Three Notions of a Variable
The concept of a variable is not simply described in most algebra
texts. A clarification follows. This clarification is not for the
expert, but for the novice. The specialized use of the term
variable should not be the first one given in an algebra text or
dictionary, mathematical or not.
A First Notion: Variables Without Symbols. We can talk about
numbers and quantities, and among them identify those which are
changing or varying, and those which are constant, known, unknown,
given, confidential and so on. Here a number and quantity which may
vary, or take many values in the circumstances of interest, is called a
variable. We can talk about variables without using the
shorthand notation, that is, letters and symbols, employed in
algebra. Examples follow below.
Second Notion: Variables with Symbols. Formulas use shorthand
notation, symbols or letters, to represent numbers and quantities. This
suggests that when a symbol or letter is the shorthand notation for a
number or quantity which may vary, we may also call that symbol or
letter a variable.
Remark 1. The association of symbols and
letters with numbers and quantities which may vary is so much a
taken-for-granted part of the algebraic way of writing and thinking
(amongst the mathematical adept) that the observation that we can
talk about variables apart from symbols has been overlooked. But this
symbol free notion clarifies and refines the concept of a variable in
mathematics.
Remark 2. The notion that a variable may be
given by a symbol, that is shorthand notation (or a place holder) for
a number or quantity which may change, relies on our ability or
skill (i) to talk about numbers and quantities and also on our
ability or skill (ii) to employ shorthand notation (symbols) for them
in and possibly outside calculations.
Third Notion: Variables and Computer Memory Locations: Computers and calculators may be used to store
the values of numbers, amounts and quantities in named or labeled
memory locations. Computers or calculators may be
programmed to use or change the stored values of numbers or quantities,
values that may vary. The values, the memory locations where
they are stored, and the names or labels for them may be all be called
variables.
Three Skills for Algebra
-
We can talk about numbers and quantities. The words or
adjectives used here may be used in mathematics after
arithmetic. There is more to mathematics than just doing
arithmetic.
-
We can describe calculations that might be done (or postponed)
with words alone or with an (algebraic) shorthand notation. The
description of calculations that might be done is also part of
mathematics after arithmetic. There is more to mathematics than
just doing arithmetic.
-
We can change the way a number or quantity is computed. Some
rule-based reason is required here. There is more to mathematics
than just doing arithmetic.
Talking about these skills and emphasizing them in examples shows
there is more to mathematics than just doing arithmetic.
Constants, Variables, Parameters and Data
added June 23, 2005
When a number or quantity is a constant in one direction of
change (over time, in space, between examples) we say it is
constant in that direction. If the direction is understood, we may call
it a constant.
When a number or quantity is a variable in one direction of
change (over time, in space, between examples) we say it is
variable in that direction. If the direction is understood, we
may call it a variable.
When a number or quantity is a variable in one direction of change
(over time, in space, between examples) and constant in another
we say it is parameter in that direction. So a
parameter is a number or variable that may vary or be constant
depending on which direction we look.
If an observed number or quantity may be used in a table or in further
calculations, we the number or quantity is question is part of the
data of for the table or further calculations.
Talking about numbers and quantities being constants, variables,
parameter or data is part of words before and besides symbols and
arithmetic in mathematics.
|
|
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.
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:
-
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.
|
|
Return to Page Top
Home < Algebra Starter Lessons << 6 Three Notions of What is a Variable
[1] [2] [3] [4] [5] [6] [7]
All trademarks and copyrights in this are owned by their
respective owners.
Copyright to comments & contributions are owned by the Poster.
The Rest
© 1995-2011, by site author, Alan Selby, Ph. D., Montreal,
All Rights Reserved ---
Skype
or Email to contact.
|