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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 1, part ii,
Two Barriers
Previous: Chapter 1, Entrance or Intro
The first barrier opposing math education has been the lack of words to
introduce the algebraic or symbolic ways of writing and thinking.
One way of symbolic thinking is recorded and represented in
arithmetic. Further ways are recorded and represented in the solution of a
system of equation and in the comprehension of the properties of real numbers.
Algebraic reasoning, more so than arithmetic figuring, is better seen and
written than expressed verbally. Many today find this visual and nonverbal
aspect of algebraic thought intimidating and mystifying. Words have to explain
this situation. The discussion and illustration at length of the three skills
for algebra, one of which is prealgebraic or symbol-free can remove or lower
this first barrier.
Alongside the first barrier, the second barrier opposing mathematics
instruction stems from the most secure and attractive features of formal
mathematics, its strict deductive hierarchy of concepts and assertions. With the
strict hierarchy, comprehension of later terms and ideas requires a precise
comprehension of previous ones. This has meant that the concepts described at
the end of a mathematics course are incomprehensible to the student just
starting that course, or even in the middle of that course. The hierarchy
implies a student, even one who mastered all the concepts so far, has no inkling
or vision of what comes next. Thus for students past and present, including many
teachers, mathematical ideas beyond the last course passed or taken remain
unseen and unfathomable.
The strict hierarchical description of mathematics provides a hard and
rigorous route which only the most patient or the naturally adept follow far.
This strict route leaves the image of mathematics and its logic incomplete and
mysterious. Ease of exposition has not been the guide in higher mathematics. Yet
ease of exposition can be the guide in extending the common knowledge of
mathematics provided in elementary school before any rigorous deductive
exposition begins. Following a command of arithmetic, counting and the use of
simple formulas, there are wordy lessons which offer a simple command of
algebraic thought, and further lessons which offer a math-free command of
deductive reasoning: how to use rules and patterns one at a time, or one after
another to arrive at conclusions or decisions. Such lessons offer strands of
thought which are easily described and repeated. Putting them first provides a
context in which more deductive accounts can be presented and discussed. In it,
previously mastered strands of thought can be threaded or rethreaded together
with more and more rigour.
More Chapter 1 Sections: [ Up ] [ 1 Two Barriers ] [ 1 Lowering Barriers ] [ 1. Keys to Success ] [ 1 Units & Decimals ] [ 1 Chapter Guide ]
Next: 1 Lowering Barriers -
three skills for Algebra and words before symbols
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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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