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 < Archives < Mathematics Education Essays << Lessening Algebra Difficulties

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How to Lessen Algebra Difficulties

Three remedies for algebra difficulties likely to be effective and involving a greater and clearer use of words are proposed for digestion and refinement. The remedies (a)  describe and illustrate three skills for algebra; (b) describe and illustrate the forward & backward use of formulas and equations,  proportionality relations included;  and (c) re-introduces the concept of what is a variable with words that can be understood  first before & then besides the shorthand roles of letters and symbols. The three skills and the two equivalent phrases  "Forward and Backward Use" and "Direct and Indirect Use" vocalizes a unifying and previously unspoken themes in the use of formulas and proportionality relations. Examples showing how appear in chapters 8 to 14 in Volume 2, Three Skills for Algebra.  See too site areas on solving linear equations and on fractions. 

Before or besides the simple use of  formulas in primary school and of the  mastery via numerical examples of  methods for addition, subtraction, multiplication and division of fractions, the algebraic description of these  operations on fractions (rules for them)  may give a taste of the shorthand role to come of letters and symbols in describing  associative, commutative and distributive properties for arithmetic with real and complex numbers.

The unavoidable occurrence of arithmetic and algebraic expressions, better seen in silence, too awkward to read aloud,  has been a huge barrier to the role of words in  understanding and describing algebraic skills and concepts. Formulas and equations like pictures are worth a thousand words. So the exposition or development of skills and concepts has been too quiet. Site pages provide a more vocal, a more audible path. 

Silence in the exposition or explanation of algebraic skills and concepts has twisted or complicated mathematics from first use of simple formulas to full-strength use of algebra in advanced calculus. In retrospect, there has been a domino effect.

While introducing more words into the exposition of mathematics, to lower one barrier and not raise another, we need to remember that  skill development and perfection requires some drill and practice - not too much but enough. The algebraic way of writing and reasoning before or during its development. requires mastery of arithmetic with fractions without a calculator - arithmetic should be repeatable, reproducible, verifiable and automatic. Calculators and spreadsheets may remove the burden of arithmetic in complicated situations, handling data, but a careful command of exact arithmetic with fractions remains a base for operations in algebra, trigonometry and calculus. Calculations with units of measurement and measurements themselves should also be developed and maintained.

Mathematicians: The student need to understand and have an operational command of what is a variable  before any mention of functions and sets in their instruction.  The neat function views may be a set-based codification of a concept that needs to be understood & mastered  before the codification begins.

For secondary mathematics, the easily understood and used remedies above lead to a greater use of words in development of algebraic skills and concepts, and in that, recognition of a unifying theme - the forward and backward use of equations.

The above vocal development of skills and concept can and should be helped by geometric perspectives. See the site starter lessons for (i)  solving linear equations,  (ii) the distributive law and  multiplication of both decimals and polynomials and (iii) complex numbers. See too the algebraic and geometric previews of calculus. Calculus is the subject which requires the algebraic way of writing and reasoning at full strength. The previews prepare for that full strength use, may be met in pre-calculus courses, and aim to ease or avoid algebra shock in calculus. (iv) A decimal viewpoint of limits, or error control in the evaluation of functions and limits,  also helps avoid algebra shock.

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Parents: Help your child or teen learn:

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 < Archives < Mathematics Education Essays << Lessening Algebra Difficulties

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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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