|
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
|
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. |
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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.
|
// _ _ \\
/\ /\
<| (o) (o) |>
| |
| |
\
/
\ = /
|
Caution: Site advice is approximately
correct, for some circumstances, not all. That leaves room for thought |
-/[]\-
||
_ / \
||||||||||||||||||||||||||||
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.
| |
Chapter 2
For and Against Mathematics
Previous: Chapter 1, Introduction
Motivations from philosophy, daily life and science or technology or
business can be offered for the study of mathematics. They are described
next with some links to reason. This chapter ends with paragraphs which
discusses why students avoid mathematics.
1. For some past philosophers and thinkers, the definitions and
conclusion reaching methods in Euclid's books on geometry provided the most
certain model for rule-based reason, or how to argue if you must. Geometric
knowledge was based on rules, patterns and definitions which seemed self-evident
once said or described. Euclid's books or elements were composed two thousand
years ago. Modern translations of them exist. The translations are recommended
to specialists in geometry only, as today newer presentations of geometry and
other ideas of mathematics are favoured in the classroom. For students wanting
to understand whatever they may be doing, if not why, the Euclidean organization
of mathematical disciplines can be attractive. This is a link to the love of
rule-based reason, if not knowledge.
2. For the person in the street a few centuries ago, writing, reading and
figuring skills were signs of knowledge or education. Before the seventeenth
century A.D., if not later, the absence of a good notation for arithmetic
made figuring hard, except perhaps in areas where the abacus was readily
available. Since the seventeenth century the development of the printing
press and of arithmetic based on decimal notation, the skills of writing,
reading and figuring have become common. For the person in the street, these
skills are useful in correspondence and in the buying and selling of goods,
property and services. Counting, decimal arithmetic and the use of simple
formulas provide people with repeatable and therefore verifiable rules for
arriving at conclusions. Figuring on paper or in the head is also part of
rule-based thought and reason. This link to rule-based reason needs to be
remembered.
3. The decision of what or how to calculate in business, science and
technology often depends on an algebraic way of writing and thinking for
describing (or changing) calculations, numbers and quantities. The latter
appears as a reasoning tool. But this tool is part of the mystery surrounding
mathematics and reason for many people in school and out. Implication rules with
the algebraic way of writing and thinking, if clearly explained, provide a
foundation for both mathematical thought and also rule-based reason in all
disciplines. A student may be encouraged to study mathematics in the hope of
understanding whatever he or she might be doing and why, and to have the option
of mastering numerical disciplines in science, technology or commerce. For
better or worse, this is a link to rule-based reason in modern life.
At least four further motivations for studying mathematics exist. First,
teachers and writers in all disciplines may have the goal of identifying and
imparting some worthwhile knowledge - an incentive for this writer. Second,
researchers in mathematics may identify the goal of extending the boundaries and
form of mathematical thought. Third, some people were attracted to mathematics
instruction and research just as a means to a livelihood in some shape or
manner. Fourth, some find an enthusiasm for mathematical thought or mathematical
recreations sufficient.
No single motivation can satisfy everyone. A motivation that is meaningful for
one may seem vacuous to another, and some motivations are not positive.
In modern society or times, science and technology are used to justify
ecological and ethical acts which appall some students. Students see that the
environment is under threat. Many leading elements of our society busily
trying to survive today have the attitude that tomorrow does not count.
Students see the use of technology and science in the creation of war
machines. Students fear that there will be no jobs regardless of what they
study. And students excelling in literature and word-based subjects may find
the symbolic and algebraic exposition of mathematics itself and of
quantitative disciplines alienating – an abstract art. Their teachers in
schools and colleges may be powerless and insecure cogs in bureaucracies that
go forward without allowing initiative. Not all is well. Schools may be like
assembly lines, impersonally processing students or livestock to be moved on
and out. Given the fears that students may acquire, many rational students
will turn away in despair from studies or planning for the future. [1]
[1] As a student and then as an adult I
had a fear and despaired of the ever-present possibility of nuclear war. Much
to my own surprise, thirty years later in 1996, I am still alive but have
refrained from having children. This refrain may continue due to my ecological
pessimism.
In education and society, give students hope or pay the consequences in the
classroom, in the streets and in the morgue – circumstances hopeless or
lacking purpose may lead students to harm (glue sniffing, drugs, crime or
suicide). Compulsory education is absurd and pointless without care for these
other factors.
Next: Chapter 3, Algebraic
Thought- What is it
| |
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.
|