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Have your gifted students read logic
chapters 1 to 5 in online volume Three
Skills for Algebra for greater skills & confidence
in work
and study.
tell students to read notes and textbooks like a lawyer, so that no nuance, no
subtlety and no clause escapes their 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
Tell students that 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. In Volume 2, Three
Skills for Algebra, a 4th skill for algebra appears in Chapter 14. It
provides a unifying theme for high school mathematics - equations and formulas
may be used forwards and backwards, directly and indirectly, numerically in arithmetic
solutions & with literals in algebraic solutions.
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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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Chapter 6
Rule-Based Reason in Math
Previous: Chapter 6, part i, The
Unruled Origin
Part II, Axiomatic Codification
From the mid-nineteenth century, efforts mostly in Europe began to make the
reasoning in the applications of algebra (including calculus) more certain like
that in Euclid’s work on geometry. Briefly put, from the mid-eighteenth
century to the 1920’s, the formulation of rules for arithmetic, more precisely
for set formation, provided a more certain, thought-based, set-theoretic
foundation [3] for the algebraic way of arriving at
conclusions about sets, numbers and calculations, and also for geometry.
[3] Technical Note: Fraenkel in the 1920
modified the 1905 set-theoretic assumptions of Zermelo. This modification
defines what is now called Zermelo-Fraenkel set-theory foundation for
mathematics. Minor variants of this foundation and some alternative
codifications have been considered since. For a more precise image, advanced
college students in mathematics may consult the non-technical and historical
passages in the work The Logical Foundations of
Mathematics by W. S. Hatcher, 1982, Pergamon Press, ISBN 0-08-025800-X.
The foundation is based on several assumptions, also called axioms, about set
existence and formation. Today, there is a set or arithmetic-based algebraic
approach to geometry that is intellectually more certain than the treatment of
geometry in the work of Euclid. The standards of reason or persuasion in
mathematics have been raised and made more strict with the passage of time and
the accumulation of knowledge of what to do and avoid.
Thus previously unruled exploration and command of algebraic thought was
axiomatically codified and reorganized in a logical, rule-based fashion. This
rule and thought-based codification further includes methods for arriving at
conclusions used only in mathematical disciplines. The principle or method of
mathematical induction provides a first example [4].
[4] Further examples are given by the axioms
of choice
The codification provides a restrictive framework for algebraic and
mathematical thought, less free but more certain or reliable than before. This
almost legal codification provided a more strict logical or thought-based
construction and organization for pure mathematical knowledge. Logos is a
Greek word for thought.
Within the codification, whole numbers, rational numbers and real numbers are
precisely represented by sets, and not by decimals. In the codification,
decimals have no special role nor place. The decimal representation of numbers
can be introduced and their properties derived from the assumptions. But strange
as it may seem, the framework allows for a precise decimal-free discussion of
computations.
The need or desire for precision in the logical codification led to the
writing and selecting of definitions, assumptions and associated chains of
implications, removed from everyday thought and language and remote from the
primary or common knowledge of mathematics. Ease of exposition and the extension
of the common knowledge of mathematics were not concerns of the codification. Finding
a logical, thought-based sanction for mathematics was[5].
[5] In retrospect, what was found was not absolute
sanction, but a codification, a thought-based framework.
The algebraic way of writing and thinking is present in all
mathematics after arithmetic. Yet in contrast to the wide reach of algebraic
thought, the subject of algebra in mathematics today is more focused. In
college, the study of algebra may drift from the high school description of
operations on real numbers to a description or study of the various forms of
rule-based calculations. Here there is no concern for the type of numbers,
arrows or quantities etc., which appear in those calculations. The guiding
questions in algebra in the late nineteenth and most of the twentieth
centuries have concerned (i) what is implied by the specification of rules or
properties for operations, and (ii) what kind of objects or numbers, if any,
would satisfy the specified rules or properties.
Before the nineteenth century, algebraic thought and algebra
included calculus and the equations of all mathematical disciplines –
everything that was not in geometry and not in the recently developed decimal
arithmetic. But the concern about the justification for calculus (slope)
related computations and the concern of how to speak about sets, while
avoiding inconsistencies or contradictions, led to the birth of analysis, a
subject separate from algebra. Within this new subject of analysis, the
codification has yielded a rule-based foundation for the precise description
and handling of set and number based operations. For further information on
the division of mathematics, see the VNR Concise Encyclopedia of
Mathematics. See also the earlier discussion of what is algebra.
Postscript (Oct 1, 2006): Three Kinds of Reason in mathematics
There appears to be at least three kinds of reason in mathematics: (i)
Pattern Recognition, (ii) Use of methods with repeatable and reproducible, and
thus verifiable results; and (iii) deductive based chains of implication rules,
direct or indirect.
- The recognition of patterns, geometric or arithmetic, is part of
elementary mathematics. There we use patterns. A pattern is tested by seeing
whether or not it fails. If it succeeds we tend to use. If it fails we
reject. Inductive (hands-on, constructive) instruction in elementary school
appears to be based on offering students patterns to recognize or construct,
and then test in the way just described.
- The use of arithmetic and then deductive methods with repeatable and
reproducible results is a basis for learning by rote or with comprehension
in mathematics. Results which are repeatable and reproducible are
verifiable. That leads to confidence in the methods. If a method does not
lead to repeatable and reproducible results, the method or the user's
mastery of it is false.
- The use of deductive reason in mathematics requires two gambles. The first
gamble lies in the assumption of deductive logic, the assumption that chains
of reason, direct or indirect, can be employed to imply results in a
repeatable and reproducible manner. The second gamble lies in assumptions
about numbers. their use and properties, and in the case of applied or mixed
mathematics, their connection to geometry and/or physics. Every theory
provides a good yarn or story or piece of theatre to follow to fictional or
non-fictional conclusions. Theories are refuted (falsified) when their
consequences fail. That being said, tried and tested, arithmetic,
geometric and deductive methods of mathematics appear to give repeatable and
reproducible results.
Mathematics education is a multi-faceted entity. The formal statement of
axioms (arithmetic or geometric patterns) in terms of letters and symbols , and
the consequences of those axioms assume logic and the shorthand role of letters
and symbols in arriving at conclusions, arithmetic, geometric or
deductive. Mathematics instruction need not be rushed. Students need to learn
carefully how to use rules and patterns, or follow methods, one at a time
and one after another, alone or in combined, to arrive at arithmetic, geometric
or formally deductive results and conclusions. Once students have acquired the
patience and ability to follow or apply methods in a repeatable and reproducible
manner, they are ready to combine methods to arrive at further methods, and they
are ready, patience permitting, to follow theories or explanations in which
understanding why is based on combining methods and patterns (axioms or
assumptions included) to provide explanations. Yet the ability and patience to
carefully follow the steps of methods , one step at a time and one after
another, with an awareness that an error in one step leads or most likely leads
to incorrect (non-reproducible) results is required for at least two of the
three kinds of reason mentioned above.
End of Postscript.
Next: Chapter 7, Geometry, 2 Ways
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www.whyslopes.com
Volume 1B, Mathematics Curriculum Notes,
Foreword + Chapters 1 to 10 + 12
Book Entrance
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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