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.
-
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.
-
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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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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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