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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 3
Algebraic Thought
Previous: Chapter 2, For
& Against Math
The Barrier
In the absence of other data, one resorts to personal experience – the
small sample of one – and then hopes that the experience is objective or
representative. As a high school student, I mastered the algebraic way of
writing and thinking, but many of my fellow students and some of my science
teachers had not. I was frustrated in some courses by the slow pace. I
attributed this slow pace to the difficulty others had with algebra. Moreover
the algebraic way of writing and thinking was required to understand courses in
algebra, physic and chemistry, but it was never directly explained.
As a college student in the 1970s, I tried to help a high school student.
When I tried to explain or demonstrate the algebraic way of writing and
thinking, she would say give me numbers, not letters. The algebraic way
of writing and reasoning was a powerful engine, but I lacked words to describe
it, and the descriptions in math texts seemed too brief. The student later said
that the algebraic way of writing and thinking, or what I was trying to say,
only became clear when she took calculus. [1]
[1] Calculus in the first instance
deals with slope-related calculations, their applications and interpretations.
Its explanation employs and illustrates the symbolic or algebraic way of
writing and thinking at full strength.
Through my studies of mathematics in high school and college I was aware of
this difficulty of explaining algebraic thought, but could not then conceive of
how to resolve it. After becoming a full-time instructor in 1983, I thought of a
simple remedy.
Postscript (2005). The algebraic barrier
described above is made more difficult by the lack of emphasis on fraction
sense and skills. Too many students and teachers at the limit of their
knowledge believe calculator usage removes the need to master fraction sense
and fraction skills. But the latter fraction sense and skills are
prerequisites to the algebraic way of writing and reasoning. The
introduction to algebra in Solving
Linear Equations With Stick Diagrams visually develops or re-enforces
fraction sense and skills. Try it with students 10 plus in your care.
Chapter sections: [What is Algebraic Thought] [ 3 Three Skills For Algebra ] [ 3 Words and Concepts Before Symbols: Variables ] [ 3. What is Algebra? ]
Next: Skills For Algebra -
Defining the or an Algebraic Way of Writing and Reasoning
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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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