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Protest: The site author, a McGill University,
1983 Ph. D in mathematics, failed a McGill Faculty of Education B. Ed pgm 2003-5
due to
YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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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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Four principles offer an inductive philosophy for the explanation and
comprehension of math and reasoning skills. Three of the principles were met in
a course on how to teach Nordic, that is cross-country skiing. The course was
taught one weekend early in 1981, by an instructor-trainer from CANSKI, the
CANadian association for Nordic SKIing in Flin Flon, Manitoba. Nordic ski
instruction may begin with a lesson on how to put on the boots and attach them
to the ski and also how to hold the ski poles – to be precise one holds not
the poles, but their straps in way that will guide the poles.
There is a technique here, one that is not obvious. The course gave minute
attention to the details which novice and even experienced skiers might not know.
In this course on ski instruction, the more complicated movements or skills were
deliberately preceded by simpler motions. Each of which was easy to describe,
master and/or review separately. This course turned Nordic ski instruction into
an art. The four principles follow.
1. Each discipline needs to be presented, so that students understand what
they are learning and why. Without a knowledge or an opinion of why, students
may lose interest and not go further. The why could be approximate — a
little uncertainty leaves room for thought.
2. Pathways through easily described and repeated ideas may extend
knowledge of any discipline, area of thought or belief. One or more paths
through easily described and easily repeated topics may allow those who travel
further to tell others willing to listen, what to expect and again possibly
why. Of course, differences of opinion exist on which disciplines should be
taught or what pathways in them should be followed.
3. Awkwardness with an idea or skill often signals difficulty with previous
ones. It may indicate at least one earlier skill has been missed or forgotten.
When an awkwardness is felt or seen, learners should go or be taken back to
practice the missing skills, more precisely the ones just before them. This
retreat aims to restore confidence and build skills, so that the learner can
go further. This requires a diagnostic skill – a knowledge of or opinion on
how the topics in question can be organized and taught. Here again opinions
may differ.
4. Each collection of mental and physical skills should be organized into a
ladder-like sequences of steps with the basic ones first and the more advanced
ones second. Learning in any subject stumbles when a first or succeeding step
is not easily reachable from those before them. [1] To
climb a ladder, the initial steps must be reachable, and each further step
must be reachable from the one or ones before it, else failure occurs.
Explanations should follow chains of reasons or persuasion which begin at the
level of the student.
In mathematics education there are two barriers to comprehension to be
lowered or removed. First, the algebraic or symbolic way of writing and thinking
is better seen and read silently than read aloud or spoken. This has been an
obstacle to the comprehension and communication of mathematical thought. Second,
the deductive nature of formal mathematics exposition with its long chains of
reason and preparation implies that concepts appearing at the end of a course
are not comprehensible to students in the middle of the course nor at its
beginning. Mathematics beyond the last concept mastered may seem impenetrable
and mysterious.
To lower both barriers, students may be given lessons, easily described and
repeated, which require a minimal formal comprehension of mathematics and logic
while presenting ideas essential to deductive and to algebraic or symbolic
thought. Recognizing, collecting and offering first such lessons may extend the
common knowledge of mathematics beyond the mastery of arithmetic, counting and
simple formulas that should be obtained in elementary school. This work
identifies such lessons and indicates ideas for math and logic instruction from
primary school to the start of college. Some of the ideas may be worth reading,
repeating or refining – the three Rs that this author hopes for.
Next: Fairness
in Course Design - impossible when methods to directly and clearly explain are
missing.
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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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