A Mathematician’s Lament, by Paul Lockhart

[MacCruiskeen]

This is a brilliant diatribe about the terrible state of maths teaching:

[...]In fact, if I had to design a mechanism for the express purpose of destroying a child’s natural
curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being
done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing
ideas that constitute contemporary mathematics education.

Everyone knows that something is wrong. The politicians say, “we need higher standards.”
The schools say, “we need more money and equipment.” Educators say one thing, and teachers
say another. They are all wrong. The only people who understand what is going on are the ones
most often blamed and least often heard: the students. They say, “math class is stupid and
boring,” and they are right.

Mathematics and Culture

The first thing to understand is that mathematics is an art. [...]

http://www.maa.org/devlin/LockhartsLament.pdf

High School Geometry: Instrument of the Devil

There is nothing quite so vexing to the author of a scathing indictment as having the primary
target of his venom offered up in his support. And never was a wolf in sheep’s clothing as
insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it
is school’s attempt to introduce students to the art of argument that makes it so very dangerous.
Posing as the arena in which students will finally get to engage in true mathematical
reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative
rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject,
and permanently disabling them from thinking about math in a natural and intuitive way.

The mechanism behind this is subtle and devious. The student-victim is first stunned and
paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly
and painstakingly weaned away from any natural curiosity or intuition about shapes and their
patterns by a systematic indoctrination into the stilted language and artificial format of so-called
“formal geometric proof.”

All metaphor aside, geometry class is by far the most mentally and emotionally destructive
component of the entire K-12 mathematics curriculum. Other math courses may hide the
beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured.
(Apparently I am incapable of putting all metaphor aside.)

What is happening is the systematic undermining of the student’s intuition. A proof, that is,
a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof
should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted
argument should feel like a splash of cool water, and be a beacon of light— it should refresh the
spirit and illuminate the mind. And it should be charming.

There is nothing charming about what passes for proof in geometry class. Students are
presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted— a
format as unnecessary and inappropriate as insisting that children who wish to plant a garden
refer to their flowers by genus and species. [...]

http://www.maa.org/devlin/LockhartsLament.pdf

[Elihu]

http://professorfekete.com/articles%5CA ... umber0.pdf

...The evolution of the number concept through the phases N, Z, Q, R, C is hereby described
through a uniform method: the construction of quotient sets. The picture that comes along is easy and
lovely. A complex number no longer appears to be the odd man out. The mystery is taken out of
“imaginary numbers” by pointing to the organic relation between algebra and Euclidean geometry.

Nevertheless, the “office of inquisition” does not allow a textbook into the classroom which
uses this method of introducing complex numbers — presumably because of the prevailing anti-Euclid
ethos of our age. The teacher is forced to toe the line, or out he goes. The student is forced to go
through the rigmarole of introducing complex numbers in a wholly artificial and unilluminating way,
making them appear to have arrived from the Mars. As they have an “imaginary” component, they can
never be fully integrated in the community of “real” numbers. Students are forced to learn new
paradigms for the arithmetic operations as applicable to complex numbers, instead of being told to do
something they already know how to do: add and multiply them as if they were polynomials in the
variable i, and in the end simplify by making the substitution i2 + 1 = 0.

Lest the reader thought it was too far-fetched to suggest that the “office of inquisition” could
remove a teacher from the classroom for the offence of putting an unauthorized textbook into the
hands of students, let it be stated here that the author of this blog suffered that fate. As a tenured
professor at a Canadian university he was defrocked midstream, in full view of the students, for no
better reason then refusing to make his students buy the prescribed textbook. Tenure notwithstanding,
he was intimidated and threatened to be fired. The case took five years to resolve and the university
teachers’ association had to be called in to mediate. The university never apologized.

The way complex numbers are being taught is a scandal and a blot on our “enlightened”
educational system. In 1949 the new Chinese government in one of its first decrees outlawed the
ancient practice of maiming and deforming the feet of little girls — an age-old cruel custom. But the
maiming and deforming of the minds of talented young boys and girls via a thoroughly outmoded way
of teaching complex numbers still continues in China, as it is in the rest of the world!

http://professorfekete.com/maths.asp