A Quote by Paul Lockhart
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.
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Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
I deny that art can be taught, or, in other words, maintain that art is completely individual, and that the talent of each artist is but the result of his own inspiration and his own study of past tradition.
What is a body that casts no shadow? Nothing, a formlessness, two-dimensional, a comic-strip character. If I deny my own profound relationship with evil I deny my own reality. I cannot do, or make; I can only undo, unmake.
If somebody says, 'I am a gay person, and I want to get married,' is their own family going to deny them that? Are our own fellow citizens going to deny them that?
Mathematics is about problems, and problems must be made the focus of a student's mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work.
[The error in the teaching of mathematics is that] mathematics is expected either to be immediately attractive to students on its own merits or to be accepted by students solely on the basis of the teacher's assurance that it will be helpful in later life. [And yet,] mathematlcs is the key to understanding and mastering our physical, social and biological worlds.
What fabrications they are, mothers. Scarecrows, wax dolls for us to stick pins into, crude diagrams. We deny them an existence of their own, we make them up to suit ourselves -- our own hungers, our own wishes, our own deficiencies.
Most people have some appreciation of mathematics, just as most people can enjoy a pleasant tune; and there are probably more people really interested in mathematics than in music. Appearances suggest the contrary, but there are easy explanations. Music can be used to stimulate mass emotion, while mathematics cannot; and musical incapacity is recognized (no doubt rightly) as mildly discreditable, whereas most people are so frightened of the name of mathematics that they are ready, quite unaffectedly, to exaggerate their own mathematical stupidity
Like a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.
To regret one’s own experiences is to arrest one’s own development. To deny one’s own experiences is to put a lie into the lips of one’s own life. It is no less than a denial of the soul.
Because many people deny the Palestinian struggle. They deny them everything. They deny them humanity, they deny them the right to be on the land they were born in. They deny them the right to return to the homes that were stolen from them, to build Israel.
To deny this is to deny our own heritage, our own spirit, and our own souls.
Those who deny their own feelings tend to deny other people's.
Now, we deny not, but that politicians may sometimes abuse religion, and make it serve for the promoting of their own private interests and designs; which yet they could not do so well neither, were the thing itself a mere cheat and figment of their own, and had no reality at all in nature, nor anything solid at the bottom of it.
I don't think that everyone should become a mathematician, but I do believe that many students don't give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.
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