Paul Lockhart
  1. 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, soul- crushing 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.

    (A Mathematician's Lament, p.20, Bellevue literacy press, 2009)
  2. The first thing to understand is that mathematics is an art.
    (A Mathematician's Lament, p.22, Bellevue literacy press, 2009)
  3. Nevertheless, the fact is that 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 depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
    (A Mathematician's Lament, p.23, Bellevue literacy press, 2009)
  4. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary. [...] That’s what math is — wondering, playing, amusing yourself with your imagination.
    (A Mathematician's Lament, p.24, Bellevue literacy press, 2009)
  5. 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.
    (A Mathematician's Lament, p.25, Bellevue literacy press, 2009)
  6. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work.
    (A Mathematician's Lament, p.26, Bellevue literacy press, 2009)
  7. By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. How am I supposed to be inspired by that? (And of course it’s actually much worse than this — at least it’s understood that there is an art of sculpture that I am being prevented from appreciating).
    By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity — to pose their own problems, 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. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

    (A Mathematician's Lament, p.28, Bellevue literacy press, 2009)
  8. Math is not about following directions, it’s about making new directions.
    (A Mathematician's Lament, p.31, Bellevue literacy press, 2009)
  9. 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? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand — a self-fulfilling prophesy if there ever was one.
    (A Mathematician's Lament, p.31, Bellevue literacy press, 2009)
  10. How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers — the kind of thing a real mathematical education might provide.
    (A Mathematician's Lament, p.33, Bellevue literacy press, 2009)
  11. Better to not have math classes at all than to do what is currently being done. At least some people might have a chance to discover something beautiful on their own.
    (A Mathematician's Lament, p.35, Bellevue literacy press, 2009)
  12. There is surely no more reliable way to kill enthusiam and interest in a subject than to make it a mandatory part of the school curriculum.
    (A Mathematician's Lament, p.36, Bellevue literacy press, 2009)
  13. The mathematics curriculum doesn't need to be reformed, il needs to be scrapped.
    (A Mathematician's Lament, p.37, Bellevue literacy press, 2009)
  14. 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 breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it. Remove this from mathematics and you can have all the conferences you like; it won’t matter. Operate all you want, doctors: your patient is already dead.
    The saddest part of all this “reform” are the attempts to “make math interesting” and “relevant to kids’ lives.” You don’t need to make math interesting — it’s already more interesting than we can handle! And the glory of it is its complete irrelevance to our lives. That’s why it’s so fun!

    (A Mathematician's Lament, p.37, Bellevue literacy press, 2009)
  15. 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.
    (A Mathematician's Lament, p.39, Bellevue literacy press, 2009)
  16. The main problem with school mathematics is that there are no problems.
    A good problem is something you don’t know how to solve. That’s what makes it a good puzzle, and a good opportunity. A good problem does not just sit there in isolation, but serves as a springboard to other interesting questions.

    (A Mathematician's Lament, p.40, Bellevue literacy press, 2009)
  17. I can understand the idea of training students to master certain techniques — I do that too. But not as an end in itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process — that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.
    (A Mathematician's Lament, p.41, Bellevue literacy press, 2009)
  18. So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant mathematical criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject.
    (A Mathematician's Lament, p.43, Bellevue literacy press, 2009)
  19. The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Of course it’s easier to post a set of rules than to guide aspiring young artists, and it’s easier to write a VCR manual than to write an actual book with a point of view.
    Mathematics is an art, and art should be taught by working artists, or if not, at least by people who appreciate the art form and can recognize it when they see it. It is not necessary that you learn music from a professional composer, but would you want yourself or your child to be taught by someone who doesn’t even play an instrument, and has never listened to a piece of music in their lives? Would you accept as an art teacher someone who has never picked up a pencil or stepped foot in a museum? Why is it that we accept math teachers who have never produced an original piece of mathematics, know nothing of the history and philosophy of the subject, nothing about recent developments, nothing in fact beyond what they are expected to present to their unfortunate students? What kind of a teacher is that? How can someone teach something that they themselves don’t do? I can’t dance, and consequently I would never presume to think that I could teach a dance class (I could try, but it wouldn’t be pretty). The difference is I know I can’t dance. I don’t have anyone telling me I’m good at dancing just because I know a bunch of dance words.
    Now I’m not saying that math teachers need to be professional mathematicians — far from it. But shouldn’t they at least understand what mathematics is, be good at it, and enjoy doing it?

    (A Mathematician's Lament, p.44, Bellevue literacy press, 2009)
  20. Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.
    In particular, you can’t teach teaching. Schools of education are a complete crock. Oh, you can take classes in early childhood development and whatnot, and you can be trained to use a blackboard “effectively” and to prepare an organized “lesson plan” (which, by the way, insures that your lesson will be planned, and therefore false), but you will never be a real teacher if you are unwilling to be a real person. Teaching means openness and honesty, an ability to share excitement, and a love of learning. Without these, all the education degrees in the world won’t help you, and with them they are completely unnecessary.

    (A Mathematician's Lament, p.46, Bellevue literacy press, 2009)
  21. It may be true that you have to be able to read in order to fill out forms at the DMV, but that’s not why we teach children to read. We teach them to read for the higher purpose of allowing them access to beautiful and meaningful ideas. Not only would it be cruel to teach reading in such a way — to force third graders to fill out purchase orders and tax forms — it wouldn’t work! We learn things because they interest us now, not because they might be useful later. But this is exactly what we are asking children to do with math.
    (A Mathematician's Lament, p.48, Bellevue literacy press, 2009)
  22. SIMPLICIO : Then what should we do with young children in math class?
    SALVIATI : Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.

    (A Mathematician's Lament, p.49, Bellevue literacy press, 2009)
  23. SALVIATI : A real appreciation for poetry does not come from memorizing a bunch of poems, it comes from writing your own.
    SIMPLICIO : Yes,but before you can write your own poems you need to learn the alphabet. The process has to begin somewhere. You have to walk before you can run.
    SALVIATI : No, you have to have something you want to run toward. Children can write poems and stories as they learn to read and write. A piece of writing by a six-year-old is a wonderful thing, and the spelling and punctuation errors don’t make it less so. Even very young children can invent songs, and they haven’t a clue what key it is in or what type of meter they are using.

    (A Mathematician's Lament, p.52, Bellevue literacy press, 2009)
  24. Mathematics is not a language, it's an adventure.
    (A Mathematician's Lament, p.53, Bellevue literacy press, 2009)
  25. Mental acuity of any kind comes from solving problems yourself, not from being told how to solve them.
    (A Mathematician's Lament, p.53, Bellevue literacy press, 2009)
  26. How many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. We teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone needs is to be trained.
    (A Mathematician's Lament, p.54, Bellevue literacy press, 2009)
  27. The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. 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 same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?
    (A Mathematician's Lament, p.58, Bellevue literacy press, 2009)
  28. Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher. If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it?
    (A Mathematician's Lament, p.62, Bellevue literacy press, 2009)
  29. It is the story that matters, not just the ending.
    (A Mathematician's Lament, p.65, Bellevue literacy press, 2009)
  30. 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.

    (A Mathematician's Lament, p.68, Bellevue literacy press, 2009)
  31. A proof should be an epiphany from the Gods, not a coded message from the Pentagon.
    (A Mathematician's Lament, p.75, Bellevue literacy press, 2009)
  32. The point is you don’t start with definitions, you start with problems.
    (A Mathematician's Lament, p.79, Bellevue literacy press, 2009)
  33. Efficiency and economy simply do not make good pedagogy.
    (A Mathematician's Lament, p.80, Bellevue literacy press, 2009)
  34. But even working within the conventional framework a good teacher can guide the discussion and the flow of problems so as to allow the students to discover and invent mathematics for themselves. The real problem is that the bureaucracy does not allow an individual teacher to do that. With a set curriculum to follow, a teacher cannot lead. There should be no standards, and no curriculum. Just individuals doing what they think best for their students.
    (A Mathematician's Lament, p.82, Bellevue literacy press, 2009)
  35. School has never been about thinking and creating. School is about training children to perform so that they can be sorted. It's no shock to learn that math is ruined at school; evertything is ruined at school!
    (A Mathematician's Lament, p.91, Bellevue literacy press, 2009)
  36. The only thing that matters in mathematics is what things are, and more important, how they act.
    (A Mathematician's Lament, p.94, Bellevue literacy press, 2009)
  37. Being a mathematician is not so much about being clever (although lord knows that helps); it's about being aesthetically sensitive and having refined and exquisite taste.
    (A Mathematician's Lament, p.104, Bellevue literacy press, 2009)
  38. The only thing I am interested in using mathematics for is to have a good time and to help others do the same. And for the life of me I can't imagine a more worthwhile goal. We are all born into this world, and at some point we will die and that will be that. In the meantime, let's enjoy our minds and the wonderful and ridiculous things we can do with them.
    (A Mathematician's Lament, p.106, Bellevue literacy press, 2009)
  39. Math is not about collection of “truths” (however useful or interesting they may be). Math is about reason and understanding. We want to know why. And not for any practical purpose.
    (A Mathematician's Lament, p.110, Bellevue literacy press, 2009)
  40. We are biochemical pattern-recongnition machines and mathematics is nothing less than the distilled essence of who we are.
    (A Mathematician's Lament, p.115, Bellevue literacy press, 2009)
  41. Anyway, the point is not whether mathematics has any practical value - I don't care if it does or not. All I'm saying is that we don't need to justify it on that basis.[...] To say that math is important because it is useful is like saying that children are important because we can train them to do spiritually meaningless labor in order to increase corporate profits. Or is that in fact what we are saying?
    (A Mathematician's Lament, p.120, Bellevue literacy press, 2009)
  42. [...] Mathematical structures are designed and built not so much by us, as by ou proofs.
    (A Mathematician's Lament, p.131, Bellevue literacy press, 2009)
  43. If you don't have a personal relationship to your subject, and if it doesn't move you and send chills down your spine, then you need to find something else to do. Il you love working with children and you really want to be a teacher, that's wonderful - but teach something that actually means something to you, about which you have something to say. It's important that we be honest about that. Otherwise I think we teachers can do a lot a unintentional harm.
    (A Mathematician's Lament, p.140, Bellevue literacy press, 2009)