Joe Pagano Article Blog

Joe Pagano

Mathematics and the KISS Principle

Keep it simple, stupid!  That phrase resounds in your brain while you contemplate how it could ever be applied to your current course in mathematics.  You remember hearing that if you kept things simple, then things would keep you.  Yet you’re mired in difficulty, you need to get a decent grade in your current course, and all you can think of is failure, failure, failure.   What do you do?

First and foremost, you need to stop panicking.  Going into this mode will only sink you deeper into difficulty.  The perfect analogy is what we have come to believe—largely because of TV and the movies—of what happens to the person who falls into quicksand.  Although the reality is quite different, we have come to believe that if you struggle in quicksand, you will only sink deeper.  Thus panicking in your current situation will only mire you more deeply.  What to do?  First stop!  Next think more clearly.

You are bigger than any obstacle that a math course—indeed life—can throw at you.  One of the reasons why I love math so much is because of the feeling I have as a result of having overcome this most difficult subject.  You see, I too, was once in panic mode and did not know what to do with the curves and sliders that math was throwing me.  I was stuck in a college calculus course and sinking deeper by the day.  My grade point average was going to suffer greatly because I was looking at a D grade in this course—at best.  Although 60% of my grade was still undecided, I was getting worse, not better. 

Then one day, I decided I would fight back.  I would go back to basics and apply common sense to this seemingly “un-common-sense" subject.  Gingerly, I opened the text book to the problems that were assigned on this newest of topics: relative maximums and mininums.  Who would have thought that one day I would be teaching this stuff!  I told myself that I could understand this, that I would keep it simple, stupid.  Thus I put on my thinking cap and went to work.

Reading the first problem, I started to apply basic principles.  Fear came out of the equation and confidence went in.  And then it happened.  The light went on and clarity of understanding broke through the misty haze of clouds that obfuscated my horizon.  I understood.  What an amazing feeling. I knocked out one problem after another and got everyone right.  I now had the confidence to approach any problem.  The end result: I ended up getting an “A" in the course,  and this outcome would decide my future:  I ended up becoming a math major and math educator, the author of more than a dozen math ebooks (see here www.mathbyjoe.com/page/page/2908604.htm Cool Math Site and www.mathbyjoe.com/page/page/2924777.htm Cool Math Ebooks) and the creator of the Wiz Kid teaching methodology. 

If you find yourself in panic mode with your current course of study, stop!  Take a step back and apply the KISS principle.  Forget all the mumbo jumbo and ask yourself, “What does this mean?"  It really is quite simple.   If you still don’t get it, email me and I will show the way.  For nothing is more frustrating then not being able to see the light at the end of the tunnel.

Fractions- Why Are They So Hard?

Not even signed numbers cause as many problems as those two-headed monsters called fractions.  But why so much trouble?  Maybe because fractions have a dual nature, that is, they consist of the numerator and denominator, and most of us are poor at multitasking.  But did you know that once you master fractions, you learn to handle multiple tasks as well? This is one of the benefits of mastering these pesky little creatures.

In my humble opinion, I would venture to say that any kid who is good at fractions is definitely going to be good at math.  Why can I say this without reservation?  Well, working with fractions is a task that requires an individual to deal with a complex object.  You see, unlike other numbers, fractions consist of two parts: the top part, or numerator, and the bottom part, or denominator.  Because of this duality of nature, fractions cannot be treated as ordinary numbers like the integers such as -2, 3, or 6; or the irrational numbers such as pi or e.   When dealing with a fraction like 2/3, you must deal with a quotient of two integers.  You can no more separate the 2 from the 3 than you can the ring finger from the hand. 

Because the character of fractions is determined by both the numerator and denominator, and not by each one separately, ordinary arithmetic operations become a real headache.  For example, when performing multiplication, you are not simply multiplying two numbers together, but two pairs of numbers together; moreover, you might be confronted with large numbers to multiply, which further complicates the operation.  When adding or subtracting fractions, you cannot simply add numerator to numerator and denominator to denominator.  Oh no!  Not in fraction land. 

In order to add or subtract, you must first be sure you are comparing apples to apples, so to speak.  That is, you must first get a common denominator, as this is what rules in fraction land.  Once you do this conversion, these creatures actually become quite tame.  But getting to this common denominator can sometimes require a little work.  

Although fraction mastery requires more work than mastery over ordinary numbers, the payoff is quite good. For once your children are competent with these interesting numerical creatures, they can conquer new horizons in mathematics.  On this fact, I stake my reputation.  And if you are somewhat skeptical, then try this out: work with your kids until they have a mastery of fractions.   Then watch how their overall math grades improve.  Fractions always lead the way.  Once the objective of fraction mastery is accomplished, school headaches should be a thing of the past.  Just think of the money you’ll save on aspirin, or should I say alleve. 

For more information, see the fractions troubleshooter www.mathbyjoe.com/catalog/item/2924777/2699924.htm Fractions for the Faint of Heart

Why Study Calculus? - How About Cube Roots Without a Calculator!

If you thought that the only purpose of calculus was to confuse you, then think again.  Continuing with my Why Study Calculus? series, I discuss yet another application of this branch of mathematics to numbers. Numbers and the operations performed on them are the linchpins to mathematics, and all higher branches are one way or another intimately linked to their inherent properties. One nice application of the Calculus to numbers is the approximation of cube roots. What is enticing is that this technique can be done without a calculator and without even a knowledge of the underlying theory.

The method to get the cube root is very similar to that discussed in my Square Root article. The theory behind this method hinges upon the derivative, which in calculus, is a special kind of limit.  The differential, which is a quantity that approximates the derivative, particularly under certain conditions, is the mathematical tool that we employ to estimate our cube root.

The way the method works is as follows: Suppose I want to approximate the cube root of 65.  This is the number which when multiplied by itself three times will give 65 exactly. Now 65 is not a perfect cube like 64. That is, there is no whole number cube root of 65. The cube root of 64 is 4, since 4*4*4 = 64, and 4 is obviously a whole number. (I have used the “*" symbol for multiplication here) The calculus, using the concept of the differential, tells us that the cube root of 65 will be approximately equal to the cube root of 64 plus 1 divided by three times the cube  root of 64 squared.  Whew!  That’s quite a mouthful.  

Let’s break this down and simplify.  The cube root of 64 is 4.  The difference between 64 and 65 is 1.  That’s where the 1 comes from in the numerator of the second part of our equation.  Now the cube root of 64 squared is 4^2 or 4*4 or 16; 3*16 = 48.  Thus according to this method, the cube root of 65 will be 4 + 1/48 or 4.0208 to the ten-thousandths place. (You can get 1/48 by doing the division by hand, or by observing that 1/48 is a little bigger than 1/50 which is 0.02.  So if you used 4.02, you wouldn’t be far off either.

Take out your calculator and compute the cube root of 65. You will see that the result is 4.0207.   The approximation is off by less than one ten-thousandth! If you need to be any closer than that—well then, take out your calculator!

See more at www.mathbyjoe.com/catalog/item/2924777/3351424.htm Wiz Kid Calculus.  Also see here for more articles (see here www.mathbyjoe.com/page/page/2908884.htm (Square Roots without a Calculator). 

What's Wrong with Education in America?

Why is education so bad in America compared to other countries?  I’m sick and tired of reading and hearing about all the bad things in the American educational system.   As a former teacher of both college and high school mathematics, I find my insides turning every time I read a report on how we are failing our children.  But are we failing, or are there other factors which need to be addressed?  Let’s take a look at these.

The educational debate has been raging for decades already, and every month or so someone comes up with the supposed solution to our educational ills.  These soi disant experts rally their call and come up with new programs which will remedy our maladjusted programs.  The “new math" is one example of these innovative constructs which only served to further bewilder an already confused educational curriculum.  

The truth of the matter is that education can never improve when the very customers are against such, when they don’t see any connection with reality, and when they can’t find any reason why they need to learn the fodder we force on them.  We need to make our children self-sufficient.  We need to show them how to thrive and prosper.  Our educational programs need to show students how to make it in the world, and yes, this means showing them how to thrive financially.  It does no good to tell them that they need to learn history and geometry and English literature so that they can ultimately graduate high school, get into college, and then hopefully find a job.  This type of persuasive speech can neither fool nor motivate our savvy kids of today, and it certainly will not get them to taking to the books and getting A’s. 

As Napolean Hill mentioned in his classic Think and Grow Rich, if the Carnegie philosophy about accumulating wealth were taught in schools, the time spent in school could be cut in half.  This should not be construed to mean that education should be all about learning to make money and such lofty ideals as being literate and well-read are not important; it’s just that what good is being lettered if you can’t make it in the world?—and yes, making it in the world means being able to make money and provide for your family.

The underlying theme of my educational philosophy, and one seen in all my teachings, writings, and ebooks, is that of the shortcut approach.  No need to spend countless time trying to learn something.  Get right to the meat and go straight for the jugular.  My shortcut mathematical methods give one a huge advantage in that they permit one to master mathematics with a minimum amount of time invested.  The rest of the time could be spent learning how to make money and how to thrive financially. 

Once students are a on firm footing with school and don’t dread the daily routine of having stale fodder crammed down their already stuffed throats, they can approach school and their studies with a much healthier attitude.  Look at it this way.  Suppose you were a gym enthusiast and really liked to have a great build.  You hated the time you had to invest to maintain your current physique. Suppose someone offered you a way to maintain that build with a program that took one-third to one-half the time.  Would you still want to do your longer, already stale program, or would you like to jump on board the new one?  I think the answer is self-evident. 

The same is true with education in America.  We need to show our kids the good shortcuts that will lead to academic success, increased self-esteem, and a healthier attitude toward school and learning.  If we don’t, we’ll just get more of the same old innovative programs that come along promising to cure our educational ills.  Much like resistant bacteria though, our savvy kids will just thwart any attempts to be overcome by the “new medicine" that the administrators and other educational gurus throw at them.  Rather than breed new strains of bacteria, why don’t we work with our kids before they themselves mutate.  After all, we don’t need any more mutations.

See more at  www.mathbyjoe.com/page/page/2924777.htm Math Ebooks

• Previous Posts:

• Hey, Who Said You Couldn't Do Math? - It's All in Your Head
• Why Study Math? Prime Numbers
• Mathematics and the KISS Principle
• Fractions- Why Are They So Hard?
• Fractions: Why Are They So Hard?
• Do Your Kids a Favor - Take Away the Calculator
• Why Study Calculus? - How About Cube Roots Without a Calculator!
• The Secret to Life - It's All in a Word

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