Logic Skills and Concepts
Site Volume 1A, Pattern Based Reason , 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.
Here are a few ideas and steps for the logic instruction etc of teens and
adults. For course design and delivery, the earlier steps are more
certain than the latter ones. The selection is left for another day.
Step I: Logic for work, school and home:
Logos is a Greek word for thought. In
every discipline including mathematics, signs of rule- and
pattern-based reason, explanations of why, are given by the word and
phrases from this, therefore, thus, because, since, as, gives,
yields etc. Their presence in any line of thought indicates a
physical or thought-based explanation of why this or that should
be.
Logic mastery is a key for enriching skills and
understanding, and a must for easing or avoiding difficulties in school
and work, difficulties due to imprecise reading and writing.
- The chapter Implication
Rules presents two logic puzzles to test or improve your reading and
writing. Each consists of a rule and five questions. Answers are
given. Answers are also provided. The puzzles show the
difference between one- and two-way implication rules.
- The chapter Deception
describes faulty and misleading ways of reason and persuasion. It
describes the hype, hype and hype approach too often used for persuasion
in advertisements and public debate. The practice of deception is
not encouraged.
- The chapter Chains
of Reason describes how to directly use rules one at a time or
chain them together, one after another, for arriving at conclusions and
judgments.
These three chapters on reason develop 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 further description of reason and logic relies on the
method described and offered in these three chapters.
Step II: More Logic for work and school
When ideas in mathematics or another discipline are described instead of
being drawn from implication rules, the role of implication-rule based
reason or logic may be forgotten or not seen.
- The chapter Longer
Chains of Reason indicates the special role of rule-based 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 and the related subject of recursive
definition provide two examples of reason used mainly in mathematical
subjects.
- 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. The phrase when and only when gives another way
of saying if and only if.
- 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.
The study of logic, that is, methods or laws for rule- and pattern-based
thought, has been motivated by the need in mathematics to reach
conclusions. In particular, proofs based on (1) mathematical
induction, (2) the contrapositive, and (3) proof
by contradiction all stem or originate from the conclusion-reaching
needs of mathematics.
Step III: Occurrence Tables and Truth Tables
The subject of logic as it is studied within college mathematics courses,
is often presented as an algebraic (or symbolic) perspective of the
methods of reason.
The algebraic description of logic further allows
algebraic methods for arriving at conclusions, in particular
mathematical induction, to be applied to the drawing conclusions about
rule-based reason and logic. The algebraic description of logic
provides models of mathematical logic. Conclusions drawn about the
models then reflect on the limitations and reach of logical or
rule-based thought in mathematics.
The next lessons present the algebraic perspective. They with the earlier
algebra-free discussion of implication rules and chains of reason give
some preparation for the description of the indirect methods.
The occurrence (or obedience) tables invented and introduced below
identify those situations in which implication rules are obeyed,
disobeyed or not disobeyed. The latter notions are intended to simplify
the explanation of truth tables. An implication rule is said to be true
in the case when it is obeyed or it is at least not disobeyed. An
implication rule is said to be false or not true when it is
disobeyed.
Truth Tables: Here is another viewpoint of implication rules
(material implications) with an attempt to explain and justify truth
tables entries.
Logic Step IV: Methods of Indirect Reason:
The Contrapositive
provides the simplest and clearest form of indirect reason.
The chapter The
Contrapositive (part I) shows the equivalence of an
implication rule with its contrapositive formulation. The analysis is
based on the three notions of a rule being obeyed, disobeyed or not
disobeyed. The language previously used to explain and justify
the entries of truth tables overuses the word true. The introduction of
the three notions of an implication rule if A then B being
obeyed, disobeyed or not disobeyed aims to avoid this
situation. Such implication rule is said to be false in situations
where it is disobeyed, and it is said to hold (or be true) in those
situations where it is obeyed or at least not disobeyed. Finally, the
implication rule is said to be always true in the circumstances of
interest provided it is never disobeyed in those circumstance.
That leads to a discussion of Vacuously True Implications in part
II of the chapter.
The chapter Direct
and Indirect Reason describes and explains direct and indirect
methods for reaching or proving conclusions. Among the indirect methods,
this chapter describes in particular, how an implication rule can be
shown to always hold by (a) showing its contrapositive form always hold
(see earlier discussion) or by (b) looking for absurdities that would
occur if the implication rule did not hold. The second method (b) is more
indirect than the first method (a).
Step V: Logic and Knowledge in mathematics, science and technology
-
Theory of Knowledge - Stories, Longer and longer
-
Formal or Informal Peer Review
-
Education in Mathematics, Science and Technology - All based on
empirical verification and empirical skill development and
verification. But in mathematics we can offer a full thought-based
development while in science and technology, we can introduce the
scientific method and introduce lab equipment, but can only provide a
full-thought based development through visits to the lab and library.
The lab alone is insufficient.
Step VI: Logic and Knowledge
Musings on what to include
Mixing Rote & Thought-Based Development
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Cultivating Intelligence - Why value careful mastery of rules and
patterns, steps and methods, practices, in a repeatable and
reproducible manner.
-
Multiply Kinds of Reason in mathematics - Essay I
-
Multiply Kinds of Reason in Mathematic- Essay II - On the
hierarchical development of rules and patterns, steps and methods, and
practices in pure and applied mathematics (mixed mathematics). What is
proof? What options are there for a thought-based development and
verification of college and pre-college mathematics?
-
Mathematics Instruction in General - Three Goals A B and C to Set
for Student, Supporting those goals and why rewrite the curriculum
-
Operational Viewpoint - Aim for an Operational Command of
Mathematics First.- For students with no immediate interest in the
know-why, a focus on the practice, an operational command of key skills
and concepts may make comprehension later of the know-why easier and
more appealing. The calculus teacher may says to students - learn to do
now and to understand later.
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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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