Chapter 3: Three Skills For Algebra
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Algebraic Thought- What is it
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.
- 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.
- 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[2].
There is more to mathematics than just doing arithmetic.
[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.
- 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.
Chapter sections: [ Home ] [ Up ] [ 3 Three Skills For Algebra ] [ 3 Words Before Symbols ] [ 3. What is Algebra? ]
Next: 3 Variables - Clarifying
the Notion
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Mathematics
Curriculum
Notes
understanding & explaining
Reason and Math
Volume 1B
Printed in Canada
ISBN 0-9697564-6-1
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Foreword +
Chapters 1 to 10 + 12
Up
Home
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Readings
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
11 Primary Math
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Book Entrance
Inductive Principles
1 Introduction
2 For & Against Math
3 Algebraic Thought
4 Why Slopes & SQRT of -1
5 Readings
6 Unruly Origins of Algebra
6. Axiomatic Civilization
7 Geometry, 2 Ways
8 Modern Instruction
9 The Two Ends
10 The Transition
11 Primary Math
10 Explaining Logic
10 Explaining Algebra
10 Why Sets in Math.
12 Four Phases
Links
Chapter 11: Primary School Mathematics
Will provide an alternative to Chapter 11 later, most likely in the Parent's
Area: Help Your Child or Teen
Learn
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