Employ an online or offline tutor at your own risk
from
AU:
tutorfinder.com.au
CDN :
findatutor.ca
CDN: .i-tutor.ca
CDN: Montreal
Tutors
NZ:
findatutor.co.nz
UK:
tutorhunt.com
UK: tutors4me.co.uk
USA:
wiziq.com
USA: ziizoo.com
or employ the site author - View
his WiZiQ profile
- Calculus students are very welcome.
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YOU are better than YOU think. Show
yourself how:
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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
Do not leave here without it - Logic
mastery will develops critical thinking, improve reading and
writing, and give a firmer base for work and studies at many levels.
Good luck.
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Caution: Site advice
is approximately correct, for some circumstances, not all.
Site How-TOs are
logically developed, but not tried and tested. That leaves
room for thought and refinement.. |
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After logic,
(a) continue reading Three
Skills for Algebra, chapters 8 to 14 and do so alongside
site area on solving
linear2007 Equations ; or (b) see this calculus
starter lesson and Volume 3, Why
Slopes & More Math, chapters 2 to 6;
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.
|
Explore collaborative whiteboards
from groupboard,
twiddla or
scriblink.
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Chapter 6
Rule-Based Reason in Math
An Unruled Origin
Decimal notation did not appear overnight. In the past four centuries, it was
invented and refined by many people along with rules for addition, subtraction,
multiplication and division. Decimal notation made the common knowledge of
arithmetic possible. Decimal notation varies from country to country. In many
countries which employ the metric system, a decimal comma is employed instead of
a decimal point.
In Europe, the algebraic way of writing and reasoning started to emerge, not
in finished form of course, about the twelfth century[1].
[1] Algebra began with examples that
provide patterns for inheritance computations. Following this the use of
algebraic shorthand notation to describe calculation and to change them
developed slowly. The full power of the algebraic shorthand notation for
arriving at conclusions was not fully recognized until the age of Leibniz and
Newton. Leibniz had the idea of a universal algebra for thought, and an
enthusiasm for it. Precursors of algebraic thought existed in ancient times,
but they did not necessarily influence the long gestation or rebirth of the
algebraic way of writing and reasoning in Europe and its spread around the
world. That the rebirth occurred in the geographic region of Europe is an
accident of history.
According to the book The Historical Roots of Elementary
Mathematics, by Bunt, Jones and Bedient, Dover Publications Inc, New York,
1976 & 1988, the word algebra is a corruption of the title Al-jebr
w’al-muqabala of an 820 work by Mohammed ibn Musa al-Khowarizimi.
His work amongst other contributions to mathematical thought, included
examples to teach or show the arithmetic patterns followed in the division of
inheritances according to Muslim law. (I suspect the algebraic way of writing
and reasoning with its description of calculations may be regarded as a
refinement of this demonstration of arithmetic patterns. The proof of this
suspicion is a matter for further historical inquiry or historical hindsight.)
According to the book Algebra began with examples that
provide patterns for inheritance computations. Following this the use of
algebraic shorthand notation to describe calculation and to change them
developed slowly. The full power of the algebraic shorthand notation for
arriving at conclusions was not fully recognized until the age of Leibniz and
Newton. Leibniz had the idea of a universal algebra for thought, and an
enthusiasm for it. Precursors of algebraic thought existed in ancient times,
but they did not necessarily influence the long gestation or rebirth of the
algebraic way of writing and reasoning in Europe and its spread around the
world. That the rebirth occurred in the geographic region of Europe is an
accident of history.
From then on, algebraic ideas developed, but the rules of procedure were not
clear. Yet the algebraic reasoning including slope calculation suggested new
results and calculations in mathematics and the sciences. Some were clearly
true, some were clearly false and some were doubtful: more could be stated or
suggested than proven with algebra. The imprecision in algebraic thought was a
source of concern. Not all was certain. In contrast to Euclid’s work on
geometry, algebraic thought was not a model for reason[2].
Despite the latter, algebra along with say physical intuition provided methods
for arriving at conclusions, where none existed before.
[2] Newton in his work employed algebraic methods to
obtain or suggest results, but he relied on geometric demonstrations to
confirm them.
Next: Chapter 6, part ii, Axiomatic
Codification of Mathematics
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www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Area Map
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Books & Articles to Read
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Chapter 11: Primary School Mathematics
11 Primary Math
11 Cue Cards
11 Counting
11 Decimals - Addition
11 Decimals -Times
11 Decimals & Subtraction
11 Fractions and Division
11 Notational Conflict
11 Reciprocals Etc
11 Decimals - Ratios
11 Size Comparison
11 Numbers, +ve or -ve
11 Rename < Sign
11 Complex Numbers
Will provide an alternative to Chapter 11 later, most likely in the Parent's
Area: Help Your Child or Teen
Learn
Most students in high school are not heading for calculus,
but most topics in high school mathematics are present due to calculus.
Preparation for calculus demands their coverage at full strength.
See too, this site 55+, Math
Education Essays. Site areas and pages provide pieces of the a Mathematics
Education, Jigsaw Puzzle, in formation.
-Inductive
principles for course design & delivery require a clear
description of where and how skills and concepts may rest on earlier ones, so
that difficulties may be explained and remedied by looking for what was
missed or forgotten in earlier studies.
Mathematics is a demanding subject. All errors in notation
and comprehension need to be identified and corrected. In
reading, spelling and writing, students have to learn all the letters in the
alphabet, not just some. and memorize spelling. Anything less implies
difficulty.
Likewise in mathematics, students have to master key skills
and concepts, one at a time and one after another. Anything less implies
difficulty.
Modern mathematics curricula introduced an inconsistency
into course design and delivery. They did not sanction the use of decimals nor
the use of diagrams in skill and concept development but decimal arithmetic
and diagrams are needed for student comprehension and for an operational
mastery of quantitative skills. That implies the need for an mixed-math
curricula based on a systematic development of operational skills, sufficient
for applications and sufficient to provide a base & context
for the very optional study of pure mathematis.
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