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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.
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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.
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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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Location: Site Entrance < Archives < Mathematics Education Essays << Lessening Algebra Difficulties
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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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Location: Site Entrance < Archives < Mathematics Education Essays << Lessening Algebra Difficulties
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