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Protest: The site author, a McGill University,
1983 Ph. D in mathematics, failed a McGill Faculty of Education B. Ed pgm 2003-5
due to
YOU are better than YOU think. Show
yourself how:
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Read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study
Learn to read notes and textbooks like
a lawyer, so that no nuance, no subtlety and no clause escapes your
attention. |
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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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A Theory of Knowledge
Science and technology develops from hypotheses
(rules and patterns) for testing directly or through the consistency of
implications (chains or reason) with observations, all in an empirical
repeatable and reproducible manner. The latter may imply the limits of
rules and patterns. Mixed or applied mathematics too is an empirical
subject built on assumed numerically and geometrical rules and patterns -
assumptions drawn from experience and consistent with the most part with
experience. While historical and pedagogical path to the
thought-based development of mathematics skills and concepts goes
through synthetic (coordinate-free) drawings in geometry, the empirical
limitations of the latter path appear in diagrams whose faults are explained
with the aid of analytic geometry, and the empirical nature of pure
mathematics appears in the absence of an absolute basis for mathematical
theories rich enough to represent the infinite set of natural
numbers. There are stories to be told and repeated here about the
development and construction of skills and concepts in mathematics. The
telling and repetition of stories to understand and explain the development of
mathematical skills and concepts in a repeatable and reproducible manner is
most likely inconsistent with post-modern, rule and pattern -rejecting
developments in educational theories favoring subjective learning and
knowledge, and indirect instruction.
We have the ability to follow and present stories on paper and
on stage. Those stories may be fiction or not. Some stories may follow
each other, one at a time and one after another, or in parallel. Each person has
his or her story to tell. Mine is brief since I have forgotten many of the
details. Now the ability to follow and tell stories echoes in the works of
knowledge and fiction met in mathematics, science, technology and society.
Non-fiction is preferred.
In mathematics, each proof or deductive chain of reason in
represents a story or a sequence of stories to be told and
repeated. The telling and repetition of stories or proofs links and
develops skills and concepts in mathematics, one at a time and one after
another, all in a repeatable and reproducible manner.. In each empirical
theory, there are stories to be told and repeated in the development,
construction and testing of skills and concepts, or skills and
concepts, subject to the limitations of rule and pattern based thought. There-in
lies a gamble. So no all certain. But many of the methods of
mathematics appear to be repeatable, reproducible and hence reliable tools in
science and commerce. So there is a chance, the methods are non-fiction.
Mathematics instruction may be given the task of providing
students with an operational command of the calculating and reasoning or proof
methods in mathematics, pure or applied or mixed, and an eventual awareness of
benefits, origins and limitations of the rules and patterns involved in the
subject and other disciplines. In education, the empirical hope or
hypotheses that a student has an operational command of one area of proof or
figuring can be tested by observing what a student writes or produces. If a
student fails, more instruction or study is required while if a student passes
the test, chances are he or she has master some mathematics, enough to continue
instruction without review. Mathematics education is an empirical art in which
instructor may observe the work of each student, and provide feedback or
correction while the student is trying to follow the theories and methods of
mathematics in a repeatable, reproducible and objective manner, modulo the
limitations of rule and pattern based thought and processes.
Science, Mathematics and Education
Mathematics is called the Queen of Science. But mathematics is
still an empirical science. Historically, the thought-based development of
mathematics begins began with synthetic (coordinate-free) drawings in geometry
to arrive at conclusions with the aid of axioms (assumed patterns). But
the empirical limitations of the latter path, the use of drawings, appear in
diagrams whose faults are only explained with the aid of analytic geometry, the
use of coordinates. That use turns the development historical development
of mathematics upside down. Synthetic geometry is now replaced by
coordinate-based geometry - models in drawings are codified or represented by
points and sets of points, models in which the properties of real numbers are
now employed to arrive at conclusions. None the less, the empirical
nature of pure mathematics stems in the origins of its axioms - assumed patterns
which are not given, they are chosen. Here they are chosen to avoid
inconsistencies met in previous attempts to provide a consistent thought based
development of mathematics from axioms for real numbers - more precisely
assumptions about sets that give a model of mathematics in which real numbers
are represented or codified. Thus mathematics itself has an
empirical origin, albeit one sufficient to imply repeatable and reproducible,
and hence verifiable deductive chains of reason.
Hypothesis (Conjecture) Testing in mathematics: In
a mathematics theory or model based on axioms (assumed patterns),
we test of an statement or assertion by looking for a proof, that
is, a deductive chain of reason starting with and only involving
previously tested or proven deductive consequences of the axioms (assumed
patterns). If a valid proof is found, the statement is considered to be
tested and hence proven. That is subject to the comments above about works of fiction
and non-fiction, consistent or otherwise.
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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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