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
- 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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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
Introduction
Arithmetic
Geometry
Algebra
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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