Multiply Kinds of Reason in Mathematics - Essay II
First, rules and patterns may be accepted because they work in a
repeatable, reproducible and thus verifiable manner. What is right or
wrong is thus clear, or can be checked. The careful mastery of rules and
patterns, one at a time and one after another, with repeatable and
reproducible results, is a sign of intelligence and gives an
operational viewpoint of mathematics. Explanation in mathematics may be
based on giving examples to suggest or illustrate and confirm such rules
and pattern. There-in lies one motivation for rote learning. And some
students will say that there is no need for any thought-based development
as indicated below, since they assume courses are presenting reliable
rules and patterns to follow.
Second, rules and patterns in mathematics may follow and be accepted
since they stem from combining earlier rules and patterns to arrive at
new ones. The thought-based development of mathematics begins with the
appearance or development of the ability to combine rules and patterns to
arrive at results or further rules and patterns, even before the direct
and indirect use of implication rules IF A then B.
Students may appreciate the use of logic or a thought-based development
that gives new results or patterns, but the thought based development or
proof of previously accepted mathematics will be seen as redundant, at
best a confirmation of what has worked before, and not strictly
necessary.
Students, all or most, will see no need for a thought-based
development or explanation of a rule pattern which has worked or been
accepted before.
The combination of earlier patterns to obtain new ones - deductive reason
with or without explicit mention of implication rules - can be
introduced as tool for further skill and concept development. That leads
to a hierarchy of skills and concepts in which later ones depend on early
ones.
Proofs: When a statement is made in mathematics, the question of
whether or not there is a chain of deductive reason leading to it, built
on prior knowledge gives a test. If such a chain of reason exists, the
statement is accepted, and the chain of reason is recognized as a proof.
After an operational command of that deduction, the possibility or
option of an Euclidean style derivation of theorems from a minimal set of
axioms (assumed rules/patterns) - prior knowledge can be mentioned as
technical solution to the chicken and egg question of what comes first.
In the Euclidean-style logical development, derivation and
codification of a mathematics or a body of knowledge, a few key
patterns are assumed. A further pattern is accepted as (judged to be)
part of that body of knowledge if its pass a test, namely, there is at
least one chain of reason employing the key patterns which implies the
further The latter chains of reason provides a proof and give a
further reason for logic
mastery mastery - besides its development of precision writing
and reading, two must for work and study.
Mentioning the possibility of an axiomatic development may be sufficient
for many students - as a logical or axiom (assumption-based) codification
of mathematics take time and effort, and interest too. Whence some
streaming according to interest may be required. However, seeing how
rules and patterns, steps and methods, combine to give further ones
connects course material (mathematics) in a way that helps students who
find that learning with comprehension in preferable or easier than
learning by rote. There-in lies a connections or connections which may
favour meeting and mastering proofs and proof techniques.
The Question of Context and Motivation for Proofs
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Opposition: Students may say its it the job of the teacher to
give them facts and reliable methods. Therefore they view facts and
methods provided by a teacher in preparation say for final
examinations need to be mastered without need for justification or
proof.
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Counterpoint 1. Seeing and even mastering the justification
may be a course objective, one that may be required to answer some
questions on the final examination.
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Counterpoint 2. Seeing how rules and patterns, steps and
methods, combine to give further ones connects course material
(mathematics) in a way that helps students who find that learning
with comprehension in preferable or easier than learning by rote.
-
Counterpoint 3. Justification (proofs) may be required on
final examinations.
The second counterpoint above, namely
Seeing how rules and patterns, steps and methods, combine to give
further ones connects course material (mathematics) in a way that helps
students who find that learning with comprehension in preferable or
easier than learning by rote.
provides a justification for offering proof or deductive connective
arguments and support in the development of Euclidean Geometry,
trigonometry and calculus. But the full, logical codification of
mathematics can be left for post-calculus studies in undergraduate
programs covering or including some pure mathematics.
Further Reading: Logic
chapters 1 to 5 (Français)
in Volume 2, Three Skills for
Algebra introduce the Euclidean logic methods and questions in
mathematics free manner. . The use of logic in the form of direct or
indirect use of implication rules B if A or equivalently, If
A then B, informally or within axiomatic (assumed rules and
patterns) frameworks leads to further rules and patterns to accept and
use. See to the last chapters and postscripts of Volume 1A, Pattern Based Reason,
for a further discussion of consistency questions and indirect chains
of reason in general, and not just in mathematics.
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