In 1999, a certain group of people neglected some fundamental mathematic procedures, along with some of their own checks and balances, with very unfortunate and expensive consequences. They neglected some important steps that anybody learning math, including home schoolers, should follow. These people were the engineers and flight controllers at the Jet Propulsion Laboratory (JPL), part of the California Institute of Technology, under contract to NASA for the Mars Climate Observer spacecraft. So it can happen to the best!
The first problem was a lack of attention to details. One of the subcontractors had sent a data value in pounds force (English system of measurement) to JPL, who used it, thinking it was metric. JPL’s math calculations for navigations started with a number 4.45 times as big as it should have been. This introduced an additional bogus number into their calculations. This error made ground controllers think that the spacecraft was several hundred kilometers from where it actually was. They actually discovered that something was wrong, but it was too late to correct. The spacecraft, instead of neatly sliding into orbit around Mars, simply disappeared; the cost to the American taxpayer was 125 million dollars.
Mathematics is the key to understanding all the sciences, engineering, and business. Do it right, and you can have real success; do it wrong, and it can result in spectacular failures.
I have worked with math all my professional life and have found a few steps which, when followed, result in correct answers. Including these steps in any math curriculum will produce good math discipline.
- Write down the problem as stated. This action starts you thinking about it and gives you a valid starting point.
- Write down all the sub-steps needed for the solution. I have seen some home school students first read the problem and then just write an answer down. This is absolutely guaranteed to be the wrong answer. Writing all the steps down causes you to think more closely about it. Remember there are an infinite number of wrong answers and only one correct answer.
- Pay attention to details. The JPL folks did not, and let one small (wrong) number get into their calculations, which resulted in a mission failure.
- Double-check your answers by a different means. If, for example, you have subtracted two numbers, then check the difference by adding the answer and the subtrahend to get back the minuend. In algebra if you solve an equation, check your calculated answer by substituting it back into the equation. Check the answer to see if it is reasonable (e.g., units are consistent, order of magnitude of the answer is reasonable).
- Work several word problems. These really teach how to use math. In my engineering career, most problems are word problems with no answers in the back of the book.
Do not be discouraged with mathematics. Remember, more advanced math always builds on earlier, more fundamental concepts; then you can progress to the next step.
By the way, the JPL engineers learned from their mistakes. In April 2001 they launched another spacecraft (Mars Odyssey) toward Mars, and in October it was correctly placed in orbit around Mars.