Algebra Essay
Algebra is based on the shorthand roles of letters and symbols in describing
calculations that may be done; in describing numerical identifies - alternate
ways to compute the same number or quantity; and in solving equations for
practice or for solving a word problem, realistic or not. In preparing
students for college mathematics, there is a need to show students how to use
calculators, but there is also a need to develop and maintain exact arithmetic
skills with whole numbers and fractions, etc.
The Silent Thinking in mathematics
Mathematics teachers should emphasize the shorthand role of letters in giving
formula for numbers and quantities when the formula is worth a thousand words,
or where the use of words - the rhetorical description of a calculation is
becoming marginal or awkward. Here vision provides an 2 or 3D sense of our
surroundings, mathematics expressions included, while words must be spoken or
heard in sequence in a 1D manner. So our visual drawing and observation of
mathematical expressions and diagrams is more powerful and more immediate than
our sound-based speaking and hearing communication. There-in lies the onset of
silence - the advent of arithmetic and algebraic expressions, formulas, and
equation better seen and digested in a glance than read aloud in manner that
reflects the order of operations precisely and clearly.
Compensating for Visual and Silent Observations - Alleviating the Silence
Remedies involve adding or emphasizing the verbal dimensions of mathematics,
written or spoken, while exploiting mathematics silent means of recording and
developing thoughts on paper to the greatest extent possible.
On maps, we use labels and place names to locate and identify features in
our memories and in our discussion of map contents. That remains true
even when coordinates are available for same task.
Step I: Talk about three or four skills for algebra
Direct and Indirect Use of Formulas, Equations and (!) Proportionality
Relations -
Step II: Formatting Issues - Good Notation, good format is a vehicle for building
ideas and doing calculations - extends our memories, provides a longer or
permanent record.
Step III: Fractional Operations on Stick Diagrams
Step IV: Proper Use of the Equal - duck the issue or its discussion in class by
requiring students to follow teacher prescribed formats for the evaluation of
arithmetic and algebraic expressions - all for the benefit of communication,
reasoning and problem solving skills on paper
Geometric Starter Lessons for Algebra: Geometry introduces
the use of names and letters to locate points on a map or drawing and to
identify and denote lengths and areas. Instructions on how to
calculate the lengths and areas of perimeters and figures can be given in
words or with formulas. For example the edges or sides of an
pentagon need not be equal. The written or verbal instruction to find
it perimeter by adding the length of its sides could be clearer or more
efficient than introducing letters to denote the lengths of its sides, and
expressing a formula for the perimeter in terms of the letters. It can be
done, but is not always required. That being said, formulas for
areas and perimeters of squares, rectangles, trapezoids, parallelograms,
triangles, circles and half-circles continue or introduce the algebraic
shorthand role of letters to identify and denote lengths and areas, or their
measures. The foregoing and the evaluation of formulas in a required
show work format similar to the format required above the evaluation of
arithmetic expressions introduces the role of algebra or formulas in
describing calculations that may be done.
Remark: Rectangle based, geometric proofs of the distributive law AB
+AC = A(B+C) explains why calculations of the form AB +AC - A(B+C)
result in zero. The algebraic thinking skills of students might be
developed by giving them numerical expression of the above form to evaluate
directly. Then after they have got their zero result, explain how the
distributive law could have save them some work.
Geometry provides a simple venue to visually introduce the shorthand roles of
letters:
- labels or names or identifiers for points
- labels or names or placeholders for lengths, areas, volumes and even areas
alone and in formulas for the latter.
The geometric origins of algebra are indicated in how the we read 42
(4 -squared) and 23 (2 cubed) aloud. Those powers of 4
and 2 are associated with the area of a square and the volume of a cube.
Too often in mathematics, arithmetic and algebraic expressions are too
complex to read aloud in a way that indicates precisely the order of
operation necessary to evaluate the expressions correctly. Yet words (rhetoric,
short phrases) may be used along side and even in place of formulas in the
description of geometric calculations for perimeters, areas and volumes, and
more physical and geometric quantities.
- The area A of a rectangle is given by the product of its length L
and width W (or equivalent terms) or by the product of its dimensions: A =
LW.
Note how the previous sentence includes letter in its composition to explain
the placeholder, pronoun-like shorthand roles of letters A, L and W.
- The area A of a triangle is half its base length B times it height
H. In brief, A = ½ BH.
Note: some students may not know that ½ of an expression equals the
expression divided by 2 and hence may see the formula A = ½ BH as
being different from
A = BH
2
- The area A of a circle is p (pi)
times its radius r squared. That is, A = p r2.
Here elementary textbooks may say take the value of p
to be 3.14 - there-in an expression that leads many teachers
and students to falsely think p is 3.14 exactly
instead of approximately.
- The perimeter P of a circle is twice p
(pi) times its radius r squared. That is, P = 2p
r.
But in the introduction of algebra and beyond, the verbal or rhetorical
description of calculations should not be dismissed or cast aside in favor of
formulas. Examples follow where words are better or clearer than formulas:
- The perimeter P of a polygon is the sum of the lengths of its sides.
That may be clearer to students than writing that the polygon
perimeter
P = a + b +c + ... z
where a to z are lengths of sides, or clearer than writing the perimeter of
an n-gon (a polygon with n sides)
P = x1+ x2 + ... + xn.
Here the notation with its dot-dot-dots (...) may introduce confusion by
being to complicated for novice students. That being said, it does not
hurt (we hope) to give the brief and clear rhetorical direction for
computations, to make the directions are understood and then to briefly
write the complicated notation as an indication of things to come. The
complicated notation should not be the basis of the lesson.
- In statistics, arithmetic averages may be computed. The
direction to compute those average by adding all the numbers present and
then dividing the resulting sum by how many numbers were present (were
added) will clearer for students than starting with the somewhat mystifying
dot-dot-dot notation.
Whenever a calculation is done, if there is a clear phrase or phrases
describing how how to do the calculation directly or via steps, let
students know. That being said, there will also be calculations or
formulas where the shorthand role of algebra is quicker and clearer, and an
alternative to the task of writing several phrases or an essay to describe the
calculation and all its steps.
Words versus Formulas: Explain that for irregular polygons, instead of
labeling the lengths of all polygon sides and giving a formula for the
perimeter, it may be simpler and preferable to describe the computation of
perimeters via the instruction: add the lengths of all sides to get the
perimeter. In contrast, for a regular n-gon, a polygon with n sides all of equal
length, say s, the perimeter p = n x s. That is as easy as the word description
this shortcut for the perimeter calculation. There are situations where
formulas (algebraic description of calculations) are easier to grasp than word
descriptions, and vice-versa. Use each method accordingly.
Names and Adjectival Phrases to Identify Formulas: To compensate
for the manner in which formulas are easier seen and digest in a glance than
read, teacher should identify formulas, relations equations and also quantities
by descriptive phrases and names. Examples follow:
triangle area figuring formula, circle perimeter formula, trapezoidal
area calculation formula, box or parallelepiped volume formula, simple
interest formula, a distance from speed & time formula, the average
speed definition formula, compound interest formula, exponential decay
formula, quadratic formula, inverse square law, equation 53, the last
formula, the kilometer to meters multiplication factor, and so on as
these formulas appear - do not give them all at once.
Every time a formula or diagram appears, identify it by name or with a
descriptive phrase. That may lead to a greater use of words and gossip in
mathematics to compensate for algebraic shorthand notation and diagrams that are
better seen and understood in silence. That silence is a barrier to
communication and hence learning, teaching and use.
Idea: But if you work in a school where students are given an academic calendar
with mathematical notes in it, open the calendar to that page and identify
the formulas and concepts in the notes that will or will not be covered in
class.
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LAMP
(first
draft, June 2008) a program for adult
and teen mathematics education
Mathematics education standards implied by calculus should
be a factor, not the only one, yet not a forgotten nor hidden one in course design
Area Intro
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Introduction
Arithmetic
Geometry
Logic
Calculus
Musings - More Ideas
More About LAMP
Evaluation
Maths Cultural Origins
First Nation Education
Modern Mathematics
Before LAMP
Problem Solving Skills Routine to Non
Instructional Concepts
Student Cooperation
Maths Extrinsic Origins
Science Education
For further musings or thoughts see site books.
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