Appetizers and Lessons for Mathematics & Reason Français: 26 pages
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What is a variable 5 fraction operations by raising terms Solving Linear Equations: Take I Take II

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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.

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Apprentices and Masters in Arts, Trades and Disciplines, a classical view of learning and teaching

December 28th, 2007

In many arts, trades and disciplines, the apprentices learn to follow and apply rules and patterns, steps and methods, practices given or demonstrated by masters, and to do so with sufficient care to obtain repeatable, reproducible and verifiable results. The apprentice in the first instance follows the customs and conventions of the master.  In this education, rules and patterns, steps and methods or practices may be learnt with or without explanation of why they work or why are they are followed.  But over time, the apprentice may see or the master may demonstrate how to combine rules and patterns, steps and methods, or practices to compound them to form further ones in a repeatable, reproducible and hence verifiable manner.  That process of combining or compounding lends or creates a hierarchical structure to the learning and mastery of the rules and patterns, steps and methods, or practices.

The extent to which  the apprentice may meet and obtain an operational command of art, trade or discipline may be subjective - some apprentices will develop abilities beyond that of their master, while others will equal or develop lesser skills. Masters and apprentices alone and in groups may accumulate rules and patterns, steps and methods, or practices to share and extend in a repeatable, reproducible and therefore verifiable manner. Moreover some rules and patterns will fall into disuse or be forgotten while newer ones arise. Each participant in the art, trade or discipline will see practices come and go, and may in time, see why.

Intelligence within an art, trade or discipline appears and is used when or while apprentices and masters are carefully to apply their practices, carefully, while learning about the benefits, flexibility, limitations and origins of elementary or compounded (combined) practices. In this, there may some be approximations or some uncertainty known to the master, if not the apprentice, through trial and error, and/or stories of what is feasible or not.  The master of an art, trade or discipline in meeting situations that have been handled before or not, will look for practices that work, and try to duplicate or refine them, and if need-be and if-possible, invent new practices to handle new situations. Operational command of a discipline may be enhanced and advanced by stories or theories to explain why rules and patterns work.  On the other hand, the latter may also unnecessarily inhibit and limit the operational command. Whence not all is certain.

Over years and decades, the customs and conventions of an art, trade or discipline in being refined may become more and more contrived and cease to be immediately obvious to the apprentice.  Whence the apprentice needs a guide or a master to show what is possible, and self-instruction is impeded.  The operational command of an  art, discipline and trade with a history of custom and convention including a hierarchical organization in which some skills and concepts depend on earlier ones in ways that took time and effort to discover and perfect requires guidance from masters of the art, discipline or trade. The apprentice is well-advised to seek that guidance and stand on the knowledge and wisdom of others, including practical and theoretical knowledge of the benefits, limits and origins of steps and methods, to avoid an ad hoc,  incomplete and most likely slow, construction or reconstruction of the latter skills and concepts.

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7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
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10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
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4 Lines and Slopes Take 1
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11 Parallel Straight Lines and Transversals
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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.

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70 Calculus Starter Lessons


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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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