Foreword
The physicist Richard Feynman (1918-1988) gave three public lectures at
McGill University in 1976 1979. His work
on physics has been followed by many scientists and students.
In the lectures, partly tongue-in-cheek, he suggested that physics was
based on two easily described operations, namely the addition and
multiplication of arrows in the plane. His description of arrow addition
and multiplication for a general, non-mathematical audience was a model
for the informal, very visual, most adequate, presentation of
mathematical ideas. But he gave it under the guise of describing physics.
And he avoided panic among the mathematically shy by not saying that the
arrows, with their addition and multiplication, represent what pure and
applied mathematicians (since Gauss) regard as the complex numbers.
No mastery of the algebraic way of writing and thinking was required to
understand his live description of addition and multiplication.
When I attended Feynman‘s lectures, I thought his description of arrows
in the plane could be an excellent way to introduce complex numbers. The
chapters on complex numbers elaborate on Feynman’s live presentation,
although their on-paper presentation employs the algebraic way of writing
and reasoning.
With Feynman's energetic presentation as a model, I looked for and found
in 1983, a preview and simple tour of calculus (slope-related
calculations) which likewise required a minimal knowledge of algebra.
Just the definition of a slope to a straight line needs to be understood
to follow it.
The why slopes chapters extend this tour and provide a geometric
motivation for calculus, easy to describe and to repeat without a great
dependence on algebra and without requiring a mastery of the rules of
differentiation, that is slope calculation, for nonlinear functions.
This book is one of three volumes on understanding and explaining
reasoning skills and mathematics. The objective of this volume is to
complement other texts in algebra, trigonometry and calculus. Students
may be able to read the first part of this book during their high school
days and keep the rest of this work for consultation during their college
studies.
The first why slopes chapters gradually illustrate the algebraic or
symbolic way of writing and thinking. The later is employed more deeply
in some later chapters and at full strength in proper calculus courses.
The aim of the first chapters is to provide a simple image-based preview
or review of calculus. In it, dependence on symbols or algebra is kept to
a minimum. The images may help readers to see and physically grasp the
simplest slope-related ideas in calculus. The remaining chapters cover
more topics – see the table of contents. Appendices present the most
advanced topics. Theorems in first courses on calculus are often stated
without proof. The appendices state the theorems and give or indicate the
proofs. This should provide a context for the decimal-free approach
favored in advance calculus or modern mathematical analysis.
This is a book which a student could begin reading in high school and
continuing reading through further college math courses. Material
elementary to advanced is covered.
Alan Selby
Montreal
March 1996
Copyright © 1995, 1996 by A. M. Selby
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
In fall 1983, I gave three lessons to extend or complete the skills of
students starting calculus - recent high school graduates.
-
The first lesson three skills for algebra
gave a remedy for olde gaps in the high school introduction of
mathematics Exercise for students: Find the fourth skill
for algebra.
-
The second lesson two logic puzzles fostered
precision reading and writings skills, and hinted at the role of
logic in maths. Exercise for math and English teachers:
Present this puzzle in senior high school classes.
-
The third lesson why slopes - a geometric calculus
appetizer gave a starter lesson for calculus. It explains why
slopes may be met in high school maths, and non-algebraically informs
students where calculus will head after a coming review of high
school maths and a discussion of limits and continuity.
Chapters 2 to 14 in the 1996 site Volumes 2, Three Skills for Algebra ,
and chapters 2 to 6 plus 14 in the 1996 Volume 3, Why Slopes and More
Maths , present these three lessons and add to them. In doing so,
they provide words and stories to introduce logic and provide a clearer
oral and geometric paths for introduction of algebra in calculus and
earlier high school maths. Newer site area on Solving Linear
Equations may offers a geometric introduction for algebra at the the
junior high school level.
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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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