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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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No Decimals
Chapter 14, subsection
Previous: Applied
Mathematics & Electric Circuit Theory apart from Modern Mathematics
Within Zermelo-Fraenkel set theory, the decimal representation of real
numbers is not special. There is no direct motivation for decimal notation or
base ten. The discussion of convergence, continuity and accuracy of computations
does not require and can be made independently of the decimal representation of
real numbers. The Zermelo-Fraenkel set theoretic framework for mathematics
provides a path for the comprehension of mathematics with no reliance on decimal
arithmetic, geometry nor physical senses and abilities – apart from the
ability to put pencil or pen to paper to record chains of reasoning and
figuring.
Yet description of the set theoretic foundation of mathematics without any
mention of decimals has separated the higher mathematics from the common
knowledge of arithmetic acquired in elementary school. This has been perhaps one
barrier to the simple explanation of mathematics beyond the common knowledge or
memory of arithmetic, counting and the use of simple formulas. Our attention now
turns to the views of mathematics found in the classroom. Some critical
observations follow. The companion work Mathematics Curriculum Notes echo
them and further offer a philosophy for mathematics education.
In The Classroom (1995)
Mastery of arithmetic, geometry and further mathematics has been regarded as a
sign of learning and possibly intelligence. Essentially, all education in
mathematics is ended by the completion of schooling or by a failure or
near-failure in a course. The latter leaves a bad impression of mathematics or
one's own ability.
Since people stop their mathematics education at different levels,
mathematics instructors socially fall in the class of people who hear admissions
or confeesions of the form: I hated math or I liked or studied math
until such and such a level. [5]
[5] In college, students typically identify themselves by
their field of studies. In one encounter, I indicated to a new acquaintance
John that my field was one that resulted in my hearing confessions. He was
left to guess the field. Amongst the choices were counseling, psychology,
religious ministry, law and so on. John felt that I had misled him with the
hint I heard confessions. But several minutes later, after one or two others
topics of conversation, John suddenly indicates that he was good in
mathematics until the end of high school. This was the confession. The hint
that I heard confessions was not misleading.
Ideas described at the end of a math course are typically not
understandable to students until a few moments before their presentation. This
observation applies to courses at all levels in mathematics from elementary
school to university. Taking a course in mathematics is for many like following
a trail with the hope of being able to comprehend what appears just around the
next corner. That has been an insurmountable obstacle for the explanation and
comprehension of mathematics and reason.
In elementary school, instruction is concrete and reassuring. Explanations of
counting, figuring, measurement and geometry may encourage students to use
physical objects and arguments to understand arithmetic and its basic
applications. Confidence may be based on methods with repeatable and
reproducible, and therefore verifiable, results. Arithmetic methods along with
some descriptive geometry [6]
[6] The recognition of simple geometric shapes: circles,
rectangles, squares, pyramids, etc; and the use of standard formulas for
perimeters, areas and volumes.
provides a first most certain view of mathematics. It is the basis of what
was the common knowledge – today some students may forget their arithmetic
knowledge as they pass through high school. And a remedy for the latter may lie
in some drill and repetition.
Most people succeed in mastering figuring with decimal numbers and the use of
simple formulas. These skills are acquired in elementary school and possibly
reinforced at the start of high school, that is, secondary education. Beyond
these skills comes a knowledge of algebra, logic, trig and/or calculus. In North
American high schools, logic unfortunately has been taught in mathematics only,
say in algebra and geometry courses, and not else where.
Chapter Subsections: [ 14 Set Theory ] [ 14 Before & After Set Theory in Pure Mathematics ] [ 14 Euclidean Model for Physics ] [ 14 Applied Maths and Electricity Apart from Sets ] [ 14 Decimals Absent From Pure Mathematics ] [ 14 Modern Mathematics Education ]
Next: Modern Mathematics Education
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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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