|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
| |
Previous: Start of Chapter 2,
Communication of Skills
Principles for Instruction
For learning and teaching in all disciplines, the following overlapping
principles appear self-evident, at least once stated.
- Each discipline needs to be taught or 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.
- 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 possibly why. Of
course, differences of opinion exist on which disciplines should be taught
or what pathways in them should be followed.
- 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, and possibly 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 and a knowledge of
or opinion on how the topics in question can be organized and taught. Here
again, opinions may differ.
- Each collection of mental and physical skills could 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. Explanations should follow chains of reason or persuasion
which begin at the level of the student before advancing further.
Remark 1. An alternative to ladder-like structure is a tree-like
structure. Here skills and ideas are represented by the branches of a tree or
bush. The tree can be climbed when those branches closest to the ground are
accessible while the higher ones are accessible from branches below. For ease of
exposition and comprehension, the organization of ideas into a flat tree-like
structures where each branch can be reached via a few lower branches or directly
from the ground is to be preferred to the case in which most branches are high
and can be reached only from the one just below it. This is to simply observe
that short chains of reasoning are better for explanation and comprehension than
longer ones.
Remark 2. These words or thoughts on communication of skills echo a
course on how to be a cross-country ski instructor. The course was taught one
weekend early in 1981, by an instructor-trainer from CANSKI, the CANadian
association for Nordic, that is cross-country, SKIing. The course gave a piece
by piece approach to instruction. The objective was to build both the confidence
and ability of students. The course emphasized that difficulty with a skill
signaled the need for a retreat to, or even before, previously mastered skills.
The detailed structure provided turned cross-country ski instruction into an
art. Arts of this kind are required in other areas of instruction.
Next: Chapter 3, Elements of Reason -
Seeing the difference between one- and two way implications, a way to improve
reading, writing and thinking
| |
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.
.
|