Volume 1 Elements of Reason
Foreword
Volume 1, Elements of Reason
The first part Pattern Based Reason (Volume 1A) of this work
Elements of Reason describes rule and pattern based thought and
processes in daily life, society, science and technology. Reliable rules
and patterns can be followed one at a time or one after another to obtain
conclusions or results. Not solved is the problem of identifying reliable
rules and patterns to employ. Instead, the empirical method of coping
with this problem is discussed.
Elements
of
Reason
understanding and explaining
reason and math
Volume 1
by
Alan M. Selby
Ph. D.
Printed in Canada
ISBN 0-9697564-1-0
|
Rule and pattern based thought and processes touch many arts and
disciplines. Awareness of the difference between one- and two-way
implication rules will
improve reading, writing and argumentation skills. Students of critical
thinking, persuasion, philosophy, mathematics, science and technology may
find this first part worth reading.
In both arithmetic and logic, rules and patterns if followed carefully
lead to results which are repeatable and reproducible, and thus
verifiable and objective: two individuals following the same rules and
patterns with the same data or in similar circumstances should obtain the
same or similar results. Arithmetic and deductive reason are but examples
of verifiable rule and pattern based thought or processes.
Verifiability, repeatability and reproducibility form a basis for the
appreciation of, if not reliance on, rule and pattern based thought and
processes. This appreciation should not be too firm. The identification
of reliable rules and patterns, or reliable data to use
with them is not certain. Further, where rules and patterns do not apply
mechanically, there is room for thought. Still, verifiability,
repeatability and reproducibility may provide a basis for the common
knowledge and informal mastery of a subject.
The second part Mathematics Curriculum Notes (Volume 1B) is
for teachers and advanced students of mathematics or a quantitative
college discipline. This part describes simply yet precisely, the role
of rule-based reason, that is logic, in providing a thought-based
framework and codification for mathematical thought. This second part
further describes how an inductive educational philosophy provides a
context for math and logic instruction from primary school to college.
Ideas which are easily repeated and understood may provide a common
knowledge of mathematics and the rule-based reason sufficient for a
more formal and rigorous comprehension.
This two-part work and its the companion volumes Three Skills for
Algebra Why Slopes and More Math stem from a project to
write a single book, namely Ideas that
Might Count for Education, Reason and Mathematics (1994). That
single book (no longer available ) was written and distributed. It
covered a vast number of topics. Some of interest to one audience but not
to another. With further writing and rewriting, this first endeavor was
divided into three volumes, the first of which, the one before you, was
divided into two parts. Writing for some is an iterative affair.
The initial aim was to report some unique idea, innovations, for math and
logic instruction. These ideas or lessons had worked well with college
students, shy or curious about one or both disciplines. But in writing
and rewriting, the aim became wider. The possibility of a consistent and
coherent scheme for math and logic instruction from primary school to
college was seen and explored. The scheme is comprehensive save for the
treatment of geometry. How to fit or emphasize Euclidean geometry in the
curriculum is not covered.
Formal mathematics can be difficult to follow for students who fail to
grasp deductive thought and the symbol-based algebraic way of writing
and reasoning. The latter like arithmetic is better seen and written
than spoken aloud. Symbols like pictures can be worth a thousand words.
Words have been missing to explain the role of symbols in providing the
shorthand notation of mathematics or its algebraic way of writing and
reasoning. The latter consists of recording and developing thoughts on
paper at least for those among us afflicted with a short or too
forgetful memory.
The absence of a verbal culture to introduce and explain the algebraic
way of writing and thinking leaves its mastery to immersion and osmosis.
Comprehension depends on one's aptitude for learning some basic ideas by
immersion. I am in the radical position of suggesting that a certain
change is possible and desirable. This work and its companions suggest
how. They have yet to be formally peer reviewed and so should be read
with caution. The discussion of math and logic instruction and the
discussion of reason and persuasion are both fraught with controversy.
Scrutiny or critical examination of this work may lead to its refinement.
Alan Selby
Montreal 1996.
December 2011 Postscript
Site chapters and steps now stands at
the sharp edge of mathematics education reform. Site material stems
from olde and continuing gaps and inconsistencies in ends and methods - there was no pleasing all.
The essay which way to
go "lightly" introduces a more detailed, five phase framework and nearly plain-language remedy.
Phases 1 to 3 focus on skills of value for adult or daily life - precision in reading-writing-figuring included.
Phases 4 & 5 focus on calculus and preparation for it. Many university programs demand calculus. Preparing for
it has value in senior high school science too, and some value for trades-professions not taught in university.
The framework addresses and remedies all the difficulties identified above, and implement most of the ideas
in the subvolume 1B, Mathematics Curriculum Notes. The framework being done sets the stage for
yet another consolidation of site material.
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.
|
|
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:
-
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.
|
|