Foreword
Volume 2, Three Skills for Algebra
Logic, that is a mastery of rule- and pattern-based reason is needed in
all disciplines. In particular, it may lead to precision reading and
writing. If you cannot read precisely, how will you understand and how
will you see errors in your own work or that of others.
The first chapter on logic or rule-based reason shows the difference
between one- and two-ways implication rules. Not seeing this difference
is a source of confusion. Seeing the difference is a first step towards
the better understanding of the implications, suggestions, rules or
information met in daily life, at work and in school or college. The
initial chapters on reason talk about chains of reason, about islands and
divisions of knowledge and about longer chains of reason.The last
chapters on logic connect the ideas of a rule being true or not with the
common ideas of a rule being obeyed, disobeyed and/or not disobeyed. (In
retrospect, there should also be a discussion of when a rule applies or
not. In the latter case, the rule is vacuously true -holds vacuously.)
Altogether, the logic chapters provide a unique mathematics-free
introduction to the direct and indirect definition and rule-based
thinking that appeared in Euclid's work a long time ago (2300 years
ago)
Three Skills
For
Algebra
understanding and explaining
reason and math
Volume 2
by
Alan M. Selby
Ph. D.
Printed in Canada
ISBN 0-9697564-2-9
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Three Skills for algebra are as follows.
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We can talk about numbers and quantities. The words or
adjectives used here may be used in mathematics after arithmetic.
There is more to mathematics than just doing arithmetic.
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We can describe calculations that might be done (or
postponed) with words alone or with an (algebraic) shorthand
notation. The description of calculations that might be done is also
part of mathematics after arithmetic. There is more to mathematics
than just doing arithmetic.
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We can change the way a number or quantity is
computed. Some rule-based reason is required here. There
is more to mathematics than just doing arithmetic.
The first skill, talking about numbers and quantities, use words to
describe them, gives a unique comprehension of numbers and quantities
apart from but parallel to the the shorthand role of letters and symbols
in mathematics. The separation here is needed for a clearer, more precise
understanding of& the shorthand, symbolic, way of writing and
reasoning that we call algebra.
Alan Selby
Montreal, 1995
Three Skills For Algebra - the Best Chapters
The following logic and algebra chapters may make the hard
easier. Chapter 7 will test and enrich arithmetic skills - catch a few errors.
Canadian Cataloguing in Publication Data
Selby, Alan M,
Understanding and Explaining reason and math
Contents: v. 1. Elements of Reason - v. 2. Three Skills
for algebra - v.3. Why Slopes and more math.
ISBN 0-9697564-4-5 (set) -
ISBN 0-9697564-1-0 (v. 1) -
ISBN 0-9697564-2-9 (v. 2) -
ISBN 0-9697564-3-7 (v. 3) -
1. Mathematics–Philosophy. 2. Reason.
3. Algebra. 4. Calculus. I. Title. II. Title: Elements of reason.
III.Three Skills for algebra. IV. Title: Why Slopes and more math.
QA8.4.S44 1995 510’.1 C95-900945-0
Reprinting may lead to new ISBN numbers
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For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
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:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
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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