"Learn the subject"
Can you explain how it is that when you divide by a fraction you get more than the original amount? I asked a high school student the same question and she looked at me like I had struck her on the head! Although she knew how to calculate the result she had no clue how this is so. Not that I am surprised, most teacher's can't explain it either. By the end of this post you will understand not only why this happens but you will learn a great deal more about how to pass your exams and never fail another one - and all that without wrecking your health or your nerves.
Now, if you are wondering what this could possibly have to do with passing your exams, you may want to pay some serious attention here. Knowledge gaps are at the heart of most academic confidence issues and a great number of failures. Although many persons will have noticed the phenomenon decribed above when dividing by a fraction, few have bothered to look any further.
We have long advocated, and, it is part of our training method that students ought to learn the subject thoroughly. The current practice of training students to answer exam questions at the expense of learning the subject is a worrying thing.
In a certain context it makes perfect sense
Let's look at that for a moment. If you had a life threatening health problem, needed surgery to correct it and found that your surgeon had been good at answering the questions in the medical exams but never really understood the whole business of surgery, had done your particular procedure several times and got it right most times, what would you think about your chances of survival? Would you feel better if your surgeon was thoroughly familiar with not only the procedure you would be doing but with the whole business of surgery? Do you think that this degree of thorougness would be useful in areas other than surgery? Which of the two surgeons would you think was the better student when learning surgery, the one that swatted and practiced to answer the questions and impress the examiners or the one that made the effort to really learn the subject? Yeah, when it comes down to it we know the better way. Its those that take the superior approach that make the real difference. Its the approach that we take and pass on to our students at 2GN Education Centers.
Not much more effort required
One of the practices that cannot be over recommended is to "learn the subject". This sometimes amounts to a very small difference in the amount of effort you make as you study but it will usually make an enormous difference in your results. In fact, any self improvement program will eventually get to telling you that the difference between major success and the disappointing results is usually very small differences in the mix of inputs. One of the most respected names in personal development, Bob Proctor, said, as he worked with insurance sales persons, that "it takes no more effort to sell a $1000 policy than it takes to sell a $10,000 policy". The more successful sales agents are the ones who sell more $10,000 policies.
In the same way, it really does not take any more effort to learn the subject properly than it takes to learn it partially and then try to figure out the answers to every possible question that might turn up in an examination. These same principles apply to every area of life. It may not be necessary to learn everything about a particular matter, but the part that you need to know, make sure to know it well. Otherwise, the knowledge gaps will turn up to haunt you sooner.
The solution
So back to my high school student who did not know why dividing by a fraction produces a larger result than the original value. The standard mathematical procedure is to invert the fraction, so that the part that was at the botton goes to the top (so 1/2 becomes 2/1) then multiply the original amount by the top part and divide the product of that multiplication by the amount at the bottom of the fraction. Many people note that dividing by half (1/2) produces the same result as multiplying by two (2). But are they really the same? To determine this we have to convert this to a social problem. Humans are pretty good at understanding social problems, not so good at acdemic abstractions.
Let's say you got a bill for $10. Multiplying it by 2 has the effect of you getting another bill for $10, so now you owe $20. Dividing by 2 has the effect of splitting the bill between you and someone else. So its the same bill, but now you owe half - in this case $5.
On the other hand, what do you get when you divide by half? This is the same as asking how many bills would you get if each bill valued half of each of the dollars in the original $10, and the answer is, of course, twenty. You see, in a social context the questions that these mathematical issues answer are quite different. Let me share a real life example of this.
As you suspect, a story
When I was barely more than a toddler I had a major altercation with my kid sister who was a bona fide toddler and two years younger than me. Our mother had given me a brand new pencil and I had put it down to attend to something else. My kid sister got hold of it and proceeded to make a meal of it. Now, ignoring the debris, you know, the part she chewed up, she now had two pieces - a veritable division by 2. You can imagine the screaming and excitement when I tried to retrieve my pencil, or rather, two pieces of pencil.
I was horrifed, and not a little annoyed with her. Our mother intervened and saved us from ourselves. But I was still peeved about the broken pencil. In her great wisdom, mom sharpened both pieces, eliminating the teethmarks, and now I had two pencils, each being half the length of one original pencil. The wonder of a different perspective!
Applying the principle in other areas
This same idea of converting an academic situation into its social equivalent can be applied to any subject and it puts you in a better position to understand and resolve it. I was talking with a history student who was certain that there is nothing more boring than history and cursed the creator of such a subject. I was amazed because, like everybody else that I know, she likes stories. So I asked, what is it about history that you hate so much? "The dates", she responded without missing a beat! Well now, there is the issue. Most people with this view of the subject point to the difficulty of remembering the dates. So I suggested that she convert the history she was studying back to a story with a timeline and make the connections like you would in real life.
Take the voyages of Columbus, for example. You could look at it like this. Life in europe was bad so he managed to persuade some rich folks to hook him up with a fleet to go buy stuff that they could make big profits from selling. The normal routes were a problem due to the massive customs duties and extortion rackets going on so, his selling point was that he knew a route that would avoid this problem. Unable to resist the offer anymore they gave him a startup investment and sent him off.
He got lost because of a massive knowledge gap in his geography but he did hit upon the caribbean. He did what he could with what he found and headed back with a very convincing argument - he had found new land that nobody owned! (I never said anything about his arguments being true.) You realize the first one that got him the first outfit turned out not to be true either, but nevertheless there was opportunity. That was 1492. By the time he got back to Europe, sorted things out and was ready again two years had passed. His next trip was 1494 and the other would have taken about the same amount of time.
So we have eliminated the date issue and made the whole thing more meaningful. There is no end to the number of way you can do this and have no problems dealing with an examination. What is bottom line important is "learning the subject". Making that extra effort to bring the subject into what works best for you while at the same time bringing you into the real knowledge of the matter. People who do this, as a rule, do not fail exams.
Other Issues
Now there are some other issues that can affect your passing your exams such as psychological fears of doing exams and we gave you a fix for the most common one in an earlier blog. We will review that in another post but for now, you can put what we have suggested in the last two blogs to work and you may expect to pass your exams and never fail another one.
So we looked at:
- The importance of "learning the subject
- The fact that not much more effort is required to do a lot better
- The question of dividing by a fraction
- Applying the principle to other areas
In closing, let me share with you something a wise person discovered. He said, "Opportunities are often missed because they come in dirty overalls and look like work!"
Two questions I have for you. What would you say is the social context of multiplying by a fraction? And, are you putting this to work?
2GNEC
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