Appetizers and Lessons for Mathematics & Reason Français: 26 pages
A 1100+ page site with math-free logic chapters and wordy algebra chapters.
For comprehension, study site chapters and steps. Go beyond rote learning.

Logic mastery strengthens comprehension and so improves home, work & study abilities .
Logic 5 Chapters Arithmetic 10 Steps Algebra 12 Starter Steps & 5 Advanced Steps
Work & Study 23 Tips Geometry 15 Steps Calculus 70 Lessons

Ages 15+: Why study slopes Polynomials Quadratics Why factor polynomials Logarithms Functions
What is similarity Euclidean geometry leanly Coordinates + complex no.s Vectors DC Electric Circuits

Ages 14+: Prime factorization Written work formats Decimal place value Extend arithmetic skills orally
What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

Online Volumes: 1 - Elements of Reason, 2 - 3 Skills For Algebra, 3 - Why Slopes and
More Math
, 1A - Pattern Based Reason, 1B - Skill Development Principles + Troubles
Forewords + leading chapters give original reasons, still valid, for site content & growth.

Site Review: Mathphobics, this site may ease your fears of the subject, perhaps even help you njoy it. ... unintimidating, sometimes funny and very clear. ... . Read all. Continue with Volume 2, Three Skill for Algebra.

Site Review. Math resources ... span ... arithmetic, logic, algebra, calculus, complex numbers, and Euclidean geometry. Lessons and how-tos .... provide a good foundation ... Read all. See site books as well.

Teachers & Tutors: Site material uniquely explains common troubles in terms of steps too large or missing. Plus, this December 2011, 5-phase framework offers a context for mathematics & logic education. Phases 1 to 3 may focus on skills with actual or potential local value for adult & daily life. College-oriented phases 5 & 4 focus on calculus & preparation for it. Phases 1 to 4 may also serve trades & professions not dependent on calculus.

Location: Site Entrance << Volume 1 Elements of Reason


Volume 1 Elements of Reason

Foreword

Volume 1, Elements of Reason

The first part Pattern Based Reason (Volume 1A) of this work Elements of Reason describes rule and pattern based thought and processes in daily life, society, science and technology. Reliable rules and patterns can be followed one at a time or one after another to obtain conclusions or results. Not solved is the problem of identifying reliable rules and patterns to employ. Instead, the empirical method of coping with this problem is discussed.

Elements
of
Reason

understanding and explaining
reason and math
Volume 1

by
Alan M. Selby
Ph. D.

Printed in Canada
ISBN 0-9697564-1-0

Rule and pattern based thought and processes touch many arts and disciplines. Awareness of the difference between one- and two-way implication rules will improve reading, writing and argumentation skills. Students of critical thinking, persuasion, philosophy, mathematics, science and technology may find this first part worth reading.

In both arithmetic and logic, rules and patterns if followed carefully lead to results which are repeatable and reproducible, and thus verifiable and objective: two individuals following the same rules and patterns with the same data or in similar circumstances should obtain the same or similar results. Arithmetic and deductive reason are but examples of verifiable rule and pattern based thought or processes.

Verifiability, repeatability and reproducibility form a basis for the appreciation of, if not reliance on, rule and pattern based thought and processes. This appreciation should not be too firm. The identification of reliable rules and patterns, or reliable data to use with them is not certain. Further, where rules and patterns do not apply mechanically, there is room for thought. Still, verifiability, repeatability and reproducibility may provide a basis for the common knowledge and informal mastery of a subject.

The second part Mathematics Curriculum Notes (Volume 1B) is for teachers and advanced students of mathematics or a quantitative college discipline. This part describes simply yet precisely, the role of rule-based reason, that is logic, in providing a thought-based framework and codification for mathematical thought. This second part further describes how an inductive educational philosophy provides a context for math and logic instruction from primary school to college. Ideas which are easily repeated and understood may provide a common knowledge of mathematics and the rule-based reason sufficient for a more formal and rigorous comprehension.

This two-part work and its the companion volumes Three Skills for Algebra Why Slopes and More Math stem from a project to write a single book, namely Ideas that Might Count for Education, Reason and Mathematics (1994). That single book (no longer available ) was written and distributed. It covered a vast number of topics. Some of interest to one audience but not to another. With further writing and rewriting, this first endeavor was divided into three volumes, the first of which, the one before you, was divided into two parts. Writing for some is an iterative affair.

The initial aim was to report some unique idea, innovations, for math and logic instruction. These ideas or lessons had worked well with college students, shy or curious about one or both disciplines. But in writing and rewriting, the aim became wider. The possibility of a consistent and coherent scheme for math and logic instruction from primary school to college was seen and explored. The scheme is comprehensive save for the treatment of geometry. How to fit or emphasize Euclidean geometry in the curriculum is not covered.

Formal mathematics can be difficult to follow for students who fail to grasp deductive thought and the symbol-based algebraic way of writing and reasoning. The latter like arithmetic is better seen and written than spoken aloud. Symbols like pictures can be worth a thousand words. Words have been missing to explain the role of symbols in providing the shorthand notation of mathematics or its algebraic way of writing and reasoning. The latter consists of recording and developing thoughts on paper at least for those among us afflicted with a short or too forgetful memory.

The absence of a verbal culture to introduce and explain the algebraic way of writing and thinking leaves its mastery to immersion and osmosis. Comprehension depends on one's aptitude for learning some basic ideas by immersion. I am in the radical position of suggesting that a certain change is possible and desirable. This work and its companions suggest how. They have yet to be formally peer reviewed and so should be read with caution. The discussion of math and logic instruction and the discussion of reason and persuasion are both fraught with controversy. Scrutiny or critical examination of this work may lead to its refinement.

Alan Selby
Montreal 1996.

December 2011 Postscript

Site chapters and steps now stands at the sharp edge of mathematics education reform. Site material stems from olde and continuing gaps and inconsistencies in ends and methods - there was no pleasing all. The essay which way to go "lightly" introduces a more detailed, five phase framework and nearly plain-language remedy. Phases 1 to 3 focus on skills of value for adult or daily life - precision in reading-writing-figuring included. Phases 4 & 5 focus on calculus and preparation for it. Many university programs demand calculus. Preparing for it has value in senior high school science too, and some value for trades-professions not taught in university. The framework addresses and remedies all the difficulties identified above, and implement most of the ideas in the subvolume 1B, Mathematics Curriculum Notes. The framework being done sets the stage for yet another consolidation of site material.

      
      
      
             Canadian Cataloguing in Publication Data 
       Selby, Alan M, 
         Understanding and Explaining reason and math 
      
          
      Contents: v. 1. Elements of Reason - v. 2. Three Skills 
       for algebra - v.3. Why Slopes and more math. 
      ISBN 0-9697564-4-5 (set) - 
      ISBN 0-9697564-1-0 (v. 1) - 
      ISBN 0-9697564-2-9 (v. 2) - 
      ISBN 0-9697564-3-7 (v. 3) - 
      
1. Mathematics--Philosophy. 2. Reason. 3. Algebra. 4. Calculus. I. Title. II. Title: Elements of reason. III. Three Skills for algebra. IV. Title: Why Slopes and more math. QA8.4.S44 1995 510'.1 C95-900945-0

Reprinting may lead to new ISBN numbers.

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Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills.

Mathematics Skills For Ages 3 to 14

Skills with take home value

Basic skills include time-date-calendar Matters; money matters; map, plan and scale diagram matters;counting, measuring and figuring; decision making with logic and likelyhood; being careful and being aware of the domino effect of mistakes; reading and writing with precision.

Is your child able to add, subtract and multiply amounts of money, work with fractions, work with clocks and calendars, work with maps and plans, and measure length, weight-mass and volume? Schools may promote your son or daughter without providing basic skills in reading, writing and arithmetic.

Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons


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Location: Site Entrance << Volume 1 Elements of Reason


Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity
Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards
More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers
Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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