Appetizers and Lessons for Mathematics and Reason (www.whyslopes.com)
||Définition d'une variable || Algèbre || Arithmetique || Logique ||La raison basée sur les règles et modelés||

Online Volumes
1,  Elements of Reason.
1A. Pattern Based Reason 
1B. Math Curriculum Notes
2. Three Skills for Algebra
3. Why Slopes & More Math

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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

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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.

Three Remarks (Postscripts)

Previous:  Foreword

1. Society and Pattern  Based Reason  

Rules and patterns need to be read and written with precision.

  • If you do not read precisely what is meant, how will you understand?

  • If you do not write precisely  what you mean, who will understand you? 

Volume 1A, Pattern  Based Reason, read slowly, will test and develop precision reading and writing while pointing to the benefits, origins and limits of rule and pattern-based methods in thought and deed. Not all is certain. The question of what rules and patterns are reliable remains open.  But in daily life we do assume or rely upon many rules and patterns. There is no escape. 

2. Science and  Pattern  Based Reason

In science and technology, rules and patterns range from being reliable to speculative.  Engineers and scientists need to know the benefits, origins and limits of their rules and patterns, that is, when they may be applied  or not. 

Mathematics offers the most certain area of rule-based thought. Physics, chemistry and biology include rules and patterns that can be tried and tested in lab work (experiments).  Biological descriptions of anatomy can be verified in the lab via dissection - direct verification. . In contrast, quantum physic and cosmology offers speculations on the origins and evolution of the universe at the subatomic and macroscopic level- these speculations provide  calculations that sometime work regardless of justification. Finally, evolution theory in biology is based on or the fits the clues that come from geology, the dating of fossil layers and fossils,  and the similarity of organisms, adult or embryonic, at the anatomical and microscopic level.  If you accept the dating methods or how they were established, you will come to a conclusion that the earth is old and that life did not appear quickly.  

There was a dream that empirically discovered rules and patterns would provide answers. But such rules and patterns have their limitations.  

In society, religion, commerce and science, the follower of rule- and pattern-based thought and methods needs to the know the origins, benefits and limitations of those rules and patterns, and so see when to apply them and to when further thought is required. 

3. Scale Factors in Science and Society - Ecology and Pattern  Based Reason

Questions of scale appear in society, science and technology.  Processes that appear beneficial and harmless on a small scale may on a larger scale bring ill-effects.  Think of towns on a river.  Untreated sewage may leave river waters drinkable when the towns are small. But the growth of towns and population may pollute the waters and require remedies.  City planners and designers of industrial processes need to consider the effects of scale.

The unchallenged right to pursue methods or processes that do no harm to others and yourself, may need to be revisited when a method or process is practiced on a larger and larger scale, as side effects initially considered harmless accumulate. Ouch

Rule and pattern based methods with results that appear to be repeatable and reproducible on a small scale may be rotten on a larger scale due to accumulated side effects. 

Next: Chapter 1, What is Reason. - Different kinds of Reason

 
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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.

Vol 1A Postscripts
- online only

+Proof by Absurdity alias proof by contradiction
+How the demand for consistency supports the law of the excluded middle

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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a 1983 McGill. Ph. D. in mathematics
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