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SAMPLE LESSON 1

Joseph Buczek - Instructor  
Course Documents  
                  LESSON 1 - Why Do We Study Mathematics?

 This lesson describes the reasons why students need to study mathematics.

Why Do We Study Mathematics?

Many students have had bad experiences with mathematics that left them unhappy about the subject. However, most students do not realize that mathematics is really an understandable subject that can easily be learned if it is properly presented to a student.  Students often encounter hardships when they study mathematics because, usually, they are not taught a sufficient amount of overview, or background material to help them understand the math. Another reason that students often have trouble when they study mathematics is that they are not shown the reasoning behind the math studied and what they can do with the math after they learn it. The purpose of this overview is to provide to students a means for overcoming these difficulties.  The theme behind this course is that of presenting the central reasons why mathematics is studied, why it was originally developed and how it can satisfies student's needs.  Before we go into the details of mathematics, let us look at some of the reasons why people wanted to learn about math in the first place.

Why Mathematics Was Developed?

- Learning is important because it can provide a framework of knowledge, or basis from which a person can extract inferences, or conclusions from a problem they might be working on for the reason of gaining understand of the problem.  People know from their experiences that gaining knowledge helps them to solve problems.  Learning helps a person know about things and, therefore, helps the person to understand what to do to make things right when they have gone wrong.

Mathematical symbols can represent (be a likeness to, or be the equivalent of) actual objects and phenomena (things we observe).  Some people look upon the manipulation of mathematical symbols as a system means for predicting and/or controlling events.

Mathematics is Known as The Science of Prediction

Mathematics can be used to predict the outcome of events and phenomena. For this reason mathematics can be called the science of prediction.  A reason why people study math is that it can help them to calculate, or estimate, the results of an event before it occurs.  This is a cause of much wonder for informed people who are always enticed by the possibility of using mathematics for predicting the outcome of physical events (phenomena).

For example, a customer can find out how many items she can afford to buy at the grocery store by adding the prices of the items needed before she goes to the store.  Then, when she actually gets to the store she can have confidence that the items she picks out will not exceed the cost for them at the check-out counter.  If the customer's math is correct, the customer is not going to be embarrassed at the check-out counter because the items' total to more than the customer has money for.

In like manner, when people at NASA decided to send some astronauts to the moon, they performed all the math calculations that allowed them to infer, or conclude, from the facts and premises, what would happen during the moon voyage.  Then, when the systems were built and sent off into space, the space ship systems performed exactly as the math predicted.

As a third example, by doing some simple calculations, a carpenter can conclude and, thereby, predict how many 2' by 4' studs he will need to construct a frame for a house.  If his calculations are right, he knows how many studs to order from his distributor and how much they will cost him before he starts construction of the house.   Good things to know when you are working on a budget.

As a final example, a floor covering installer, by measuring the sides, i.e., the length and width of a floor, can accurately determine the area of the floor and, therefore, how many floor tiles are required to cover the floor.  Thus, from his conclusion, he knows how many to purchase when he goes to his distributor for materials.

If, indeed, mathematics is the science of prediction, then, upon becoming aware of this, most people would want to learn mathematics so that they could use it to help them predict what is going to occur before it happens!  This is surely be an important resource (ability) for people to possess.  When people are surrounded within a world environment in which they don't have the power to see around a corner, or instinctively know what sizes of beams are needed to hold up a building that they want to construct, it is little wonder that they become interested in mathematics - a field of study that helps them to uncover solutions to problems they otherwise could not solve.

If math is really a subject that can be used to predict the outcome of an event before it occurs - then, upon realizing this, most students would have a strong interest in math.  Most people want to be able to have more control over their lives and their environment.  This is probably the main reason people take the time to learn anything.  This is the reason why the old masters who discovered math took such an intense interest in the subject. It helped them to better understand their world so that they could better handle the problems they encounter!  Thus, a conscientious person will want to learn mathematics so as to become more practical in his/her affairs, and have more control over his/her environment.

Mathematics is a Tool That Helps Us Understand What is Already Here

With the above understood, let us delve a bit deeper into what mathematics can do for us.  Whether we thought about it or not everything around us has been there when we were just born. Light, sound, matter, electricity, velocity, acceleration, impact, momentum and distance have existed since the beginning of time.  We didn't understand what heat was so Pierre Laplace (1749 -1827) analyzed what it was and wrote his famous Laplace's equation which effectively tells how a heat field pattern will distribute itself in, say, a frying pan or an oven. Remember, it can be important to know this because sometimes, because of variations in conductivity, a hot frying pan can have high temperatures on some areas of its surface and lower temperatures on other areas of the same surface.  So, if a person is trying to cook something on a hot frying pan some parts of the food may burn and other parts of the same food may not brown up properly.  These temperature variations also can hold for an oven.  Some sections of an oven can be considerably hotter than other sections of the same oven.  These are circumstances that we should be aware of.  Why? Well, try to serve burned food to a customer in a restaurant - or to your mate. The point here is: the properties of heat were in existence since time immemorial. Laplace's equation was developed to help us to calculate and better understand how the heat patterns look when heat travels through something hot like a frying pan or an oven.  Later, people using Laplace's equation brought out an important insight when they showed that the electric field patterns from electric charges resembles the heat field patterns.  When positive and negative electric charges are replaced by corresponding hot and cold bodies the resulting forms of the heat patterns are identical to the electric  field patterns.  So, learning how heat flows teaches us how electric fields flow. Because of this likeness, or analogy, we are able to understand the flow of radio, TV and radar waves from an antenna.  Light and radio waves have been around for millions of years.  When James Clerk Maxwell, a Scottish physicist discovered the curl (rotation) and divergence (change in gradient) relationships between magnetic and electric fields and expressed them in his very famous Maxwell's equations, he developed the mathematical foundation that made radio, TV and communications possible.  As stated, electromagnetic (radio & TV) waves were here for millions of years.  So, Maxwell's equations were developed to help us understand how these invisible electromagnetic waves can be used, that is, how they are generated, sent through space and received as a signal with voice or picture information on it.

What is the point, or essence, of the above?  The main point to be understood from the above is that mathematics is only used to describe and interpret what is already here.  Any mathematics that we learn can assist us in understanding problems that confront us, right now.  All too often students get an idea that mathematics is something that exists all by itself without reference to anything else.  Some people do try to play games with math alone, like shuffling cards and seeing what combinations they uncover.  But, here we are interested in applied mathematics, that is, mathematics that can be used to solve practical problems.  When we add up the prices of groceries that we intend to buy, we are adding up the prices on groceries that already exist.  When we vectorially add up forces to find a resultant force, we are trying to find a resultant force that already exists.  When we analyze the conditions necessary to send a man to the moon, we are working with phenomena like the earth's gravity and centrifugal forces that already exist.

By now the reader should be getting the idea that a background in mathematics can help a person to better understand the world he/she is living in.  The next question that arises is "Who cares?"  Well, there are some people who are intrigued over the things and phenomena in the world and how they work.  Some people are intrigued by the fact that they can study the heat field patterns in a frying pan or an oven and use that knowledge to prepare first-class fried foods and cakes.  Some people are intrigued by the fact that they can study how an electronic amplifier can be altered to become an oscillator which, in turn, can be connected to an antenna, that will send a replica of their voice across the ocean to Portugal.  There are people who want to use their lives to achieve results for the benefit of others.  There are people who want to live the rich life, a life filled with intrigue, a life in which a person has confidence in his/her abilities perform and do things, and have the power to control the factors in one's environment, as opposed to the poor life, wherein one is content doing nothing.  This overview was written for students who wish to take positive steps to find out about things so they might better understand the factors that they will have to contend with in their personal and working lives.

A Notable Amount of Mathematics Was Developed Because of a Need to Understand Wave Motion

One essential point is clear: a large part of intermediate and advanced high school and college mathematics had its origin from people trying to understand and interpret mechanical vibrations and their corresponding wave motions. This means that a quick way to learn mathematics is to learn the principles of wave motion. Once the concepts of wave motion are understood, the mathematics that describes the wave motion becomes obvious. Readers should be aware of this fact:  Those students who studied math by rote, i.e., by memorizing formulas and methods, i.e., without attention to meaning, have been at a great disadvantage when it came time to apply the mathematics they learned.  On the other hand, those students who were presented a well explained, or cogent, theory of math with attention to the meaning of the principles from which the mathematics was originally derived, namely the study of wave motion, had a superb advantage when it came to understanding and applying college level mathematics.  When students had something to which they could relate the mathematics, they were in a superior position to use the math to solve practical problems that they encountered in their lives

The following lesson is intended to provide students with a sound understanding of the first basic and main subject behind the principles of practical or applied mathematics.  The theory of waves is the main subject that is behind a large amount of mathematics presented in college.  From a study of this subject students can more easily understand the underlying concepts and relationship of mathematics to the real world problems.

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