A Theory of Knowledge
Science and technology develops from hypotheses (rules
and patterns) for testing directly or through the consistency of
implications (chains or reason) with observations, all in an empirical
repeatable and reproducible manner. The latter may imply the
limits of rules and patterns. Mixed or applied mathematics too is
an empirical subject built on assumed numerically and geometrical rules
and patterns - assumptions drawn from experience and consistent with
the most part with experience. While historical and
pedagogical path to the thought-based development of mathematics
skills and concepts goes through synthetic (coordinate-free)
drawings in geometry, the empirical limitations of the latter path
appear in diagrams whose faults are explained with the aid of analytic
geometry, and the empirical nature of pure mathematics appears in the
absence of an absolute basis for mathematical theories rich enough to
represent the infinite set of natural numbers. There are
stories to be told and repeated here about the development and
construction of skills and concepts in mathematics. The telling
and repetition of stories to understand and explain the development of
mathematical skills and concepts in a repeatable and reproducible
manner is most likely inconsistent with post-modern, rule and pattern
-rejecting developments in educational theories favoring subjective
learning and knowledge, and indirect instruction.
We have the ability to follow and present stories on paper and on stage.
Those stories may be fiction or not. Some stories may follow each
other, one at a time and one after another, or in parallel. Each person
has his or her story to tell. Mine is brief since I have forgotten
many of the details. Now the ability to follow and tell stories echoes in
the works of knowledge and fiction met in mathematics, science,
technology and society. Non-fiction is preferred.
In mathematics, each proof or deductive chain of reason in
represents a story or a sequence of stories to be told and
repeated. The telling and repetition of stories or proofs
links and develops skills and concepts in mathematics, one at a time and
one after another, all in a repeatable and reproducible manner.. In
each empirical theory, there are stories to be told and repeated in
the development, construction and testing of skills and
concepts, or skills and concepts, subject to the limitations of rule and
pattern based thought. There-in lies a gamble. So no all
certain. But many of the methods of mathematics appear to be
repeatable, reproducible and hence reliable tools in science and
commerce. So there is a chance, the methods are non-fiction.
Mathematics instruction may be given the task of providing students with
an operational command of the calculating and reasoning or proof methods
in mathematics, pure or applied or mixed, and an eventual awareness of
benefits, origins and limitations of the rules and patterns involved in
the subject and other disciplines. In education, the empirical hope
or hypotheses that a student has an operational command of one area of
proof or figuring can be tested by observing what a student writes or
produces. If a student fails, more instruction or study is required while
if a student passes the test, chances are he or she has master some
mathematics, enough to continue instruction without review. Mathematics
education is an empirical art in which instructor may observe the work of
each student, and provide feedback or correction while the student is
trying to follow the theories and methods of mathematics in a repeatable,
reproducible and objective manner, modulo the limitations of rule and
pattern based thought and processes.
Science, Mathematics and Education
Mathematics is called the Queen of Science. But mathematics is still an
empirical science. Historically, the thought-based development of
mathematics begins began with synthetic (coordinate-free) drawings in
geometry to arrive at conclusions with the aid of axioms (assumed
patterns). But the empirical limitations of the latter path, the
use of drawings, appear in diagrams whose faults are only explained with
the aid of analytic geometry, the use of coordinates. That use
turns the development historical development of mathematics upside
down. Synthetic geometry is now replaced by coordinate-based
geometry - models in drawings are codified or represented by points and
sets of points, models in which the properties of real numbers are now
employed to arrive at conclusions. None the less, the
empirical nature of pure mathematics stems in the origins of its axioms -
assumed patterns which are not given, they are chosen. Here they are
chosen to avoid inconsistencies met in previous attempts to provide a
consistent thought based development of mathematics from axioms for real
numbers - more precisely assumptions about sets that give a model of
mathematics in which real numbers are represented or
codified. Thus mathematics itself has an empirical origin,
albeit one sufficient to imply repeatable and reproducible, and hence
verifiable deductive chains of reason.
Hypothesis (Conjecture) Testing in mathematics: In a
mathematics theory or model based on axioms (assumed patterns),
we test of an statement or assertion by looking for a proof, that
is, a deductive chain of reason starting with and only involving
previously tested or proven deductive consequences of the axioms
(assumed patterns). If a valid proof is found, the statement is
considered to be tested and hence proven. That is subject to the
comments above about works of fiction and
non-fiction, consistent or otherwise.
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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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