Jul 02 2009

Review: A Mathematician’s Lament

by chrisb

In an effort to Continuously Increase my Effectiveness, I have spent quite a bit of time this summer reading various books and articles about teaching. I recognize that I am by no means an expert teacher; part of the reason I wanted to move to a charter school was that I knew that they would help me develop in my profession in a meaningful way. I can’t help close the achievement gap if I just stay as effective as I am now. As a teacher, there’s always more you can do, and so here I am, spending my summer working. (And I love it, by the way!)

I was sent A Mathematician’s Lament through a TFA-Bay Area listserv that I am a member of. Since I’m only going to be teaching math to second graders next year, I figured it was highly relevant to me. What I read humbled me.

The author, Paul Lockhart, starts with a hypothetical situation about a musician trapped in a world without music. Children are being taught to write music in sheet form, but they are never allowed to hear music or taught to play it. The beauty and art has been sucked out of music. This, Lockhart claims, is exactly what has happened to math in our public schools.

Sadly, our present system of mathematics education is precisely this kind of nightmare. 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.

Well, I certainly agree that most math classes are boring. In fact, I’m pretty sure I slept through much of my high school math education. I remember often feeling like my teachers were kidding me when they tried to convince me that “I would need to know this later.” Yeah, right. I’m pretty sure that I use the quadratic equation every single day- thank goodness I spent all that time learning it!

To be fair, though, I can’t remember much of my lower elementary math education. I remember doing multiplication tables in third grade and hating every second of it. Is it possible I enjoyed first and second grade math?

Lockhart continues by explaining to us that the reason nobody sees math as an art is because nobody understands what mathematicians do. He claims, “[M]athematicians sit around making patterns of ideas. [...] If there is anything like a unifying aesthetic principle in mathematics, it is this: simple is beautiful. Mathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.”

O…k. So, I’m supposed to teach my students about imaginary things? Right. It’s hard enough to get some of them to stop daydreaming about fairies and princesses as it is. But, maybe I’m missing some key piece of information about math. It’s not, after all, my favorite subject. I’ll give him the benefit of the doubt for now and accept that math is about imaginary things.

Lockhart takes us through a discussion of how to derive the formula for discovering the area of a triangle: A=1/2BH. (Yep, I remember that from my geometry days.) He has chopped a triangle inscribed in a rectangle in half, thereby discovering that the triangle fills exactly half of the rectangle. He says, ” But it’s not the factthat triangles take up half their box that matters. 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.”

Ok, fair enough, I see his point. He’s arguing for more of a self-discovery process of math- of appreciating its beauty and using your own natural curiosity to learn about the mathematical world. It’s similar to making a painting- children should be given artistic freedom to create something beautiful on their own. I can appreciate this point of view.

But, here’s my problem with it. When I place myself back in my younger self’s shoes, sitting in a desk in a classroom in front of a math teacher, I remember that I never once cared what the answer to a math problem was. I would do the work and find the answer, but I never thought that it was fun or beautiful. Maybe this is a result of the fact that I was simply handed all the formulas I ever needed and told to use them. There was no self-discovery process for me. Is it possible that if I had had a teacher who showed me a triangle inscribed in a rectangle and asked me, “How much of the box does the triangle take up?” that I would have found that interesting? I’d have to say probably not. I (and I’d be willing to bet, most of my classmates) would have responded, “Who cares?” I simply fail to see the beauty in this problem.

Lockhart uses the rest of his article to give a scathing report on how horrible everyone involved with mathematics education is, from the top of the government right down to the classroom teacher. Nobody gets math, he claims, so of course we are failing to teach it to our kids. The very establishment designed to impart math knowledge has, in essence, killed it. “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.”

Now there is a statement I can get on board with! I may not have ever seen the fun or beauty in math, but I have always loved reading. I think this statement applies to all subject areas- we have become so focused on the results of some meaningless test that we just force our students to test prep all day long. In the end, what have we taught them? How to fill in bubbles? Awesome. I’m sure that will help them in life.

Anyway, Lockhart does go on to suggest improvements to math instruction.

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.

Certainly, that’s a great ideal to strive towards. But I think it sort of makes the assumption that, as the math teacher, you see math as beautiful and artistic. Since the education establishment is so horrible, how many teachers out there really see it this way? Everyone in my life who actually does see math this way absolutely did not go into teaching. They went into engineering. Who is left to do the teaching? People who were brought up by and buy into the very system that Lockhart condemns.

So, Mr. Lockhart, I will do my best to see math as an art from now on, to push my students to reach their own mathematical conclusions, and to discover the beauty of it on their own. But I don’t think that any of us should be surprised if I’m not successful in every single unit. Tell me, where is the beauty in subtraction with borrowing? I can see the beauty in 3-D shapes or in multiplication. But subtraction? It’s pretty ugly, if you ask me.

It would be awesome if I could get my students to see math in this way. How much fun would they have, and how much more would they actually learn? I am sure the possibilities are limitless. But if there’s one thing I’ve learned during the past two years, it’s that you have to believe what you are teaching. If I lacked the artistic mathematical instruction from my education, how am I going to bring that to my students?

Frankly, articles like these make me feel hopeless. If seeing math as art is truly the way to go (though I’m not convinced it is) then how are we supposed to get there? The problem is so systemic that unless all those mathematicians who do see math that way come out of their high-paying jobs to teach, the problem will persist.

There must be another way.

 

Categories: Teach For America, Third Year · 14 Comments

14 Responses

1. Gene Shiau

   How can we see math as art? How can we stimulate students to care about the process they use to solve a math problem and the answer to that problem? Fortunately, there is an answer. It’s not a cure-all, but it is a significant step toward mathematical curriculum reform. The answer I am talking about is a three-week summer institute for MS/HS math teachers call Park City Math Institute (http://pcmi.ias.edu/).

   Let me describe a typical math problem coming out of PCMI (some teachers and school districts have started adopting curriculum models that incorporate math problems like this): Suppose you have to assemble a train of length n, and you can combine train carts from length 1 to n to make up the required length. How many ways can you assemble this train?

   A simple problem like this can easily take a whole week to teach. “Isn’t this very inefficient?” Well, not if you consider all the math and learning skills that goes into solving this problem: tactile learning, enumeration, tree diagram, number sequence, algebraic proof… The best part is, there are multiple levels and points of entry, so students with different ability levels can all gain accomplishment by different means.

   I only wish every teacher could go to PCMI at least once… and it’s much more relaxing than TFA Institute!

   on July 5, 2009 at 11:07 am · Reply
2. Jerry Bennett

   I think that to play music one would need to understand which notes the various signs stand for. Once that is understood, the student would need to recognize which key corresponds to which note or sign on the music sheet. I would think that these first two steps would be done using rote learning methods. After that knowledge is ingrained into the student the art and the beauty could come. The trick for the teacher would be to provide the impetus for the student to memorize the basics. It seems that you have always been good at challenging your students and making even their testing exciting for them.

   In the last year I have just started to understand some of the formulae that seemed totally illogical to me almost a half century ago. In fact, the area of a triangle just became comprehensible to me a few minutes ago. You can do it.

   on July 5, 2009 at 1:03 pm · Reply
3. Connie

   Try reading “Young Children Continue to Reinvent Arithmetic” by Constance Kamii, a professor at the University of Alabama. Her theories are based on Piaget. Math lessons include both talking about problems and playing math games—not endless worksheets. Second graders doing math this way can learn to calculate more quickly than adults—even visiting college math majors.

   The Kamii books are available through Prospector at the public library.

   on July 10, 2009 at 9:11 pm · Reply
4. Daniel

   “Is it possible I enjoyed first and second grade math?”

   I would say yes. If you give context to the math or include it in some kind of game, crosswords for example:

   Is it possible I enjoyed first and second grade math?

   http://neoparaiso.com/imprimir/crucigramas-de-tablas.html

   on August 15, 2009 at 10:21 pm · Reply
5. Stacy

   At the second grade level you need to balance trying to teach conceptually with teaching process and memorization. While to really understand math, kiddos need to eventually connect the dots and understand why it makes sense conceptually, they also need to have automicity with the operations.

   Teach both. Don’t neglect one for the other.

   on August 21, 2009 at 5:17 am · Reply
6. Stacy

   I wanted to add, memorization and process and conceptual understand are symbiotic. One reinforces the other.

   on August 21, 2009 at 5:19 am · Reply
7. Eve

   Subtraction with borrowing? (I realize I’m a bit late in replying, but here it is anyway.)

   You have containers and each container can only hold 10 items. Say you have 17 items, so you need to fill one container (you’ll have to make it a rule to full one first, so they don’t divide them 8 and 9, as opposed to 10 and 7,) and you want to take away 9 objects.

   Take away everything you can from the smaller one, since our goal is to use as few containers as possible. But… you can only take 7, so you need to dip into the full one and pull out two more to give you 9–leaving 8. There you have borrowing, and it’s exactly what you’re doing with the numbers. Alternatively, to make it even more like the typically taught process, literally combine the two groups, since they know that 9 is bigger than 7. Be creative.

   This is a very simple example, but hopefully it may show you that, even with basic subtraction, there’s more to it than “cross out the tens place, reduce by one, stick a one in front of the ones place.” (I’m using two digit number for simplicity.)

   - Somebody who loves the process (usually I only work with students algebra and higher as a tutor.)

   on December 5, 2009 at 10:45 pm · Reply
8. Mike Hardy

   Here’s the beauty in subtraction:

   (1) Euclid’s algorithm for finding the greatest common divisor (GCD) of two integers is based on the idea that if you subtract one from the other, you don’t change the GCD. So 1989 – 867 = 1122, so GCD(1989,867) = GCD(1122,867). Repeat as needed until you get it down to GCD(51,51) = 51. Thus to reduce the fraction 1989/867, divide both by 51, getting 39/17.

   (2) Modular arithmetic is based on subtraction. Why is it that a number is divisible by 3 if and only if the sum of its digits is divisible by 3? Modular arithmetic answers that right away.

   Don’t mistake the _algorithm_ for subtraction (involving borrowing, etc.), which is a _process_, for subtraction itself, which is not a process.

   Elementary school children can understand what prime numbers are. How many prime numbers exist? Do they go on forever, or not? The patternlessness of primes seems to put the answer out of reach of mortals. But Euclid succeeded in proving that there are infinitely many. One seeking beauty in arithmetic shouldn’t miss that one.

   Don’t seek to be successful in every unit. Don’t teach “units”. Don’t require those not interested in math to work on it. At most, tease them. I suppose standardized tests imposed by Caesar might get in the way of that, but that’s the ideal.

   (What I remember of my first- and second-grade math instruction is that when I mentioned negative numbers, my second-grade teacher said “THERE’S NO SUCH THING AS NEGATIVE NUMBERS IN THE SECOND GRADE!!!”)

   on December 27, 2009 at 5:40 pm · Reply
9. Mike Hardy

   Here’s the beauty in subtraction:

   (1) Euclid.s algorithm for finding the greatest common divisor (GCD) of
   two integers is based on the idea that if you subtract one from the
   other, you don.t change the GCD. So 1989 – 867 = 1122, so GCD(1989,867)
   = GCD(1122,867). Repeat as needed until you get it down to GCD(51,51) =
   51. Thus to reduce the fraction 1989/867, divide both by 51, getting
   39/17.

   (2) Modular arithmetic is based on subtraction. Why is it that a number
   is divisible by 3 if and only if the sum of its digits is divisible by
   3? Modular arithmetic answers that right away.

   Don.t mistake the _algorithm_ for subtraction (involving borrowing,
   etc.), which is a _process_, for subtraction itself, which is not a
   process.

   Elementary school children can understand what prime numbers are. How
   many prime numbers exist? Do they go on forever, or not? The
   patternlessness of primes seems to put the answer out of reach of
   mortals. But Euclid succeeded in proving that there are infinitely many.
   One seeking beauty in arithmetic shouldn.t miss that one.

   Don.t seek to be successful in every unit. Don.t teach .units.. Don.t
   require those not interested in math to work on it. At most, tease them.
   I suppose standardized tests imposed by Caesar might get in the way of
   that, but that.s the ideal.

   (What I remember of my first- and second-grade math instruction is that
   when I mentioned negative numbers, my second-grade teacher said .THERE.S
   NO SUCH THING AS NEGATIVE NUMBERS IN THE SECOND GRADE!!!.)

   on January 3, 2010 at 11:16 pm · Reply
10. Jer

    I just finished reading “A Mathematician’s Lament”. I am a math teacher, and at first Lockhart’s ideas really appealed to me. But when he started to elaborate on the “ideal” classroom, I just moved farther away. Trying to ask kids to create their own math in these ways may work for very small classes, but the reality I face is that there are many kids who aren’t even interested when we say, “Let’s play a game.” Some games are fun to some kids, to other kids they are boring. If we let kids do what they want, I think the default for many kids is nothing.

    Most of all, I appreciate his statement “Algebra is not about daily life, it’s about numbers and symmetry – and this is a pursuit in and of itself.” I think we should be honest with our students when they ask, “When will we use this in real life?” and say, “You probably won’t, but I like it anyway.” Face it, most of the “real world problems” you get in an algebra text are pretty contrived.

    On the same note, don’t be afraid to get sidetracked by an inquisitive student. That’s where we learn the most. My most exciting times of teaching actually come during one-on-one tutoring when the kid just asks what comes to mind. I’ve even had classes where this is the overall attitude.

    The second problem I see is one of inefficiency. Yes, there is a benefit for kids to see the value of what they are doing before we give it to them, and I try to do that as much as possible. But this would be like asking kids to build a house and deny them the tools to do it, asking them to go and develop the necessary tools on their own. It’s like ignoring the benefit of centuries of development. Ultimately, these tools are useless if kids don’t understand what they are for, but I don’t need to spend a year hitting nails with my shoe to understand the value of a hammer.

    And ultimately, if as a teacher I choose to throw out the curriculum, I shouldn’t be surprised when my school district throws me out too.

    on March 2, 2011 at 7:44 pm · Reply
    • Ms. Math

      Serious math educators I work with would never suggest having kids have tons of freedom or to have them ignore that others have already developed powerful tools over centuries. In fact the carefully plan their lessons with specific meanings they hope to promote. Sometimes these meanings are discussed as a class or with peers. Sometimes these meanings come out as kids answer carefully crafted problems. Sometimes the meanings are clearly and repeatedly spoken in lectures.

      Also, although it is obvious people get by without understanding Algebra, I would say that they can’t be a scientist, an engineer, a doctor, an analyst, a statistician, etc without understanding high school math. I’m taking a Geology course, a Engineering Course and a Social Statistics course and in every single one of these courses I use substantial amounts of high school mathematics. In the Geology course, my understandings of three dimensional calculus are pushed to the limits as I try to understand gravity. It’s not that it is essential to understand science or statistics or business, but the kids might want to be prepared for those careers.

      on September 26, 2012 at 3:13 pm · Reply
11. lily

    I think there might be a clue in this sentence:
    ” So, I’m supposed to teach my students about imaginary things? Right. It’s hard enough to get some of them to stop daydreaming about fairies and princesses as it is.”
    Getting them to imagine one thing over another doesn’t seem that huge of a leap, although i don;t really know.

    on February 10, 2012 at 8:31 pm · Reply
12. Ms. Math

    I can understand why you can’t see math as beautiful. You probably have been taught very little math of the kind Lockhart describes.

    Here is a question that might be beautiful.
    Can you find any patterns in a 5 by 5 multiplication table?

    Subtraction might not seem very beautiful, but it can be meaningful. Here is an article about 4th graders who did some amazing thinking about the meaning of subtraction (a difference). Their thinking is beautiful, because they are truly trying to figure something out and solve a problem. Beautiful math isn’t likely to happen if kids are left to invent things on their own, but can happen if they are encouraged to answer interesting problems.
    http://www.patthompson.net/Publications.html
    Go to this article:
    Thompson, P. W. (1993). Quantitative reasoning, complexity, and additive structures. Educational Studies in Mathematics, 25(3), 165-208.

    on September 26, 2012 at 3:05 pm · Reply
13. Ms. Math

    Here is another book that might help you better imagine how Lockhart’s dream would work in elementary school.
    http://www.amazon.com/Out-Labyrinth-Setting-Mathematics-Free/dp/0195147448/sr=8-1/qid=1167859040/ref=sr_1_1/002-8958891-7740062?ie=UTF8&s=books
    It is easier to read than the article I posted above. It describes asking kids questions like “how many numbers are there between 0 and 1″ and seeing what they come up with. You should have structured, well-planned instruction in your class, but the book could give you ideas for questions to ask that kids just might find interesting. I swear that such questions exist!

    on September 26, 2012 at 3:44 pm · Reply