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Developments in Mathematical Education from PART II - THE INVITED PAPERS
Some of the following passages are literally quoted or translated, others are paraphrased (condensed, modernised, …). [Square brackets are used to indicate inserted words in the quotations and comments following the quotations.] I tried not to distort too much the meaning intended by the authors. At any rate, as I read them, these quotations greatly helped me to clarify my opinions and they may find responsive readers.
1 The ideas should be born in the student's mind and the teacher should act only as midwife.
Socrates
2 … we should give no small share of the credit to Democritus who was the first to state the result though he did not prove it [just guessed it] … The method I used did not furnish an actual demonstration [just a suggestion, a guess … Yet] I foresee that this method, once understood, will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
Archimedes
[First guess, then prove – that's the way to do it.]
3 Intuition is the conception of an attentive mind, so clear, so distinct, and so effortless that we cannot doubt what we have so conceived.
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