|
YOU are better than YOU think. Show
yourself how:
|
// _ _ \\
/\ /\
<| (o) (o) |>
\ | | /
-/[]\-
||
/ \_
||||||||||||||||||||||||||||
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;
|
// _ _ \\
/\ /\
<| (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.
| |
Appendix E
How to Study Math and Why
Previous: D. What to do in School and Why
A Before You Stop
In school before you stop studying mathematics, please do the following.
- Look at the chapters Implication Rules and
Chains of Reason. These two chapters may help
you to use and understand precisely rules, instructions, patterns,
definitions and recipes in every subject and every area of skill or
specialization including mathematics.
- Read the first chapters on algebra. The description in them of the three
key skills for algebra and the algebraic examples will, I hope, help you
step from arithmetic to algebraic way of writing and thinking plus a little
beyond. These chapters try to explain and describe with everyday words, how
(algebraic) shorthand notation is used to describe and do mathematics after
arithmetic. There is more to mathematics than just doing arithmetic well.
B Why Study
Mathematics courses are preparation for business calculations, for handling
your money sensibly and for courses in sciences, engineering and technology. You
should view mathematics as an opportunity to strengthen your thinking skills.
In mathematics courses you should not only meet calculations to do but also
the chains and threads of reason and persuasion which justify them and links
them together. Understanding and following the rules and patterns of
mathematics, practices and nurtures an ability to think and reason well.
Mathematics provides a neutral territory for the practice of rule and
pattern-based reason and logic. The opinions and views you meet in daily life
say and care little about what mathematical conclusions should be.
If you find yourself in a course which gives formulas and numbers to use in
them but does not expect you to use algebra, you are wasting your time. Your
time would be better spent studying algebra, and then taking a more advanced
course that respects your intelligence. Similarly in college, if you find a
course which gives you formulas and numbers to use in them and also talks at
length about rates of change without expecting you to understand calculus,6
then a calculus course would be of better use of your time.
6
Calculus in the first instance provides formulas for the slopes of (nonlinear)
curves and for the rates of changes of numbers or quantities.
C When to Study
Look at the description of courses you will take in and outside mathematics.
From their description, see what mathematics course(s) you are expected to take
with them (co-requisites) or before them (prerequisites). Methods taught in a
co-requisite mathematics course are too often covered after they are required in
another course. So take a mathematics course before there is any possibility the
methods in it will be needed in another course. Then you may master the methods
before they are required and not after.
D More Advice
If you follow the advice and the cautions below, you should have a
mathematical foundation for any subject requiring calculation. In mathematics do
the following.
- Master arithmetic. Also master weights and measures. After you have
mastered the rules of arithmetic, learn to use a calculator.
- Master the algebraic way of writing and thinking. Also master the use of
rules and patterns to arrive at conclusions. Mathematics after arithmetic
builds on our abilities to talk about numbers and quantities, to describe
calculations and to change the way calculations are done. Mathematics after
arithmetic also depends on our ability to follow and understand rules and
patterns. See the first chapters on reason and algebra in this book.
- After algebra, take trigonometry and geometry.
- Learn about money matters. Take a course on money calculations, preferably
after a course in basic algebra. Most of us handle money for credit or
investment. In your last year of studies before starting work, take a course
on the mathematics and arithmetic of personal finances. This course should
include balancing of budgets, description of typical household expenses for
individuals or families in rented or mortgaged properties, problems
involving simple and compound interest, and mortgage/pension calculations.
Traces of such calculations appear in elementary and high school
mathematics, but they are forgotten years before you need them. A course
like the one described should be offered in schools and colleges for
students in any art or science. Ask for one to be given, if it is not
already offered. All this is practical mathematics. It should be more
widely known.
- If you go to college, take a year or two of the mathematics subject called
calculus, a year of probability and statistics, and a year of matrix
computations (or linear algebra). Calculus courses usually have trigonometry
and algebra as prerequisites. Calculations in many trades, including
business, engineering, computer technology, physics and health science,
require calculus.
Mastering the rules and patterns of mathematics and reason (there is a
connection) is good practice for mastering the rule and patterns of all
disciplines. To master mathematics you need to read your course notes or course
textbook carefully. Examples, solutions and proofs show you patterns to follow
or imitate. Here every step not understood hides an idea from you.
The problems you find easy to solve should be done to restore and build your
confidence and to reassure yourself that you have understood what they require.
But after you have done a few such problems, you should look at the ones which
appear harder. The problems which appear to be too hard should be noted and
remembered. You can return to them later by yourself or with help from another.
What is hard for you to solve may be easy for another, and vice-versa.
E Cautions
When I taught a remedial algebra course, one of my students was a high school
gym teacher. One of his past assignments was to teach algebra.7
7This
should not occur, but in many school systems it does. When it does, it shows a
lack of respect to students and a lack of purpose for education. It also
suggests circumstances beyond the control of students and teachers.
In some schools due to circumstance beyond their immediate control, some
instructors are required to explain ideas outside their own specialties. When or
if you meet such an instructor, be polite and do not become a troublemaker. If a
teacher sees you as a threat or troublemaker, you may suffer. When you meet a
misplaced instructor, politely and diplomatically try to transfer to another
class in the same subject or read the course textbook yourself and get a tutor.
F More Keys to Better Learning
Here are several more comments on learning mathematics or another subject.
- How you find a solution to a problem is not important provided you
understand fully the solution. (Some teachers may disagree.)
- If you have to copy solutions blindly then you will not understand ideas
well enough to pass tests and the final examination.
- You should ask another to check that your written responses or solutions
are both understandable and well-written. Mistakes brought to your attention
in any manner improve your understanding. If such checking improves your
ability to avoid mistakes in the future, then such checking should I believe
be encouraged. Again, some teachers may disagree.
- Students who know and identify in their solution those steps which are
doubtful deserve more respect than students who don't. Knowing exactly where
one is sure and where one is not is the sign of an alert mind.
- Correct answers obtained accidentally, for instance by canceling errors in
a solution should not be given full marks. Errors in a solution show that
the subject is not carefully mastered.
- Learning is better done in a cooperative atmosphere where students help
each other to understand instead of a competitive one, where the success of
one student is at the expense of others. (But you can not always choose your
environment.)
- Seeing two or more approaches to a subject can be better than one. What
appears hard in one approach may appear easier in another.
G Calculators and Computers
Calculators lessen the need for us to do arithmetic but, in using them,
mistakes can be made. Here you need to know in advance what kind of answer a
computation will yield. If you think you have made a mistake in entering numbers
or instructions, you need to reenter them again. If a different result appears
from before, at least one of your efforts, the original or the check, will be in
error. (Logic Question: What can you say for sure if the results agree?)
Suggestion: remember or learn how to do arithmetic by hand and how to estimate
the expected size of results for addition, subtraction, multiplication and
division.
Computer programs can perform arithmetic and algebraic or symbolic
operations. They can also draw graphs and solve some equations rapidly. These
programs do not provide solutions to all possible problems. For the solutions
they can provide, you have to understand the statement of the initial problem.
Beyond this, a computer (or another student) cannot understand the chains of
reasoning for you. Understanding is an personal affair. No computer and no other
person can do this for you. But if you know what to expect from a calculation,
calculators and computer programs can help you check your expectations and
explore mathematical ideas. Here you can learn from your mistakes. In some
cases, computer software can tutor you. They can tell what to expect in various
circumstances. Today there are computer programs and on-line computer books
which may help you master mathematics and other subjects. More are appearing
everyday. I know of them, but I have no experience with them.
Appendices with (repetitive) advice for Students: [ B How to Learn ] [ C. How to Read ] [ D. What to do in School ] [ PS. Study Tips ] [ PS: Time and Effort ] [ E. How to Study Math and Why ]
| |
www.whyslopes.com
Volume 2, Three Skills for Algebra -
Preview, starter & further lessons for logic and algebra
to (i) improve work & study skills; (ii) to to ease or avoid
algebra (math) fears & difficulties; and (iii) to fill gaps in the
exposition of mathematics.
Foreword, Chapters and Appendices follow.
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
Complex number: starter lesson
Solving Linear Equations:
A. Letters and Lengths
B. & C. Solving Linear Eq'ns
with stick diagrams.
(i) x + 20 = 29
(ii) 2x + 5 = 20
(iii) 3x + 10 = 32
(iv) 5a + 16 = 3a+ 24
(v) (½)x + 8 = 24½
(vI) (¾)a + 16 = (¼)a+ 24
(vii) (¾)q + 17 = 32
(viii) 13 =[2/3]x +7 twice
(x) Animated Examples
(i) Integral Coefficients (A)
(ii) Integral Coefficients (B)
(iii) Fractional Coefficients
(iv) With
Parameters
Problem Solving with Linear
Equations in one or many
unknowns, and in essentially
one unknown - Symbols before
words.
C. Solving Linear Eq'ns
without
Stick Diagrams
D.
Problems in
essentially one unknown
E: 2D Systems - Sub Methods.
F. Larger Systems
|