Chapter 1
Elements of Reason
Previous: Foreword
To reason with someone often means to persuade them of the need for an
idea or action. Persuasion and reason can take many forms. Methods for
arriving at conclusions and judgments in all skills disciplines are, or
should be where possible, based on the use and recognition of reliable
rules and patterns, and also data to use with them.
Each of us needs to understand fully or as much as is
possible, whatever we might be doing or learning. In reasoning, some
rules and patterns are reliable. Others are guidelines. Each of us
needs to know which is which.
Mathematics, in and besides arithmetic, depends on rules and patterns,
which are used one at time or one after another, to obtain conclusions.
The aim of the next chapters is to introduce and provide an image of rule
and pattern thought in mathematics and other disciplines.
When ideas in mathematics or another discipline are described instead
of being drawn from implication rules, the role of rule-based reason or
logic may be forgotten. But in every discipline including mathematics,
signs of rule- and pattern-based reason are given by the word and
phrases such as from this, then, if, therefore, thus, because,
since, as, gives, yields etc.
About the Next Chapters
The next chapters describes some basic elements of rule- and
pattern-based thought. In particular, four chapters, Implication
Rules, Chains of
Reason, Longer
Chains of Reason and Islands and
Divisions of Knowledge describe basic ideas, keys for reason and
logic.
- The chapter Implication
Rules presents two logic puzzles. Each consisting of a rule and five
questions. Answers are also provided. The puzzles show the difference
between one- and two-way implication rules.
- The chapter Chains of
Reason describes how to directly use rules one at-a-time or chained
together, one-after another, for arriving at conclusions and judgments.
- The chapter Longer Chains of Reason
starts to indicate the special role of reason in mathematics. It
describes, in a very non-mathematical fashion, the concept of induction,
a method used in mathematics to arrive at conclusions. This concept of
induction is an example of a method of reason employed mainly or only in
mathematical subjects.
- The chapter Islands and
Divisions of Knowledge describes how rule and pattern-based bodies of
thought may be organized. Here different starting points, first
principles or assumptions, may lead to the same body of rule-based
knowledge.
In philosophy, the discipline that is literally the love
of knowledge, perhaps an infatuation, Euclid's logical or rule based
arrangement of geometry provided a model for reason. This chapter with
words and images apart from geometry describes the model and the
variations possibly within it.
These chapters develop thinking and reasoning skills needed in daily
life. They provide a standard or model for arriving at conclusions and
making decisions: how to argue politely if you must. They also strengthen
basic skills needed in mathematics, science, technology, writing,
persuasion and communication. Reason and persuasion touch all skills and
all disciplines.
The chapter A Change of Language
introduces the conventional if-then and iff forms for
writing one- and two-way implication rules. The one- and two-way
implication rules in this work have been identified with condition and
bi-conditional statements. But the terminology one and two-way employed
here draws on the present-day common experience of one and two-way
roads.
Next: Chapter 2, Implication
Rules
To Learn More About Logic, see
chapters 26 to 31 in this Volume
Chapters 19 to 24 in Volume 1A,. Pattern Based
Reason . Volume 1A describes the benefits, origins of rule and
pattern based thought, deeds and hopes in greater detail, and still
leaves room for thought. Online postscripts in the Volume 1A site area
discuss further the methods and context for indirect reason in and
outside of mathematics. Finally, Appendices
Story Telling includes a theory of knowledge based on our ability to
collect, invent and tell stories with words and symbols, written, spoken
or drawn.
Multi-Level Logic Skill Mastery - What it Means
Postscript: August 28th, 2011.
Before the study of implication rules of the form A IF B and the form A
IF and ONLY IF B, older children and young adolescents may be shown how
written work can be done and recorded in steps that can be seen as done
or later by the doer, a peer or teacher. Steps observed as done or as
recorded allows work to checked and in that, the domino effect of
mistakes to be seen. In advanced mathematics, the concept of checking
goes further: In derivations and proofes, steps are not only done and
recorded for later inspection, reasons for them, where not obvious, are
recorded as well, again for later checking. However learning to do and
record steps, minus reasons for them, provides the first form of proof -
empirical and preductive - in which steps are checked. Empirical reason
in science and technology goes further by calling for results to obtained
by procedures that are recorded for the sake of confirming that the
results are repeatable and reproducible. So logic occurs in many
different ways.
To repeat, two kinds of logic children and adolsecent may accept or
understand consists of the following:
- Learning to do work that be done and recorded in steps or pieces that
can be seen and thus checked as done or later.
- Learning to follow methods or processes where the steps are not
necessarily recorded, but the results are repeatable and reproducible,
and hence verifiable.
Both kinds of logic are empirical. The former is a special case of the
latter. In it, advanced mathematics and further disciplines which employ
deductive reason in theirs proofs, records both steps and reasons for them
for later checking - that adds some rigour of the deductive kind.
Beyond and besides the foregoing mechanics of empirical and deductive
methods for arriving at conclusions, the underlying rules and patterns
may form smaller and large islands and bodies of rule and pattern based
practices and theory. Different entry points into the bodies and islands
will make entry and exploration easier or difficult. Some trial and
error, or experience, may be needed to find the easiest routes to follow.
For example, site logic and mathematics material includes some easier
entry point or easiet routes to follow, but simpler or clearer routes may
be yet be found. The latter is not surprising. Only time and comparision
will tell. The composition of site material was an iterative affair.
Another iteration will likely improve it. Bon Appetit.
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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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