Day 2, Mat194: Mathematical Sciences Seminar

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Announcements:
- I hope that you've read your "Mathematician's Lament". We'll start with a short quiz over it, to make sure that you read it.
- Your assignment for next time: type up a two-page dialogue of your own, using Lockhart's characters. Insert an experience of your own, to bolster the claim of one or the other. Try to see the other side's point of view, however!
Today we'll be discussing A Mathematician's Lament, by Paul Lockhart.
- Last time we briefly discussed some positive experiences we have had with mathematics:
  - Some of us love math because it is cut and dried; some because it's like playing with puzzles, and because there's always something interesting and astonishing to discover; another likes how one can use "an obnoxious amount of data" to learn really interesting things about our world (even baseball!).
  - Some of us had great experiences with teachers -- that's frequently what inspires students to study math.
  - I confessed my love of Archimedes, and was pleased to see that he received a positive mention in Lockhart's story.
- Now let's look at why Lockhart thinks that teachers (and school districts, government, etc.) may be killing our interest in math -- and his prescription(s) for solving the problem.
  - First, we should just take the class "temperature" for this "lament". Do we find ourselves agreeing, or disagreeing, with Lockhart?
  - What leaps out at you about the writing?
  - I'll be sharing some quotes with you, and asking you respond to any that strike a chord with you.
    Before I do, however, perhaps you have some that you'd like to point out?
    
    Perhaps it would be good to keep in mind our questions for the course as we delve into this:
    - In what ways have I loved mathematics?
    - What do I want to do (and what have I done) with mathematics?
    - Why is it important in my life, and to human civilization?
  - The choice of the names Simplicio and Salviati is an homage to Galileo's book Dialogue (Concerning the Two Chief World Systems), in which Simplicio is the proponent for the old Ptolemian and Aristotelian view, and Salviati speaks for the Capernican position (which was Galileo's as well).
  Some of my favorite moments from the Lament:
  - "Do you think carpenters are out there using trigonometry?"
  - "Better to not have math classes at all than to do what is currently being done."
  - SIMPLICIO: So you would remove mathematics from the school curriculum? SALVIATI: The mathematics has already been removed!
  - "There is surely no more reliable way to kill enthusiasm and interest in a subject than to make it a mandatory part of the school curriculum. Include it as a major component of standardized testing and you virtually guarantee that the education establishment will suck the life out of it."
  - "...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."
  - "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 triangle takes up half its box. What about a pyramid inside its three-dimensional box? Can we handle this problem in a similar way?"
  - "School mathematics concerns itself with many things that have nothing to do with the ability to get along society -- in algebra and trigonometry, for instance."
  - 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.
  - "Mathematics is not a language, it's an adventure."
  - "The most striking thing about this so-called mathematics curriculum is its rigidity. This is especially true in the later grades. From school to school, city to city, and state to state, the same exact things are being said and done in the same exact way and in the same exact order. Far from being disturbed and upset by this Orwellian state of affairs, most people have simply accepted standard mode math curriculum as being synonymous with math itself."
  - SIMPLICIO:Do you really think that most students are capable of operating on such a high level as to create their own mathematics?
    SALVIATI:If we honestly believe that creative reasoning is too "high" for our students, and that they can't handle it, why do we allow them to write history papers or essays about Shakespeare? The problem is not that the students can't handle it, it's that none of the teachers can. They've never proved anything themselves, so how could they possibly advise a student? In any case, there would obviously be a range of student interest and ability, as there is in any subject, but at least students would like or dislike mathematics for what it really is, and not for this perverse mockery of it.
  - "High School Geometry: Instrument of the Devil": Never was a wolf in sheep's clothing as insidious, nor a false friend as treacherous, as High School Geometry.... 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.)
  - Rigorous formal proof only becomes important when there is crisis -- when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind. But such excessive preventative hygiene is completely unnecessary here -- nobody's gotten sick yet!
  - [About a proof in geometry]: Could anything be more unattractive and inelegant? Could any argument be more obfuscatory and unreadable? This isn't mathematics! A proof should be an epiphany from the Gods, not a coded message from the Pentagon.
  - SIMPLICIO: But then how can schools guarantee that their students will all have the same basic knowledge? How will we accurately measure their relative worth?
    SALVIATI: They can't, and we won't. Just like in real life. Ultimately you have to face the fact that people are all different, and that's just fine. In any case, there's no urgency. So a person graduates from high school not knowing the half-angle formulas (as if they do now!) So what? At least that person would come away with some sort of an idea of what the subject is really about, and would get to see something beautiful.

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