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Start with pattern
based reason or Logic
[Online Books and More Site Areas]
[Study Tips] [Directions for High School Mathematics -
Calculus Preparation] [Curriculum Shifts - Shorter, Better, Stronger]
[References]
Site innovations for mathematics and logic education
were initially developed to fill skill and concept gaps and flaws
sensed in the high school exposition of modern mathematics
curricula prevalent from mid-1950s to the 1980s in schools and
colleges. However, exploration and refinement of ideas for learning and
teaching points to an alternative thought-based development of
high school mathematics (algebra, geometry, trig and functions) needed
for calculus. The net result may be fewer but more effectives hours in
high school mathematics.
These curriculum shifts could be the basis for a leaner and more
effective mathematics instruction.
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Two Three Shifts - clearer and
effective ways to develop algebra and fraction skills and sense:
The puzzle of how to introduce the algebraic way of writing and
reasoning clearly and directly was first met by in
high school days 1965-70. Difficulties of fellow students and
instructor in understanding and explaining algebra slowed the
site author's education. The first algebra chapters in the 1995-6
Volume 2, Three Skills for
Algebra, point to a solution - a greater verbalization in
mathematics in which the overlooked ability of describing or talking
about numbers and quantities is recognized and emphasized. That is
before and then besides the introduction of letters and symbols
in algebra as placeholders for numbers and quantities in calculations
or their description. The spring 2005 site area Solving Linear Equations with fractional
operations on stick diagrams also introduces algebra in a
parallel approach to the foregoing, which comes first is a
matter of taste, while consolidating fraction sense and skills.
The two approaches together provide a solid base for algebra
for students starting their teenage years, or later remedial
instruction. Algebra
self-instruction alone or with help allows
student to benefit immediately. For self-instruction,
the algebra
chapters in Volume 2 are recommended first.
There is a fourth skill for algebra in Three Skills for
Algebra, namely a development of the ability to talk about or
describe the numerical and algebraic use of formulas and equations
with short descriptive phrases: (i) forward and backward use (or
direct and indirect use) and (ii) algebraic and arithmetic
(numerical) solutions. These phrases appear in Chapter 14. can be used through
out high school mathematics to identify recurring themes - key
objectives - and to provide another fresh perspective on the
algebraic way of writing and reasoning.
In mathematics, I would like to see the first two years of secondary
school consolidate arithmetic and introduce algebra skills. Then I
would like the third year to be given as a reward. That is, I would
like it to provide applications, one at a time, and one after
another, to develop a favorable impression for students who have
begun to dislike the subject and may drop out, but an
impression that need not be terminal as it includes motivation for
further studies.
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Third Fourth Shift - Complex
Numbers & Easy Consequences: Vectors &
coordinates, polar & rectangular, are used in a very
simple, logical development of complex numbers., one that implies a quick,
logic-based development of senior high school mathematics (and the
use of complex number methods with ei in technical and
engineering schools.)
Technical note: Assumption that the head to tail addition
of vector described displacements in the line or plane is independent
of our choice of rectangular coordinate systems implies the
distributive law for real and complex numbers. In other
words the geometric assumption that the coordinate description
of sum of displacements gives a new logical development of the
properties of real and complex
numbers in ways that simplify and provide a base for analytic
geometry and trigonometry - that favored in university program
without explanation. This logical development based on geometry
covariance, an idea that appears in relativity, provides
an axiomatic shift for mathematics education with consequence
for high school and college studies. See the logic
chapter Islands and
Divisions of Knowledge for thoughts on multiple starting or entry
points in the deductive arrangement of ideas.
Self-instruction in complex
numbers alone or with help allows student to
benefit immediately At the college level in engineering and
physics, the properties of complex numbers and benefits for
trig via the cis function were often presented as efficient shortcuts
without proof. Here is a justification that may accelerate college
and high school instruction.
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A further shift - calculus re-arranged.:
Calculus demands full mastery of logic, fraction skills and sense,
algebra, analytic geometry, trig and functions. That demand provide a
standard and goal for high school mathematics instruction which needs
to be emphasized as the coverage of more and more topics in high
school may distracts learning and teaching from the full
mastery.. Even with that full mastery, calculus employs the
algebraic way of writing and reasoning at full strength. The
site calculus introduction
employs geometric and algebraic previews, and decimal view of error
control in computations, to develop the multiple full
strength uses of the algebraic way of writing and
reasoning gradually and systematically in ways that should eliminate
or avoid some calculus perils, and allow more to go further. Calculus
self-instruction alone or with help allows student to
benefit immediately. Note in a recently seen discussion of
the modern mathematics curricula of the 1960's, there is mention of a
slope-oriented analysis which site geometric and algebraic previews
may duplicate. If that is the case, site previews are re-inventions
and not new.
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Expert Instruction (Mastery Learning): In
classes, grades of 50%, 65% or 80% in a sequence of assignments and
tests say how well you are doing, but do not say what you have
missed. If the teacher or marker identifies and correct all mistakes
in your answers, you can learn from your mistakes, and you know what
you missed. In my classes, I intend to make a checklist of
skills and topics, so that I can record which ones have been mastered
to report to student a grade - the percentage of skills and topics
which appear to be mastered, and to track and report what remains to
be reviewed by the student or re-taught. Efficient learning
(more gain for less effort) might follow. But I am advocating
here what I have yet to do in class, an expert approach to learning
and teaching. Tutors too can be hired to follow this approach instead
of being hired to improve marks.
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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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