`sin^2x + cos^2x = 1`
`y = frac (-b +- sqrt(b^2 - 4ac)) (2a)`
`(x + 2)^2 + (y - 3)^2 = 16`
`slope = m = frac (\text(change in y)) (\text(change in x)) = frac (Deltay) (Deltax)`

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Feature Article `(lim_(x rarr 0) frac(1) (x))`

Calculator Syndrome

There is a horrible disease, of which very few people are aware, sweeping our nation's youth. Many people have witnessed the effects of it without even realizing it. This disease is not contagious, in the normal sense of the word, but it strikes in an addictive manner with devasting results. You see, the victims of this silent disease are often unaware of any problem or need for help that they have. And yet this disease affects one of our body's most vital organs, the brain. It takes control of the victim's brain in such a way that the victim is eventually left unable to think for themselves. They're brainwashed, without even knowing it, into relying on an outside source of 'knowledge,' rather than calling on their own abilities. The disease, to which I'm referring, is called Calculator Syndrome, or CS for short. Here is it's story...

An example, all too familiar

One day after coaching soccer, I stopped into a local convenience store with my daughter. My daughter wanted one of those 69¢ frozen drinks, and I got a newspaper. We approached a young man working behind the counter and placed the paper and drink on the counter. The young man, who I would say was approximately 20 years old, punched some buttons on the register, then 'Sale,' and promptly said, "That'll be $8.24." Without hesitation and with somewhat of a blank look on his face, he looked me in the eyes as he held out his hand to retrieve my money. Now my first thought was, "Hot dog! There must be some really good news today!" Then my five year-old daughter tugged on my sleeve and said, "Wow, daddy, they must have brought this ice in all the way from Canada or something! Unless they're throwing in a winning lottery ticket, that's too much money for day-old news and a cup of artificially flavored, crushed, frozen water." Then I realized she was right. The drink cost 69¢, and the paper was 75¢. So, with tax, it couldn't be any more than $1.55, yet this guy, was asking for more than 5 times that! So I said, "How much is that drink again? That seems like way too much money to me." His blank looked changed to a dumb, blank look as he responded, "Huh?" He then checked the receipt tape and said, "Oh, sorry. That last total got added to yers. Uh, give me a minute. Hey, Marlene! What'd ya' do with the calculator?" That's when it hit me! This poor guy was suffering from CS! I immediately warned, "Watch out, sweetie, don't get too close! It might be contagious!"

The Reality of CS

Well, I've learned a lot about CS since that time, and now know it's not something to fear. It's just a disease that many people have, and in many cases, through no fault of their own. It turns out that a lot of our youth contracted this disease as a normal course of 'doing the right thing' and attending school to try and make a better future for themselves. During the process, some authority figure in a math class somewhere put a calculator in their hand, and told them it was a good thing. They led them to believe that it would make math easier for them. And through a blind trust, these unknowing victims allowed CS to take control of their brains, rendering their ability to think for themselves a thing of the past. Some people believe that CS is a cousin of the disease RCS (Remote Control Syndrome), which makes otherwise perfectly capable human beings, incapable of manually pressing the power, channel, and volume buttons on the front of a TV, no matter how close they may be to the set.

In all my years of tutoring middle school, high school, and college students in math, I've come across several recurring themes with the students that have serious problems understanding mathematics. One of the most prevalent theme has been a reliance on calculators. It's amazing to me how dependent people can become on these things and consequently, how helpless they can become without one. Many students experience a total brain shutdown when their calculator is taken away. One example that comes to my mind was an initial tutoring session I had with a sixth grader several years ago at the beginning of the school year. This student's most urgent deficiency at the time involved the long division of rational numbers. When I first met her and sat at the table with her, I noticed that she had her calculator lined up right next to her math book, along with a pencil and some paper. After talking to her for a few moments to get some background, I asked to work the first problem, a long division problem. The first thing she did was to pick up the calculator. I stopped her and told her to put it down and not to use it. She then stared at her paper for a good 10 seconds, and then said, "How do you do it without a calculator?" I'll never forget the look on her face when she asked me that question. It was one of total helplessness. I was asking her to do something that, at that moment, she had no idea how even to start. So what, you say? Who cares as long as she knows how to get the answer with her calculator. Why waste time on something so mundane as long division when there is a much quicker and accurate method? Just let her use the calculator. The field of mathematics is so deep and there are so many other things to explore and learn in mathematics that her time could be much better spent learning more advanced concepts than spinning her wheels on something that out of date. Let the calculator do the work for her while she expands her thinking on the higher level ideas and applications of mathematics.

That's an argument I've read, and that's an argument that I've seen come from people who are involved in developing standards for mathematics education in the United States. I also believe that's an argument that someone without a mature understanding of mathematics might make. Someone without the mathematical maturity to know that believing such a justification of calculators is dooming oneself to mathematical medocrity at best and probably worse1. If you've explored any part of my ChrabbyMath Site, you've no doubt come across my belief that math builds upon itself. Long division of rational numbers is one of those basic concepts in math that so many more advanced concepts rely upon. The relationship may not jump out at you at first, but it's there and will reveal itself as true understanding is reached. Take polynomial long division for example. One will encounter this in high school algebra. A true understanding of the long division of rational numbers can make polynomial long division a snap. It's just an extension of the basic long division that is covered in elementary school2. You have to rewrite the numbers in their expanded form using place values to see the connection. And when you do this, you'll see that polynomial long division and long division of rational numbers is the exact same concept. How can anyone be expected to understand and perform polynomial long division without first having a good understanding of the long division of rational numbers. And you won't achieve this understanding by using a calculator to do your division for you. Who cares? Why does anyone need to know how to do polynomial long division anyway? Well, most people probably will not need to know this. But if you want to go into any scientific field (math, physics, chemistry, engineering, ...), you'll have to master higher level mathematics. Which means that long division of polynomials as well as a lot more mathematical topics will be involved. And if you don't have the basics of mathematics mastered, you'll have a rough time with the more advanced topics. Remember, math builds upon itself. Without the basics, you've got nothing to build upon, and there goes your dream of being an aeronautical engineer at NASA!

What the schools are really doing with calculators in my opinion is that they are trying to make math easier for the masses. But in the process, they're actually watering down the subject such that many students who would have a chance to really excel in mathematics and related sciences later on won't because they won't have a good enough base to do it.... (more later)

1 Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education, by H. Wu, Department of Mathematics, University of California at Berkley

2 The Role of Long Division in the K-12 Curriculum, by David Klein, Department of Mathematics California State University, Northridge and R. James Milgram, Department of Mathematics, Stanford University





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