Chapter 3. Algebra Difficulties
Mathematics Curriculum Notes, Volume 1B
Algebraic Thought, The Barrier
In the absence of other data, one resorts to personal experience – the
small sample of one – and then hopes that the experience is objective or
representative. As a high school student, I mastered the algebraic way of
writing and thinking, but many of my fellow students and some of my science
teachers had not. I was frustrated in some courses by the slow pace. I
attributed this slow pace to the difficulty others had with algebra.
Moreover the algebraic way of writing and thinking was required to
understand courses in algebra, physic and chemistry, but it was never
directly explained.
As a college student in the 1970s, I tried to help a high school student.
When I tried to explain or demonstrate the algebraic way of writing and
thinking, she would say give me numbers, not letters. The
algebraic way of writing and reasoning was a powerful engine, but I lacked
words to describe it, and the descriptions in math texts seemed too brief.
The student later said that the algebraic way of writing and thinking, or
what I was trying to say, only became clear when she took calculus. [1]
[1] Calculus in the first instance deals with slope-related calculations,
their applications and interpretations. Its explanation employs and
illustrates the symbolic or algebraic way of writing and thinking at full
strength.
Through my studies of mathematics in high school and college I
was aware of this difficulty of explaining algebraic thought, but could not
then conceive of how to resolve it. After becoming a full-time instructor
in 1983, I thought of a simple remedy.
Three Skills For Algebra
The word IT, capitalized, will refer to the algebraic way of writing
and thinking or reasoning. As indicated above, for many in school and out,
IT, the algebraic way of writing and reasoning, is part of the mystery or
nonsense of mathematics. While some people quickly absorb IT from examples,
others don't. IT is better seen and understood silently than spoken aloud
or described with words. The silent comprehension or miscomprehension of IT
has been an obstacle to the further explanation of mathematics. But talking
about the following three key skills should be enough to directly
introduce, support and introduce IT, the algebraic way of writing and
thinking/reasoning. The first skill is pre-algebraic.
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We can talk about numbers and quantities. We can say
when numbers and quantities are known, unknown, forgotten,
confidential, changing, varying, non-changing or constant. There is
more to mathematics than just doing arithmetic.
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We can describe calculations that might be done. The
description of the calculations can be done with words, or much more
briefly with a shorthand notation that uses arithmetic signs (<+, -,
×, and ÷ say), common letters and other symbols. Like a
picture, the notation is worth a thousand words. The notation
is better seen and read quietly than spoken aloud. The description of
calculations that might be done is the first service of the shorthand
notation to all arithmetic based subjects.
There is more to mathematics than just doing arithmetic. [2]
[2] Formulas for the areas of rectangles, for the roots
of quadratics and compound interest computations provide examples of
two calculations that are better described with algebraic shorthand
than words alone. For the rectangle, words alone may or may not be
better than the use of algebraic shorthand notation (a formula) to
describe its area calculation.
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We can change the way that numbers and quantities are computed
(or how calculations and their results are described). The
rules for this can be used one at a time, or one after
another. A second service of the shorthand notation to all arithmetic
based subjects lies in describing rules for changing the way numbers
and quantities are computed: saying when two different calculations or
formulas give the same result[3]. There is
more to mathematics than just doing arithmetic or being given formulas
and numbers to use in them. [3]
On the Third Skill. In
the description and performance of calculations and subcalculations,
one calculation that yields the same result as another may be
interchanged with the other in a computation and a symbol which
represents the result of a calculation may replace the calculation, or
vice-versa. The algebraically described properties of real numbers say
when two different computations give the same result. More will be said
on this later. See the discussion of intermediate level instruction in
the chapter The
Transition.
The phrase there is more to mathematics than just doing arithmetic
or being given formulas and numbers to use in them could be repeated
in classrooms as is or in short form as a chorus after each skill is
introduced. Some showmanship appears here. The symbolic way of writing and
thinking is an artificial skill. Talking about the three skills and adding
the following message give a first simple image for mathematical thought
which is easily grasped by a person with a knowledge of elementary
mathematics.
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.
The skills build on the common knowledge of reading, writing,
arithmetic, counting, geometric shapes and simple formulas. They divide the
mastery of algebraic thought into smaller more accessible steps. Talking
about them and illustrating them with examples may provide words to
introduce or reinforce the algebraic way of writing and reasoning.
Two 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.
Words before and then besides
symbols
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. 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.
The notion that a
variable may be given by a symbol, that is
shorthand notation (or a placeholder) for a number or quantity which may
change, relies on our ability or skill to talk about numbers and
quantities and also on our ability or skill to employ shorthand notation
(symbols) for them in and possibly outside calculations.[4]
Two Notions of A Constant
The term constant is a reference to a number or
quantity which will remain unchanged, or is fixed in the situation of
interest. A symbol or letter is called a constant if it
stands for a number or quantity not expected to change in the situation of
interest. There are two notions of a constant, one with and one without
symbols.
[4] Pages 44 and 46 in the 1965 book
Secondary School Mathematics by J. J. Kinsella, published by
The Center for Applied Research in Education, Inc., New York,
indicate efforts in modern math instruction to define a variable. The
definitions described formally introduce a variable as a
symbol acting as a place holder for a number, specified
or not, and possibly restricted to some set of allowable values. The
latter notion associates the notion of a variable with a symbol, and does
recognize the second symbol free notion identified above. Moreover,
understanding the latter requires a previous mastery of IT, the algebraic
way of writing and thinking, and possibly some set-based concepts, and so
cannot be an introduction to IT. (The efforts essentially describe the
role of algebraic thought with the formality and precision demanded by
modern mathematical thinking. I suspect the efforts also represent the
historical understanding of algebraic thought by osmosis. The three
skills represent an advance on this situation.)
[5] Kinsella's work also provides a
perspective, still pertinent today, on some of the problems and issues
facing mathematics instruction. It will be mentioned again.
Postscript (Sept 96): See also the article On the
Meaning of Variable, pp 420-427, Mathematics Teacher, September
1988, by A. Arcavi and A. H. Schoenfeld. Included is a conclusion that
the meaning of variable is variable. Several meanings of the
term have to be mastered.
What is a Variable?
There are many facets of algebra, but for most people, any subject is
defined by their first introduction to it. A broader, further ranging and
more refined definition can come later. In this light, the description and
illustration of the three skills for algebra provides an introduction and
thus an initial definition of the subject. For this author, algebra starts
with the symbolic description and formulation of calculations and continues
with their symbolic manipulation. It could be preceded by a pre-symbolic
comprehension of the answer to the question:
Decimal notations and operations can be regarded as part of algebra. They
are symbolic. Yet there is more to the algebraic or symbolic than this
arithmetic and concrete, most definite, use of symbols. The algebraic or
symbolic process of further mathematics involves another abstraction,
namely the repeated use of symbols as placeholders and pronouns for
numbers and quantities, known or not, constant or not. The question
What is algebra? can be answered in different ways. In past
centuries, algebra may have meant all of mathematics besides geometry. In
the 20th Century, high school and college math courses, those taught in
North America especially, give the initial impression that algebra is a
mathematical subject besides say trigonometry, calculus and possibly
geometry – geometry is not often taught. But algebra is represented by
all subjects it touches, in mathematics and out.
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Algebra
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Calculus Starter Lessons
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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