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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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Curriculum Shifts - Undoing the Damage
The constructivist view of education calls for authentic, realistic, engaging
activities in the classroom - that call is great except for my lack of
imagination, so details how are required. Many arts and discipline
develop by trial and error, with nature as direct teacher correcting the
reasoning and methods of the developers. The element of the constructivist view
of education which calls for mathematics, science and complicated rule- and
pattern based arts and disciplines in general to be understood in the
classroom via student discovery and construction of the underlying
concepts without the teacher being an authority figure, so that student
chains of reason or their consequences are accepted as valid and not
judged nor corrected is empirically absurd.
Instructors of long developed arts and disciplines, arts and disciplines
corrected by nature, need to identify and empirically verify student
mastery and comprehension of previously discovered methods, so that students
empirically learn how to arrive at results in a repeatable, reproducible and
therefore verifiable fashion. Where instruction is an empirical art, teachers
judge and guide students by observing and correcting their writings or
activities. In complicated arts and disciplines, long-developed, there is
insufficient time for students to rediscover and verify the rules, patterns and
working practices of the disciplines. So direct summaries and skilful direct
instruction with skill practice and empirical concept verification is
required. The instruction may range from learning via practice and rote to
learning via practice and Euclidean, logic-based, developments.
In some primary and secondary classroom, mathematics lessons are giving
students concrete metaphors as building blocks for their comprehension of the
subject. Those metaphors are not incorrect. They complement the views I have met
as a students and teacher of elementary to advanced mathematics. But if
mathematics mastery is to be art that is repeatable and reproducible, there is
also a need for an operational and logical command of key skills and concepts in
high school, say those needed for calculus, in more old-fashioned manner, at
least until the metaphors provide a complete path.
Further Readings
Euclidean Model for development of an art or
discipline: More than two thousand years ago, the works of Euclid in
Geometry gave a model for a clear, full, logical development and
application of rules and patterns in an or a nearly authoritative
manner. But, But, But, he Euclidean model for reason and codifying
a domain of knowledge does not represent objectivity. It represents a striving
for objectives. That is, Pattern Based
Reason is not always authoritative due to gambles or approximations
in it. Skilful and empirical mastery of rule and pattern based thought and
action, modulo limitations, is one aim, we hope, of school and college
education in rational "method-based" arts and disciplines.
Learning to apply rules and patterns in a repeatable and reproducible manner
is or should be part of education. Students should know that some subjects
strive for objectivity.
Teachers & Parents: Compare and
combine site ideas with those available elsewhere. For example, the Mathematics
portion of English
National Curriculum, a site for teachers, is well-written and
well-put. Yet content shifts
might improve it. An empirical focus on what works or well or should
would improve education. Site material stemmed from a perception of
older gaps in course
materials and design, gaps not deliberate but still present. Asking
students to discover ideas by themselves is fine as long as the ideas in
question are not critical for further instruction.
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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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