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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
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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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Multiple Kinds of Reason in Mathematics - Essay I
There are three kinds of rule-based intelligence in mathematics,
logic and most pattern-based subjects.
The first kind met in primary school arithmetic consists
of skills with repeatable, reproducible and therefore verifiable results -
results that are then considered right or wrong.
The second kind also met in primary school consists of pattern
or rule recognition. The development or exploitation of the ability to recognize
or suggest simply patterns in order to predict the next element in a sequence.
If the prediction fails, another pattern is required.
The third kind, assumption-based, deductive reason, appears
after inductive mastery of logic, that is mastery of implication rules If A then
B and their use. The third kind follows the use of implication rules and
definitions and assumptions, one at a time and one after another, to arrive at
logical conclusions. Here chains of reason how to be posed in a readable,
repeatable, reproducible and therefore verifiable manner.
For third kind of thinking in mathematics, there was a search
for secure assumptions, so that deductive reason could proceed in a
consistent and reliable manner. Unfortunately, uncertainty results
in mathematical logic imply more can suggested than proven in mathematical
theories which are not finite. So the assumptions made for the third kind of
reason stem from experience or trial and error over time. That identifies modern
pure mathematics as another empirical art. But mathematics by providing a
format for measurement and calculations remains the queen of
science, a queen in the hierarchy of empirical arts.
Pre-coordinate Euclidean geometry, the original model for
pure reason in mathematics, with its assumptions and deductive chains of reason
is still worth presenting in part if not in full in high school mathematics in a
selective manner to build algebraic-deductive skills and geometric skills and
sense. However, the empirical nature of pre-coordinate and hence coordinate-free
Euclidean Geometry is implied by diagrams with subtle faults that imply
incorrect conclusions - subtleties detected with the use of coordinates in
advance mathematics courses.
Further Reading: Logic
chapters 1 to 5 (Français)
in Volume 2, Three
Skills for Algebra introduce the Euclidean logic methods and questions in
mathematics free manner. . The use of logic in the form of direct or
indirect use of implication rules B if A or equivalently, If A
then B, informally or within axiomatic (assumed rules and patterns)
frameworks leads to further rules and patterns to accept and use. See to the
last chapters and postscripts of Volume 1A,
Pattern Based Reason, for a further discussion of consistency questions
and indirect chains of reason in general, and not just in mathematics.
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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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