The first time I ever derived any enjoyment from a math class was calculus. Although the coursework was heavy and resembled Lockhart's descriptions, my teacher was excellent. She always tried to show us the why of things, not just the how, even though there wasn't much room for that in the class, and those parts were the ones that really stimulated my interest.

What's hubski's experience with math education?


caeli:

Okay so I've been drinking more than a little bit and this post might be incoherent. But I feel like most subjects are already being taught fucking horribly in schools, not just math.

I actually always loved math and had no problem with it. I just happened to be one of those kids who loved math so much that I managed to love it even with the boring-ass way it's taught in schools. But I had a horrible experience with other subjects in school, and particularly history. History is so goddamn boring the way it's taught. Why the fuck should I memorize seemingly meaningless names and dates? I always crammed the last minute right before tests so I wouldn't fail and then forgot everything, and I'm paying for it now by knowing fuckall about major historical events. Now that I've graduated high school and can learn history that I'm interested in on my own, history is fucking interesting as shit and I love it. But it was taught in such a terrible way in school (or at least, the schools I went to).

In fact, I always hated science up until I had a fantastic science teacher in high school who made things interesting. Science is literally what I do for a living now. I can't even imagine where I would be if I had a bad high school science teacher.

    Now where did this idea of mine come from? How did I know to draw that line? How does a painter know where to put his brush? Inspiration, experience, trial and error, dumb luck. That’s the art of it, creating these beautiful little poems of thought, these sonnets of pure reason. There is something so wonderfully transformational about this art form.

This is exactly how I felt the first time I did a proof in high school geometry. I was sitting there struggling, and then all of a sudden it came to me like a stroke of lightning. It fucking felt like I was creating something. Of course I wasn't, I was proving some boring old proof that tens of thousands of other kids had done before me, but it felt new and fun. I wanted more.

    What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you.

Man, have we ever stopped to think about how mathematicians think about mathematics? This has actually been studied by cognitive scientists. One of the most important parts of thinking about a math problem is drawing shit on the board and talking aloud about it. Not doing endless exercises, not sitting alone in isolation like fucking Hollywood would make you believe. No, it's conceptually drawing stuff out and talking about it with another human being. Mathematical curiosity, list most other artistic and scientific curiosities, is a thirst for truth and beauty that you crave to share with other human beings.

    Suppose I am given the sum and difference of two numbers. How can I figure out what the numbers are themselves?
      Here is a simple and elegant question, and it requires no effort to be made appealing.

    Okay this is probably the only thing I truly disagree with in this article. I don't believe anything is inherently interesting, and I'd bet that there are a lot of people who aren't particularly interested in this question for one reason or another. I think it must be recognized that not everyone can be interested in everything. I have a great deal of love for most things, including science, math, and many forms of art, but if I were asked in school to draw something I would not enjoy it. That's just a fact of my personality. Assuming that all kids would be inherently interested in mathematical questions is not a safe assumption. But I think it is a safe assumption that most kids would be more interested in math were it taught in a different way, which is the main point of this article anyway so I shouldn't nitpick too much.

      It is far easier to be a passive conduit of some publisher’s “materials” and to follow the shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students. We are encouraged to forego the difficult task of making decisions based on our individual wisdom and conscience, and to “get with the program.” It is simply the path of least resistance

    Ooh this is so true damn. Can I give this some snaps or is that reserved for poetry? But like seriously though, how do we solve this problem? Truly investing oneself in making sure that every student is thoroughly engaged with math is so hard and feels like it's beyond the call of duty for your standard teacher.

      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?

    It's funny how deep culture goes because I hadn't even thought about this before.

    As an aside though, I disagree with some of the author's dismissal of teachers and how teachers are trained but I'm not going to spend the next 5 hours quoting every single line and responding with paragraphs ;)

      think the far greater risk is that of creating schools devoid of creative expression of any kind, where the function of the students is to memorize dates, formulas, and vocabulary lists, and then regurgitate them on standardized tests—“Preparing tomorrow’s workforce today!”

      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.

      This is intimately connected to what I call the “ladder myth”— the idea that mathematics can be arranged as a sequence of “subjects” each being in some way more advanced, or “higher” than the previous. The effect is to make school mathematics into a race— some students are “ahead” of others, and parents worry that their child is “falling behind.” And where exactly does this race lead? What is waiting at the finish line? It’s a sad race to nowhere. In the end you’ve been cheated out of a mathematical education, and you don’t even know it.

      I: 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.

    Fuck man, this paper is so full of truths.

      In place of discovery and exploration, we have rules and regulations. We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no mention of the aesthetics behind this choice, or even that it is a choice.

      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?

    One of the most annoying things about specialized fields is the specialized terminology that comes with them. I complain about this all the fucking time about my own field, linguistics. We use so many stupid ass terms that no one understands and no one becomes engaged with (except for linguistics, of course). We want people to become engaged with ideas, not memorize dumb terms, and this is what I was reminded of when I read that passage. Syntax and semantics are the worst with this, I swear...

      The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”

    I loved proofs but seeing as I was the only one in the class who did this is probably a problem.

      ! A proof should be an epiphany from the Gods, not a coded message from the Pentagon

    See I always felt like I was having an epiphany, and then the only remaining thing was to write it out in the formal language we were taught in school. So fun!!!

      SIMPLICIO: Now hold on a minute. I don’t know about you, but I actually enjoyed my high school geometry class. I liked the structure, and I enjoyed working within the rigid proof format.

    This guy is seriously predicting my thoughts right now!

    ---

    Thanks for sharing, this is something that spawns a lot of thought.

    (This post has been brought to you by four ciders)


    posted by dingus: 1183 days ago