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Evaluation of the North American Mathematics Curriculum - Hook, Line
and Sinkers
The modern mathematics curricula, say 1955-80 inconsistently
introduced ideas from higher level mathematics but provided a
nearly expert, discipline-based, discipline-centered approach to course
design and delivery, with a few awkward elements. The olde problem of too
many symbols and not enough words in the introduction of algebra was not
recognized and so persisted. The decimal-free nature of modern
mathematics - its lack of dependence on the decimal representation of
real numbers - meant the common use of decimals, required in high school
arithmetic, was not sanctioned and implied the decimal viewpoint of error
control and continuity, a view that lingers with the study of scientific
notation a * 10k for measurements (with 0.1 < a < 1),
was otherwise avoided. The discussion of ratios a :b and multiple a:b:c
also continued in an awkward manner. The sprit of the modern mathematics
curricula was not child-centered. It was discipline centered. It focused
on the elements of mathematics which appeared, which would be needed for
comprehension of high level mathematics in a context-independent
matter. That focus provided a hard route to follow due to the
lack of a clear introduction of algebraic concepts and due to the
avoidance of decimals - the sanction of their use in daily life (weights,
measures and calculations) and the absence of any dependence in the high
school & college development of mathematics. That focus
made learning and teaching harder than need-be. The new fashioned
(context-free) description of functions as sets of ordered pairs
that satisfy a vertical line property appeared too suddenly and too
absolutely. The companion concepts of - how one number
depended on others - and function notation y = f(x) should be emphasized
first. The modern mathematics curricula selection and introduction
of skills and concepts was not optimal. Its introduction was nearly
expert, but not expert enough - too much enthusiam, not enough thought.
In recent decades, factors outside of the discipline
led to curriculum reforms 1989 onward that have ignored and compounded
the earlier difficulties in course design and delivery
First, the end of streaming in course design and delivery, the
merging of course content for enriched instruction into general
instruction added topics not essential into the high school education of
every student. Second, the rich treatment of Euclidean Geometry was
judged too hard, too intimidating for the general student, so it was
dropped - site pages indicate a leaner, minimal treatment of Euclidean
Geometry, one that depends on direct use of logic. Third, arithmetic
drill, practice and correction was considered not important and so
de-emphasized in North America and UK(?) schools in favour of calculator
use and spreadsheet use. But students need to have an automatics,
efficient command of exact arithmetic with whole numbers and fractions,
one that does not require them to reach for a calculator for every simple
calculation, if they are to master algebra, trig, functions and
calculus That is a discipline-based view. Anything less delays or
dilutes high school and college level mathematics - changes the
discipline in a way that earlier masters would not understand - and so
undermines any reason for the study of mathematics, year after year in
high school. So course content needs to be maintained and protected
by discipline experts.
Mastery of the skills and concepts through their ability to do
calculations and follow rules and patterns in a repeatable, reproducible
and hence verifiable manner. That requires care and precision. It
can be a struggle to understand precisely the chains of reason, verbal
and symbolic, in a mathematics text due to steps to large,
Not every one has the patience for it. The high school and college
exposition of mathematics from algebra to calculus may make that
mastery harder than need-be with algebraic skills and concepts introduced
awkwardly. Site pages point to a remedy for that.
In the past, mastery of arithmetic, figuring skills, was regarded
as a sign of intelligence. In brief, it meant a student or a
worker had the wits or ability to follow rules and patterns in a
repeatable, reproducible, verifiable and reliable manner.
But factors not expert in mathematics, the soft science in the form
of psychology and theories of learning may call for critical thinking and
independent judgment but oppose the mastery of rules and patterns, alone
and in sequence, for the sake of repeatable, reproducible, verifiable and
reproducible results. That points to a conflict or inconsistency
between expert views of mathematics and hard sciences - how university
professors in the hard sciences and mathematics might value and define
their disciplines - and the anti-rule, anti-bureaucratic but still
bureaucratic applied and developed theories and practices for education
reform. Factors who are not expert in mathematics may influence and
control course design and delivery
Mathematics course design and delivery should identify what skills and
concepts are essential to provide a curricula which is learn but
effective. The advance in site pages for the exposition of the
mathematics suggest how. Those advances and the question
how to select topics to interest students - can we design a sequence of
courses so each one if it was the last taken by a student, would leave
a satisfying image of the discipline and with or through that an
invitation to further studies?
provides mathematics education committees in schools and colleges with
opportunities to make learning and teaching simpler and more
effective. Course design and delivery with some variations may be
built on the collection and development of appetizers and lessons easily
understood and repeated by teachers and effective in the classroom.
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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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