Algebra: The Language of
Science
by
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James
Nickel,
B.A., B.Th., B.Miss., M.A.
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Summary
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The misdirection of High School
Mathematics Education.
Biblical Christian direction (or
framework) for reformation of Algebra.
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Acknowledgement
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I am grateful to Morris Kline’s Why the
Professor Can’t Teach: Mathematics and
the Dilemma of University Education (St.
Martin’s Press, 1977) for much of the
following analysis.
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20th Century High School
Mathematics: Scope and Sequence
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9th grade: Algebra I
10th grade: Geometry
11th grade: Algebra II
12th grade: Pre-calculus, Calculus, or
Solid Geometry
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Analysis
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Note: As I analyze this Scope and
Sequence, I am going to play the “devil’s
advocate.”
I am going to be somewhat cynical with
a purpose in mind.
Let’s see if we can view this material
from the eyes of the typical student; i.e.,
what I experienced in the 1960s.
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Grade: Algebra I
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Almost entirely a series of disconnected,
atomized processes, a mélange of topics.
Note, in the early 21st century, there is a
move to cover Algebra I in the 8th grade.
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Algebra I Topics
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Factoring.
Operations with polynomials such as x2
+ 9x + 20.
7x  8
Operations with fractions such as
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5x  3
Laws of exponents.
Solution of equations of various degrees
and with one or more unknowns.
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Algebra Perspectives
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Algebra is a means to something else,
not the end in itself.
Algebra is the spelling, grammar, and
rhetoric of most mathematics, but it is
not literature in and of itself.
This maze of letters that represent
number symbols that represent numbers
was fully developed in the 17th century.
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Two Levels of Abstraction
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Level 1: Numerals are symbols of
number.
Level 2: Letters are symbols of numerals.
For example, instead of 3x = 5, we have
ax = b.
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Two Levels of Abstraction
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But, generalizations about men and women are
meaningful only to those who have had lots of
experience with individual men and women.
The analogue for Algebra is lots of experience
with various manipulations of numbers.
Unfortunately, few students have more than a
nodding acquaintance with arithmetic at the
time they are propelled into Algebra.
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Grade: Geometry
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The Geometry study is Euclidean or
Plane (two dimensions).
Here, the student is confronted with
dozens of theorems all proved in a
logical sequence and, to most students,
the goal seems to be to prove as many as
possible.
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Grade: Geometry
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Students are confined to learning
inflexible and obtuse trains of thought
that permit no derailment.
Just why anyone wants these theorems
and how these theorems and their proofs
were ever conceived is not treated.
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Grade: Algebra II
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Repeats the 9th grade subject matter
because students did not learn that
material.
Adds more algebraic processes.
Introduces trigonometry, a subject in
which, among other topics, the students
learn many identities at the risk of losing
their own.
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Grade: Various Schemes
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Coursework has not been stable.
But the material, whether solid geometry
or the beginnings of calculus, is taught
dogmatically and has been no more
enlightening or enthralling than the
preceding subjects.
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Who is Responsible for this?
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Who generated this curriculum?
It was developed in the 19th century by
university professors.
Before this time, these high school
subjects were only taught in college.
They gradually filtered down into high
school keeping the same college content.
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Rationale for this Program
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Utility in daily life.
Value of mental discipline; i.e., mathematics
teaches you how to think.
Learning of higher, nobler truths; i.e., a2 – b2
= (a + b)(a – b) would uplift any soul.
Preparation for college and professional life.
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State of Things since WWII
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1950s: the ineffectiveness of this
traditional curriculum (made evident by
the needs of WWII) stimulated some
university professors to reform the
process.
This reform resulted in the “new math” of
the 1960s.
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1960s New Math
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Emphasis on theory, generalization, rigor,
and structure of number systems.
Focus on set theory and symbolic logic.
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1960s New Math
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These “innovations” may have ensured the
stability of the several mathematical
edifices, but they also pulled the students
so far down into the dark earth that they
could no longer see the surface. e.g., 2 + 3
= 3 + 2, but does 2 + 3 = 5?
It is now been recognized (in hind-sight)
that this reform was not an improvement;
it was a disaster.
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Summary
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The contents of all four years, as
presented, are abstract, dull, boring, and
intrinsically meaningless.
It consists, at best (to use Alfred North
Whitehead’s phrase), of inert ideas.
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Analysis of Rationale
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1.
2.
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Utility in daily life.
Value of mental discipline.
Learning of higher, nobler truths.
Preparation for college and professional
life.
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Utility
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Does the average educated person use this
knowledge in daily life?
Do even mathematics teachers, who know
the subject, ever use the quadratic formula,
the Pythagorean Theorem, or the
trigonometric identities outside of the
classroom?
The honest answer is no.
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Utility
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Of course, the students learn that a straight
line is the shortest distance between two
points.
That is useful.
But even a donkey knows that: put some
food at a distance from him and watch the
path he takes.
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Utility
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The subject matter of high school
mathematics–that is, the subject matter per
se–is worthless knowledge.
One or two topics, such as the calculation
of compound interest, may prove useful.
But the exceptions do not alter the general
assessment.
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Thinking
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High school courses have not taught
thinking, pedagogue’s assertions to the
contrary.
The traditional Algebra teaches
memorization of processes.
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Thinking
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Geometry purportedly emphasizes proof
and therefore thinking.
However, because the proofs are arranged
in a logical sequence and this sequence is
not natural, the rationale eludes the
students and they are forced to memorize
here, too.
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Thinking
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The students’ appreciation of proof in
geometry is epitomized in their oftrepeated remark that geometry is where
you make proofs in two columns.
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Nobler Truths
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All the preaching and rhapsodizing will
not ennoble such ugly ducklings like
factoring, adding fractions, and the
quadratic formula.
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Preparation
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Of those who go to college, at most 10%
will need the techniques and theorems
now taught in the high schools.
The others either take no mathematics in
college–many colleges no longer require a
mathematics course for the bachelor’s
degree–or take a liberal arts course that
does not use most of the currently taught
high school mathematics.
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Preparation
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Note the growing number of
“mathematical development” (i.e.,
remedial math) courses in most 2-year and
4-year colleges  even the basics are not
being learned by many high school
students.
About 5% to 15% of high school
graduates will use mathematics
professionally.
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What to do?
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Don’t teach any math in high school.
And the students roar, “Hip, hip, hooray!”
And math teachers have to get a real job.
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The Purpose of this Cynicism
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These arguments against the teaching of
high school mathematics are arguments
against the framework:
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The kind of mathematics that has been taught.
The justifications traditionally and currently
given.
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Reformation in Framework
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The Bible gives direction for viewing
reality.
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Reformation in Framework
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The infinite, personal, and Triune God
made the heavens the earth and they
reflect a rational order based upon His
Wisdom.
Man, made in God’s image, can thus
reason with this order.
Operational science is man reasoning with
this order in terms of observation and
experimentation.
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Reformation in Framework
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Biblical Christianity is a faith that is
grounded in history.
God commands man, male and female, to
cultivate the garden; to develop the
implications of creation relationally.
Hence, history and culture are important
pieces of the framework.
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Implications for Algebra
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Place the instruction of Algebra in the
context of history, culture, and science.
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Biblical Christian Education
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What are the objectives of a Biblical
Christian high school education?
The courses in literature, history, science,
economics, and foreign languages are
intended to enable future adults to live
more insightfully, wisely, and enjoyably.
In short, they are an introduction to our
culture and the role of Biblical
Christianity therein.
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Implications for Mathematics
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The mathematics courses should serve the
same objectives.
Hence, the mathematics we teach should be
worth knowing for the rest of the lives of all
students.
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Implications for Mathematics
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Mathematics courses should contribute to a truly
liberal arts (freedom in Christ) education wherein
students get to know not only what the subject is
about but also what role it plays in our culture and
our society.
We must teach not just what mathematics is but
what it does as a Biblical Christian “tool of
dominion.”
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The Arabic Connection
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Algebra is an Arabic word.
In the 7th century AD, the followers of
Mohammed began ransacking the nations
of North Africa, the Middle East, and
Europe.
In spite of some destruction of knowledge
and books, Muslim leaders generally
respected and absorbed the manuscripts of
the ancient Greek mathematicians.
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The Hindu Connection
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The Arabs were also responsible for
introducing to Europe several valuable
concepts that they borrowed from the
Hindus of India:
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The base 10 decimal system.
Positional notation.
The number zero.
Negative numbers.
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The Decimal System
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The decimal system of writing numbers (e.g., 1,
2, 3, 4, etc.) proved to be much more efficient
than the old, cumbersome Greek letters or Roman
numerals.
This method of number writing acted as an
indispensable asset in later developments of
European mathematics and science.
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Algebra: Meaning
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In 830 AD, the astronomer Mohammed ibn
Musa al-Khowarizmi (ca. 780-ca. 850),
working in Baghdad, wrote a book titled
Hisab al-jabr w’al-muqabalah, which
literally means “science of reunion and
opposition” or more freely “science of
transposition and cancellation.”
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Algebra: Meaning
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Al-jabr, used in a non-mathematical sense,
was something a barber did in those times.
A barber not only cut hair, he mended
bones.
Thus, an algebrista was a bonesetter
(chiropractor).
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Algebra: Meaning
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In Galatians 6:2, the Apostle Paul states,
“Brethren, if a man be overtaken in a fault,
ye which are spiritual restore such a one in
the spirit of meekness, considering thyself
lest thou also be tempted.”
The Greek word for restore means “to join
back together” (as in setting a broken bone)
or “to mend” (as in repairing torn nets).
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Algebra: Meaning
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The Greek word for restore means exactly
the same as the Arabic word algebrista.
In a mathematical sense, solving an equation
(determining values of a variable that make
the equation true) is like restoring, mending,
or resolving it.
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Transition to Europe
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This Arabic text became known in Europe
through Latin translations and was shortened
to the one word, “algebra,” which became
synonymous with the science of equations.
Arabic mathematicians did not use simple
symbols like the Symbolic Algebra we used
today (e.g., x, y, etc.); they used words, or at
best, abbreviations of words (syncopated
Algebra).
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Transition to Europe
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The decimal system needed Simon Stevin
(1548-1620) and François Vieta (1540-1603)
to acquire the conceptual precision demanded
by systematic scientific work.
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Soul of Science
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The very soul of science consists in
theoretical generalizations leading to the
formulation of quantitative laws and systems
of laws.
India had practicality, craftsmanship and
organizational talent, but science in the above
sense did not exist in India prior to its
invasion by Alexander the Great.
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Soul of Science
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Tobias Dantzig (1884-1956), “Algebra...
enables one to transform literal expressions
and thus to paraphrase any statement into a
number of equivalent forms ... it is this power
of transformation that lifts algebra above the
level of a convenient shorthand ...”
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Soul of Science
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“the literal notation made it possible to pass
from the individual to the collective, from the
‘some’ to the ‘any’ and the ‘all’ ... It is this
that made possible the general theory of
functions, which is the basis of all applied
mathematics.”
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Soul of Science
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One and the Many: Biblical view of the
world enabled man to harness the incredible
power of Algebra.
Robin G. Collingwood (1889-1943), “The
possibility of an applied mathematics is an
expression, in terms of natural science, of
the Christian belief that nature is the
creation of an omnipotent God.”
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Culture and Science
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Christ is the Lord and Savior of both.
Hence, Algebra must be seen in a Christocentric perspective.
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Gate and Key of Science
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English philosopher and scientist Roger
Bacon (ca. 1214- ca. 1294), nicknamed the
“Admirable Doctor,” said in his Opus Majus
(published in 1267), Part 4: “Mathematics is
the gate and key of the sciences…. Neglect
of mathematics works injury to all
knowledge, since he who is ignorant of it
cannot know the other sciences or things of
this world.”
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Gate and Key of Mathematics
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Algebra, for it prepares us not only to
understand the sciences, but also such
subjects as medicine, art, statistics, and
economics.
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Christ: the Gate and Key of Algebra
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From the perspective of the Biblical
Christian worldview, learning Algebra is
important to our worship because Algebra is
the language of science, science is the study
of what God in Christ has made, and what
God in Christ has made is an unveiling of
His glory.
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What History Teaches
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It is beyond dispute that, in history,
mathematicians were inspired by real (i.e.,
physical and scientific) problems and found
the meaning of mathematics through them.
Equally beyond dispute is that applications
to real problems are a pedagogical necessity.
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Pedagogical Implications
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However, there is some question about
which applications will be interesting and
meaningful to students.
Only experience will enable teachers to
determine the best choices.
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Pedagogical Implications
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Fortunately, the choices are so numerous
and so varied, and interest in the real world
is so much more widespread than interest in
abstract mathematics, that attractive
applications can surely be found.
At the very least, it is easier to arouse
curiosity about real problems than about
mathematics.
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Pedagogical Implications
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The use of real and, especially, physical
problems serves not only to answer the
question of what value mathematics has but
also gives meaning to it.
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Examples
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Negative numbers are not just inverses to
positive integers under addition; they are the
number of degrees below zero on a
thermometer.
The ellipse is not just a peculiar curve; it is
the path of planets and comets.
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Examples
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Functions are not sets of ordered pairs; they
are relationships between real variables such
as the height and time of flight of a ball
thrown into the air, the distance of a planet
from the sun at various times of the year,
and the population of a country over some
period of years.
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Implications
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To rob mathematical concepts of their
meaning is to keep the branches of a fruit
tree and throw away the fruit.
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Proposal
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The proposal that the first two or three years
of secondary mathematics contain all sorts
of genuine applications may seem radical,
but a little perspective may correct this
impression.
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What History Teaches
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In the 17th century, mathematics courses
comprised astronomy, music, surveying,
measurement, perspective drawing, the
design of optical instruments, architecture,
and the design of fortifications and
machines.
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What History Teaches
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In the intervening centuries some of these
topics lost importance and were dropped
from the mathematics curriculum.
Also, the expansion of mathematics itself
and of knowledge generally has compelled
educators to drop other topics.
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Isolation is Not the Answer
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But the isolation of mathematics from all
applications relevant to our times cannot be
tolerated, even if the inclusion of
applications necessitates covering less
mathematical topography.
It is better to teach a little well and with
understanding than a lot poorly and with no
understanding.
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Ramifications
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This approach would present what is
interesting, enlightening, and culturally
significant, with the inclusion of only those
concepts and techniques that will serve to
further the liberal arts objective.
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Specifics
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No technique for the sake of technique
should be presented in the first two or three
years.
In other words, the material should be
objective-oriented rather than subjectoriented.
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Final Question
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But what do we do for the future
professional user of mathematics?
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Answer
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Admittedly a small but appreciable
percentage of the students will become
mathematicians, physicists, chemists,
engineers, social scientists, technicians,
statisticians, actuaries, and other specialists
whose work requires mathematics.
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Answer
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Of course these students, too, should know
the cultural significance of mathematics.
Moreover, students who are already inclined
toward a specific career will certainly take
an interest in mathematics if they see how
the subject is involved.
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Recommendations
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The textbooks written by Harold R. Jacobs
incorporate this pedagogical thrust.
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Mathematics: A Human Endeavor
Elementary Algebra
Geometry: Seeing, Doing, Understanding
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The only draw back is that the lifesustaining ingredient, the Christo-centric
perspective, is missing from these works.
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Summary
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The misdirection of High School
Mathematics Education: disconnection
from the world of God’s making.
Biblical Christian direction (or
framework) for reformation of Algebra:
connection to the world of God’s
making.
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Philosophical Principles of Mathematics