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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
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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:
Lowering or Removing Barriers
Previous:1 Two Barriers
Mathematics courses from primary school to college consist of chains of
reason, some quite long, but not all deductive. The mastery of elementary
mathematical thought may be inductive, that is drawn from examples by the
primary school teacher. Primary school students of mathematics can be given
methods for computation and numbers to use in them, along with suggestive
reasoning to introduce and sometimes justify the computational methods. All this
may be contrasted with the algebraic and deductive rigour of formal mathematics
met in advanced instruction.
Primary Instruction
Primary or elementary instruction now defines the common knowledge of
mathematics. This instruction introduces and explains arithmetic from whole
numbers to fractions and decimals in a pre-algebraic and pre-deductive fashion.
Physical reasoning, groupings and analogies are employed to explain and justify
ideas and methods. The counting and computational methods taught in primary
instruction are repeatable, reproducible and thus verifiable. Arithmetic is rule
based. From it, student may learn that steps must be taken carefully. Two
different people with the same arithmetic expression to compute are expected to
obtain the same result, apart from small round-off errors. Primary instruction
provides a secure, often satisfying, rule and pattern based command of
mathematics. The security is based on methods which are repeatable and
reproducible, and thus verifiable, apart from any deductive account.
Primary instruction in preceding the deductive exposition of mathematics,
provides the first image of the discipline and more generally, of rule based
figuring and reasoning. The latter touches many subjects. Advanced mathematics
instruction presently ignores and thus discards this first image. It derives
mathematics from decimal-free axioms or assumptions about points, numbers and
sets without any mention or consideration of the student’s first command of
the discipline. The knowledge of mathematics and its pattern based reason is
thus not cumulative. It presently starts once in a pre-deductive fashion and
then again separately in a deductive fashion. This represents a gap and
discontinuity in the exposition of mathematics.
Between advanced and primary instruction, intermediate instruction is
assigned the task of providing a smooth and continuous transition. The task
includes directly but gradually describing and explaining deductive logic and
the algebraic way of writing and thinking to mathematical novices.
Intermediate level instruction may place first those ideas easily grasped to
extend the common knowledge, and put second the technical details. Here the
ideas put first can be selected to ease the comprehension of those put second.
Those put first can also chosen to give a command of mathematics and logic
valuable in itself just in case the deductive exposition and codification is not
reached. This design may give students immediate satisfaction in their studies
and thus provide an invitation to continue. This perspective puts the student,
and not the discipline first.
One way to introduce the deductive thought process is through math-free
lessons on implication rules (two logic puzzles), chains of reason, longer
change of reason, islands and divisions of knowledge given in companion
works to this one. This introduction can be given in mathematics courses or in
reading, writing and composition courses. The explanation of deductive thinking,
while required in mathematics, should be across the curriculum alongside and in
support of writing across the curriculum. [2]
[2] This author has only one comment about
writing across the curriculum. Just as some people are today math phobic, say
mathematically challenged, some are essay phobic or challenged despite good
intentions.
The common knowledge, to be extended and developed further in intermediate
instruction after primary instruction, can continue to be built first on
rule-based methods that are repeatable, reproducible and therefore secure or
verifiable. For ease of exposition or to create a more easily described image of
mathematics, physical and geometrical arguments may be employed to support or
suggest conclusions while deductive strands of reason are introduced as well.
Example, see below, can be provided by discussing and illustrating three skills
for algebra, and by presenting expositions of complex numbers and why slopes.
All communicate essential, if not abstract, ideas in mathematics while requiring
a minimal mathematical background.
More Chapter 1 Sections: [ Up ] [ 1 Two Barriers ] [ 1 Lowering Barriers ] [ 1. Keys to Success ] [ 1 Units & Decimals ] [ 1 Chapter Guide ]
Next: Keys to
Success
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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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