Site Material: Key Notes and Themes
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Online chapters on logic and pattern based reason may
entertain and inform. Precision reading, writing and speaking are
useful in work and studies. The logic chapters may
lead to them. Good luck.
To improve your work and study skills, start with with
math-free logic chapters. Read
them in any order you like. Logic mastery may teach you to read and
write more carefully. That care will ease or avoid difficulties
and confusion in studies and work. The logic chapters
also hint of the role of logic (rule-based thought) in connecting and
organizing mathematical skills and
concepts.
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Words have missing in algebra from the first use of
formulas to calculus. Online Chapters 8 to 14 in Volume
2, Three Skills
for Algebra, and its online postscript what is a
variable show how and doing so enrich, clarify and extend
skills and concepts for students and teachers, novice to expert.
Chapter 14 in introducing the direct and indirect use of
formulas, and presenting, comparing and contrasting
arithmetic and algebraic solutions for the indirect or backward
use of formulas verbalizes, hitherto unifying themes in secondary and
college level mathematics. Teachers: The determination of
proportionality constants for direct, inverse and joint variation
etc would provide an occasion for the annunciation of these
themes.
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Fraction skills are a must for algebra. Words problems can be
difficult. Solving linear equations in one or several unknowns may
be difficult. The site area solving linear equations
digested in full may be used to ease or avoid phobias and enrich or
extend skills and conceptsvery early in secondary school if not in
primary school. Recognition that words problems in secondary I
and II mathematics which require the writing of one equation in one
unknown are equivalent to a system of equations in essentially one
unknown will avoid the absurdity of doing or requiring mentally,
operations best done with algebra on paper.
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For calculus, a geometric preview,
and online chapters 2 to
6 plus 11 to 18 in
Why Slopes and More
Math may speed studies and give motivation or a context for
the study of slopes and factored polynomials before calculus. This
material shows students and teachers how to make the full-strength use
of algebra more accessible! (Question: Where is the modern
mathematics curricula which introduced similar ideas in all or
part.?)
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The law of signs and the existence and properties of complex
numbers may be learnt without comprehension in secondary and
college mathematics. Yet in Euclidean plane, a definition of addition
of points with rectangular coordinates and a definition of
multiplication via polar coordinates would lead to a geometric comprehension.
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What comes first, the chicken or the egg? Before modern
mathematics hatched, matters were met in a less formal manner,
but still understood. Can the egg reappear in primary instruction?
Modern mathematics and modern mathematics curricula may build or derive
algebra and geometry from assumed patterns or axioms for real numbers
(or sets) and the codification of geometry via coordinates.
Before this chicken hatched, that is the codification, visual geometric
arguments and tacit counting principles suggested manipulatively
or hands-on, the properties of numbers whole to complex. There-in
lies the egg. This site treatment of number theory points to a high level
development of the chicken from the egg. account. Yet in
retrospect, the counting, geometric and decimal strands of primary
school school might be organized and rephrased so that hands-on
experience with manipulatives, a primary school representation of the
egg, leads to a thought-based development of the axioms. Poincare might
appreciate that. The that may provide the substance of a forthcoming
site area.
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In mass education, the ends of mathematics instruction are obscure,
not yet fully transparent. The ends of mathematics instruction need to
be defined and clearly explained, so there more to learning and
teaching than preparing for the next final examination. Calculus, the
key to the comprehension of methods and formulas in accounting,
engineering, science and technology, provides one end. But
development of practical numerical and quantitative skills and
illustration od reason, inductive to deductive, provides a few further
ends in societies where numerical and quantitative skills and concepts
for better or worse appear in the home, in buying and selling, in
technical trades, accounting, technology, engineering and
sciences. Mathematics itself may be out of context in societies
where formal measurement systems for distance, time and quantity are
recent encounters. Apart from that in pollution-age societies,
students en mass may be best served by a lean path preparing for
calculus, which weaves in or also emphasizes practical skills and the
mastery of skills and concepts, one at a time and one after another,
alone or in combination, while eliminating artifacts
(evolutionary appendices) inherited from before and developing skills
and concepts in a spiral, yet just in time manner. That being said, the
form and content of course design from counting to calculus could be
revisited, Different paths or expositions compared and contrasted to
make the hard easier, to see the benefits and limitations of different
paths, and to take into account physical and mental difficulties. That
will require many heads.
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Making the hard easier may lead to the return in leaner form of
topics deemed to be too hard for student egos. In my
high school days 1966-9, I suspected difficulties in mathematics
came from steps too large and words missing in the introduction of
algebra. Then, a decade and a half later, in fall 1983 as an
instructor, I invented three lessons three skills for
algebra, why slopes and
two logic puzzles to make
algebra alone & in calculus simpler to understand and
explain; to strengthen reading, writing & reasoning; and to
hint at the role of logic in mathematics. Those lessons and
further site ideas stem from inductive principles for
course design and delivery met in 1981 outside in mathematics; and
from the earlier example of guest speakers, mathematicians and
one physicist 1975-80 at McGill University. Those speakers made what
was hard, easier.
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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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