Three Goals to Set for students.
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A. Master the rules, methods or patterns in arithmetic, algebra, trig
and calculus so that in your hands, they lead to the same results as
others - repeatable, reproducible and hence correct results. If
you belong to a group of students whose results differ after using
the same method to arrive at them, you or the group have problem.
More drill and practice will be required alone or with help.
First Sign of Intelligence: The patience and ability to figure
well, to follow multistep method in arithmetic carefully,
precisely, to a right answer, points to and develops the
ability to read and write clearly, with precision and not with
confusion in many arts and disciplines.
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B. Watch for the use or combination of rules and patterns, one at a
time and one after another, in sequence, which gives or suggest new
rules and patterns.
Second Sign of Intelligence: If you see how
rules and patterns can be combined to get results or more rules and
patterns, you have found the key to the thought-based development of
more skills and concepts, those to come and if you like, those you
have seen.
Third Sign of Intelligence: If you develop
ability and interest to see and know the limitations of rule and
pattern based reasoning in theory and practice at home, at
school, at work and in society in general, you are becoming a
critical thinker. Good luck. This third sign of intelligence is
not always appreciated.
In arithmetic and beyond, students need to learn to apply
rule and patterns one at a time and then in combination, one after
another, in repeatable, reproducible and hence verifiable manner. In days
gone by, precision figuring skills were taken as a sign of intelligence
or potential to follow, if not bend, rules and methods, with precision to
meet the needs at hand. Rules and patterns with repeatable,
reproducible and therefore verifiable results literally provide a base
for society to function, but there is a caution. Rules and patterns
once found or given need not be fair, nor sustainable in the
long-term. Their assumption always involves some risk. The
knowledge of how to use rule and patterns in a precise, repeatable
and reproducible way, and the knowledge of the benefits, origin and
limitations are both needed for critical thinking and
intelligent problem solving at many levels.
II. Supporting Aims A and B
After arithmetic, an operational command of quantitative skills
sufficient for mathematics to pre-calculus level may follow from
lessening algebraic difficulties as indicated the site entrance, from mastering logic
and from the assumption and geometric interpretation of the
properties of real and complex numbers, from the easier consequences of
those properties, and from the assumption of that all real numbers have
decimal expansions, finite or infinite. The geometric introduction of
complex numbers only requires and involves the junior high school level
familiarity of coordinates, translations, reflections and rotations in
the plane, and may be use to develop that familiarity.
See site areas: 2. Linear Equations
& Fraction Skills 3. Fractions Ratios Rates Proportions
Units 5. Analytic
Geometry 8. Complex Numbers 10.
Secondary IV(?)
math; the online volumes 2. Three Skills for
Algebra and 3. Why Slopes
(A Calculus Intro) & More Math and the site area 7.
More Calculus.
The material here can be presented as rules and patterns to use and
combine with confidence in results coming from their repeatable,
reproducible, and therefore verifiable nature. The coverage of logic
here aims to develop precision reading and writing, and an
understanding of how implication rules B IF A can be used and combined
directly, one at a time and one after another.
The support of aims A and B advocated here may be simple, short and
effective enough to allow more students to start calculus while also
serving the needs of students heading for business and technical trades
(surveyor, plumber, carpenter or electrician) for which calculus is not
normally required.
In starting the development of trigonometry after the
introduction of complex numbers AND the
assumption of the field properties of complex numbers, turns the
development into an algebraic exercise and so make trigonometry and the
properties of vectors in the plane easier and more accessible.
(Complex numbers are not in mathematics 436)
III. Supporting Aims B and C
Axioms
and postulates in
mathematics are labels for rules or patterns that have been assumed in
order to secure a base for deduction. Further rules and patterns
are then tested in mathematics by looking for direct or indirect chains
of reason (arguments) that imply them. That provides a proof. Rules and
patterns thus proven may then being used in further tests or proofs.
The weakness of this deductive (more rigorous) style of reason lies in
the choice of initial axioms and postulates. Chains of reason
provide stories to follow one at a time and one after another. But
these stories, rigorously or deductive put together through chains of
reason, become works of fiction when the initial axioms or postulate
are not true. On the other hand, if there are elements of truth in the
original axioms and postulates, these stories may be
non-fictional.
The mixed mathematics development of synthetic (coordinate-free) 4.
Euclidean
Geometry in site pages inductively suggests and clearly identifies
geometric rules and patterns, those assumed for use in deductive
reasoning. There is motivation here for the indirect statement of the
parallel postulate as given by Euclid, namely the assumption that two
lines segments extended will meet on side of a transversal will if
the interior angles on that side of the transversal sum to less
than two right angles. This coverage of Euclidean geometry with the
selection of a unit length and assumptions about coordinates and
their decimal representation to imply the field properties of real and
complex numbers taken as assumptions in the support for aims A and B
above.
Most, if not all, of the deductive chains of reason offered here
will be direct. Ease of exposition, making the ideas more accessible, is
the objective here. That being said, in the development of
Euclidean and then Analytic Geometry here, there is focus on the possible
origins of assumptions - how they can be suggested by and extrapolated
from experience. Besides this, there is a focus on deductively deriving
the consequences.
IV. Modern Set-Based Mathematics
Modern mathematics with its set-based description of axioms for real
numbers given (or derived from assumptions about sets or natural numbers)
provides a geometry-free, model for understanding, describing and
developing the properties of real and complex numbers, and also
properties of functions which appear in calculus, all apart from the use
or mention of decimals. Geometry-free means there is no dependence
on diagrams or suggestive reason. But there is a great dependence
on direct and indirect deductive reason, and on the shorthand role of
letters and symbols to codify, record and develop concepts and results.
In North America, if not elsewhere, modern mathematics curricula from
birth in the late 1950s to abandonment in the late 1970s or 1980s with
their set-based presentation of axioms for real numbers aimed to prepare
students for the more rigorous treatment met in university programs of
study in pure mathematics. But the introduction of modern mathematics
curricula used geometry to introduce and illustrate the properties of
real numbers, avoided all mention of decimals in the representation and
properties of real numbers, and required arithmetic skills with decimals
in examples involving calculations or coordinates, and used geometric
diagrams to introduce functions. There are was also no systematic
development of the algebraic way of writing and reasoning. It was just
assumed. So the modern mathematics curricula while preparing
for pure, geometry-free and decimal-free, mathematics did so
inconsistently and awkwardly.
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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:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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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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