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
Your task is watch for examples and read them if need-be whenever a
theory is presented. Theories full of abstract or remote ideas
without examples to illustrate or apply them provide a vacuous or empty
knowledge.
Skill has to be seen to belived
Skill mastery in mathematics has to be seen
to believed. To that end, learn or teach how-to write and
draw the steps in mathematical figuring or reasoning clearly.
Do not try to save space by doing a sequence of step in one place.
Instead, do or record the steps in sequence on a separate lines to make
each step obvious and verifiable.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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