Chapter 17, Objective Ways
Trial and Error Discovery
Yours Objectively
Rule-based reasoning is in part subjective and in part creative. Results
depend on the rules and the facts you choose to use or happen to know.
Rule-based reason becomes subjective when the starting points and
assumptions required are not obvious.
Each of us may make a different sequence of decisions. Each of us is a
person with a limited and different knowledge of what has been done
before. As a result we may be ignorant of methods helpful in the
situations we face. As a result, no matter how rational (or deductive) we
try to be, the decision of what to do, or how to do it is subjective.
Knowledge of what others have done or tried to do may help and guide our
actions. Without previous know-how and knowledge, we need to improvise
and look for patterns, rules and recipes that work. This is where the
search for objective reason, or simple rules to follow, becomes
subjective. Each may have a different idea of where to look. This is
because each person has a different background and varied preferences.
The road to objectivity is in part subjective and creative.
When new situations occur, experimentation, with a little caution to do
no harm, is needed. The discovery of new objective processes (recipes and
guidelines) is subjective. It depends on our experience. Again, each of
us has a different idea (or no idea) of what to do. When approximations
or errors are made, the results in question become more subjective. They
depend on the choice of the approximation. In time, conventions or
standards may be adopted to govern the approximation and make the
procedures in question more reproducible.
Rule-based reason is incomplete. The rules or guidelines for handling
some situations are missing. People make rules or see patterns in the
situations familiar to them. New situations may go beyond the reach of
the suggested patterns and rules. Uncertainty begins where previous
rule-based knowledge ends. So rule-based reason has limits.
- Different people get different conclusions. Subjectively, that is,
from varying individual choice, interest and experience, each may use a
different subset of the available implication rules. As long as the
people in question can correctly describe their reasoning or procedure,
their creatively found results are objective. That is, a result or
conclusion becomes objective, more precisely repeatable, if the
instructions to get it, once written, can be followed successfully by
others.
- Different starting points or assumptions may lead to disagreements
between logical and otherwise objective people. Talk between disagreeing
parties can sometimes get people to agree on a common starting point for
their reason. If this occurs, the rule-based reason in question can be
followed and repeated, so that reproducible results and conclusions are
obtained. That is, subjective reason may become objective, or at least
agreeable to several parties. In some circumstances, but not all, we can
get firm, sure and reproducible results and conclusions.
The Discovery Process
Rules and laws often begin as a suggestion or a possibility that someone
wanted to accept or investigate. The direction of exploration and one's
inclinations on what to examine is subjective. It depends on personal
experience and knowledge. But the reproducible and repeatable results of
any such exploration and examination are objective.
Putting patterns or implication rules together is like putting together
the pieces of a jigsaw. Except that some, if not all, of the pieces of
this jigsaw puzzle are upside down; you are not sure that all are from
the same puzzle, and you are not sure that some subset of them will form
a complete picture. The organization of the pieces and the discovery of
results from them take time.
The process of discovery of a new idea and the putting of all the pieces
together sensibly may be long and painful. It may require imagination or
acts of desperation. At many points in the discovery process, we may want
to quit. Moral may be low. The chances of success seem low. The moment of
discovery puts the past in perspective. Once the way to accomplish
something is discovered, repetition often appears trivial (easy). This
appears as an anticlimax. A large difficult step has suddenly become
small. It is time for a new problem.
The discovery of useful new chains of reason is not obvious, especially
when you are the first to explore the problem area. Once found, the
repetition of a chain of implications is often much easier. For example,
all of us have looked for a lost object. The lost object is in the last
place we look, and once it is found we may think to ourselves I
should have looked there first. Beforehand, we could not have known
better. Similarly, when we look for useful chains of implications, the
search itself may be difficult or harder than we wanted, thought or
expected. Once the useful chain has been found, we may feel with our
usual hindsight that the search could have been simpler. The split or
dichotomy in our thoughts is as follows. What is unknown is hard. What is
known is easy or trivial. The whole process of learning and exploration
consists of making a hard puzzle easier, or a previously unknown path,
easy to follow or repeat.
With implication rules, when we try to reach an attractive or useful
conclusion that has not been proven before, there is no guarantee that we
can reach the desired end. Implication rules are clues in a detective
mystery. In unfolding or unraveling such a mystery, the trail of
implications can lead to unexpected places or nowhere. Persistence,
intelligence and luck are required.
|
|
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:
-
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.
|
|