Chapter 8. Three Skills For Algebra
Volume 2, Three Skills for Algebra
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.
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. 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
or being given a formula and numbers to use in it.
-
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. This shorthand
notation 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
or being given a formula and numbers to use in it.
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. These
replacement ideas, illustrated below with examples, allows us to
compute or describe different ways to calculate a single number or
quantity.
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. 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.
Notes
- 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.
- A number or quantity which may change in the circumstances of
interest to us is called a variable. The common idea that all
variables have to be given by letters has mislead 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.
-
How to compute the area of a rectangle
can be described with words alone or with a formula A =
WL. 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 an 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.
Learn More - Online Postscript for this chapter
See the essay
What is a Variable. It will show you
how to talk about variables and constants before and besides symbols. That represents the first skill in action.
Chapter 14 here on the compound interest formula introduces the forward and backward use
of formulas. All the rules and patterns met in mathematics, science and logic will be used forwards
and backwards. You may think of that as unifying theme. Talking about it recognizes it
and points to its appearance, again and again in mathematics, science and logic.
Using formula backwards deliberately is a sign of intelligence.
The following chapters will talk about the above three skills, including the
forward and backward use of formulas. I am not sure whether to classify the latter
as part of the third skill, or as a fourth skill. You decide.
Talking about the three or four skills above, and talking about finding counts and sums by forming and
adding subcounts and subsums, and talking about finding products by forming and multiplying subproducts
expands the role of words in both arithmetic and algebra. Earlier is better than later in mathematics
skill development.
|
|
For home-tutoring or -schooling, or for schools or colleges
with course content control: Secondary
Mathematics for Ages 11+, A Practical Approach.
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:
-
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.
|
|