Rightly Understanding the Problem

He could count to 100. He could count by twos, by fives, and by tens—if he always started at one. If you required him to start at any other number, such as thirteen, he could not count very well, even by ones, and was totally lost in counting any other way.

Gary (name changed) was a very hard worker, exceptionally so for his eleven years. His attention span was longer than most eleven-year-olds’, too. He came to me because he was scheduled to go into the sixth grade and was having trouble with math. He was unable to carry and borrow (regroup). He was unable to carry and borrow because he did not understand our number system.

A great number of people do not understand the wonderful ingenuity of the place value aspect of our number system. Most folks pick up on it enough to become operational in the basic math areas: addition, subtraction, multiplication, and division. As is true of so many other things, this marvelous arrangement of digits can only be really appreciated when it is compared with other number systems that are inferior.

To illustrate: count to one hundred in Roman numerals. Then count to one-hundred in our Arabic number system. Try to add 17 + 93 in Roman numerals. Right away, you may realize the superiority of a system based on place value. Try to subtract 62 - 39. It is much easier in our place value system, is it not?

A great deal of attention had been expended on Gary as his needs became manifest. He had been drilled to a fair-thee-well on his addition facts and subtraction facts. He was used to long hours of practicing, practicing, practicing. Those who taught him had made a fundamental mistake. They had failed to realize that he did not grasp the key principle beneath all the drill. He was trying to memorize everything. This is impossible, of course. How can anyone memorize all the possible number combinations in addition and subtraction? He had memorized a lot—more than most young people his age. And he was a very hard worker—heroic—battling against impossible odds.

Gary came from a divorced home and lived with his mother during the school year. He attended a public school. His passiveness and diligence made it easy to assign him work for the practice it seemed he needed, and consequently, he stayed at a basically first grade level of math understanding through the end of the fifth grade, even though he could work certain kinds of subtraction and addition problems of about a third grade level. During the summer when he came for tutoring, Gary lived with his father and stepmother.

Gary and I started by counting. I drew a drawing of a number which revealed its place value, such as the number 241. The hundreds were a new frontier to him. (Just think, if you had carefully memorized all the numbers up to ninety-nine, carefully labeling each with its position in the counting hierarchy in your mind, how daunting it is to face all the hundreds, to say nothing of the thousands….) He knew they were there, but he did not understand them. The entire concept of “roll over” was perfectly new to him, as well. “Roll over” is our term for what occurs when you run out of digits in one place value. For instance, when you reach 19 (one ten and nine ones) and count one higher, then all the nine ones vanish into a single ten which is completed by your new count and you end up with two tens and zero ones. This concept occurs everywhere in counting, and really gets dramatic when you roll over a number of place values as a result of counting up only one more. In the accompanying illustrations, the number, 2999, will roll over the ones, the tens, and the hundreds place values at the next count. The first drawing is before, the second is afterwards.

It was very difficult for him to follow what happened when “roll over” occurred. I could draw it, before and after, but it was extremely hard for him to imagine what happened to produce the change.

I realized that what was needed was a computer program that actually did the “roll over” right before his eyes—over and over, with all kinds of number place value combinations.

I had never seen such a program, although I could visualize it. It is possible that such a program exists (as the wise man said, “There is no new thing under the sun”* (Ecclesiastes 1:9)), but I did not know where to obtain it, and had a suspicion that it would be too costly for me to afford if I did stumble across it.

I have a good bit of experience writing business programs, but I had never had need to write a graphical representation and was therefore completely unacquainted with the techniques involved. Yet here was a great need. I prayed and asked the Lord to help me. Then I worked. I am sure that what I learned is elementary to programmers who work commonly in this area, but it seems miraculous to me, that in a week’s time, I was able to devise a program that illustrated counting up.

The effect on Gary was dramatic. He could see what he had not been able to imagine. He would count: 120… 121… 122… 123… 124… 125… 126… 127… 128… 129… and then he would set up 129 on the computer program and watch carefully as the program “rolled over” into 130. Every school day that summer, Gary spent his afternoons doing this until he could count up and tell you the next number with a great deal of assurance and accuracy. (His hardest challenges were in the combinations 499 and 5999, etc.) Then we started counting backwards, which involves the concept of borrowing from place values to the left.

Finally, the counting concept was matched to actual numbers in addition and subtraction, and the effect was striking. I do not suppose I will ever forget the triumph on this boy’s face as he successfully subtracted numbers such as: 2000 - 678 without even referring to the “circle” program. His self-esteem had taken such a beating over the years as he fell farther and farther behind. It was good to see him getting some success under his belt.

The principle is a very profound one. No matter how hard we work or try, if we are fundamentally mistaken in our approach, the problem is not corrected. Sincerity and great effort are not enough in themselves; we must rightly understand the problem (judge correctly). How patient God is with us! How much He waits on us and desires to help us! He wants to bless us.

“And therefore will the Lord wait, that he may be gracious unto you, and therefore will he be exalted, that he may have mercy upon you: for the Lord is a God of judgment: blessed are all they that wait for him.”* (Isaiah 30:18)