Welcome.
If you like Volume 2, Three
Skills for Algebra (chapters 1 to 12, 14, 16 & 17) then site exploration
is recommended.
| Work
Ethic: When a child says the alphabet has too many letters to
master, we say no, we say all the letters need to be met and mastered to
read, write and spell. Likewise in mathematics when the student or worse
course design says mastery of arithmetic methods are no longer needed due
to the presence of calculators, we still have to say no. The
ability to follow steps, one at time and one after another, begins in
arithmetic, and shows the need to do so precisely and carefully not only
in arithmetic, but in general. That is character building.
In the
first years of schooling arithmetic mastery strengthens the work and
study values and habits implied by learning to do arithmetic on paper in
an observable and correctable manner. Indeed, mistakes in arithmetic
force better work and study values and habit. Seeing the need to avoid
mistakes forces greater care and precision becomes a sign of intelligence
not only in arithmetic, but in general. In later years of
schooling, logic mastery in English and in mathematics
courses may develop lawyer like reading and writing skills to
understand the small print, the nuances and subtleties needed for work and
study, and for communication & reason in general.
|
Three motivations for reading Chapers 1 to 5, 8 to 12 and 14
in follow.
- First: The greater precision in
reading and writing that comes with logic mastery will (we hope) ease or avoid
learning difficulties in and outside of mathematics, at school and
work. See chapters 1 to 5, or if you would like to read less,
the two logic puzzles in chapter 2.
- Second.: Expressions and
rules in arithmetic and algebra are best read in silence. Because of
that, not enough words are used in explaining and understanding the
algebraic shorthand way of writing and reasoning in
mathematics. Talking about three skills for algebra provides
(we hope) a
remedy. See chapters 8 to 12.
- Third. In retrospect, there is a fourth silent skill in
algebra. In senior high school and college mathematics, the forward and backwards use of each and
every formula appears. It has been a unifying but silent thread in
mathematics education. But the phrase forward and backwards describes in words that
theme and ends the silence. See Chapter 14.
Identifying and describing four skills for
algebra provides a missing link in mathematics education. Professor
Whyslopes would not joke about that, would he?
Book and Related Site Content
|
Algebra Starter, Re-Starter Lessons
Proper
Use of Equal Sign (Very Important) |
8
The Three Skills
9
First Skill
What
is a Variables
10
Two More Skills
11 Why Shorthand
12
Shorthand Usage
13
What's Next
14
Compound Interest
15_Solving_Linear_Equations
16
Painless Proofs
17
Pythagoras
|
Fractions
and Solving Equations with stick diagrams (spring 2005
postscript) - |
A.
Letters & Lengths
B.
Stick Diagrams
C.
Beyond Sticks
D..Almost
One
E.
Binary Systems
F.
Larger Systems |
Stick
Diagram Examples
2x + 5 = 20
3x + 10 = 32
5a
+ 16 = 3a+ 24
(½)x
+ 8 = 24½
(¾)a + 16 = (¼)a+ 24
(¾)q
+ 17 = 32 |
| Stick
diagrams give a visual introduction to algebra which develops
fraction sense and skills. Try them before or besides chapter 15, Solving
Linear Equations, whatever works |
| Volume 2, Three Skills for Algebra, was misnamed. A fourth skill, the
forward and backward use of equations, appears in Chapter 14 on the Compound
Interest Formula. My aim in writing the chapters was to expand the
algebraic way of writing and reasoning. Only later, much later, did I realise
that the forward and backward use of formulas in that chapters could be
identified as unifying theme for secondary mathematics in the use of all
equations and formulas and in working with proportional relations or equations.
Before the latter can be applied directly, they have to be used backwards or
indirectly to obtain the value of the proportionality constant.
|
Still More Motivation for the first half of Volume 2, Three Skills for
Algebra
| |
Three Skills
For
Algebra
understanding & explaining
Reason and Math
Volume 2
Printed in Canada
ISBN 0-9697564-2-9
|
[ Back ] [ Next ]
Chapters and Appendices
Home
Postscript: The 4-th Skill For Algebra
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6 Old Language
7 Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I. Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19 Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning
Words Before Symbols:
What is a Variable?
Introduction
Variation between Examples
Variation of Letters
A letter denotes a variable
Cases of Double Variation
Three Notions of a Variable
Constants, Parameters
& Variables
Talking about numbers
Dependent
or Independent
Variable, a Matter of Choice
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