A New Mathematics Curriculum
Volume 1B. Math
Curriculum Notes, 1996, begins with progressive or inductive
principles for observable and verifiable skill development, identifies
barriers to skill development and then offers ideas to lower the
barriers and to make skill development easier and richer, ideas
implemented in site pages.
Volume 1B, Mathematics
Curriculum Notes, begins with inductive principles for the step by
steps in arts and discipline where skill mastery is the aim.
The leading chapters in 1B describe and reflect on mathematics education
difficulties - nuances or small inconsistencies - that affected the
modern mathematics curricula 1960 to 1990 in Canada and the
USA.
- Mathematics education world wide may be progressive in part, but not
full. In particular, from first use of formulas to calculus, the
shorthand algebraic roles of letters and symbols is a great mystery
to more than need-be. The fault lies with conventions that avoid or
do not use words in the introduction and rationalization of that
shorthand role. This algebra guide points
to remedy, one that is continued in the site ideas for senior high school
mathematics and calculus.
- Since the advent 1990 onward in North America of skill-adverse
educational theory , theories that say true knowledge is located in the
mind apart from any skill development or focus, the exact and
efficient command of arithmetic is not provide in most primary schools
and not emphasized in secondary schools. The site area Help your Child/Teen Learn
identifies 18 Booklets for preschool to grade 3, and grade 4 to 8, to
help parents how to check and if need-be develop primary school
mathematics skills, arithmetic included, in a lean and efficient
manner. The same booklets show primary teachers what skills and
concepts they need to include. That should provide a lower bound for
primary school mathematics. There would be harm in completing the grade 7
or 8 booklets before the start of grade 7 or 8.
For primary and earlier mathematics instruction in the process of
trying to identify what should be taught and when, I found 18
books, eleven for preschool to grade 3, and another seven for
grades 4 to 8 which provide a lean but sufficient preparation for
high school mathematics in progressive manner - many students should be
able to cover booklet by the end of grade 6 or 8 before the site
program for junior high school mathematics begins.
- The algebraic way of writing and reasoning requires a coherent and
efficient command of arithmetic with whole numbers, fractions and
rational numbers. The site arithmetic guide
includes most of the usual skills that requires, but also adds a few
methods or skills that will complete the command or its
development.
- Late primary school and early secondary mathematics may
introduce the use maps and plans without and then with coordinate systems
for squares and then points. Showing students how to draw and use
figures and plans drawn to scale on a map for the measurement of angles
and distances, and areas provides a practical introduction to
geometry. Except for the few instances where students may survey
and measure real-objects, geometry for the most part is done on
paper. The methods of drawing figures or plans to scale provides a
very practical method for solving problems even before any formal mention
of similarity theory or trig functions. To learn more, see the
site geometry guide, and
the first third of the site section Maps,
Plans, Similarity & Trig,
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Goals: Mathematics and Logic Skills with Take Home Value)
Logic and money mastery is a
must for consumer self-defense.
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Arithmetic skill development in
showing students that a mistake in one step of a calculation
or process leads to further errors implies the care
or attention to detail needed to ease to avoid many
difficulties in work and studies. Arithmetic skill development
done well introduces or provides ends, values
and methods for work and study.
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Contracts, agreements and instructions at home,
at work and in studies require precision in reading and
writing. Many contracts involve money matters. The
in-school mastery of arithmetic alone or with some algebra,
and too, would have immediate and long-term take
home value for students for understanding and negotiating or
avoiding agreements.
Logic and
arithmetic,
skill development together should lead learners to care and
greater precision in reading and writing. Thus logic and
arithmetic skill development has take-home value.
Arithmetic mastery by rote if need-be is possible in the last
years of primary school and the first years of high
school. Arithmetic mastery should be pushed in secondary
school to include all likely money matters students and their
families meet now or later. With logic mastery
there is no harm in trying to develop it in all or part
as early as possible as long as that is done with the
understanding that what is hard at nine years of age
should be simpler at fourteen or sixteen years. For
self-defense and sharper skills, there is also no harm
and some benefit in understanding the origins, benefits and
limitations or rule and pattern based reason.
Reference: Volumes 1A and 2 and this arithmetic
guide.
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The foregoing material accompanied by these
ends, values and methods for work and study provides a very solid
base for senior high school mathematics. Logic mastery itself
may be part of senior high school mathematics - all depends on the school
system and student ability.
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Senior High School or
First Year College Mathematics - three ends or three
bases for further instruction
A first
common, base part gives
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a natural stopping point for students who would
like to would end their mathematics, with some topics and skill
that have take-home value - serve common need - while a quick
view of the role of logic in mathematics. There is more to
mathematics than being given a method and data to use in
it; and
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a base for further studies for students who plan
to pursue intermediate or advance studies in mathematics,
science, engineering and commerce at an intermediate or
advanced level.
This second
middle part gives
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preparation for a light form of calculus.
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a light form of calculus sufficient as end in
itself, or as an
appetizer for those going on to the strong form
This third
and last part (incomplete) gives or will
give
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Calculus with proofs
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preparation for calculus with proofs.
This part includes rigorous proofs (elementary for
mathematicians at least) of all the usual theorems from a decimal
viewpoint.
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The site introduction to
logic in chapters 1 to 5 of Volume 2, Three Skills for Algebra, is
mathematics-free except for a wordy introduction to mathematical
induction - the coverage of which can be left for later. Avid readers in
school and out may enjoy the exposition. What is hard at eleven may be
easier at fifteen and easier still at seventeen. The foregoing
logic chapters may be covering in the development of reading and writing
skills. These could provide a firmer base for senior high school
mathematics before or after the latter begins.
In recent years, North American course design has omitted Euclidean-Geometry
from senior high school mathematics, nominally because it is too
difficulty. The site coverage of this topic employs only direct
use of implication rules and develops the topic in a lean manner, that
sufficient to support a geometric development of trigonometry and
complex numbers. To give students a light sense of the role of logic in
mathematics, this treatment of Euclidean Geometry may be included in
the common part for all, or the middle part for fewer. I would
prefer the former. Strictly speaking, the operational assumptions
tacit in the use of maps and plans with coordinates provide an
alternative base for the introduction of trigonometry and the
development of complex numbers. The Chinese square dissection
proof the Pythagorean avoids the need for harder Euclidean Geometry
style proofs.
The site introduction of complex numbers gives a visual or
geometric introduction of complex numbers in which the commutative,
associative and distributive laws or properties are obtained from
corresponding properties of real numbers and the observation that image
under a rotation of the midpoint of a line segment between two endpoints
is the mid-point of the line segment between the image of the two
endpoints. This introduction can be done at the high school level of
mathematics rigour before the development of periodic trigonometric
functions. The early and trigonometric free development of complex
numbers shortens the exposition of periodic trigonometry functions and
immediately implies trigonometric formulas for the dot- and
cross-products of points or vectors in the plane. It is bit of
mystery to me why this shorter route did not appear in the mathematics
curricula, earlier.
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Remedy for Inconsistent Skill Development:
The modern mathematics curricula of the 1960s introduced the
properties of real numbers as axioms - assumptions that came
from no where. While they required decimals in calculation and used
decimals in illustration of advanced ideas, course design and
textbooks did not explicitly sanction the use of decimals. That
separated the common knowledge of arithmetic and the decimal
representation of whole, rational and irrational numbers.
Secondary mathematics education depended on decimals but not
sanction nor mention them. The remedy for that in the
third and last part of the site proposal for senior high
school mathematics is to imply the properties from geometric
practices present in the use of maps and plans with
coordinates. The provides a low level route sufficient
and accessible to advanced students at the secondary
level. Students who take pure mathematics courses in set
theory may see the higher level route.
Remark: A thought-based development of skills and
concepts is possible and should be available as a reference for
gifted students in primary or secondary school, and may be
included in advanced high school mathematics.
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The third and
last part is presently incomplete. The site coverage of
functions and proofs for the theorem have yet to be embedded into third
part. Sets and set notation remains to be used. Set notation is
useful in the description of intervals and functions. It is useful too
in the development of counting methods in combinatorics or the for sake
of probability calculations. It useful in the exact or precise
description of probability skills and concepts. The operations of
intersection, union and complements correspond to the operators AND, OR
and NOT in logic. Set theory was introduced with great keenness in the
implementation of modern mathematics curricula. Presently I am
satisfied with the arithmetic, algebra and geometric guides of the
lower bound for mathematics course design described above. They
prepare well for senior high school mathematics. But the optimal form
of the senior high school mathematics is still open to debate. While
the mathematicians involved in the 1950 to 1960 discussion of education
focused on the preparation of students headed for university studies in
mathematics, science and technology - the cold war had started, the
question of how mathematics education may serve common needs was not
addressed. Mathematics instruction should serve common need to
the greatest extent possible, and put the service of common needs
before preparation for university mathematics. The question of
how to end mathematics education in manner that its leaves a good
impression and does not become a source of discomfort is remains to be
addressed.
Lamp an earlier program, these
Mathematics education
essays, and site volumes all provide prequels to the above
effort. All being said, for the last four decades, the
professor of mathematics due to the competitive environment in which they
live and are formed spend five minutes or less thinking about the content
of mathematics. It remains to be seen whether or not the technical
changes proposed here will be of interest to university professors and
within the reach of mathematics education professors.
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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:
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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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