“If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.” (Paul Lockhart, “A Mathematician's Lament", 2009)
"Mathematics is not a language, it's an adventure." (Paul Lockhart, “A Mathematician's Lament", 2009)
"Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity - to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs - you deny them mathematics itself." (Paul Lockhart, "A Mathematician's Lament", 2009)
"Mathematics is the music of reason. To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion - not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a break-through idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.” (Paul Lockhart, “A Mathematician's Lament”, 2009)
"Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.” (Paul Lockhart, “A Mathematician's Lament”, 2009)
"There is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depends heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)
"A proof is simply a story. The characters are the elements
of the problem, and the plot is up to you. The goal, as in any literary fiction,
is to write a story that is compelling as a narrative. In the case of
mathematics, this means that the plot not only has to make logical sense but
also be simple and elegant. No one likes a meandering, complicated quagmire of
a proof. We want to follow along rationally to be sure, but we also want to be
charmed and swept off our feet aesthetically. A proof should be lovely as well
as logical."
"And art is always a struggle. There is no systematic way to create beautiful and meaningful paintings or sculptures, and there is also no method for producing beautiful and meaningful mathematical arguments." (Paul Lockhart, "Measurement", 2012)
"Do not ignore symmetry! In many ways, it is our most powerful mathematical tool." (Paul Lockhart, "Measurement", 2012)
"Essentially, engaging in the practice of mathematics means that you are playing around, making observations and discoveries, constructing examples (as well as counterexamples), formulating conjectures, and then - the hard part - 'proving them'." (Paul Lockhart, "Measurement", 2012)
"It turns out pi is different. Not only is it incapable of
being expressed as a fraction, but in fact pi fails to satisfy any algebraic relationship
whatsoever. What does pi do? It doesn’t do anything. It is what it is. Numbers
like this are called transcendental (Latin for 'climbing beyond').
Transcendental numbers - and there are lots of them - are simply beyond the power
of algebra to describe."
"Mathematical reality, on the other hand, is imaginary. […] Mathematical reality is an infinite jungle full of enchanting mysteries, but the jungle does not give up its secrets easily. Be prepared to struggle, both intellectually and creatively." (Paul Lockhart, "Measurement", 2012)
"Mathematics is an art, and creative genius a mystery. Of
course, technique helps - good painters understand light and shadow, good musicians
have a thorough knowledge of functional harmony, and good mathematicians can
untangle algebraic information - but a beautiful piece of mathematics is just
as hard to make as a beautiful portrait or sonata."
"The solution to a math problem is not a number; it’s an argument, a proof. We’re trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying." (Paul Lockhart, "Measurement", 2012)
"The tangling and untangling of numerical relationships is called
algebra. […] The point of doing algebra is not to solve equations; it’s to
allow us to move back and forth between several equivalent representations,
depending on the situation at hand and depending on our taste. In this sense,
all algebraic manipulation is psychological. The numbers are making themselves known
to us in various ways, and each different representation has its own feel to it
and can give us ideas that might not occur to us otherwise."
"There are thousands of apparent mathematical truths out there that we humans have discovered and believe to be true but have so far been unable to prove. They are called conjectures. A conjecture is simply a statement about mathematical reality that you believe to be true [..]"(Paul Lockhart, "Measurement", 2012)
"There is no systematic way to create beautiful and meaningful paintings or sculptures, and there is also no method for producing beautiful and meaningful mathematical arguments."
"This is what it means to do mathematics. To make a discovery (by whatever means, including playing around with physical models like paper, string, and rubber bands), and then to explain it in the simplest and most elegant way possible. This is the art of it, and this is why it is so challenging and fun." (Paul Lockhart, "Measurement", 2012)
"What is a math problem? To a mathematician, a problem is a probe - a test of mathematical reality to see how it behaves. It is our way of 'poking it with a stick' and seeing what happens. We have a piece of mathematical reality, which may be a configuration of shapes, a number pattern, or what have you, and we want to understand what makes it tick: What does it do and why does it do it? So we poke it - only not with our hands and not with a stick. We have to poke it with our minds."(Paul Lockhart, "Measurement", 2012)
"What makes a mathematician is not technical skill or encyclopedic knowledge but insatiable curiosity and a desire for simple beauty. […] Part of becoming a mathematician is learning to ask such questions, to poke your stick around looking for new and exciting truths to uncover." (Paul Lockhart, "Measurement", 2012)
"Whenever you create or define a mathematical object, it always carries with it the blueprint of its own construction - the defining features that make it what it is and not some other thing." (Paul Lockhart, "Measurement", 2012)
“A good problem is something you don't know how to solve. That's what makes it a good puzzle and a good opportunity.” (Paul Lockhart)“Teaching is not about information. It's about having an honest intellectual relationship with your students.” (Paul Lockhart)
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