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Volume 2, Three Skills for Algebra
1 Words are Not Enough
Imagine describing a picture by saying there is a patch of blue, a bit of
red or orange there and yellow here. Words alone do not usually describe
a picture fully. The picture must be seen. A picture seen conveys or
presents information which words alone cannot capture. Similarly,
formulas and arithmetic calculations are better seen than described with
words alone. Pictures and formulas are often worth more and may be easier
to understand than a thousand words. Words cannot replace pictures and
formulas.
2 Decimal Notation in Arithmetic
Decimal notation is used in the representation of whole numbers,
fractions and irrational numbers. Decimal notation allows us to write
down these numbers easily. Further, before decimal notation appeared,
addition, multiplication, division and subtraction were complicated
operations. People had to use the abacus or Roman numerals or other means
for calculating. Imagine, if you can, how to do arithmetic without
decimal notation. Decimal notation is the key to our doing arithmetic
easily. Decimal notation is itself a shorthand notation for the
representation of numbers.
Since the 16-th century A.D. popularization of decimal notation, rules
and methods have been invented or found to say how to write, add,
multiply, divide and subtract numbers and fractions with this
notation. Today these methods or variants of them are taught in
elementary school.13
Good or understandable shorthand helps people to describe calculations
and even to do them. The ease with which we write or record numbers and
calculations affect the way in which calculations are done and if they
are done. Before the use of decimal notation, basic arithmetic operations
were difficult to do and record, few people mastered arithmetic.
In talking and writing, our language and vocabulary and experience limit
and guide what we can say or even think of. Similarly, in mathematics the
language (shorthand conventions) and previous experience limit and guide
what can be said or attempted.
3 Shorthand Notation in Algebra
The use of shorthand notation (letters or symbols) in the description of
calculations is only several centuries old in Europe. Prior to the 12-th
century A.D., calculations were described with words alone. The use of
shorthand notation for describing calculations is not old. In geometry
letters have been used by some Greeks, two thousand years ago, to label
the sides or vertices of triangles, but not to describe calculations.
As in arithmetic, algebraic shorthand notation records and describes
calculations in a written, visual and non-verbal manner. Here words are
not enough to describe simple calculations. For instance think how you
would describe the calculation of the area of a rectangle or a triangle
to a young child with words alone. But in the description of more
complicated calculations, words become inconvenient and awkward to use.
The description is best given with the so-called algebraic shorthand
notation of mathematics. For examples of more complicated calculations,
consider the compound interest formula and the quadratic formula.
-
- The compound interest formula:
- The quadratic formula:
You may try to explain with words alone, the calculations described by
one or both of these formulas. Also explain with words alone, why each
calculation is done. These tasks are hard. Just to make it harder, the use
in your explanation of the shorthand notation including the letters that
appear in these formulas is forbidden. (All of this is too difficult and
too awkward for me to think about further. The same may hold for you.)
Complicated formulas are written to be seen. They are not easily read
aloud. Their meanings are not easily described without the use of
shorthand notation. The shorthand notation in mathematics gives a written
code in which calculations are efficiently and briefly described and/or
changed. It is a code which is better seen than read aloud.14 The
algebraic shorthand provides a language and environment in which
calculations are changed or manipulated into new ones.
4 Seeing is Better Than Hearing
The saying seeing is better than hearing usually means that seeing
and witnessing an event is better or more reliable than hearing about it.
In this section, we will give a new meaning to this saying.
Our ability to see is more powerful than our ability to hear. Words are
typically understood one at a time. After that they need to be
remembered. In contrast, a picture, a calculation or a formula can be
seen all at once. They will remain in front of you for further thought
and observation while you are looking. No memory is required. Once a
picture, formula or calculation has been put on paper, the details can be
seen all at once. The paper used this way, serves as an immediate or
quick extension of our minds or memory.
Eyesight together with algebraic shorthand notation and geometric
diagrams provide more powerful ways for the communication and recording
of mathematical thoughts than words alone. All this explains why
mathematics is better seen and written than spoken aloud. Words alone are
not enough for the communication of mathematics.
Footnotes:
13The development of arithmetic during the
period 1300 to 1637 A.D. is described in Chapter XI of the book A
Short Account of the History of Mathematics written by W. W. Rouse
Ball (Publisher Dover, New York 1960). A copy should be in your town or
school library. The book was written in the last part of the 19-th
Century.
The book A Source Book in Mathematics by David Eugene Smith,
first printing 1929, Dover reprint 1959, has a chapter on the 1585 A.D
contribution of Simon Stevin (1548-1620) to the popularization of
decimal notation. More recent accounts of the history of arithmetic and
decimal notation may modify or correct the historical impression in the
above references.
14We need a code which could be read aloud
more easily.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
groups and develop skill building exercises & activities to share. Start now. The effort here is the best I can do.
Others are welcome to refine or exceed it. Please do.
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
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See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
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justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
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due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
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gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
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Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
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Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
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