25. Convergence to Axiomatic View
Ends and Values - a matter of choice
The ends and values of mathematics education, or logical and quantitative
skill development may vary between students. Teachers who have been
suddenly assigned a mathematics courses, despite a lack of background in
mathematics or a quantitive discipline, may not value the logical
development of mathematics. For many students and teachers, mathematics
appears to be collection of facts and methods to learn and teach without
any attempt to obtain or provide a thought-based development.
One grade 8 student on observing my attempts to explain and justify
mathematical methods instead of teaching them as facts told me that
mathematics teachers were hired only to present mathematical correct
methods, and thus I was doing my job properly, the methods I was covering
would not need explanation nor justification. For many students, the
notion that mathematics has a logical structure, one in which ideas are
developed and derived in place of being given, is odd and not necessary.
Indeed, they may be partially correct. The common know-how in mathematics
may be met and mastered with comprehension or thought-based development,
but for students or the common person in the streets, the take-home value
of an operational command of counting, figuring and measuring skills with
take home value in a repeatable and reproducible manner is more important
than full or partial comprehension of why methods work in the first years
of mathematics education and for students who may or may not go further
in mathematics.
The site approach and Choice
Logic chapters 1 to 5 in Three Skills for Algebra end with a discussion
of Islands and Divisions of Knowledge. The latter provides a metaphor
for the organization of mathematics and the possibility of having
different starting points for its development.
Site pages show with two or more paths how the existence of real numbers
and their arithmetic properties can be derived from common practices
assumptions about numbers and geometry with maps and plans. The
demonstrations appear to be empirically and pedagogical sound, given the
need to introduce skills, patterns and even axioms in an inductive
manner. Site pages also provide a systematics introduction to algebra, or
the shorthand role of letters and symbols.
Nostalgia or attraction to the rigour of modern mathematics means the
demonstration were written in a thought-based manner with as much rigour
as possible. But there is a difficulty. Too much explanation may
overwhelm skill mastery. Moreover, mastery of skills with care to avoid
the domino effect of errors has great take-home value in which full or
partial comprehension of why is optional. The foregoing suggests ends and
values for instruction that support a rigourous development of skills,
step by step, because of the take-home value with explanations why being
available and present where they do not overwhelm.
Site pages are part of a two level approach POMME. The first level and
part of the second are dedicated to providing skills and concepts with
take-home value, by rote if need-be. The second level, what is left, is
dedicated to a thought-based development that does not begin with the
modern mathematics mid-way axioms for secondary mathematics, but implies
them. See site slow paths, computational and geometric, for the
thought-based development of numbers and their properties from counting
to the properties of real and complex numbers. The paths may not be given
in classes where students have mixed ends and values - some wanting
mathematics with take-home value only - some wanting to continue onto
college programs in disciplines requiring or best taught with a command
of calculus. The second level in full, as presented here, values thought-
or pattern-based development of skills and concepts as possible
preparation for college studies in mathematical fields.
The thought-based development of numbers and their properties from
counting to the properties of real and complex number, with the
subsequent assumption of those properties as axioms for the further
logical development of mathematics implies a partial convergence of the
site two level approach POMME for quantitative and logical skill
development with modern mathematics curricula.
The modern mathematics curricula I saw began well at the start of senior
high school mathematics, but soon departed from pure mathematics with the
employment of a diagrams in the introduction of trigonometry, analytic
geometry, and calculus to develop methods and prove theorems. The
diagram-free development, a possibility in university mathematics, would
be too difficult and have no context in senior high school mathematics.
Whence some departure is needed - in for a penny, in for pound. The site
development provides a departure in a two-level manner, with one level
focusing on empirical rigour in skill mastery and the second level offer
a thought-based development consistent first the need to sanction and
extend common skills and know-how, with numbers, maps and plans.
Modern Mathematics Curricula,
In the modern mathematics curriculum, circa 1955-1990, the existence of
the real numbers and the satisfaction of above properties were given as
assumptions or axioms. That provides a simple starting point for a
logical development of secondary and college mathematics. A justification
of the axioms might then be seen by students who enter mathematics
studies in university. In particular, assumptions for set existence and
"safe" set construction provide an axiomatic codification, Euclidean
style, for pure mathematics. For rigour, the approach sould be context-
and diagram-free, a rigour not possible before university level studies
in pure mathematics. As said, the modern mathematics curricula depart
with the employment of diagrams in the introduction of trigonometry,
analytic geometry, and calculus to develop methods and prove theorems.
For all students, and many teachers, axioms for real numbers and within
them, rational numbers, integers, natural numbers and whole numbers, the
axioms will appear and will have to be accepted without explanation. But
the axioms were not chosen to continue and sanction common knowledge and
practices with decimals and diagrams which would have had take-home
value. The axioms for real numbers provided a view of numbers that did
not explicitly sanction and support common skills in counting, figuring
and measuring with maps, plans and decimals. The modern mathematics
curricula was not designed to meet the needs of students who would have
benefited from mathematics with take-home value. The modern mathematics
curricula was designed to prepare students for college programs that
required calculus or beyond, with context-free development being an
objective. Axioms and further development of mathematics did not sanction
earlier number skills and sense with fractions and decimals.
For many students and many of their teachers, the modern mathematics
curricula was further flawed in that the secondary level axioms were
described algebraically with out a systematics introduction of the
shorthand role of letters and symbols. Whence the deductive axiomatic
development of mathematics was beyond the reach of students and teachers
for whom the algebraic way of reasoning with letters and symbols on paper
was not a natural talent. The slower and more detailed systematic
development of algebraic reasoning in site pages points to a remedy, one
that requires less natural talent.
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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:
-
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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