Should We Retreat? Should We Lower The Bar? The Answer Is No.

Think Archimedes Takes Initiative As South Carolina’s Education Department Proves Incompetent In Providing Real Solutions To Real Problems

Think Archimedes - The Student Development Company

I am 27 years old – it’s not my job to be political. I can’t even run for the presidency until I am at least 35 years old. At my age, it’s supposed to be my job to continue educating myself, to give to my fellow man in any way that I am able, and to continue growing my business so that I can afford to have a wife, so that I can start a family, so that we can, as a team, add to and benefit society. My job is not supposed to be to think for the government and do their jobs for them. Nevertheless, I have already written one short proposal for how to intelligently modify the current broken school system, and have provided some additional solutions in this piece.
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The Anatomy of a Number

Article: Should Fractions Be Confined To The Dustbin Of Math History?

Habitually we scan the top of the morning paper, “USA TODAY – THURSDAY, JANUARY 24, 2008 – 11D”. Scan the page, let’s see…Mayan sacrifices…amphibian extinction…fractions to the DUSTBIN?! In the first place, what is the deal with all of these articles concerning death? And second, what branch of academic pursuit has actually brought up the question, “Should fractions be confined to the dustbin of math history?” (<–???!!)

Who Needs Fractions Anyway?

If you have ever seen a 1/1000 scale model of a ship in a jar; if you have ever measured 3/4 of a cup of flour while baking chocolate chip cookies; if you have ever given the time saying something like, “Quarter ’til Eight;” if you have ever earned 8% on an investment; if you have ever found an average; if you have ever split anything into equal groups; then you understand the value of fractions in an educated and fast-paced world. Furthermore, if you have ever studied number theory at any level then you know that to challenge the position of the fraction at any theoretical level is to challenge one of the very four primary operators and the very foundation of numbers themselves.

Ever Driven In Miles Per Hour?

Fractions are a fundamental part of the study of mathematics and absolutely a necessary component to the modern day curriculum. They allow us to compare two unrelated measurements (distance divided by time – miles/hour as an example – to understand the idea of speed). Fractions are an integral part of understanding everything from rates of change to models of growth and decay. We use them in nearly every profession whether reading and understanding your electric bill in “kilowatts PER hour” or studying the proportions of the human body to create a masterpiece like Michelangelo’s David.

Fractions Have Always Been An Integral Part Of Technological Advancement

In the late 1600s Gottfried Leibniz and Sir Isaac Newton showed the world that a fraction could be challenged to the very limits of its definition. Dividing by the actual value of zero we call “Undefined” as in “there is not a way to combine words, thoughts, or ideas in such a way that the human mind can conceive of this operation.” The discovery of Calculus, without a doubt, has been one of the single greatest mathematical achievements mankind has ever accomplished and it could not have been done without the concept of a fraction.

Fractions Let Us Use EXACT Values

While the idea of teaching children only the decimal system appears to be wise on the outside, it robs children of gaining an early understanding of “number”. Fractions allow me to be EXACTLY correct in representing certain numbers like 1/3, since no matter how many .333333 decimal places I can extend out to, I will always be estimating. Furthermore, as I count to ten I begin with one digit values (i.e. 1, 2, etc) until I get to ten and then a new digit appears to the left (the tens place). This is an illustration of what its like to work in base-10 mathematics: with every group of ten I add another increment to the next place to the left.

Fractions Allow Us To Easily Understand Multi-based Mathematics

Fractions are one of the main ideas to working in multi-base mathematics. As an example, if we are working in terms of thirds, this is like working in a base-3 system where every time we count three more we add one more to the place immediately to the next place to the left. If we are working in 7ths (base-7 i.e. days of the week), then every time we count 7 7ths we add one more to the ones place (by the way, this concept is tested on the SAT). Simple as this might seem, it is perhaps more important than ever to ensure that the modern student understands these classic fundamentals.

Dumbing Down American Curriculum Is Not The Answer To The Problem

We are at a point in history where we define “technology” to a large degree in terms of programming, web aps, and database management. Historically the most technologically advanced civilizations have always had an edge in global commerce and it certainly seems as if the internet will continue to be the cutting edge of technology for years to come. It really doesn’t make sense to discourage the use of fractions, and hopefully this article will serve as a great refute the this idea.

Even In Sports We Go Back To The Fundamentals

Ask 1,000 students if they know what a number is and you will get 1,000 affirmatives. Now ask those students if they can properly dissect a number and explain it piecewise … Sadly, in the modern school curriculum’s race to Calculus many students cannot explain this fundamental question about mathematics. Any great coach knows that when a player gets into a slump, one of the best places to start in bringing him/her out of it is to go back to the fundamentals and break the dynamics down, step-by-step, analyzing each component of a move, swing, or play until the root of the problem is solved. Thus, let’s reconsider the idea of what a number is.

The Anatomy Of A Number

Numbers only exist because we say they do, and it is important to recognize that mathematicians, over thousands and thousands of years, have come to define numbers as having 5 components to them (there are actually 6 since every number has an imaginary component also, but for practical purposes we shall disregard the ±bi).

Every number has five components: a base (b), a coefficient (c), a divisor (d), a power (p), and a root (r)

As a simple example, 24 is a number: It exists as an independent, specific, and unique value and has eight “factors” or “numbers that evenly divide” into 24: 1, 2, 3, 4, 6, 8, 12, and 24. The amount of 24 is the same as 1 x 24, and this is such an important idea in mathematics that we have a name for it: The Multiplicative Identity. And by the way, this is the idea we are able to exploit in mathematics when adding fractions with different denominators, etc. as we use The Master of Disguise. So whereas we started with 24, now we have:

We call a number that sits just out front of the a set of parenthesis a “coefficient”. If you break down that word it gives you more of a clue to what it means: “with” + “efficiency”. It is a simplified way of dissecting and thinking of a number. As an example, rather than thinking of the number 24 as a solid, concrete value of “24″ it might be better to think of it as 8 x 3. In this case, I could think of it two different ways: 8(3) where 8 is my coefficient and 3 is the base number or 3(8) where 3 is my coefficient and 8 is the base number. This is all, of course, very simple – and that is exactly the point. This is the first step in understanding what numbers are. We have covered 2/5 of the components of a number which brings us to our third idea: the divisor.

Every Number Is A Fraction Anyway

Any number you can imagine is a fraction, because at the very least every imaginable value is always “over 1″. The number 5? It’s really 5/1. The number 14? It’s really 14/1. -8? You guessed it: -8/1. Even fractions (i.e. 3/5 = (3/5)/1), constants (i.e. π = π/1), imaginary numbers (i.e. 2i-3 = (2i-3)/1), or even algebraic variables (i.e. x = x/1) may ALWAYS be written as [insert every number you can imagine]/1. In the case of our number 24, it is equivalent to write it as:

1(24)
—–
1

Symbolically I might code this as:

c(b)
—–
d

where d = divisor

Even the obelus (”old school” division sign: ÷) looks like a fraction! Mathematicians are so literal, but that is the advantage of mathematics! There really is no guessing involved.

The Power of Working With Roots

The last two components of numbers are the power and root. Powers and roots are used in financial equations as a way of looking into the future to see what an investment will grow to, and as a way of seeing into the past to see what amount we must have started with in the first place.

Of course, most people are familiar with where powers are written – as superscripts to the right of the base number (ie, the 2 in 4²). But what might cause a bit of confusion is the location of the root value. The radical sign was used back in the day before computers and calculators took over as a method for solving root values (square roots, cube roots, etc). The way it was done looks similar to how long division is performed now, so it made sense to write the root value inside the v shape of the radical sign √.

It looks beautiful, so we still use it today when writing formulas, but for all practical purposes it is much easier to place in the denominator of the exponent, so that the √4 can be written as 4^1/2. It’s just really convenient to have both the power and the root values listed together, so that in very long equations you don’t have to look back and forth across the page to see what the roots and powers are – it keeps everything in one place. And anyway, the modern exponential value, itself, is a fraction!

So Our Final Dissection Of Numbers Gives Us Our Final Structure Of How They Really Look

c(b)^p/r
—–––
d

Where c = Coefficient
b = Base Number
p = Power
r = Root
d = Divisor

What is important is to walk away understanding another reason why not teaching fractions in school is just a bad idea. Contained within the very definition and understanding of “what it means to be a number” is the concept of fraction. To stop teaching this is to hinder students from understanding the foundation of mathematics itself. We rest our case – the answer is no.