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Ryan Creedon
Dr. Haspel
ENGL 138T
29 Mar. 2013
A2 + B2 ≠ High School Mathematics.
“There are then two mathematics. There is the real mathematics of the real
mathematicians, and there is what I call the ‘trivial’ mathematics, for want of a better word. The
trivial mathematics may be justified by arguments which would appeal to Hogben [an early
1900’s medical statistician noted for his disinterest in mathematical aesthetics], or other writers
of his school, but there is no such defense for the real mathematics, which must be justified as art
if it can be justified at all.” (Hardy 139)
How gratifying it must be for a mathematician in the 21st century to witness the latest
renaissance in mathematical thought—to see new theorems arise out of the dawning information
age! Certainly, G.H. Hardy, one of the more prolific mathematicians of the previous century,
would agree. Yet how sad Hardy, or any one of the great mathematicians from yesteryears,
would be to see his subject so liquidized in the minds of the public. The culprit for this
widespread disillusionment in mathematics is not hard to identify. In fact, Hardy himself hints at
many possible sources above. In today’s times, however, high school mathematics is the main
culprit, for it rebuffs all advances of mathematical imagination in classrooms of fully capable
math students. As a result, both the aesthetics and rigor involved in mathematics, two of the
largest and most important portions of its serious study, are lost and with it the student’s ability
to reason in a way that is mathematically sound. Therefore, to claim that high school
mathematics compares in any way to real mathematics is a false, amateur impression and gross
misrepresentation of what mathematics really is; it is synonymous with saying A2 + B2 = C2 is
used for a right triangle only! Instead, I propose one can make a more accurate simile. High
school mathematics is like a shadow; it is a dull, inferior remnant of a great subject whose
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actions distract from the artisanship of its namesake and do not challenge students to think
beyond petty exercises and preconceived word problems.
1983—The National Commission for Educational Reform published A Nation at Risk.
America read in disgust, her lackluster mathematics program exposed globally for the first time:
Average [SAT] mathematics scores dropped nearly 40 points [from 1963
to]….Only one-third [of seventeen year olds] can solve a mathematics problem
requiring several steps….Between 1975 and 1980, remedial mathematics courses
in public 4-year colleges increased by 72 percent. (A Nation at Risk 11)
Thirty years later, some progress has been made. For nearly a quarter of a century, the nation has
operated under the guidance of the National Council of Teachers of Mathematics, a group of
individuals whose intent is to add real-world purpose behind K-12 mathematics courses. The
results of these standards are mixed, but at the high school level, America still falls short of her
competition. (See Figure 1.) This is so because the NCTM standards fail to raise the ceiling on
high school mathematics (Wu 4). Consequently, many adequate math students must be content
with watered-down concepts, where intuition, inquiry, and rigor are all but existent.
To understand more concretely what NCTM expects of all American students, consider
the following example extracted from McGraw-Hill’s popular Algebra 1 student workbook:
Peg is planning a rectangular vegetable garden using 250 feet of fencing material.
She only needs to fence three sides of the garden since one side borders an
existing fence….Write an expression to represent the area of the garden if she
uses all fencing material….Find the vertex of the equation. (Ch. 9 24)
Problems like these are the norms in most algebra textbooks today, and they all follow the
NCTM format. If unconvinced, look no further than the attached student workbook page to
verify my claim: Every question has an application, ranging from interior design to falling
objects to the planet Mars to even frogs! (See Figure 2.) These cutesy backstories, though a
noble attempt to motivate students, betray otherwise reasonable and thought-provoking questions
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in mathematics. In this case, rather than seeing the beauty of quadratic optimization, intelligent
math students wonder why Peg was not satisfied with a smaller garden or with extra fencing
material to use another time.
Now consider the same type of word problem expressed in a different manner:
Comment on the shape of a rectangle whose perimeter cannot be changed and
whose area must be maximized. Justify your conclusions.
The motivation for students to complete this new problem is synonymous with that of the
former: They are expected to apply their knowledge of quadratic functions to find the maximum
area of the rectangle. Notice, however, that these two problems carry out their motivations in
vastly different ways: Where the first word problem utilizes gardening as a mode of real-world
application, this problem investigates a question of pure mathematics. There are no Peg’s, no
gardens, and no dimensions to distract the reader from the question at hand. Additionally, note
this problem does not “spoon feed” students the solution; it does not tell them to write an
equation for the area of the rectangle and find that equation’s vertex. It is implied that students
must reach this conclusion on their own or find an alternative route to the answer. Herein the
proper balance of intuition, inquiry, and rigor thrives: The student guesses that the shape is a
square, proceeds to find a convincing argument for their conjecture, and then formalizes these
thoughts in a more rigorous fashion. Consequently, the student gains more mathematical insight
and appreciation in his struggles to find an answer as opposed to having it handed to him
initially, making NCTM applications further obsolete to good math students.
Along with their textbook counterparts, many math teachers have fallen for NCTM’s
dirty tricks, too; they use calculators and numerical examples to teach their students “proof”
techniques during lectures. For instance, because √2√3 = √6 and √3√4 = √12 on a
calculator, math teachers who follow NCTM will come to the conclusion that, for any positive
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real numbers a and b, √𝑎√𝑏 = √𝑎𝑏. How are they able to make such hasty generalizations? In
truth, they cannot. In practice, “‘to coax the students into proof, [they] call [these
generalizations] explanations” (Wu 5). While these explanations suffice students not looking to
pursue mathematics beyond high school, they are extremely dangerous to the otherwise
scientifically minded. Proofs are the poetry of logic; they sway from one truth to the next. Just as
poetry enhances literacy and vocabulary so, too, do proofs build mathematical maturity and
understanding. Therefore, to deny the scientific student proofs is to starve him of knowing, to
send him unprepared to connect and justify major mathematical observations at university.
In truth, it is not solely NCTM’s fault that many high school math teachers depreciate the
value of rigor in their classrooms. There is actually another disturbance that places far more
weight on their shoulders—state standardized tests. Though unfortunate, whether students take
Maryland’s MSA, New Jersey’s ASK, or Pennsylvania’s Keystone exams, math teachers follow
not too far behind, serving as caretakers of student proficiency and advocates for government
funding. There is no time to “think and, therefore, be” in their classrooms as Descartes once said,
only time to memorize formulas, algorithms, and exam questions.
To illustrate this ideology more clearly, consider a case presented by Dr. Paul Lockhart in
A Mathematician’s Lament:
I might imagine a triangle inside a rectangular box:
I do see something simple and pretty:
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I can see that each piece is cut diagonally in half by the sides of the triangle. So
there is just as much space inside the triangle as outside. That means that the
triangle must take up exactly half the box! (Lockhart 4-5)
Of course, standardized tests provide a formula for this problem—𝐴 =
1
2
𝑏ℎ, but by doing so,
they take something as simple and elegant as finding the area of a triangle and “reduce [it] to a
sterile set of ‘facts’ to be memorized” (Lockhart 6). How difficult it must be to take the few extra
minutes in class for students to prove this formula! Is it too much to ask that state education
departments grant students the right to engage in real mathematics once in a while?
Perhaps their hesitance to do so stems less from ignorance than from doubt. Rigor and
creativity are crucial to real mathematics, that case I have made certain, but to what extent should
they be implemented at the high school level? Will it ever be possible to satisfy the needs of both
the mathematician and non-mathematician? The truth is that educators now are more divided
than ever, and the severity of this fracture has left only two camps—the traditionalists, rigor
intensive and specialized, and the reformists, example driven and real-world bounded. Here I
combine highlights of both parties which I find work best together.
Math teachers need proper credentials: “Nearly one-fifth of high school students are
enrolled in math classes whose teachers neither majored nor minored in mathematics” (Drew 9).
Those teachers who satisfy the required criteria should be permitted freedom to employ their
own teaching philosophies, not necessarily those suggested by state standards. If teachers have
classes with gifted math students in them, the mathematical ABC’s—algorithm, beauty, and
creativity—should guide instruction (Whitcomb 15). Otherwise, the NCTM standards suffice.
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Gifted students should carry out the ABC’s according to the rigor-relevancy framework.
(See Figure 3.) First, they should acquire and assimilate knowledge. The depth of knowledge
acquired during these processes must shift from merely identifying and describing mathematical
concepts to justifying them and making connections to other known knowledge. (See Figure 4.)
This transition between identifying and justifying should be organic and filled with sustained
inquiry, creativity, and proof. Thoughtfully posed questions, stories from history, and projects
can all help stimulate this development of mathematical imagination. Later, as students work
their ways up the rigor-relevancy paradigm, real-world applications and adaptations may be
added to round out and polish their understanding of mathematics.
And so a renaissance is born, for there are two students of mathematics instead of just
one. The first shall master the standards of NCTM and be made aware of math’s real-world
applications. Provided he works hard and honestly, the first student will impress the world with
his own talents after high school. The second student, on the other hand, experiences a much
more salient—more rigorous, creative, and true—mathematics program; he will thrive in the
beauty of Euclid’s infinitely many primes and Pythagoras’ existence of irrational numbers.
Though he will be misunderstood by a world hostile to his profession, he will prevail. He will
become a new beacon for mathematics. His expertise and artistry will pay tribute to the many
teachers who prepared him well during high school, and his research, demeanor, and integrity
will change the way society views his subject. No longer will the chains of high school
mathematics blind us from the truth: A2 + B2 = C2 will live on as a theorem to celebrate, not to
memorize. It is only if we give our students this chance to imagine and think critically that we
can be liberated from the current debauchery of mathematics and never again to return to 1983,
for “the moving power of mathematics is not invention but imagination” (Whitcomb 15).
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Attachments
Figure 1: Math assessment score rankings for U.S. over the last fifty years. Source: STEM the
Tide, David Drew.
Algebra 1 Textbook
Figure 2: Scroll to page twenty-four of chapter nine.
Figure 3: The rigor-relevancy framework. Source.
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Figure 4: The depth of knowledge chart. Source.
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Works Cited
A Nation at Risk: The Imperative for Educational Reform. The National Commission on
Excellence and Education. Apr. 1983. Washington, D.C.: United States Department of
Education, Apr. 1983. Print.
Drew, David. STEM the Tide. Baltimore: The Johns Hopkins University Press, 2011. Print.
Hardy, Godfrey. A Mathematician’s Apology. London: Syndics of the Cambridge University
Press, 1967. Print.
Lockhart, Paul. “A Mathematician’s Lament.” Maa.org. Web. 01 Mar. 2013.
Whitcomb, Allan. “Mathematics Creativity, Imagination, Beauty.” Mathematics in School. 17.2
(1988): 13-15. Print.
Wu, H. “The mathematics education reform: What is it and why you should care?” Berkley.
Web. 10 Mar. 2013.