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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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.
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