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Students need an operational command of fractions, logic, algebra,
geometry, trig and calculus. Seeing how to follow multi-step methods to
obtain and present results in a repeatable, reproducible, readable and
therefore verifiable (right or wrong) manner may be source of confidence
in the reliability of mathematics and a source of abilities or a sign of
intelligence for further work and studies.
For students with no immediate interest in the know-why, a focus on the
practice, an operational command of key skills and concepts may make
comprehension later of the know-why easier and more appealing. For
high school mathematics and calculus, fraction sense and an efficient
command of arithmetic with fractions is more important than full
comprehension of why the calculation methods work, but seeing the why,
depending on the student, may help with the practice. Logic or chains of
reason appear in many places in mathematics. A mix of logic and
rote learning may be optimal.
In mathematics, skill and confidence begins
with arithmetic methods with repeatable and
reproducible results - so answers are right or wrong, and so that
student acquire the discipline to follow steps carefully. Drill and
practice in whole numbers and fraction arithmetic and meaning
(number sense) until second nature is needed in
moderation. Explanations why arithmetic methods work
may be presented in part as aids to their mastery, where not too
complicated nor too alienating for students. The fact that a
method works may be sufficient empirically and hence intellectually for
many. The further development of mathematics (algebra, logic, geometry,
trig, calculus) after arithmetic provides opportunities to
emphasize the thinking part of the subject, at which point some, not
all, may revisit arithmetic methods to see why they work, but for
marking and evaluation, an operational command is sufficient.
Tutoring, or checklist approach to skill and comprehension development
and tracking for each student, may go further.
In skill and concept based subjects, I would like to see
instructors track for each student, which skills and concepts have been
mastered and to what level, so that students and teachers have a
clearer guide for what needs to be reviewed or learnt. Keeping such a
checklist might allow a teacher to say to student, you may skip these
question, but you have to do or try those. That may lead to more
thought in direction of studies and less work in marking in mathematics
or science courses alone and in sequence. That may also provide
an objective evaluation of a students skill and concept
level.
The thought-based development of applied mathematics present in site
pages may stand alone or be seen as platform for further studies in
modern mathematics, pure or applied. High school mathematics with
its reliance on diagrams and coordinates for its comprehension and
development is mixed rather than pure mathematics.
The exposition of mathematics, the introduction of algebra or the
shorthand roles of letters and symbols has been confusing in the past for
many literate students, skilled and intelligent outside of
mathematics. Innovations in site material may make existing
mathematics courses easier while setting the stage, trust but
verify please, for expositional or content changes in high school
mathematics and calculus
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Talking about three skills for algebra and what is a variable may
provide a remedy.
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The role of logic in mathematics may be seen more easily if logic
ideas are introduced and clarified alone.
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Algebra shocks in calculus can be eased or avoided by re-arranging
calculus to put some easier ideas first. Rigour can come later.
The implications or scope of site material grew from just a few
ideas to submit to educational authorities to a full, self-contained
theory for changing and improving mathematics education, all driven by
the inductive
principles for instruction and by reports of poor results in
high school mathematics. Writing began in 1991 to report and
develop further fall 1983 starter lessons for logic, algebra andcalculus which had been
useful in easing or avoiding difficulties in college classrooms 1983-89,
lessons which had been motivated by a sense of incompleteness in the
introduction of skills and concepts.
More:
For an operational command of arithmetic, logic, algebra, trig, complex
numbers and calculus, students need not see a logical development of
geometry. Students may obtain some of the algebraic-deductive
maturity needed for calculus with or through site coverage of algebra and
logic if the arithmetic (field) properties of complex numbers and the
decimal representation of real numbers are assumed. For gifted or
interested students, the field properties can derived.
For that, the axiomatic, development of Euclidean geometry with
rulers and compass instead of coordinates, and assumptions about vector
sums are independent of the choice of unit length and orientation of
unit directions, combined with the decimal representation of
numbers, may provide a geometric-decimal development of the field
properties before the introduction of pure mathematics.
In mixed mathematics curriculum (course design) before the study, if any,
of pure mathematics, the full axiomatic development and connection of
rules and patterns is not for beginners. Courses instead may focus on a
local thought-based development in which some rules and patterns are
explicitly assumed or given, and then combined to obtain further
rules and patterns. Here the ability to combine rules and patterns is the
objective. Where a full thought based development is too ambitious, too
many details for students to follow, this local objective provides a more
accessible alternative.
Mathematics education to the level of advanced calculus may have the role
of providing a connected, geometric, physical and thought-based
development of skills, concepts and comprehension which is functional or
operational, and which provides the algebraic- deductive- geometric
maturity sufficient for an operational command of mathematics, and
sufficient to allow but not compel students to study a more rigorous,
axiomatic development and written codification of mathematics. In this
role, ease of comprehension may chosen in the initial development
of skills and concepts, so the logic and results are easily understood
and repeated. For example the easily understood area-based development of
column methods for the multiplication and addition of polynomials is only
valid when coefficients and variables are non-negative, but the column
methods once learnt, can be explicitly assumed for real- and even complex
value coefficients and variables. Advanced mathematics courses can
provide the missing rigour. Students may be content in the first
instance with mastering the use and combination rules and patterns
one at a time and one after another in a repeatable, reproducible and
thus verifiable manner apart from technical details that may alienate or
impede their comprehension. At the same time, those details should
available in appendices or references for students wanting more.
In mathematics starting with arithmetic, students need to learn or
show how to arrive at results in a repeatable and
reproducible manner, and in that learn or show a mistake in step of a
method leads to results that nor repeatable and reproducible. The
thought-based development of mathematics is possible after the arrival of
the skills and patience to needed apply rules and patterns with
care and precision, one at a time and then in combination, one after
another. The ability to recognize and combine rules and patterns,
once seen and learnt, allows for a thought-based development and
connection of skills and concepts, those to come and optionally those
previously covered. The ability to follow rules and patterns in
repeatable, reproducible and therefore objective ways needs to be
cultivated along with some caution. That is rules or patterns we apply or
follow in this manner may be wrong or off, and in need of correction or
abandonment. That is where critical thinking appears.
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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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