Sunday, April 7, 2013

A Mathematicians Lament



Lockhart, Paul. "A Mathematician’s Lament." Http://www.maa.org/devlin/LockhartsLament.pdf. Mathematical Association of America, n.d. Web. 2 Apr. 2013.

I am reading this article because I remember that it begins with the author's nightmare scenario wherein the math curriculum we use is applied to music and then visual arts.  I use a musical instrument analogy in my understanding  of how mathematics can be taught as well.  I am also reading it to buttress my ideas about the problems with teaching procedures rather than investigating and exploring.  Rather than having fun with shapes numbers and problems. 

General Reflection
I found this article difficult to read.   It was inspirational at times especially when h described mathematics as art and sharing and inspiring with anecdotes.  He was logical and compelling in imagined interviews between Simplico and Salvate.  Simplico would play the role of asking the questions on the minds of the incredulous reader and Salvate would correct his misperceptions.  I looked the names up and didn't quite have time to process the references but I am sure they are meaningful. 

Most of his time, though is not spent inspiring, it is spent railing against mathematics curriculum as it now stands and many of his positions resonate with me.  He laments the lack of real problems in math curriculum, the regurgitation of facts, the substitution of exercises instead of real honest to goodness creative problems, the over codified approach to teaching mathematics concepts, the lack of student discovery. 

Even though much of what he says resonates with me, honestly, I'm not sure what to make of it all.  I like the application of mathematics towards useful purposes and on this point we disagree.  But I find it difficult to think and read Lockhart at the same time because he is so righteous, so confident of his views that it is difficult to think critically about what he is saying lest you become part of the very thing he is raging about.  He also strikes me as close to brilliant and brilliance combined with righteous indigination is difficult to parse critically.

Another takeaway that I have from this article is that the way I wish to teach mathematics is not the way I was taught mathematics and that becoming a good problem based facilitator will require honesty and acceptance that it will take time and there are discoveries that I need to makes as well.

Musical Analogy
The analogy of a path through music that does not allow students to actually play until years after they have studied the minutia of music is an interesting one. I think think that a music instrument is an especially good metaphor for how I currently think about mathematics instruction, but I think it differs from Lockhart's view.  I think of both the play and the practice of the instrument as needing to be stressed.. Having lived for many years with a professional jazz and classical musician I can tell you that much of her time is spent practicing.  Same this for my uncle who is a professional musician.  Every morning us spent doing scales.  They are both amazing musicians and both practice all the time.  With mathematics I think about practice as something that reduces the cognitive load of more interesting problems and let's one engage in more interesting problems.  Of course it is pointless to practice scales  all day if you don't get to make music and make people love you and all the great things that go along with being a musician.   The instrument analogy also makes me think of alternately acceptable paths to understanding.  For instance I know a classically trained musician who has played in major orchestras and is considered an expert in her field, she realized at twenty something years of age that she was completely incapable of improvising.  There were kids who had only been playing for five years who could create music on the spot and express themselves in ways she could only dream of and some of them could not even read music.  This did not negate the path that she had taken through music but i made  her aware of the multiple paths and approaches. She has spent many years since mastering the art of improvisation and today is able to bring both of the approaches to bear on her art form.  This is sounding tangential o mathematics but I this about the debate about approaches to mathematic and wonder if again we are not talking about either or propositions,but instead both propositions.


Random quotes and responses

"So 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 with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard work." p4

Math for it's own sake is exciting but I wonder if it's appeal is a wide spread as Lockhart claims.  I project based learning I find that student enjoy the practical applications of mathematics. 

"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)." p5

This reminds me of the idea that the problem is not about the answers it is about the questions and the evidence of thought.  You don't read just the last page of the book you want to experience the whole thing and only then does the last page have meaning for you.

"It would be bad enough if the culture were merely ignorant of mathematics, but what is far worse is that people actually think they do know what math is about— and are apparently under the gross misconception that mathematics is somehow useful to society!"
Lockhart has a lot to say about the idea that mathematics should be studied because of its usefulness, that it is useful, is not nearly the most significant and beautiful about his conception of mathematics.  Although, he contradicts himself when talking about the beauty of story of the circumference of the circles.


"SIMPLICIO: So you blame the math teachers?
SALVIATI: No, I blame the culture that produces them. The poor devils are
trying their best, and are only doing what they’ve been trained to do.
I’m sure most of them love their students and hate what they are being
forced to put them through. They know in their hearts that it is
meaningless and degrading. They can sense that they have been made
cogs in a great soul-crushing machine, but they lack the perspective
needed to understand it, or to fight against it." p17

These kind of statements make Lockhart difficult to read.  He is not the only beacon of light in an otherwise dark world.  I am not sure how to summarize what I think he is doing in this statement.  He is suggesting that we are crushing the souls of the students we love - with out question.  But he is offering us an out if we admit our lack of perspective and admit that we are powerless cogs in a greater machine. He also claims to know my heart.  I dunno, I think if he was talking directly to  me at this point I might yawn.


"SIMPLICIO: So we’re supposed to just set off on some free-form mathematical
excursion, and the students will learn whatever they happen to learn?
SALVIATI: Precisely. Problems will lead to other problems, technique will be
developed as it becomes necessary, and new topics will arise naturally.
And if some issue never happens to come up in thirteen years of
schooling, how interesting or important could it be?
SIMPLICIO: You’ve gone completely mad.
SALVIATI: Perhaps I have. 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." p23

I think this section is pretty powerful.  It seems to have vision for the future and for the present.  Maybe we can work towards a point where mathematics can be a freeform excursion.  But if that is the unreachable we can still 'teach' mathematics so that it is a process of discovery.


1 comment:

Stacey said...

Bryan~
I liked reading about the music analogy and how you are making it your own. I wonder if you would find the research on deliberate practice relevant to what you are wondering - especially the work by Erikkson and Cushman. (You can check out Mark Anderson's DP or book for some great sources since he focused on this in his AR, which was focused on band!)

I have been wanting to read this book for some time, and now I really want to! Thanks for such a thoughtful analysis and reflection on this work. I'm curious, what do you think this could look like in your own classroom? Do any next steps come to mind?

If you haven't already, I think Bryan Meyer at HTHNC would be a fantastic living resource for you to reach out to! He could point you toward some interesting stuff!

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