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1 " Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them. "
― Paul Lockhart , A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form
2 " A good problem is something you don't know how to solve. That's what makes it a good puzzle and a good opportunity. "
3 " Why don't we want our children to learn to do mathematics? Is it that we don't trust them, that we think it's too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon. Why not about triangles? "
4 " [Math] curriculum is obsessed with jargon and nomenclature seemingly for no other purpose than to provide teachers with something to test the students on. "
5 " SIMPLICIO: ... You have to [learn to] walk before you can run.SALVIATI: No, you have to have something you want to run toward. "
6 " Doing mathematics should always mean finding patterns and crafting beautiful and meaningful explanations. "
7 " 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. "
8 " ... That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free! "
9 " In any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide -- a relief from daily life, an anodyne to the practical workaday world. "
10 " ... This is a major theme in mathematics: things are what you want them to be. You have endless choices; there is no reality to get in your way.On the other hand, once you have made your choices then your new creations do what they do, whether you like it or not. This is the amazing thing about making imaginary patterns: they talk back! "
11 " If teaching is reduced to mere data transmission, if there is no sharing or excitement and wonder, if teachers themselves are passive recipients of information and not creators of new ideas, what hope is there for their students? "
12 " So [in mathematics] we get to play and imagine whatever we want and make patterns and ask questions about them. But how do we answer these questions? It’s not at all like science. There’s no experiment I can do ... The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work. "
13 " It is the story that matters not just the ending. "
14 " I don't see how it's doing society any good to have it's members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. "
15 " Mathematics is the art of explanation. "
16 " And I'll go even further and say that mathematics, this art of abstract pattern-making — even more than storytelling, painting, or music - is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are. "
17 " Be honest: did you actually read [the above geometric proof]? Of course not. Who would want to? The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious by insisting that it be 'rigorously proved' ... is to say to a student 'Your feelings and ideas are suspect. You need to think and speak our way. "
18 " Of course, being a mathematician, I'm always looking for ways to avoid tedious mundane labor. Especially if it means I get to do a bunch of interesting abstract thinking. That's kind of the whole math thing: working hard to find ways to get out of working hard. If you are really, really lazy, and also happen to be really, really clever, then math just might be the life for you (assuming you also have no interest in wealth, fame, or popularity). "
― Paul Lockhart , Arithmetic
19 " Perhaps the most surprising and powerful aspect of place-value arithmetic is how it reduces any calculation to a set of purely abstract symbolic manipulations. In principle, I suppose, one could even be trained to perform such symbol-jiggling procedures without any comprehension whatever of the underlying meaning. We could even (if we can possible imagine being so cruel) force young children to memorize tables of symbols and meaningless step-by-step procedures, and then reward or punish them for their skill (or lack thereof) in this dreary and soulless activity. This would help protect our future office workers from accidentally gaining a personal relationship to arithmetic as a craft or enjoying the perspective that outlook would provide. We could turn the entire enterprise into a rote mechanical process and then reward those who show the most willingness to be made into reliable and obedient tools. I wonder if you can imagine such a nightmarish, dystopian world? Let's try not to think about it. "
20 " No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a "mixed number " while 5/2 is an "improper fraction." They're EQUAL for crying out loud. They are the exact same numbers and have the exact same properties. Who uses such words outside of fourth grade? "
