Chapter 1, Introduction
To reason often means to persuade someone of the need for an idea or action.
That someone could be yourself. In the latter case, reasoning may mean following
a line or pattern of thought to arrive at a conclusion, action or decision.
Persuasion or reason can take many forms. There are fair and unfair ways of
persuasion. There are sensible and absurd ways as well. Methods for arriving at
conclusions and judgments in all disciplines are, or should be where possible,
based on the use and recognition of reliable rules and patterns. Where ever
there is a presentation of ideas, there is an element of reason or persuasion.
Reason and persuasion are met in the home, in the print and television media,
in the classroom and in the work place. Rule-based reasoning, that is logic, and
departures from it can be described in and outside of mathematics. The
recognition of rules and patterns, methods with repeatable, reproducible and
thus verifiable results, provides a basis for science, technology and even
accounting.
The first chapters on reason give two logic puzzles to show how rules and
patterns can be used to arrive at conclusions or judgments in all subjects,
mathematical or not. Logos is the Greek word for thought. The puzzles
show the need and so reinforce the ability to precisely read and understand the
statements of rules, patterns, instructions and definitions. The two logic
puzzles in particular show the difference between one- and two-way implication
rules
A one-way implication rule says that when one event occurs, so
should another. A two-way implication rule says that when either of two events
occurs then so must the other. The terminology of one-way and two-way
implication rules may be new to this book. It is a plain language replacement
for the more traditional phrasing which speaks of conditional and
bi-conditional statements.
Not seeing the difference between one- and two-way implications or
suggestions is a source of confusion and false expectations in everyday life,
contracts, instructions and technical areas. Recognizing the difference between
one-way and two-way rules gives an initial step in mastering rule- and
pattern-based thought. Seeing how reliable rules and patterns can be used
one-at-a-time or one after each other to arrive at conclusions gives another
step.
In mathematics courses, logic is often met as the algebraic or symbolic
description and analysis of rule and pattern-based methods used in the
discipline (math) for arriving at conclusions. Some rule and pattern reasoning
methods developed in response to the conclusion reaching needs of mathematics.
The last chapters in this work introduce the algebraic or symbolic
description of logic while leading to an explanation of direct and indirect
methods of reason. The description innovatively employs the simple notions of a
rule being obeyed, disobeyed or not disobeyed, or never disobeyed to
clarify the technical truth-table description of one-way (material)
implications. The very last chapter describes the direct and indirect chains of
reason and persuasion met in mathematical proofs. Indirect methods are also of
service perhaps in the writing and resolution of detective and mystery stories.
In all fields of endeavor and inquiry, the main obstacles to the use of
reliable rules and patterns for arriving at conclusions lie first in
their identification and second in the identification of reliable
information to use with them. To understand or cope with these obstacles, a
knowledge is required of the origins of rules and patterns in daily life as well
as in science and technology. Science, engineering and technology have
empirical, that is experience-based methods, for coping with or circumventing
the two obstacles. Here rules, patterns and procedures which give repeatable and
reproducible results appear to be the most reliable or trustworthy, although not
always optimal. Some rules and patterns appear to be more reliable or secure
than others, but not all is certain.
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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:
-
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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