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Extrinsic Origins [ Back ]
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Extrinsic Origins
Mathematics has an extrinsic or external origin. Over time, we have
learnt to describe physical quantities in terms of coefficients -
counts and numbers - of a unit. Modern mathematics with its assume
patterns (axioms) as starting points for a deductive arrangement
and codification of mathematical operations with numbers - whole to
real or complex, etc - gives an instrinsic (non-extrinsic)
development independent of the extrinsic origins. In the past,
Euclidean geometry with its definition, theorem and proofs provide
an axiomatic model for the rigorous, logical development of
mathematics and further subjects. But the development of Euclidean
geometry depends on generic drawings, drawings inspired say
extrinsic, approximate and precise use of maps, plans and designs
and construction and navigation, and generic drawings may be
faulty. So there has been a movement in mathematics for the sake of
greater rigor and certainty to a more secure intrinsic and abstract
development and organization of concepts apart from drawings and
the geometric and physical assumptions there-in. That movement
provides the content of graduate and undergraduate studies in pure
mathematics. That movement provide motivation for modern (pure)
mathematics curricula in the period 1955-80 or so which in aiming
to represent the axiomatic foundation clearly and properly
introduced some inconsistencies or incompleteness in its secondary
school development of mathematics from arithmetic to
calculus. Modern mathematics curricula did not sanction and
so was inconsistent with the use of decimals in arithmetic, the use
of drawings in Euclidean Geometry to arrive at results, the further
use of drawings in trigonometry and calculus to define and an
analyze calculations. Further more, following earlier
traditions, it expect mastery of the algebraic way of writing and
reasoning by exposure instead of explicit development.
Furthermore, applications of mathematics and even instruction in it
applied subjects (geometry, trig and calculus) require an extrinsic
viewpoint to facilitate skills and concept development. Thus an
extrinsic view is unavoidable.
LLAMP aims for a consistent, accessible extrinsic development of
geometric and quantitative skills and concepts in an empirical and
thought-based manner. Due to the possibility of faulty drawings,
instruction offers a drill and practice based development for
solving a wide variety of culturally relevant problems in routine
and then perhaps more adventuresome, non-routine ways,
repeatable and reproducible, if not well-described, recorded
and observable, for the sake of verification or correction.
Verification and testing of solutions remains an empirical part of
applied mathematics despite and besides all deductions or logic in
it that suggests the methods and results in question. For
students, their fellow-students, their teachers and tutor are
(optional) part of the peer review process present in skill
and concept development during instruction. Part of the empirical
development of science and technology is based on methods which in
practice produce repeatable and reproducible results alongside
theories, dare we call them stories, to describe and connect the
pieces of the practice and to provide a framework for
comprehension absolute not, and further repetition of the
practices. While the empirical development of science and
technology requires labs and equipment too expensive for in school
use, LLAMP provides an operational command of mathematics
that may generated and verified in the classroom, and also
accompanied by a nearly full-thought based, extrinsic, development
of its skills and comprehension. The development will be
nearly full except for tables of values for key functions and/or
the use of electronic calculators to also provide and combine
function values. In schools, the development of
mathematics and its applications may be self-contained and peer
review an immediate possibility
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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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