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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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Chapter 1
Units and Decimals
two missing links
Previous: 1. Keys to
Success
Mathematics courses, besides preparing students for the deductive exposition
of advance courses, should also provide students with the deductive and
quantitative reasoning skills required in other subjects. Quantitative and
algebraic reasoning in science, technology and commerce involves units of
measurement or quantity and the decimal, if not binary, representation of
numbers. The set theoretic axiomization of pure mathematics is free of both
units and decimals. Primary and intermediate level courses can be assigned the
further task of sanctioning the use of both units and decimals in the
quantitative or algebraic reasoning of other subjects. Presently, the
presentation of axioms for real numbers makes no mention and offers no sanction
for the use of decimals or the use of units.
Primary instruction provides, one hopes, a thought-based command and
investment in the decimal representation of numbers. The latter representation
is indeed adequate for those who will never see in full the decimal-free set
theoretic codification of modern mathematics. Until the presentation of the
latter, the decimal representation is also adequate for students of mathematics.
For continuity between primary and further courses, added to the axioms or
assumptions about real numbers in intermediate level courses may be two
assumptions, first that real numbers have decimal expansions, and second, that
infinite decimal expansions define a real number. This reflects the common
knowledge or belief. Some sanction for it should be provided in mathematics so
that the common knowledge and axiomatic perspective do not need to be
reconciled.
Indeed, the decimal concept of convergence, with or without set theoretic
wrapping, is sufficient for students not meeting the decimal free alternative. [3]
Its discussion, see the chapter Error Control, Continuity and Limit in the
companion work Why Slopes and More Math, can further provide a background and a
context for the understanding of the decimal free approach – part of the
motivation or explanation why. Abstraction by itself, without concrete examples,
provides the student a vacuous knowledge. The vacuum is abhorred.
Primary instruction in quantitative reasoning introduces units of quantity or
measurement. Their absence in the algebraically described axioms for real
numbers, and in intermediate courses apart from trigonometry, separate
mathematics courses from the quantitative reasoning required in other courses.
While the need for units of measurement can be circumvented by the dimensionless
(unit-free) development of formulas and equations, that circumvention is a
complication not needed in elementary mathematics. In the first instance, the
algebraic way of writing and thinking can be employed with calculations
involving decimals and units.
Axioms for real numbers can be augmented with more axioms or assumptions
about quantitative and algebraic reasoning with units, monetary or physical.
Here for instance mathematics courses could present the innermost axioms, those
for real numbers and for their decimal expansion, while other courses in
quantitative reasoning, if not those in mathematics, could add further axioms,
those involving units. For students, this would yield an axiomatic framework for
numerical computations within a broader axiomatic framework for quantitative
reasoning. A consistent discussion of units would be welcome in mathematics or
adjunct disciplines. [4]
With the insertion of assumptions about decimals and units along with the
discussion of implications rules and three skills for algebra, mathematics
instruction could reach for or even achieve the following goals:
1. A continuous extension in high school and college of the common knowledge
acquired in primary schools of decimal arithmetic, counting and the use of
simple formulas.
2. The formal and/or informal provision of a logical framework for
computations in all mathematical disciplines with and without units.
3. The description and demonstration of a simple logical structure or
codification for mathematics and quantitative reasoning in other disciplines.
The consideration and retention of both decimals and units in the exposition of
mathematics would make it continuous and cumulative pedagogically from primary
school to college. The common knowledge of calculation and logic, developed in
primary and intermediate level courses, can be axiomatically codified in
advanced courses, and provide a context for the discussion of the codification.
[3] The college book Calculus by
Lipman Bers (Holt, Rinehart and Winston 1969) heads in this direction by talking
about the decimal representation of numbers and other equivalent
representations. The discussion of areas under a curve further begins with area
approximation of a region based on its coverings by small squares. This
approximation was taught in my primary school.
[4] Physicists and standards bureaus with their conventions
on physical units may be of assistance here.
More Chapter 1 Sections: [ Up ] [ 1 Two Barriers ] [ 1 Lowering Barriers ] [ 1. Keys to Success ] [ 1 Units & Decimals ] [ 1 Chapter Guide ]
Next: 1 About
the following chapters
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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
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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