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An educated mind is an opened mind. An opened mind is a liberal mind. Teachers don't have to intend to create liberals, it happens naturally.

On the inside:

  • On Beauty
  • Links to other education-related stories.
  • As always, the topics will be whatever you want to discuss.

Door's Open...

On Beauty

In an interesting confluence of events, I was thinking about the beauty of mathematics the other day and lo and behold, a discussion about it breaks out in Marion Brady's diary.  It's probably still going on, but I had to leave for awhile to get something written for Teacher's Lounge.  So why don't I just write a bit about mathematics?  I am, after all, an abstract algebraist [If there are others out there, I'm a Frank Anderson student out of the University of Oregon].

I am a studier of rings and their modules.  A ring is a nonempty set with two operations, roughly equivalent to addition and multiplication (in that we require the addition to be commutative, associative, and have an identity and inverses and that multiplication be associative and distribute over the addition.  Sometimes we also require multiplication to have an identity.  Sometimes not.  Sometimes we require multiplication to be commutative, but not me.  I study what are called non-commutative rings, but what is true when the commutative law is not assumed is still true if we assume it.

A left module (handedness matters) over that ring, on the other hand, is like a vector space (think matrices)...sort of one that you left on the radiator overnight.  It's an abelian group (oops!  I mean it's a nonempty set with an operation roughly akin to addition (commutative, associative, identity and inverses)), and a "scalar multiplication of elements of the ring times elements of the module (in that order) which produces other elements of the module.  We require that the distributive law holds...and if there is an identity 1 in the ring, that 1 * x = x.  Math is hard if we don't assume that.

My specific area of expertise could be said to be in the possible formation of fractions in such rings.  The modules are studied to shed light on the problem.  It's hard to study fractions when x(1/y) and (1/y)x may not be the same element.  Or when 1/y may or may not exist (remember, 1/0 doesn't exist and so if xy = 0 for some element x, which happens more often than you might think, then an element 1/y = x/xy = x/0, which isn't going to exist either.  So I studied topologies on rings and torsion theories in the categories of the rings' modules.  I won't go further.  But the intricasy and beauty of the structures are awesome to behold!  And discovering levels of depth to meaning that eventually inform our life philosophies is incredible.

Wow!  What a screed!  I probably lost most of you there.  I'm sorry that I tend to do that, caught up in the beauty flowing through my brain.  What saddens me is that there are so few people I can talk about this with, this artistic pastiche called mathematics.  I did my best for 25 years trying to share the joy with people, as their algebra teacher (I feel the urge to expound on what an algebra is, but I'll save that for another day...other than to note that I said an algebra).  I taught in college, remedial level through grad school level.  Now I have moved on to programming languages, art and poetry.

But I feel the need...I have the defend my field and promote its well-being.  No place in my screed up there did I refer to any numbers other than 0 and 1.  Mathematics is about words.  It's about logically pasting thoughts together to see what is true and what is not.  It is assembling building blocks provided by other people and adding our own blocks and the paste to put them together to create new knowledge.  We have a very difficult time trying to get students to think that way.  You know why?  Because in grade school, while their curiosity is being stomped out of them somewhere between grade 2 and grade 6, the students are told that arithmetic is mathematics and mathematics is arithmetic, that mathematics is all about the numbers and if they don't learn the numbers right, they can't go on.

Who said so?   Where is it written?

If we want to improve education, which I believe is the goal, then we have to think about it critically.  That requires that we think deeply about all of our assumptions with an open mind.  I try to have an open mind about people who have good enough verbal skills to understand and master challenging mathematics being kept out of my subject because they can't master skills they will not need in order to understand that mathematics.

I understand that other people don't have my agenda of preserving the beauty of the past and passing it on to future generations, adding to and embellishing that creative structure and presenting its results to the rest of humanity, that they may be applied wherever by whomever.  To me it is all about the beauty.  My side doesn't get expressed a lot.

--Robyn Elaine Serven
--Bloomfield College, NJ

Education Round-up:  I've categorized.
Philosophy and PoliticsStories:  Ourselves and OthersFreedom on CampusNCLB/Department of Education/Standardized Testing/AssessmentMoneyAction, Advocacy and Information
I'll be hanging around most of the day, actively waiting for your comments (actually, I'll be working in another program, but I'm close by), so at least one person will be here to discuss whatever anyone wants to discuss.

The Not-so-many Rules
  • No general bashing of administrators, politicians, etc, just on general principles. If you want to bash them, have a point and a plan.
  • No bitching about students unless you're talking about what you are going to do to alleviate the problems you think the students have.
  • Introductions are encouraged, but not essential.
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  • Teacher's Lounge can be "slow blogging" if you want it to be.  You don't get quality writing if you demand velocity.  It doesn't have to be the case that something posted today is dead by tomorrow. I would like it to eventually be up and active 24/7, but that may have to wait until I have developed an independent blog site.

Every Saturday I'll post a clean slate, between 10am and 12am EST.

Originally posted to Robyn's Perch on Sat Feb 10, 2007 at 08:53 AM PST.

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Comment Preferences

  •  A little of my beauty: (25+ / 0-)

    Art Link

    The numbers dance
    by the symbols
    terpsichorean delight
    in my mind
    I hear the music
    see the beauty
    as phrases expand
    and collapse
    evolving into simplicity
    and meaning
    In my soul
    I feel the beating
    of the heart
    of creation

    --Robyn Elaine Serven
    --February 2, 2006

    Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

    by rserven on Sat Feb 10, 2007 at 08:51:01 AM PST

    •  Beauty, like mathematics, (10+ / 0-)

      is something I appreciate but rarely understand.

      •  I understand. (8+ / 0-)

        The difficulty I have in a discussion like this is that I had three paragraphs in there with a discussion on my concept of beauty that probably lost most of the potential readership.  How many people would give up before getting to my point?

        Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

        by rserven on Sat Feb 10, 2007 at 09:14:28 AM PST

        [ Parent ]

        •  I suffered.... (4+ / 0-)

          ..and still suffer from an arithmetic deficiency.  Throughout primary and secondary schools, the dreaded "math" course was my only "B" or "C" grade; then I got to Geometry!  The shapes, the solids, but most of all the logic of proofs made such immediate sense to me - once I got "rid" of the damn arithmetic.
           After I had graduated from college, I took a year to take the science and math classes I needed for medical school;  I did not dare take those courses at my alma mater (Rice University), so I went across town to the public university.  I had a graduate student for calculus.  I don't know if it was my older brain, my improved study habits - but I think it was his love of the subject, but a light went off in my brain, and I started "seeing" calculus as a language, as was a miraculous transformation for me, to see mathematics as completely separate from arithmetic, and I fell in love with that course.
           I am no longer a student of mathematics, but thanks to that graduate teaching assistance, I could follow your second, third, and fourth paragraphs and feel the wonder of the subject.
           Thanks, Robyn!

          In a time of universal deceit, telling the truth becomes a revolutionary act. - George Orwell

          by drchelo on Sat Feb 10, 2007 at 09:39:13 AM PST

          [ Parent ]

          •  You are welcome. :-) (2+ / 0-)
            Recommended by:
            cookiebear, drchelo

            My "Aha!" moment came in Introduction to Abstract Mathematics, when we learned how to prove things were true.  Another came when I realized what I learned in one area of mathematics had applications in others.


            Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

            by rserven on Sat Feb 10, 2007 at 09:49:28 AM PST

            [ Parent ]

          •  geometry was my Waterloo (5+ / 0-)

            algebra, on the other hand ... perfect sense

            kos, my daughter keeps writing on the wall with crayons. I'm tired of it. Do something about it NOW, dammit. -MJB

            by cookiebear on Sat Feb 10, 2007 at 09:50:04 AM PST

            [ Parent ]

            •  Never quite got algebra (5+ / 0-)

              NEVER got calculus

              But I LOVED geometry because of the written proofs.  (I'm told they don't do those any more in high school...a pity).

              •  it's interesting to me (5+ / 0-)

                i've even toyed with dividing the world into algebra people and geometry people

                symbolic logic has always been my strength. by the same token, i'm a total contradiction in that i'm very earthbound - yet the physical world is like a mystery to me in the same way geometry was. eg, solar dehydrators. people keep telling me i should give up trying to buy one and just make my own for $2. what they seem to miss is that i could easily spend weeks and weeks lost in thought in building a plan - and have a completely fabulous time doing it - yet that plan, even though it would be gorgeous and beautiful and intricate and absolutely accurate in theory would be completely unbuildable. :=D

                i took a sculpture class once and spent the entire class, in fact, trying to design unbuildable and illogical sculptures for corners. the teacher tried to explain to me if it's unbuildable and physically illogical, it can't be built. never quite sunk in ...

                kos, my daughter keeps writing on the wall with crayons. I'm tired of it. Do something about it NOW, dammit. -MJB

                by cookiebear on Sat Feb 10, 2007 at 10:03:06 AM PST

                [ Parent ]

              •  I'm odd in that I like math (3+ / 0-)
                Recommended by:
                algebrateacher, rserven, drchelo

                but am not a visual person.

                In calculus, the dreaded epsilon delta proof made perfect sense.  But applications! Gevalt!

                What are you reading? on Friday mornings
                What have you got to learn? (or teach) on Saturdays

                by plf515 on Sat Feb 10, 2007 at 10:08:46 AM PST

                [ Parent ]

        •  If it's any consolation (4+ / 0-)

          I actively wished that I could understand it.  Especially when you said this:

          I try to have an open mind about people who have good enough verbal skills to understand and master challenging mathematics being kept out of my subject because they can't master skills they will not need in order to understand that mathematics.

          I was told by my college calculus professor that I ought to major in math; my response to that was "huh?  I'm an English major!"  I always figured that math beyond a certain level would be lost on me, despite being good at the basics... but in a way I wish somebody would have said, nonsense, you can approach these concepts through your verbal skills.  Trust me, they are beautiful!

          (Of course, now I'm in science, but always my eyes glaze over reading statistical analyses...)

          My five-year-old daughter is already incredibly talented in math, at least how I see it.  I don't sit around making her division worksheets, though-- I try to explain the concepts behind how we work with numbers, when she expresses interest or it happens to come up.  (Occasionally she does demand I make her worksheets, since they don't do that in kindergarten...)  Anyway, then we can talk about her real questions, like how old the earth is or how big the universe.  Talking about those things while taking a drive in the dark under the stars really is beautiful.  :)

        •  I gave up... (3+ / 0-)
          Recommended by:
          cookiebear, annetteboardman, rserven

          ...a long time ago.

          Quantitative analysis and rigorous logical reasoning do not come easily for me.  On the other hand, I'm really good at mindplay with text and with social constructions.  I like what I've come to think of social algebra:  Given the patterns of behavior I observe among the people around me, what are the underlying social scripts that we don't explicitly acknowledge?  What are the implications in terms of power and moral behavior?

          I have to work very carefully, very deliberately, with no distractions, in order to develop working understandings of mathematical concepts.  I'm a little lazy, I guess, but I'm also contending with the limited number of hours in a lifetime.

          I have to admit, though, that every time I've been required to wrestle with mathematics I've benefited from it.  I still have a tiny bit of a crush on my college algebra professor, even though he's a colleague now, just because of the glimpses he offered of entire models of thought I had never encountered before.  I had similar glimpses of something amazing when I saw the similarities between sine waves and squirrel tracks during Trig.  Even with statistics, I had the "aha!" moment (granted, not until I got to multiple regression analysis) of realizing the implications of it all-- in that case, realizing that correlation is still not causation, no matter how elegantly you model the correlations.

          I do wish I had grown up with an expectation that everyone would develop some fluency in the language of mathmatical concepts.  I don't know how to offer that to my kids.  I don't expect them to get it from anyone teaching at less than the university level, unless they just happen across the rare individual who "gets it" and loves helping kids get it, too.

  •  I gotta go out for a bit (6+ / 0-)

    but I will DEFINITELY come back to discuss this

    Excellent stuff.

    What are you reading? on Friday mornings
    What have you got to learn? (or teach) on Saturdays

    by plf515 on Sat Feb 10, 2007 at 09:01:44 AM PST

  •  I think the beauty of their subjects... (12+ / 0-)

    is what motivates many teachers to teach.  I find beauty wrapped all up in words and sentences.  I hope to help my students see that beauty.  Social studies teachers see beauty in history and documents and social trends.  Science teachers see beauty in cells and the Periodic Table.

  •  art and math--escher (5+ / 0-)

    As the semi-innumerate daughter of a mathematician, I grew up appreciating something about the beauty of mathematics, but seldom quite grasping it myself.
    The closest I've ever come is in the works of m.c. escher--the math, the wit, and the art all come togetehr so beautifully1

    •  I feel a poem coming on: (5+ / 0-)

      Escherize your mind, as it says over my desk (the art is my homage):

      Art Link
      Perspective Be Damned

      Escher had it right
      when he showed us that
      it was all about perspective
      Up and down are relative notions
      as are darkness and light
      Is this reality simply
      the reflection of another?

      But even Maurits
      would be hard-pressed
      to find validity
      in the fabric
      of truth and untruth
      woven to justify
      this war

      --Robyn Elaine Serven
      --December 7, 2005

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 09:54:53 AM PST

      [ Parent ]

  •  rambling Happy Beth Day (9+ / 0-)

    Forgive the lack of linear thought or any substantive contribution, please; I'm having an open-ended mind-wandering kind of morning.

    This week is the ninth anniversary of my little sister's suicide.  She was Einstein-quality smart; she was almost finished with double majors in biology and physics when she died.  Her main area of interest was biomechanics.  She found transcendent beauty in the mathematical description of living things.

    She was also bipolar and such a beautiful misfit in this world.  I suspect that she ultimately couldn't reconcile a universe of such perfect beauty full of human beings of such finite decency.  I miss her.

    Okay.  So as an educator, I'm back at my/her alma mater working on curriculum design for the university as a whole, as well as plugging along within my own discipline.  I have this mission in mind of throwing open the world of liberal arts and sciences to incoming freshmen in a way that honors centuries of traditional higher learning, the groundedness of practice disciplines, and that elusive transcendent experience of beauty that was my sister's life.

    Modest, huh?  If anybody has brilliant insights along these lines, please do share.  Mostly I read everything about education that shows up in dkos just to borrow ideas and get inspired.

    •  You wrote: (7+ / 0-)

      Modest, huh?

      If we didn't try to climb the big mountains, where would be the challenge?

      For your sister, I'd like to pay special acknowledgment, from someone who was on the point of suicide at least four times in this life.  Please understand that if any of us make that choice, it was because the pain had become too much, whatever the pain was.  I am alive because I keep being able to say today is not the day.

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 10:01:20 AM PST

      [ Parent ]

      •  "the pain had become too much" (4+ / 0-)
        Recommended by:
        Cali Scribe, annetteboardman, cfk, rserven

        Thank you.  My family of origin has a strong shared tendency towards depressive disorders.  My low point was in October of 2001, watching all the nonsense-- the complete loss of all sense-- among the people around me.  Beth was gone, but our brother the sociopathic pedophile was alive and well.  A judge was saying that Andrea Yates (drowned the kids in the bathtub) couldn't possibly be insane because she had such a high IQ.  Everyone, everywhere, seemed to be so focused on revenge and wreaking mayhem on small brown people on the other side of the planet.

        But... my kids were still beautiful, even if the world they came into was chaos.  The full moon and wildflowers were still beautiful.  The moon and the flowers have brought me back every time.

        I had to be angry at Beth for a little while, for not giving me an opportunity to try and talk her out of it.  Really, though, I had to admit to myself that I couldn't make life bearable for her.  I'm such a big advocate of personal self-determination-- how could I begrudge her this decision?  I hate that she felt like that's what she needed to do, but it was her decision to make.

        Now she's part of the beauty.  The loss of her has made me a better person.  I'd give almost anything to have her back, of course, but I've learned to cherish the terrible knowledge the loss of her brought me.  How's that for a liberal education?

    •  "Borrow ideas" (2+ / 0-)
      Recommended by:
      world traveler, rserven

      Any teacher worth his or her salt steals every good idea as it comes by.  Even if we must visit a dump to find spare metal and parts, we must armor ourselves for battle and self-preservation.

      Not that I'm equating dkos with a dump. (snark, dammit)

  •  as a 19-20 something (4+ / 0-)

    i actually dreamed of being a mathematician

    i loved algebra in high school and discovered a kind of meditative ecstasy (for lack of a better term) in it

    i found similar in linguistics, esp. with Native American languages - there were times i would be working on them and feel a kind of fear and awe that i'd discovered the primeval void. seriously!

    but i'm a Taurus. :-D i am forever doomed to the material and mundane. :=D and the dirt is always calling us Taureans.

    Sophie's Choice.

    although i've wondered before if i would have chosen mathematics had my background been stronger.

    kos, my daughter keeps writing on the wall with crayons. I'm tired of it. Do something about it NOW, dammit. -MJB

    by cookiebear on Sat Feb 10, 2007 at 09:45:34 AM PST

    •  mathematics, linguistics, programming... (5+ / 0-)'s all the same really.  For some, throw in music. :-)  And the artists.  Communication.  Hear the hum of the universe.  

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 10:04:09 AM PST

      [ Parent ]

      •  the universe hums for me in botany and childbirth (3+ / 0-)
        Recommended by:
        cfk, algebrateacher, rserven

        Strange, huh?  Something about plant structures as living things just sends me right out of my head.  sigh  Some days I wish I'd stuck with ethnobotany.

        Mostly I teach nursing students how to take care of childbearing women.  It's basically the same thing, in my mind.  It has to do with witnessing the overwhelming beauty of being human in this world.  Being present for the birth of a baby, recognizing the resonance with flower/fruit/seed throughout the living world, and being an element of the human meaning-making in that moment-- what a trip.  My favorite thing in the world.

        I love seeing a student walk out of a birthing room with the look on her face that tells me it's happened.  She saw the two-headed mommy monster and glimpsed the foundation of the universe.

      •  Hmmm... (1+ / 0-)
        Recommended by:

        maybe that's why I was good at music (at least when my fingers didn't lock up)...or did the music skill help my mathematics skills? 'Tis a puzzlement...

  •  My first lesson on Strange Attractors (1+ / 0-)
    Recommended by:

    I needed to understand the fundamental concept, to be able to apply it to placing the archetypal figure at the center of the dynamical field of a work of art, an artist's corpus of work, a genre, a time period, and on up the creative chain (Complexity Theory got in there, too, in discussing the proliferation and gradual restriction of critical perspectives into a dynamic equilibrium, as scholarship on a work, an author, etc., developed over time).

    For my first lesson, my mathematician friend (specialization in Complex Analysis) put up a black-on-white geometric figure on the screen. He told me to look at it.  He said, "Isn't it beautiful."  The rest of the lesson was looking at the figure.  He was right.

    Abigail, I'm sure if there is something out there, looking down on us from somewhere else in the Universe, they're wise enough to stay away from us. --Grissom

    by world traveler on Sat Feb 10, 2007 at 10:04:54 AM PST

  •  Math and beauty (4+ / 0-)
    Recommended by:
    cookiebear, cfk, algebrateacher, rserven


    I sort of followed what you wrote....but I think there are plenty of examples of mathematical beauty, or 'neatness' that require much less of a background.

    The best example I know of is Euclid's proof of the infinitude of the primes.  Only arithmetic needed.

    Some other neat things (adapted from the wonderful book Math Charmers by Alfred Posamentier (and other places)

    1*1 = 1
    11*11 = 121
    111*111 = 12321
    1111*1111 = 1234321
    11111*11111 = 123454321

    Start with any two numbers (aside to the advanced folk - any two REALs). Add them.  Then add those two. the Fibonaccis

    1   1   2   3    5   8   13  21 ......

    where each number after the second is equal to the sum of the preceding two.


    102  -4.3    97.7   93.4    191.1  284.5   .....

    now, divide pairs

    top series:
    1/1  = 1  2/1  = 2   3/2 = 1.5 .....21/18 = 1.61

    bottom series:
    4.3/102 = 0.374
    97.7/-4.3 = -22.72
    but go for a while, and you will ALWAYS get to the same number....but never get all the way there.

    Proof that the square root of 2 is not rational
    Cantor's diagonal proof

    1 = 1^2
    1+2+1 =  2^2
    1+2+3+2+1 = 3^2
    1+2+3+4+3+2+1 = 4^2

    Take any 3 digit number where the units digit is not the same as the hundreds
    e.g 153
    reverse the digits: 351
    subtract the smaller from the larger:351-153 = 198
    reverse the digits:  891
    Add the last two numbers: 198 + 891 = 1098

    start anywhere.  That's where you'll end up

    Palindromic numbers are those that read the same backwards as forwards (e.g 121  or 1881)

    start with any number:   172
    reverse it:   271
    add:  443
    reverse that: 344
    add:   787

    eventually, you will always get to a palindrome number


    There's LOTS of beautiful math.

    What are you reading? on Friday mornings
    What have you got to learn? (or teach) on Saturdays

    by plf515 on Sat Feb 10, 2007 at 10:07:02 AM PST

    •  No doubt! (3+ / 0-)
      Recommended by:
      cookiebear, algebrateacher, plf515

      But my beauty lay in the words and the symbols, rolling onward to Truth.  Algebra is not the only beauty to be found.  Number Theory is equisite in it's own way.  I was taught that by one of the best, Ivan Niven.

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 10:12:35 AM PST

      [ Parent ]

      •  exquisite (1+ / 0-)
        Recommended by:

        Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

        by rserven on Sat Feb 10, 2007 at 10:13:03 AM PST

        [ Parent ]

      •  Sure (3+ / 0-)
        Recommended by:
        cookiebear, algebrateacher, rserven

        but if one wants to show the beauty of math to people who don't know much math.....I think number theory is more accessible.  At least, the PROBLEMS are more accessible.  The proofs can be another matter.

        What are you reading? on Friday mornings
        What have you got to learn? (or teach) on Saturdays

        by plf515 on Sat Feb 10, 2007 at 10:19:07 AM PST

        [ Parent ]

        •  That's true. (1+ / 0-)
          Recommended by:

          But I find beauty in such as this:

          Theorem:  1 ≠ 0 (i.e. the multiplicative identity is not equal to the additive identity).

          Proof:  First, the Ordering Axiom states that there is a nonempty positive set N such that for every integer x, either x ∈ N or x = 0 or - x ∈ N (the Trichotomy Principle).  Thus 0 ∉ N.  If - 1 ∈ N, then for ever x ∈ N, - x = (-1)* x ∈ N, violating the Trichotomy Principle, since N is closed under multiplication (an exercise left to the reader...consider the cases).


          Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

          by rserven on Sat Feb 10, 2007 at 10:34:44 AM PST

          [ Parent ]

          •  Oops: Trichotomy Principle addendum: (0+ / 0-)

            Exactly one of the three possibilities is true.

            Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

            by rserven on Sat Feb 10, 2007 at 10:35:54 AM PST

            [ Parent ]

          •  OK (1+ / 0-)
            Recommended by:

            I'm glad you find that beautiful


            Let's compare, though, to music.  Many find beauty in 20th century classical music, or in Indian sitar music, or in Chinese music, or in any other of the huge variety of music out there.  Fine.  Excellent.


            If one were to try to 'sell' the beauty of 'art' music (as opposed to 'popular' - although they don't really oppose each other) then Mozart would be a better bet for most western ears.

            No one can find beauty in things they can't begin to comprehend.

            What are you reading? on Friday mornings
            What have you got to learn? (or teach) on Saturdays

            by plf515 on Sat Feb 10, 2007 at 11:32:16 AM PST

            [ Parent ]

  •  I really wish I'd taken more mathematics (2+ / 0-)
    Recommended by:
    algebrateacher, rserven

    I've always been fascinated with the subject -- and actually tested out better in mathematics than I did with English, which is what I've always considered my strong suit. (In junior high and high school, I typically was placed in the regular English classes, but in "A-lane" math and sciences, one step below Honors...I wonder if my shitty handwriting had something to do with that.) But in high school, I fell in with the "wrong crowd" -- no, not the dopers or the gangs (though I had friends in both camps), but the "cool girls" who thought that if you were good at math you'd never get a boyfriend. (I don't know how I ended up in that group -- I never had a boyfriend till I got to college anyway.)

    I find myself thinking about, of all things, Norton Juster's The Phantom Tollbooth -- and the fight between the kingdoms of words and numbers. There is a certain simplicity to numbers, a certain symmetry -- I tried closely to follow what you were saying up above, and on some gut level I got it (but don't f-ing ask me to explain it!).

    On the other hand, we've seen just this week how words can be twisted and misused. Just as in the cop shows, we've learned that anything you say CAN be used against you, not just in a court of law, but in the court of public opinion.

    Hmmm, maybe I should head up to the community college and take the placement test, and see what level I'd be able to start 50 too old for Introductory Calculus? ;)

    •  There are people who say... (3+ / 0-)
      Recommended by:
      annetteboardman, cfk, algebrateacher

      ...we do our best mathematics in our twenties and thirties.  It may be true.  I'm of the firm belief, however, that as long as one is learning new things, one isn't succumbing to old age. :-)

      A secret among grad school faculy in mathematics:  we look at the verbal GRE score as being more indicative of success than the quantitative.

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 10:56:09 AM PST

      [ Parent ]

      •  interesting (2+ / 0-)
        Recommended by:
        algebrateacher, rserven

        I did really well on my GRE language (went down 30-40 pts. from the SAT math, up 60 on language -- how do I know that 20 years on?).  But hadn't done math since sophomore year of high school.  I got really confused with bad teacher in soph year, and then got sick and dropped math in junior year.  Nothing more.  I never did calculus.  I would rather have done stats at some point -- still maybe think I should.

      •  well yeah (1+ / 0-)
        Recommended by:

        let this statistician interject

        The GRE quant section (not the Math SUBJECT test) is not designed for math majors.  In fact, the GRE quant section is easier than the SAT math section, because a lot of people never take any math in college, so they are rustier.

        The math subject test is designed for math majors - I don't know if you use that at all.

        Also, IIRC, math majors don't have the highest scores on the quant section - in my year, I think it was engineering students.  

        What are you reading? on Friday mornings
        What have you got to learn? (or teach) on Saturdays

        by plf515 on Sat Feb 10, 2007 at 11:43:26 AM PST

        [ Parent ]

        •  I barely remember my GREs. (0+ / 0-)

          The night before it snowed and since I lived on the top of a hill, I had to go to the bottom and catch a bus downtown (15 miles or so) and sit in a restaurant all night to make sure that I could get to the test in the morning.  I was also dosed on acid.  Oh, well...

          Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

          by rserven on Sat Feb 10, 2007 at 11:48:05 AM PST

          [ Parent ]

          •  I don't remember taking them (1+ / 0-)
            Recommended by:

            I have really lousy episodic memory

            I do remember my scores.....I even remember my PSAT and SAT scores, from 30 years ago.  

            cockamamie brain stores stuff I don't need (the theme song to Gilligan's Island) and forgets stuff I do need (where did I put that damn coffee cup????  How do I do what I need to do in R?)

            What are you reading? on Friday mornings
            What have you got to learn? (or teach) on Saturdays

            by plf515 on Sat Feb 10, 2007 at 12:06:57 PM PST

            [ Parent ]

  •  Algebra and Geometry are languages, but they are (0+ / 0-)

    not taught that way.

    In the past seven years, I have taught:
    Pre-algebra to eighth graders who did not succede learning pre-algebra as seventh graders,
    Pre-algebra to seventh graders who had trouble learning arithmetic in sixth grade,
    Algebra to high school students who have "failed" Algebra at least once,
    Geometry to high school students who have "failed" Geometry (probably Algebra, too!) at least once,
    Algebra to eighth graders whose English skills are not yet "proficient"...

    and other teachers are given the "Honors" students,

    but I am sought out to mentor the school's eighth grade "Pentathlon" team, which was the winner last year in my county.

    I have been told, by students, parents and colleagues, that I teach unlike anyone else.  Both of my daughters were lucky enough to have a recent county "Teacher of the Year" when they were in junior high school; both have said, "He teaches like you do, Dad."  I do not use the textbook because it was written by a cabal of illiterates which, of course, did not stop it from being adopted by the State of California and my school district.  I am so good at what I do that they keep sending me shit so that I can shine it into gold.

    This year, for the first time, I have seriously considered using my seniority to grab an open History position because I am really a History teacher.

    Now that I have stamped my foot and shook my fist, I should go have another cup of coffee.  I have lesson plans to work on.  

    •  I have stacks of grading... (3+ / 0-)

      ...and not a single elve has showed up to do it while I wasn't looking.

      Teacher's Lounge opens each Saturday, sometime between 10am and 12 noon EST

      by rserven on Sat Feb 10, 2007 at 11:30:50 AM PST

      [ Parent ]

    •  Tell me more (2+ / 0-)
      Recommended by:
      algebrateacher, rserven
      I agree, they are languages.

      I agree, the Official Curriculum that I was taught algebra with (Saxon) made it feel unnatural and convoluted. Now I can feel myself using these concepts as easy as breathing.

      I am fairly pleased with what I've seen my daughter (now in first grade) working on - they're doing graphs and "number families" showing all the addition and subtraction problems you can make with the numbers 3, 5, and 8 for example. Problems are set up with the answer blank in all three positions and she understands that if she knows that 5+3=8 and that 4=3+1 that she can derive that 5+4=9.

      As I watch her do her homework, I feel the truth of what you say, describing it as a language, even more.

      So tell me, how does your approach differ from that of the previous teachers these students had? Can you teach it to other teachers?

      Fry, don't be a hero! It's not covered by our health plan!

      by elfling on Sat Feb 10, 2007 at 07:17:50 PM PST

      [ Parent ]

      •  Where were you when Robyn needed an elf? (1+ / 0-)
        Recommended by:

        There is no quick way that I can explain.  I am not running from that responsibility, however, it's just that I don't have all the words yet.  Yet.  I continue to dream of how to write the better book.  I'll do what I can to have it written by the time your daughter is in fifth grade or so. ;)

        I will say that Algebra and Geometry have been, and still are, taught as if they were just a long series of equations and facts that can be memorized.  Alternatively, Algebra and Geometry are taught as if they were some form of "revealed truth;" some people get it, some people don't and I hope you're one of the lucky ones.  Such an approach can be compared to teaching reading without comprehensive skills or teaching history as facts and dates without narative, trends, conflicting theories and viewpoints and more.  There are, in all texts I have seen, attempts at "real life" applications that are either so simplistic or so bizarre that they are useless.

        It is no wonder that the typical adult will say, "I took Algebra and Geometry.  I don't remember any of it and I don't use it in what I do."

        To understand Algebra, and the Algebraic Geometry now taught (includes Trigonometry in the second semester) at least in California, students need to understand the following at the beginning:

        x does not mean "times,"

        fractions show division (as represented by the classic two dots and a line division symbol which is not on my keyboard) and comparisons or ratios, not "parts of a whole,"

        numbers are movements on a number line where zero is the center and origin, not "how many apples,"

        if two numbers have the same sign (they are on the same side of the number line), they are alike and may be added together and keep the sign,

        if two numbers have different signs (they are on opposite sides of the number line), they work against each other through subtraction with the apparently larger number winning and thus giving its sign,

        human beings add, subtract, multiply and divide two numbers at a time, therefore grouping is essential,

        and there is the rule of one negative: if one, and only one, number in multiplication or division is negative (see it with you eyes, students), then the answer is negative.  Otherwise it's positive.  Period.

        Sorry for the lesson.  What I'm trying to show is what I have to teach first, before anything else, in order to move students from the applications arithmetic they've been taught to symbolic algebraic language.  There tends to be a communications breakdown and resistance at first because I tell them that what they've been taught in the past has other meaning.

        Could I teach it to other teachers?  Let me give you an example: add 1/2 to 4/5.  Do this by multiplying the first number, 1, by the last number, 5.  Then multiply the second number, 2, by the third number, 4.  You will have multiplied diagonally.  Now add those answers, 5 and 8, for 13.  That goes on top.  Now multiply the bottom two numbers, 2 and 5.  That answer goes on the bottom to create 13 line 10.  How many tens in thirteen?  One with three remaining that are still divided by ten.  This three-multiplications approach works for addition and subtraction of both positive and negative numbers.  What you are really doing is adding or subtracting the answers to two division problems: One divided by two plus four divided by five (try it on a calculator).  Algebraically, you are solving by using the inverse operation to division: multiplication.

        I showed this when challenged (I'd been showing others at my table) at a training conference.  My colleagues, the teachers in the room, loved it and wanted to know more.  The district-level personnel and the Professor of Mathematics Education were, however, up in arms and scandalized.  One even declared, "You can't teach that to kids!"

        •  My mom was a math teacher (2+ / 0-)
          Recommended by:
          algebrateacher, rserven
          And her last job was working at a continuation high school, where she had basically every student at a different level, and almost all of them with some pathological math issue that had been ignored or brushed over. She was (and is) very good at the one-on-one tutoring of kids and them bringing them forward to wherever she could get them in the time that she had.

          The "x doesn't mean times" issue is one of those utterly regrettable and unnecessary problems that we shouldn't saddle our kids with. We can just use a * like we do for computers and solve a lot of problems down the line.

          I had the same problem when I started calculus, because of a strange hiccup in my education, I had been taught differentiation using ' and hadn't been introduced to the d/dx notation. So I wasted at least 5 weeks grappling with "why don't the d's cancel out again?" which put me way behind in my science classes. Grrr.

          Fry, don't be a hero! It's not covered by our health plan!

          by elfling on Sun Feb 11, 2007 at 08:08:36 AM PST

          [ Parent ]

  •  Warning! Danger! Aquinas ahead!! (2+ / 0-)
    Recommended by:
    algebrateacher, rserven

    According to The First TA, Beauty consists of (ST-I, q39, A8):

    Integritas: aka unity, wholeness,
    Consonantia: aka harmony, proportion,
    Claritas: aka brightness, vividness, bright-colored, radiant.

    A mathematical proof certainly has the first two characteristics. Does it have Claritas, which TA saw as the critical component of the three?

    Philosophers have disagreements on this, but since I have a deep understanding of TA on Claritas, you can ignore the rest of them and trust me ;-)

    Radiance, luminousness, from a source of light.  For TA, all being exists because of the grace and love that the god radiates into and through it.  A work of art radiates that luminousness because, in its Integritas and Consonantia, it opens up a portal (my word) that allows that luminousness through the work and into our mind (soul), where it strikes harmony with the beauty in us.

    Think of the stained glass windows of a Gothic cathedral (TA did): they open up a portal for the light and light-warmth of the sun to fill the space of the cold building with the vivifying warmth of sun-god.

    So also, a mathematical proof radiates a luminous consonance that resonates with the sense of harmony and proportion that is embedded in our brain-mind, which sees the wholeness and harmony out there as resonant with and illuminating our internal beauty. And so we are illuminated.

    Mathematics is Beauty is Art. q.e.d.

    Abigail, I'm sure if there is something out there, looking down on us from somewhere else in the Universe, they're wise enough to stay away from us. --Grissom

    by world traveler on Sat Feb 10, 2007 at 11:46:08 AM PST

  •  I thought this would be about Keats... : ) (1+ / 0-)
    Recommended by:

    Oh, well.  I enjoyed taking tests back when, but probably not now.  The college kept trying to get me to take math, but I just couldn't imagine it.

    I know they do better teaching math, now, than when I was a child.  On the other hand, blind panic may come from my genes...not a snark.

    I love Escher. When I look at one of his pictures, the creative side of my brain turns on.  

    I am the child who argued that dolls and drums could be added and called toys. Now, I would be fine doing sets...sigh.  

    I was only good at the logic puzzles in geometry and I love the fold up boxes laid out flat where you are asked what they will look like when put together in 3-D.  

    I am very right-brained so I am much faster at adding 2+?=6 than what does 2+4= ?  

    Ah, me...I see beauty in biology and nature and music, but words are my first love.

    Great poems, Robyn!!!

    "Other cultures are not failed attempts at being you. They are unique manifestations of the human spirit." Wade Davis

    by cfk on Sat Feb 10, 2007 at 06:12:01 PM PST

  •  Emmy Noether! (1+ / 0-)
    Recommended by:

    Brief bio, but I had to just note that, since I did take a math class (real analysis, not abstract algebra) at Bryn Mawr when I was a student at Haverford many years ago.

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