Why study or master SOME mathematics
The aim here is to say why and how to study mathematics without
giving too little nor too much motivation. Education, yours or that of others,
is not yet a tidy affair. The advice below may appear with some repetition at
this website.
Most people learn mathematics until circumstances force to them
to stop, or until the subject becomes too hard or until they lose interest.
Failure or near failure is one way to halt learning in a subject, and leave a
last impression not worth repeating. Mathematics courses, being
compulsory, are not designed to leave a good last impression. Mathematics
courses, being compulsory, are designed to cover topics. One by one, the
topics need not be important or of immediate use, but altogether or
cumulatively, the topics provide or point to a skill, a mastery of
mathematics.
Despite these adverse circumstances, reasons for studying mathematics and
making it compulsory exist.
The ability to read, write and figure well is a required
for many disciplines including mathematics. Imprecision in reading leads you to
not understanding and not recognizing errors in the writings of others and
yourself. Students who read, write and figure well are easier to teach and
will do better than others in all subjects. To show that you are teachable is
one reason to learn mathematics. But there are others reason.
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Logic and mathematics starting with arithmetic onwards may
show you how to follow steps, one at a time and one after another, for
arriving at results or conclusions, one at a time and after another.
Learning that an error in one step make all the following steps and results
or conclusions wrong or a least suspect (errors could cancel if you are
lucky) is a step towards cautious wisdom or intelligence. This wisdom or
intelligence applies to all subjects.
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Mathematics or any other rule and pattern based discipline
may show through experience and trial or error, how to solve problems
first by following given methods and later, if needed, by
combining and exploring different methods, by trial and error,
opportunistically or with some advance knowledge of what may work. I
call this the jigsaw puzzle
approach.
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Mastering it in one subject, say mathematics, gives you a
wisdom, applicable to all. Understanding or least using patterns one at a
time and one after another, while figuring, while sewing, while cooking,
while building a model car or airplane, or while rebuilding a mechanical
instrument such a bike or car, all points to a jigsaw
puzzle approach to problem solving, applicable in all subjects,
circumstance permitting. Tackling easy to challenging
cross-word puzzles also demand a trial and error approach to problem
solving. Here the early clues and entries in the crossword puzzles may
help in later ones. Playing games of chance, checkers or chess may also lead
to an interest and practice in problem solving.
More Reasons For Mastering SOME Mathematics
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Mastering arithmetic by hand or with a calculator is needed
in the calculating weights, measures and amounts (money included) that
appear in daily life. If you can do arithmetic and estimate the
results of calculations in your head, then you can catch or double check the
figuring of others or your electronic calculator. Incorrect numbers
that appear in one step of a calculation make all the rest wrong. Tax
forms give step by step instructions for calculating your taxes with
arithmetic and a minimal use of formulas because government assume no
competence in algebra. Arithmetic and not algebra is required for computing
your taxes. That is good to know :) And it is possible to have
a thought-based comprehension of why methods for arithmetic work - a
comprehension I would like to see offered or given in school.
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Algebra may begin with formulas or methods for calculating
areas, perimeters and volumes of common geometric objects in the line, plane
or space. But there is more to algebra than following steps in a
calculation, evaluating a formula or programming a calculator to evaluate a
formula for you. Rules (assumptions) in algebra say when different
calculations will give the same result. Applying these rules one at a time
and one after another allows you to solve problems algebraically and to
algebraically obtain formulas for calculating numbers and quantities.
There is more to algebra than just doing arithmetic or being given a formula
and numbers to use in it. Algebra at full strength involves the
thought-based derivation of formulas, that is, of explanations why they
work.
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For life, now or in the future, you should meet and
master formulas or methods for calculating areas, perimeters and volumes,
and you should meet and understand formulas or computation methods needed
for loans, pensions and investments, for shop keeping or buying and
selling with markups or markdowns. This understanding should go beyond using
the formulas. You should understand how the formulas may be obtained or
justified. With regrets, you may take several high school and college
mathematics courses without covering the simple formulas and methods for
money computations. That leaves you unable to compare precisely different
options for earning, investing or borrowing money. Slight differences
between different options may cost you years of work. Caveat
Emptor. Understand the origins of formulas in money computations
and beyond, helps avoid costly errors in their use. Again,
algebra at full strength involves the thought-based derivation of formulas,
that is, of explanations why they work.
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Beyond money computations and simple formulas for areas,
volumes, weights and measures, algebraic calculations are not needed or not
commonly used. Most calculations can be done without comprehension of why
they work. But the further study and use of accounting (money
matters), carpentry, engineering, science and computers involves
formulas or calculating methods based on and described with algebra,
geometry, trigonometry and calculus. Here the why is important to understand
the computational theories given and why they work or don't.
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Probability and statistics are further topics in
mathematics, in fashion at the moment. The calculation of odds,
chances and probability involves algebra and may involve a knowledge of sets
and functions for modeling and calculation. Modeling and calculation
starting from assumptions may be done precisely, but an error or doubt in
the assumptions makes all the modeling or calculation suspect. Not all
is certain. The uncertainty may begin with the assumptions made to calculate
probabilities. Statistics is useful in the measurement or
estimation of numbers and the error or variation in the estimates. Estimates
may be given by average. A small variation or none in the estimate is best.
A small variation in estimate may allow you say a given number or quantity
will be near a certain value. In social situations in contrast to physical
situations, statistics for income, productivity, the price of a
car or house, does not concentrate around a single value. Large variation in
a number or quantity that is observed implies that the calculation of
averages give little or no information. And in sports, averages
without mention of variations in performance may appear as a source of
admiration for professional athletes and as possible source for computing
the odds of a team or horse winning the next game or competition. The
initial motivation for calculating probability came from gambling in games
of chance. Calculating the odds of winning might be enough discourage you
from purchasing lottery tickets but for the hope such purchases may provide.
This site author does not purchase lottery tickets, except in social
circumstances where the expectation of losing is offset by the knowledge
that the purchase benefits a charity.
Each item and skill in the further study of mathematics may not
be important or useful by itself. Yet the items and skills in mathematics
altogether, cumulatively, have a greater and greater use in obtaining and
describing calculations, and in describing the calculations and assumptions that
appear in many disciplines. Mathematics courses are designed to problem solving
skills, rote or opportunistic, and to provide a growing knowledge of ideas and
skills that altogether, if not individually, may be useful in further study. If
you follow how to obtain and justify formulas for calculations with money,
the mastery of further ideas in mathematics involves similar and further
ideas. Each method of algebraic reason can be recycled and eventually will
be if you move from topic to topic.
Theories without examples are vacuous
In some mathematics courses, you will find there is too much
theory and not enough examples. Examples, too often skipped, provide a
context. Courses or course design in haste may omit examples to illustrate
and reinforce ideas. Theories full of abstract or remote ideas, given
without examples to illustrate or apply them provide a vacuous or empty
knowledge. Education is not a tidy affair. Your task is watch for examples
and read them if need-be whenever a theory is presented without examples.
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Algebra, Odds & Ends,
1. Hints for Exams
2A. Exact Arithmetic
2B. Fractions Briefly
3. What is a Variable?
4.. Square Roots
5. Straight Lines
6. Problem Solving Methods
7. Trig and Complex No.
8. Complex Applet
9. History of No.s
10. ln(x) and exp(x)
13. Rename the > Sign
14. Problems: Quadratics
15. Problems: Algebra Test
16. Problems: Linear Eqns I
17. Problems: Linear Eqns II
18. Problem Solving Hints
20. Independent Variables
21. Why Logic
22. Why Math
23. The 15 Times Table
24. The 20 Times Table
25. Algebra Formulas
26. On Learning Maths
27. Maths in Biology
28. Navigation +Time
29 Quibble-What is Algebra
30. Logic in Maths
Odd and Ends, Essays
Constant Retirement Rate
Road Safety
3 Strikes Law in California.
Math HOW-TOs
9 Steps in Maths
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
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