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YOU are better than YOU think. Show
yourself how:
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Logic
chapters 1 to 5 re- appear not in sequence, as is or longer,
in Volume 1A, Pattern Based
Reason, Bon Appetite.
Logic
Mastery
Amazing, Amusing, Amorous, Delicious, Delightful, Edifying,
Strengthening Elixir.
It eases work & learning difficulties Makes the hard easier. Opens eyes.
Leads to greater precision.
in reading and
writing
Logic
mastery makes the hard, easier. Logic
mastery leads to better, stronger and richer comprehension. Logic
mastery improves reading and writing. Logic
mastery ease learning difficulties. Logic
mastery gives a headstart. In sum, logic
mastery will develops critical thinking, improve reading and writing,
and give a firmer base for work and studies at many levels. Good luck.
After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving
liinear Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
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What may be learnt and when depends on how skills
and concepts are developed. Making the hard easier and clearer will allow
earlier & richer development of skills and concepts.
Try the Twiddla
Whiteboard. In principle, it allows
to people to draw and chat together online on a copy of this webpage or a clean
sheet. The chat may be via text or audio. Visit www.twiddla.com
to set up whiteboards to work with the webpage of your choice.
For online automated help in senior high school maths & calculus,
visit quickmath.com For Automatic
Calculus and Algebra Help with derivatives, integrals, graphs, linear equations,
matrix algebra, visit calc101.com
With overlap, each site quickmath
& calc101offers a different range of
services, some free, some not, all based on webmathematica. Good luck.
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Making the Hard Easier but Ignoring how
and so missing the point
Teachers: Site material stems from question met in
1965-9 of how to introduce algebra directly and clearly, a question answered
in part by the two
logic puzzles, three
skills for algebra and why
slope lessons in fall 1983 eurekas, a question answered fully and
completed by the site preparation for calculus
with the a little more material found in these algebra
lesson plans, all based on inductive
principles for instruction or the communication
of skills. Mathematical
induction and inductive
principles for instruction both fail in similar ways. Such failures need
to be avoided if an instruction is to be full and complete.
Site pages address the question of how to make skills and
concepts, clearer and accessible, easier or less hard, in accordance with inductive
principles for instruction met in 1981.While writing may continue,
these Algebra
Lesson Plans, October 2005, essentially complete the site
mathematics program for skills and concepts from algebra and fractions to
calculus.
Instruction in full accordance with inductive
principles require for each skill and concept to be clearly and fully
explained - get the point, quickly. Seeking, inventing and putting first tried
and tested lessons, easily understood and repeated in the classroom,
should be first priority in course design along side some clarity in the reason
and motivation for course content. . The principles are very similar to
those of mathematical induction.
Knowledge of how induction may fail due to gaps is guide for lesson
development.
The 1981 meeting was predated by 1965-9 question of how to
make algebra accessible to my teachers and fellow students - General
lack of skill in algebra on the part of fellow students and some of my
teachers slowed learning in my classes, but while I was comfortable with the
use of letters and symbols in reasoning algebraically, I lack the words to
transfer my understanding to others, a sense of incompleteness followed, and I
did not see in courses taken thereafter, a clear and reproducible introduction
to algebra. But as student 198-1983, improving mathematics education, making
it more accessible, was the responsibility of my betters, albeit I tried to
address the algebra & logic problem in 1975 in voluntary contribution to a
MgGill University open house, and in fall 1983, I invented three lessons,
namely two logic
puzzles, three skills for
algebra and why
slopes (a calculus preview) which effective for the most part then and
thereafter. The last lesson is seen first in the name of Volume 3, Why Slopes
and More Math, and later in the site domain name. A lot of work, mostly unpaid
and unrecognized, has gone into thinking about mathematics education before
and after site construction.
Barriers to Improving Instruction.
Ideas effective in the classroom fall 1983-89 could not be
shared by this author as a college instructor. The same ideas are still
not being used in classrooms due to a Chinese wall between lessons that
work and education reform. The notion of seeking or inventing lessons
easily understood and repeated to put first in mathematics course design is
too vulgar, too much like common sense, for consideration in mathematics
education reform 1970s onward in English speaking lands. The reason for
writing December 1990 with an amateur status in education and mathematics was
to report and explore ideas that worked. This website in its
present state (October 30, 2005) not only report ideas, it defines a coherent
program for the inductive to deductive development of skills and concepts from
algebra and fractions to calculus in accordance with inductive
principles for instruction.
Walk first, Run Later:
Site material supports direct instruction. Advocates
of indirect instruction should begin with a a knowledge of how to develop
skills and concepts directly and clearly, along side a clear definition of
purpose to say what is essential and what is not in a discipline. Where a
subject has not had a direct and clear explanation, indirect instruction faces
a daunting challenge. Site material will ease that burden.
Education authorities facing a shortage of mathematics
instructors should shrink the curriculum and number of hours spent on
mathematics to focus on the needs of calculus, for the sake of effective
instruction. Teaching less and doing that well may leave a thirst for knowledge
and avoid alienating students. Hours taken from mathematics could be spent on
physical exercise.
Student centered education is best served by learning how to
explain material directly and leanly before trying to do so indirectly
Instruction has to learn to walk before it runs in the constructivist style.
Calls for engagement, authenticity, realistic, might be met in direct
instruction by a focus on practical problems. Learning by discovery with an
emphasis on problem solving is fine for teachers expert in that style. Such
expertise should be documented -described and/or filmed - so that other
teachers may follow.
While there is emphasize on indirect instruction in education
for the sake of student motivation and engagement, the practitioner of indirect
instruction needs to how to develop & define skills and concept directly and
clearly. If the latter is not known or is not feasible, I fail to see how
indirect instruction as in constructivism can succeed.
Missing the Point
Mathematics education reform and teaching training has missed
the mark in focusing on new pedagogical styles while old difficulties in
explaining concepts directly and clearly remain, where inductive
principles for instruction are not respected.
The pushing of technology - spreadsheets and calculator
programs in high school mathematics has been a distraction from good
preparation for calculus and a false cure or distraction from the discussion
of mathematics education
shortcomings. The student who cannot do mathematics without a
calculator has been misled. His or time has been wasted.
Too Much Hope, not enough reason: Architects risk
failure or costly overruns when they insist upon previous untried or unproven
methods for building their projects. The constructivist approach to education
has some great banners, great calls for action and thus a great design, but
the enthusiasm for constructivist is not tempered by the caution. It
is based on the assumption or faith that in implementation existing
teachers & teachers in training will become constructivist adepts by
edict.
Unethical: Hope or promise-based education reform today
can be compared to the early days of drug testing in which early results held
promise and the need justified speed while ethical procedures for testing were
not established. The constructivists are pushing a solution that has
never fully implemented nor tested in mathematics, apart from ethical
consideration, the consideration of the risk of failure.
While the standards or objectives were being written and
rewritten, and held out as a model before the how-to had been fully
defined. The constructivists today has saying the results since 1990 do
not truly represent their movement because of poor implementation or
improperly formed instructors does not acknowledge the incomplete state
of their program while demanding and insisting on further
experimentation.
The constructivist themselves in emphasizing the subjective
nature of knowledge, respect for individual conclusions or constructions, are
pushing aside the logical structure of mathematics in which chains of reason
in logic and computation should lead to objective result, right or wrong,
independent of the individual. A group of anarchist, irrationality, have been
in control of the mathematics education standards and despite their calls for
authentic, realistic and problem solving skills and situations, have made the
standards content-free.
More Missing the Point
The US National Council of Mathematics
Teachers, an unfortunate influence in Canada too, was taken over by
psychologists a few years before the publication of its 1989 Principles and
Standards. The latter called for a dramatic change from direct instruction to
indirect instruction in which student would be led to construct their own
deductive (?) comprehension. The NCTM calls for more realistic, more
engaging, more authentic, more deductive, more logical and more accessible
student centred instruction sound very good. But the 1989 standards and the more
recent year 2000 standards are long on talk of the Orwellian 1984 kind, and
short on action. The standard mention mathematics and call for
activities for students to develop their own comprehension. But where is
the model? Where are activities documented? Has any been testing been done and
documented, so that others may repeat the successes.
The standards have pushed aside the question of what should be taught and
replace it by the question of how mathematics should be learnt, that is,
the advocacy of indirect instruction in which teachers not only have to
understand mathematics well, a problem for many in the first place, but also
have to understand to it in a superior fashion in order to replace direct
explanations by indirect activities that are suppose to lead students to an
understanding which should be respected even if it is wrong by pre-1989
standards of mathematicians. That is folly.
Student centered education sounds great, but its introduction should not
throw-out the education. Calls for more realistic, more engaging, more
authentic, more deductive, more logical and more accessible, and thus student
friendly or if not centered education can be applied to direct
instruction.
By themselves, the standard are more complicated to read and follow for a
mathematician or a person well-versed in the subject. than old-fashioned,
textbooks which provide a full explanation of high school mathematics. Focusing
on the pedagogy while not covering what is to be taught betrays the preparation
of mathematics instruction. It is putting the horse before the cart.
For clarity in defining mathematics standards, what should be taught apart from
pedagogy, see the Mathematics
portion of English
National Curriculum,
Mathematics curriculums should identify what is be taught in one document,
and explore or document methods how in another. But the definition of the
curriculum is too complicated. There are too many cooks, too many steps,
too much, Too many Too many artificial, committee and
bureaucratically determined criteria, that individual judgment (a constructivist
aim) is suspended or squeezed in development textbooks and manuals for
instruction. Mathematics course design would be better off with the
competition between textbooks written in accordance with the judgment of
individual authors, experts in mathematics and pedagogy, a competition based on
ease of use and readability.
If the constructivist movement called for indirect method of
teaching to be invented before direct methods worked, would that make
sense.
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www.whyslopes.com
Mathematics Education Essays
57 or so
Area Entrance & Hub
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course
Design & Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in
Logic -
Occurrence
& Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition
- Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes
& (?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for
dot & cross - cosine
law
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