101, and 50 X 101\ 5,050. It's brilliant; a seemingly difficult problem reduced to just one, simple calculation.
In explaining this problem to children, it is best to start out by considering the sum of all the numbers from 1 to 10. Write the numbers in a column or a row, then cross off the pairs formed by the two numbers on each end (e.g., 1 + 10\ 11, 2 + 9\ 11, and so on). Then total the five pairs of 11 for the correct answer of 55. Now let them try 1-100.
The lesson here is that not all problems are as difficult or as complicated as they may appear. The first thing to do is what Gauss did: Think about it. Maybe you won't come up with the astounding insights of a prodigy like Gauss, but maybe you'll see something that you didn't see before - some pattern that was concealed by the apparent and distracting complexity of the problem at hand.
My father gave me some very helpful advice about how to approach the countless "story problems" that were a never-ending part of my math homework, and used to confuse me so much that I would come up with answers that made no sense whatsoever. "Substitute simple numbers first, and see if it all makes sense," he would say, and so the train that was traveling 54.92 mph in the problem became a train that traveled 50 mph, and the 13-3/8 gallons of syrup that dripped out of the holding tank every 3 hours and 45 minutes became 10 gallons dripping every 4 hours.
Now I could see what was really going on, and I could try different operations on my simple numbers until I understood how to go about solving the problem in a reasonable way. It was then just a matter of slugging in one set of numbers for another and carrying out the arithmetic. I still use this system today.- Dr. William F. Russell's books for parents and children include "Classic Myths to Read Aloud." Send your questions and comments about Family Learning to him at P.O. Box 1279, Menlo Park, CA 94026.