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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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Post Script I
Reality Versus Fiction
or Reality and Imagination
Previous: Chapter 13, Euclidean
Model for Reason
The author of a story in a book or a play creates an imaginary world for us
to visit in our minds. The story may or not be consistent with our
knowledge of real life. More and less can be suggested in a story than occurs in
real-life. Stories can be fictional, half-fictional, approximately true to
life or true.
Stories may explain or describe how things came to be. Stories
may provide lessons a for reader directly or through the words and
interpretations of another. Stories may give us ideas of what to do or
not. Stories have plots and chains of events or reasons to follow, real or
not. Stories presented on stage as plays may include not only words but also
actors and props to make the plot or reenactment easier to follow. Actors have
scripts to follow. Actors are defined by their names, costumes and
actions.
Most of us, many of us, have the ability to follow a story, its sequence of
scenes with words and events, and to recognize what is real or pretend. We
can learn stories, invent them and tell them to others via spoken and written
words. Stories can be told and retold in ways that are almost repeatable and
reproducible. Our knowledge of a culture may come from its stories and
myths.
In cooking and construction, plans and recipes give or suggest sequences of
steps or actions to take to arrive at results. The steps and the results should
be repeatable and reproducible. Technical know-how is based on rules and
patterns to follow plus some judgment as to when they can be applied. Trying to
apply rules and patterns when items they require are missing usually leads to
bad results.
In mathematics, science and technology, as in daily life, there are stories
to follow. These stories, normally called theories, describe a situation (say
what is what is assumed) and describe as a well assume methods for arriving at
results or conclusions in a step by step way. The authors of these
stories or theories would like their consistency with reality. A theory is
inconsistent with reality if it says two exclusive events occur at the same time
or if predictions based on it fail. Unfortunately, the author of theory to say
what should happen may capture a pattern in theory that works in some
circumstances, but not all. So a theory may be applicable and sufficiently
consistent with reality to be useful in some circumstances - those it reflects -
while failing in others.
Knowledge in mathematics and science and technology is based on theory and
practice. A method or procedure describe in a lab or controlled circumstances
how following certain steps will give a result. Those steps and the
results, done carefully enough, appear to give repeatable and reproducible
independent of the doer. Methods that work in practice may be described and
accumulated, and used one at a time and one after another to follow steps and
arrive at results, one at a time and one after another. A skilled
practitioner may recognize when one method can replace another because it gives
the same result or a more convenient result.
Geometry was codified in the works of Euclid, about 300 B. C. The
codification consisted of assumptions or definitions about points, straight
lines, circles, triangles and the geometric figures composed from the latter.
The resulting theory or theories was presented not on stage, but on paper (a
prop) with the aid of rulers and compass (more props) to provide construction
methods and to suggest and describe results and conclusions one at a time and
one after another. Students and teachers and philosophers could
follow explanations one at a time and one after another in way that follow
some or all of the strands of thought in Euclid's work. The codification
provides a mechanical knowledge of geometry because each of us in following the
steps should verify that the steps are valid, that the implication rules used in
each step are justly applied.
The foregoing gives rule- and pattern-based chain of reasons independent of
the followers and authors. All that provides a model for making and
arriving at conclusions with rules and patterns not only in geometry, but also
in other disciplines where rules and patterns are valued as guides. But this
model for reason has its limitations.
Rules and patterns describing what we have observed, drawn from experience,
are not absolute. We do not know if they are fully reliable, or we may not
precisely when they apply, if at all. When rules and patterns are not reliable,
a risk appears. What they suggest, one at a time and one after another, may not
be consistent with reality. None the less, recognizing rules and patterns in a
subject provides a means for accumulating know-how for arriving at results, and
a limited know-why. The latter is given by the chain of reason or
suggestion with rules and patterns, reliable or not, that led to a result.
(Implication rules and suggestions in a theory may themselves rely on the need
for a theory to be consistent. See above). Volume 1A, Pattern
Based Reason, gives a further description of the benefits, origins and
limitations of rule and pattern based thought. Not all is certain.
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www.whyslopes.com
Volume 1A, Pattern Based Reason
Chapters 1 to 24
FOREWORD
Three Remarks
1 Introduction
2 Communication
3. Elements of Reason
4 Implication Rules
5. Deception
6 Chains of Reason
7 Longer Chains
For & From Consistency
8. Language Change
9 Next Chapters
10 Responsibility
11 Accidental Patterns
12 Knowledge Islands
13 Euclidean Logic
14 Deductive
& Empirical Views of Mathematics
15 Objectivity
16 Origin of Rules
and Patterns
17 Objective Ways
18. Waking up
19. Symbols & Logic
20. Pronouns or Symbols
21. Truth Tables I.
22. Truth Tables II
22. Biconditional
22. Contrapositive
23. IF-THEN table
24. Indirect Reason Again
To reason often means to persuade someone of
the need for an idea or action. That someone could be yourself. So be
careful.
1A Logic Postscripts
- online only
+Proof by
Absurdity alias proof by contradiction
+How the demand
for consistency supports the law of the excluded middle
+Reality versus or with the aid of Imagination
+Links for reason, logic and crtical thinking
+Three Remarks
+History
Lost or Missing
There is a difference between
knowing how to spend money,
and having money to spend.
There is likewise a difference
between mastering a skill
and having meeting a situation in which it applies.
.
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