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Mathematics Should Be Meaningful And Learned Through Understanding Not Through Rote Memorization of Facts
   What Is The Suggested Way To Learn Mathematics?
     Having read the previous paragraphs the student can appreciate that mathematics can be used to represent, or describe, any phenomena, or physical happening, that can occur.  If a phenomenon cannot be related to mathematics, then  mathematics cannot be utilized to predict that phenomena.  After all, the things and phenomena were here all the time and the math was developed by people who started understanding that physical phenomena are related to physical causes.  Mathematics has the practical purpose of relating different physical factors to each other.  The great mathematician Felix Klein once said "Physics is geometry."  By this he meant that because physics is concerned with developing an understanding of all physical things, then, geometry, which is mathematics, can be related to all physical things.  In other words, mathematics is universal and can be applied to, and, apparently, be used to help students understand, or solve, any physical problem .

    Utilizing mathematics for understanding, and, solving a problem, infers a clear understanding of all of the factors or ideas involved in that problem.  The reason for this is that if one detail is not known about the circumstances in the problem, then, that very detail could be the cause of the problem and, therefore, the problem cannot be solved without knowing about that one fact. The idea to be obtained from the above is that mathematics  learned in a  rote manner, i.e., by memorizing a lot of formulas and concepts, without attention to understanding the meaning behind the physical phenomena and the basis from which it was originally derived, is not able to be put to much practical use.  This means that mathematics taught in a rote manner, i.e., by making a student memorize formulas and methods that show no relation to the real, physical, ideas that the mathematics originated from, is boring and unrelated to the applications of the mathematics.  Rote learning, or learning by memorization, causes the student to work a lot harder than if she/he learns through understand what the ideas behind the math are described in terms of the physical ideas, or concepts, from which they originated.

    The underlying concepts of the principle being described can be alternately conveyed as follows:  In order to use mathematics it is important to have a clear mental picture of all the factors that contribute to the operation of the physical system being studied.  The idea being presented here is well understood by any seasoned electronics analyst who will be quick to state that problem in a defective radio or TV is most readily found when one has a clear understanding of the circuit operation and layout of the apparatus being repaired.  

    There is one central fact widely accepted by people who work with applied mathematics.  It is the basic principle that if you know how the math originated, you will have a much better understanding of what the math is really all about, and, consequently, you will be in a much better position to know how to use the math for solving a problem.  

The suggested method for learning mathematics is to learn by understanding the background ideas behind the subject being studied, rather than learning through rote, or memorization processes, without understanding the underlying ideas within the mathematics studied.  Mathematics should be learned through understanding not by rote, or memorization.

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