Three Skills For Algebra
Chapter 8, Volume 2, Three Skills for Algebra, modifie
Talking about three skills and illustrating them with examples may be
enough to go from a mastery of arithmetic to a mastery of algebra. In
learning to talk, write, argue and possibly do arithmetic, we have
mastered harder skills. In elementary school, we mastered the first two
skills: the ability to talk about numbers and quantities and the ability
to describe calculations. The third skill depends on the first two. The
three skills are as follows.
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First, we can talk about numbers and quantities without doing any
arithmetic. For instance, numbers and quantities may be big,
small, known, measured, never known, changing or unchanging, private,
top-secret, confidential, embarrassing, or simply forgotten. A
number, measurement or quantity may be known to you but not to me. We
can speak about numbers and quantities in many ways.
The first skill, our ability to talk about numbers
and quantities, is widely known. We can say whether or not a number
is known, forgotten, unknown, small, large, changing or varying,
constant or unchanging, confidential and so on. Thus we can talk
about and describe numbers and quantities. This can be done before
the very visible, but sometimes misunderstood, symbols, letters and
written shorthand of algebra, is introduced. Talking about numbers
and quantities represents a easily-spoken element of algebraic
thought apart from the algebraic way of writing and recording such
thoughts.
Talking about numbers and quantities is an ability we all have. It is
a part of mathematics that does not require us to do arithmetic.
There is more to mathematics than just doing arithmetic
carefully.
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Second, we can describe calculations which we want to do or avoid
or have someone else do, without doing any arithmetic. The
description gives a recipe or a formula for doing a calculation. The
description can be done with words alone or with shorthand notation.
How to compute the area of a rectangle can be
described with words alone or with a formula $A = W \times L.$ In
contrast, the compound interest formula $A =P(1+i)^n$ and even more
so, the quadratic formula \[x = \frac{-b \pm\sqrt{b^2-4ac}}{2a}\]
describe calculations in a algebraic and symbolic way. It would be a
horrible exercise to describe what these formulas mean, do and
represent with words alone and no symbols. The quadratic formula is
awkward to read aloud.
The algebraic shorthand notation in formulas is worth a thousand
words.
The first service of mathematics to other subjects lies in the
description of calculations that can be done or repeated as needed.
There is more to mathematics than just doing arithmetic well.
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Third, we can change the way numbers and quantities are computed
(or measured). Rules or properties of arithmetic tell us when
different calculations or measurements give the same result. These
rules are described using shorthand notation. That gives a second
role to the shorthand notation. In the computation of numbers and
quantities, we may replace a calculation by another, when both give
the same result. And in the description of calculations, we may
replace a calculation by a shorthand symbol that represents its
result, and vice-versa.
Algebra or the manipulation of formulas is
concerned with the shorthand description of different computations
and with when one description can replace another. Description of
one calculation can replace the description of another in any
circumstance where the two calculations give the same result. Such
replacements can be made one at a time, or one after
another.
Replacement or subsitution ideas and steps, illustrated below
with examples, allows us to compute or describe different ways to
calculate a single number or quantity There is more to mathematics
than just doing arithmetic or being given a formula and numbers to use
in it.
The description of calculations that might be done is a first service
of mathematics to other subjects. The creation of new calculations by
changing old ones is a second service to all subjects using arithmetic.
Mathematics after arithmetic is based on the above three skills and the
ability to read exactly rules, patterns and definitions. For the latter,
see the previous chapters on logic.
Benefits and Limitations of Formulas
Formulas are useful when and where working with or describing a
calculation is too awkward to be done with words. Remember the saying: A
picture is worth a thousand words. Formulas like images and pictures can
be worth a thousand words. Quadratic formula even if you do not
understand it provides an example.
But pictures can be named. The two words Mona Lisa may bring to
mind a famous painting of Leonardo de Vinci. So in the other direction a
few words that name or identify an object may stand in place of that
object, and almost have equal value. In mathematics, operations and
formulas identified by name [or a descriptive phrase] can be mentioned
and discussed while talking face to face or not - say via a telephone.
While the algebraic way of writing and reasoning written on paper is
often best seen and read in silence, names and identifying phrases add an
oral element that allows us to talk about what is best seen and done in
paper.
The algebraic shorthand role of letters and symbols on paper needs to be
learnt and taught. But some calculations and some properties of
arithmetic are easier to describe with words. For example, how to
calculate the the perimeter of a polygon is briefly described by the
phrase Add the lengths of the sides. The algebraic expression or
description of this sum with letters - alone or with subscripts -
denoting side lengths raises the level of complexity. In the latter case,
the algebraic description is harder to grasp than the verbal description.
Which is best depends on the objective. Where the aim is to compute the
perimeters of polygons, the verbal description is best - easier to grasp.
Where the aim is to illustrate how formulas may describe calculations,
the letters have to be used.
What is a Variable - An Understanding before symbols
The issue of complexity needs to be consider in introducing the concept
of what is variable. We can talk about and describe numbers and quantities
before or besides using letters or symbols to identify or stand for those
numbers and quantities - in a pronoun like manner.
Counts and measures may change and vary over time and between examples. Those
that do may be called changeable or variable. This concept of changeable
or variable - variable being the standard word - can be understood
without introducing letters to denote the counts or measures.
The common idea
that all variables have to be given by letters has misled many. As just
suggested, talking about variables, that is numbers or quantities which may
change or vary, can be done without from any reference to letters and
symbols. That is, the notion of a variable can be clarified or explained
before any linkage to algebraic shorthand or symbols used to write and
record calculations and further parts of algebraic thought. See the
site essay A number or quantity which may change in the circumstances of interest
to us is called a variable. See the somewhat amusing essay, what is a variable.
to learn more.
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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.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
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
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
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
science courses, and in developing organizational skills,
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
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
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
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
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.
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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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