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YOU are better than YOU think. Show
yourself how:
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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.
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The Trouble With Algebra
People have difficulties in algebra as the shorthand role of
letters and symbols appears is required but steps for its full development
are too large or missing. Remedies are offered in site material. The
remedies (a) describe and illustrate three skills for algebra; (b)
describe and illustrate the direct and indirect use of equations,
proportionality equations included, and so vocalizes a unifying and hitherto
theme in secondary school mathematics; and (c) re-introduces
the concept of what is a variable with words instead of
symbols. The remedies are continued in site pages on solving linear
equations, arithmetic with polynomials, complex numbers and calculus.
Before or besides the early use of formulas in primary
school and of the mastery via examples of methods for addition,
subtraction, multiplication and division of fractions, the shorthand role of
letters and symbols in describing the properties of arithmetic operations with
further numbers (real or complex) can be introduced or hinted at by giving and
illustrating algebraic descriptions of these operations.
The unavoidable occurrence of arithmetic and algebraic
expressions, better seen and read silently, 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. To lower
the barrier but to circumvent it, site pages provide a more vocal, a more
audible paths. In retrospect, shortcomings in the exposition or
explanation of algebraic skills and concepts has twisted or complicated learning
and teaching from from first appearance of algebra to their full-strength use of
algebra in elementary and advanced calculus. Finally, 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 - that
arithmetic should be repeatable, reproducible, verifiable and automatic.
Course design and delivery may now follow the easier path of developing
skills and concepts in solving linear equations, of arithmetic with polynomials,
of complex numbers, from assumptions about arithmetic with real and complex
numbers.
Optional would be a mixed inductive & deductive development of
Euclidean Geometry and the properties of real and complex numbers where
decimal representation of real numbers assumed..
All the foregoing has the aim of providing (i) an operational command of
mixed mathematics and providing (ii) the algebraic-deductive maturity needed in
the further and optional study of modern mathematics, axiom based.
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www.whyslopes.com
Mathematics Education Essays
57 or so
Area Entrance & Hub
Ideas for Better Instruction
4 Ways to Improve Reform
Theory of Knowledge
Peer Review
The Trouble With Algebra
Course Design and Delivery
How Letters Appear
Sit Down & Study
Modern Education
Key Notes and Themes
Site Lesson Plans
How This Site Differs
Site Origins
Math & Logic Puzzles
Comments on site content.
Words For Instructors
Inductive Principles
Fairness Principles
Apprentices & Masters
Three Remarks
For a Leaner Curriculum
Mixed Maths Curricula
Cultivating Intelligence
Reason - 3 kinds in maths
Logic in Mathematics
Science Education
Maths Instruction in General
Operational View & Values
Standards
Ends and Values
Goals & Unifying Themes
Algebra Lesson Plans
Algebra, Geometrically
Mathematics Curriculum Shifts
Teaching Tips - Fractions to Calculus
Math Ed Perils
Talk the algebra talk
Sec I - Fraction Focus
Sec II - algebra focus
Sec III - Focus on Slopes
Maps-Plans-Drawings
Math Wall Posters
Education, Empirical Art
Damage Reversal
North American Math Curriculum
Managing Reform
Essay January 2007
Educational Follies
Contructivism Incomplete
Missing the Point I
Mathematics in Context
What and When, A Challenge
Grouping Students
Teacher Certification
Education of Math Ed. Professors
Site Eurekas
Links
Help Me Learn/Teach;
- Algebra
words before symbols
- direct &
indirect use of formula, numerical versus algebraic solutions - what
is a variable (more words)
- Arithmetic
- exercises
- with fractions
-
videos on primes, lcm, gcm,lcd, square roots etc
- Calculus - geometric
preview, algebraic
preview,
3 study guides,
much more
- Complex numbers
-starter lesson with java applet - easy
consequences for trig & vectors in the plane
- Education
- Empirical Course
Design & Delivery
- Fractions
- alone
- by rote
- with
algebra
- videos
- Functions - introduction
hindsight
- composition aka
substitution -
- Geometry, Euclidean - Correspondence
of triangles, Triangle
construction, duplication & Isometry - Failure
of ASA & the // line postulate - angle
sum in triangles -//
grams - Triangle
Similarity
- Geometry-
Analytic - functions, polynomials, complex numbers, unit circle
trigonometry
- Logic
- First Steps -
Symbols in
Logic -
Occurrence
& Truth Tables - Indirect
Reason -Indirect
Reason More
- Proportionality
- Definition
- Direct & Indirect Use - Numerical versus Algebraic Solutions
- Real Analysis
- Decimal View of concepts
and of proofs
- Rules &Patterns in Science, Technology & Society
- Pattern Based Reason
- Mathematical Reasoning, empirical, inductive or deductive
- Units
- in rates & slopes
& (?) derivatives
- in ratios
& proportions - slopes & rates included
- Complex Numbers & Vectors & Trig
- trig expression for
dot & cross - cosine
law
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