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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
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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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Education, An Empirical Art
In empirical arts, practices with repeatable and reproducible results come
first, tested via trial and error. Then theories and principles come later to
summarize, to codify, to refine and even enlighten the practices. While
practices or sequences of them in some empirical or hands-on arts in science,
technology and business, assembly lines included, may comply with
principles and standards, even be connected and organized and designed around
said principles and standards, the forerunner to such organization or
optimization should consists of methods that are tried and tested, methods
that work in a plug and play manner, methods whose benefits, origins and
limitations are described..
Education is an empirical art. We may not read a student's mind,
how a student thinks or links together skills and patterns, yet we can
observe and test student performance, skill by skill, concept by concept, and
encourage, but not guarantee, mastery of standard calculations and standard
arguments or chains of reason in algebra, geometry and beyond. In some
disciplines, not all, there are right and wrong answers due to methods that lead
to repeatable and reproducible, and thus verifiable results independent of
whom-ever applies the method. Learning how to apply and combine methods
carefully to obtain reproducible and thus verifiable results is an old sign of
intelligence in many old arts and disciplines in business, trades, science,
engineering, technology and bureaucracy. The latter is subject to the
limitations of rule and pattern based thought and practices, and the critical
knowledge that not all is certain in empirical based thought and practice.
Critical thinking in science and technology begins with an
awareness that what we hope for, dream of or construct in our minds remains
speculation or faith IF or WHILE it or its consequence cannot be observe or
tested directly to be corroborated if not confirmed. The foregoing is a rebuttal
to the constructivist theory of learning, the part which opposes testing, the
existence of questions with right or wrong answers, and which says student
knowledge, if individually constructed, should not be contradicted.
Empirically sound education must oppose wishful thinking. That being said,
constructivist methods for engaging, authentic, genuine material and the
development of critical thinking could be incorporated into education as an
empirical art.
More on Testing. Knowledge empirically
found or tested is relative and not absolute. Instruction which relies on
testing skills and concepts can only identify errors in the mastery of the
latter while correct responses only confirm, but do not guarantee mastery. But
the level of student competence in a discipline defined by skills and concept
mastery can be estimated from the degree of difficulty, the unlikelihood of
correct responses if skills and concepts have not been mastered, and
comprehensive of a test or series of test. Here individualized testing may be
informative that mass testing. Empirical soundness of instruction and testing,
the issue of lessons and associated tests with repeatable and
reproducible results locally and beyond, should not be scrutinized in an
absolute manner. Cognitive theory should look at education as an
empirical art.
While a teacher can not read the mind of a student, a teacher may see and
correct mistakes, minor to major, in the content and style of student writings
and further endeavors or products, so that the student may learn from his
or her mistakes, and possibly learn how to make fewer mistakes. In the short
span of education, several years or more, the student will meet subjects
in which individual construction or organization of skills and concepts cannot
in the first instance replace the early collective and refined products of many
minds.
Instruction is an empirical art with value judgments and decision dependent
on the subject at hand and what students produce - observable behaviors or
products only. Any else is subjective - not repeatable and reproducible.
In particular, the constructivist approach to instruction, despite fine
calls for authentic, realistic and engaging material and practices in the
classroom, calls that should be heeded and empirically supported as much as
possible, in its opposition to the testing and measurement of skills and
performance provide vacuous standards for instruction and undermines the
sequential nature of learning in which skills and concepts at one level need to
be learnt and verified before the next level begins.
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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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