They can come in all shapes and sizes. Some can be mistaken as credit cards. Others are as big as typewriters or computers. Usually the ones we use amidst our day-to-day routines can fit firmly in an outstretched palm.
Calculators. Devices to speed up, or possibly squelch out altogether the duration between sums and number crunching. Calculators weren't always the name given to a machine. They denote any person who did numeric work using anything that wasn't totally mental - and by this it refers to a process done in the head, not a procedure completely senseless. This could include options from pen and paper to an abacus.
But wait, there's something missing. In the modern world today, or to be more specific, in educational systems taught by Western techniques, almost every student is equipped with such aids. There is less attention focused on the manual, mental mathematic solutions to problems. All there is to it nowadays is: "Press this button, then follow it with your command, and hit the ENTER key to get your answer."
Kids have gotten into the habit of seeing calculators as completely foolproof methods of producing answers. Suddenly there's no intermediate stage in their thinking. Suddenly the logical pathways they followed in order to reach conclusions have disappeared. By replacing these crucial, but otherwise time-consuming stages, using our calculators - these tools of "aid" designed to help us big time - is killing our ability to be more successful in life.
Of course, it takes nearly a degree to figure out how some of the calculators work these days. Some graphical calculators have manuals thick enough to drown people, and I thought these tools were supposed to save us time. We've now got ways to roll with the best of them: every time you reach for the calculator, it's a pretty big gamble. Will we obtain our answers quicker than working them out manually? Or will we wind up losing our minds and sense of direction in the multifarious keys, and the impressive liquid crystal numeric monitor?
True, sometimes there are mathematical concepts we just can't grasp normally unless the numbers are written out before us. Like Pi, for example, the nasty little constant that decides every circle we come across. The endless decimals of recurring fractions. We can't successfully visualize these unless something regarded with awe and august - our trusty mechanical buddies - produces the figures for us.
Sounds pretty good right? But they can also play tricks on us. Some calculators, without correct use of brackets, can completely disorientate negative numbers, and produce a chain of bad habits and incorrect answers for students. The calculator can become more of a crutch than a tool, and amidst exams or any lifetime situation, people will be double-checking even the most trivial sums and calculations, for they have no experience to handle them otherwise.
This is the huge compensation required to make up for speed and efficiency. It is a big number. A tortuous sum. One no calculator - human or mechanical - will ever be able to solve.
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4 comments:
I think once you get to university level mathematics, simple algebra and large sums become negligible as you focus more on the concepts then the actual doing of problems. Solving/doing maths is for the calculator and is what we master in lower secondary/primary school, understanding it however is a different story and is what we try to tackle at university, eventually applying these ideas to real-life (applied mathematics).
...at least thats how i think it is.
Actually I heard that in first year mathematics, a calculator isn't even required.
arithmetic is not math... math is not multiplying or adding, it is the logic of deduction.
calculators are virtually necessary for real-life definite integrations. Try, for example, to integrate to just 2 or 3 decimal places e^(-(x^2)) with respect to x from x=0 to x=1. TO do so would require arduous riemann approximating, and does not enhance conceptual understanding.
I think you guys are missing the point. All the author's trying to say is that people shouldn't be so reliant on the calculator. I've known some people who have grown so reliant that they've forgotten how multiply two digit numbers in their head. Teaching children at a young age to do their math, arithmatic, etc. on a calculator may lead them to think of the calculator as a short to the "answers" they need on a test and in doing so neglecting to learn about the process that leads one to the correct answer. This may lead to result of the child not knowing how to double check his or her work in the case of human error in which one punches in the wrong numbers for a numeric problem. Whereas nowadays there are calculators that can "show work" allowing one to figure out his or her mistakes in an answer, one has to understand that such calculators are expensive and that many people still rely on the simple one rowed screen calculator that only shows an answer after pressing enter. Would one know if the answer was incorrect if one was only taught to use the calculator and not the process behind such a calculation? Do you really think the author is that stupid to imply that we should do everything by hand? Of course there are some mathematical/arithmatic processes that would be better off by a device such calculator. If it helps one understand a mathematical complex algorithm or concept, all the better. Why else would people teach students to use it? All the author is merely saying is that we should rely on the calculator to the point in which we have to use it for something as simple as 1 + 1. Guys, don't be so quick to attack the author. He's just giving off a simple message.
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