And now, finally, I give you my own response to Paul Lockhart’s A Mathematician’s Lament. Sorry it’s a bit long.
I enjoy math very much, which makes me an exception among math teachers in general. Many math teachers begrudgingly teach their subject. So many statements in Lockhart’s essay are true. I echo his sentiment as he complains about dispassionate math teachers, saying, “But shouldn’t they [math teachers] at least understand what mathematics is, be good at it, and enjoy doing it?” He encourages math teachers to be mathematicians, just like art or music teachers. And he encourages math teachers to allow their students to be mathematicians too. Students, he says, should have opportunities to play, which is what mathematicians actually do.
Admittedly, I don’t feel like I’m teaching students this “real” kind of mathematics most of the time (the creative, artistic, and fundamental aspects of what mathematicians do). I do my best, but I often do little more than lecture and attempt to keep kids awake with goofy antics. So I understand Lockhart’s point of view on math education. It needs more “art.”
That being said, I feel a bit torn between battling camps when it comes to math education. There are two (often contradictory) goals in the mind of the math educator:
(1) The math educator wants students to like math, think creatively/rationally/logically, and to understand the context in which mathematical ideas are created. This is Lockhart’s position.
(2) Second, the math educator wants students to become good at math and qualified for higher-level math, science, and engineering—ultimately preparing them for college and careers. The math educator recognizes that students going on to college and careers will be expected to have specific mathematical knowledge.
So I think that we, as educators and as a society, must decide if we want to “get students to LIKE math and enjoy it as a creative act” (the first goal) or “make students good at mathematical procedures” (the second goal). That’s the hang up. And the debate is not as easy to resolve as Lockhart would like to make us think.
Points on which I disagree with Lockhart
In my first paragraph I agreed with Lockhart. But now let me take the opposite position (number 2 above).
You see, math is plagued by something that art rarely has to deal with: it is very, very useful in the real world. Some mathematician has described mathematics in terms of this analogy: Mathematics is like a large store, in which there are many shelves housing mathematical tools. The scientists and engineers come into this store and take useful tools off the shelves. Mathematicians, in most cases, did not create these mathematical tools for scientific use. They created them because they were interesting in their own right and beautiful. In fact, most mathematicians would agree with Poincare, “The mathematician does not study pure mathematics because it is useful; he studies it because he delights in it and he delights in it because it is beautiful.” That being said, math educators still have to equip students with those “mathematical tools” the sciences find so useful. Math just happens to be both beautiful and useful.
Let me give an example:
If you’re a student who has been in my Calculus class, you can DO all sorts of things after a year of being in my class. You understand how to find the area between curves, minimize cost functions, maximize the volume of a container. You know how to apply the quotient rule and the chain rule, and you can do u-substitution with your eyes-closed. And you’re prepared for college courses in math and science. Now imagine you go on to take a Calculus-based Physics course in college (as any math or science major will undoubtedly do). Without having had a formal treatment of Calculus in high school, the task of a college professor teaching Calculus-based Physics would be impossible. The college Physics professor will simply expect that you can do u-substitution with your eyes-closed. In the Physics-class scenario I just laid out, formalism in your high-school math education was a good and necessary approach.
In that same scenario, how would one of Lockhart’s students perform? How does his version of education prepare students who will be math majors in college? If I structure my class like an art class, having my kids work on self-directed projects, standing at easels, playing with mathematics—they’ll gain an understanding of what mathematicians do and they may even learn to love math (good things!). But when they get to that college Physics class and the professor simply expects them to be able to integrate a function, will they be prepared? What if they didn’t discover how to integrate while dabbling at their easel? So in this case, Lockhart’s “artistic” pedagogy fails.
I had a professor who was well-liked. I still see him sometimes and I still think he’s wonderful. His style of teaching was exactly as Lockhart prescribes. His classes were fun and interesting. But he compromised the material in order to do that. I took a course in Modern Geometry and I can’t tell you anything about projective geometry (a central topic) or a host of other important ideas in modern geometry. We did lots of unrelated puzzles and problems, barely ever cracking the book. As I now consider graduate work, I’m a little sad I wasn’t better prepared for graduate-level mathematics.
Can you now see why goal (2) is just as important as goal (1)? For college and career-bound students, I have an obligation to teach certain mathematical skills. For them, my curriculum needs to be systematic, formal, structured, methodical, and well-planned.
Attempting to balance
Clearly a balance has to be struck. If I teach a structured curriculum, my students will be well prepared for their academic future. They might not see the beauty of math; and many won’t get the joy of discovery which is so important to the mathematical experience. However, if I teach only as Lockhart recommends, I risk students not getting the skills they will need.
It’s not an easy battle. Formalism in education is valuable in meeting objectives and preparing students for times when they will actually need certain mathematical skills. But I also agree with Lockhart that we could use a little more space in our curriculum for creativity.
I’d love to just have a math-art class. Sounds like fun. But until then, I (and all the other math teachers) have to do my best to balance a structured curriculum with mathematical play. I’m trying hard. And I hope this blog will, among all my other efforts, spur you on to mathematical play.
I like this argument. Lockhart makes some good points about the current state of math education, and I agree with him to an extent. Having done some “math art” of my own, I definitely find it more interesting, and it’s very exciting when you discover something one your own. That being said, I don’t think I could have come up with integration on my own. Newton and Leibniz discovered/invented calculus on their own, but they had no precedent or rules that they were trying to discover. If I came up with my own rules for “area under a curve,” they probably would not be considered “correct” given the precedent already set.
With the way math education works right now, finding a middle ground is really the best solution I can think of. I remember, at some point, doing a sort of “directed discovery” in math class, in which students were guided to unearthing a concept on their own. I’m not trying to say that these are the answer, but maybe they’re a step toward it. While we can’t completely switch over to Lockhart’s system, using certain forms of math art can enhance classes, particularly when introducing/exploring a new concept.
That being said, a math art class would certainly be appreciated on my part. HL Math is certainly closer to it than any other math class I’ve taken, but it still has a certain tone of “this is the way it is” about it. Perhaps a math art class could be offered as an elective subject, available to students interested in pursuing math for its own sake.
Maybe we should start a Math Art Club.
One place where I disagree with Lockhart is the idea that math is originally devised soley through abstract problems, and only then were practical solutions discovered. Many of histories finest mathematicians were also scientists. They created their math because they needed it to solve a problem in the nautral world, not because they were starving artists playing around with shapes and numbers. As it turns out, the process goes the other way. A particular solution was created and then somebody else came along later and made it more abstract and general.
We need creativity and cirrrriculum. Begin iwth argument 2 to hlep us learn and goto argument 1 to enhance the lesson.
Hi Mr. Chase! Nice blog. I think that if math teachers creatively teach a lesson in an interesting way, then naturally students will absorb the information taught.
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What is so important about being prepared for calculus-based physics? I would have vastly preferred that I learned calculus *in the context* of a combined calculus and physics course, where the topics were not removed from one another but instead seen from their historical perspective, for of course calculus was developed for the express *purpose* of solving problems in physics. The fact that they are considered separate subjects is a modern artifact, not motivated by any particular pedagogical utility. Furthermore, it wasn’t until Calc 3 that I learned the essential pieces of vector calculus that would have so brightly illuminated my struggles in Physics 1, which I tool earlier in my university program. This out-of-order confusion would have been entirely avoided had the two subjects been taught as one.
This is the essence of Lockhart’s argument; when removed from its context, learning a particular branch of mathematics becomes seemingly arbitrary and abstract. While I spent three years in calculus classes in college, my present career (as a software engineer) would have been far better served by learning linear algebra, abstract algebra and category theory, which I use every day (and I’ve never used calculus since I learned it.) When I did my masters’ in machine learning, I needed to learn linear algebra, and having learned it in that context, I’d say my understanding is far deeper than any I ever obtained from doing arbitrary exercises in calculus class.
Great contributions, Kris. Thanks for the comment.
I agree, many times ‘just in time’ teaching is the best–teach the calculus concept right when you need it. That’s not always possible, though, for two reasons: (1) the physics teacher might not *know* the math concept and (2) the physics teacher might not be able to take the time to explain it in full detail.
This happened in my Queueing Theory class this past Tuesday night. The professor quickly solved a system of differential equations, assuming we all knew how (we did). And in the homework this week, I was expected to calculate some probabilities from a joint distribution, which is pretty easy if you’ve had a Calc-based Stat course (which all of us have had).
This professor COULD have probably taught us those necessary skills, but she wanted to spend time teaching the subject of the class. I guess this all just boils down to prerequisites, right?
More importantly, I think Mathematics deserves to be its own subject, not the servant of others. Like I said, most of mathematical progress has happened without any motivation from its applications–just for ‘fun.’
Would you advocate having only application-driven mathematics education? Or perhaps, eliminate all math classes completely and out-source math instruction to Physics, Chemistry, and Engineering courses?
“You see, math is plagued by something that art rarely has to deal with: it is very, very useful in the real world.”
“He is intelligent, but not experienced. His pattern indicates two-dimensional thinking.” — Spock, commenting on Khan, Star Trek II, The Wrath of Khan — http://www.youtube.com/watch?v=RbTUTNenvCY
Similar to Kahn, your argument indicates an inexperience with the full artistic dimension, the value and usefulness in the real world of art, starting with the language arts: The Lament is a much more interesting and enticing read than your response ( or this reponse to yours, for that matter 😉 You see only the mathematical dimension and the beauty of the numbers, not the Da Vinci or the Escher. Let alone the Lewis Carol. Great artists and great mathematicians.
On the short list, Art is incredibly useful in the worlds of business, advertising/marketing & education (sit through a boring PowerPoint sometime versus a kick-ass Keynote presentation). Classical music calms me and helps me think through math problems. Hard Rock keeps me up and going through the day.
Similar to fine art on a wall in a museum, theoretical math and physics initially has very little usefulness in the real world, until an applied physicist or applied mathematician who learns about it sees that it has a use — e.g. Fractal Geometry was seen initially by the math illuminati as a bizarre artifact of those “weird machines called computers” and not useful at all (Mandelbrot was essentially denounced by his peers). Now it is recognized as one of the newest forms of math and serves as the foundation for describing the geometry of natural systems (http://www.pbs.org/wgbh/nova/physics/hunting-hidden-dimension.html). Hollywood special effects artist depend heavily on it, and their mathematically generated celluloid art makes a LOT of moolah. As do artists that are video game designers. Fractal Geometry/ART also produces some very lucrative art t-shirts for greying dead-head set. Like applied math/physics there is also fine versus applied art. If you use an Apple product, then you are using applied art and design each and every day (form follows function) it’s a melding of applied math, physics for the technology inside, and psychology (human interface design and haptics) as well as art on the exterior design as well as the marketing and advertising. Sure you can buy a cheaper Android cell phone that may have equal if not better hardware (math and physics), but it’s the attention to art and design that puts Apple’s products at the top. Art is very, very useful in the real world, as long as you have the proper scope!
All the math and science in the world will not save it from future problems if we do not teach future generations how to creatively recombine the existing knowledge into new knowledge. Creativity can indeed be taught, and this is at the core of most art and music classes in schools (versus private lessons/tutors which typically focus on the fundamentals), and at their soul are just a different form of math (drawing a building in 2D perspective in art class, M.C. Escher’s lizard tessellation and impossible staircases and drawing hands showing infinity and iteration and were one of the best mathematicians in bringing mathematical concepts to the masses… via… Art 😉
I *am* the math dad they speak about in the cartoon: http://www.winecellarcobwebs.com/2011/08/how-old-is-my-dad.html
But I find the focus on Math and STEM at the cost of Art and Music to be… inexperienced. The two are intertwined, both in practical application as well as in theoretical — they are the yin and a yang to problem solving, and creativity and imagination inspired by artistic thinking is equally important as raw calculative knowledge (mathematical thinking).
Plus, learning how to use the principles of art, specifically advertising and marketing to make math classes less boring to the *bulk* of the people in them will be critical to increase our math literacy: you’re one of the few, the ones that already love numbers for their own sake. But you don’t quite understand the dire need for art to be the marketing, the hook to bring folks on the fence and take the first steps to see the beauty of numbers.
What is the mathematical equivalent of the comic book and the graphic novels, the tools of reading specialists everywhere to entice the reading phobic into the path of reading? I use Hershey Bars, candy, toys, coloring and pocket change. Compare that to existing math and physics curricula based on standard mass-market textbooks… Math Class needs a Makeover — http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html
Thanks for stopping by and commenting!
I agree with most of what you’ve said, and if you explore my blog some more you’ll see I’m a huge fan of “math as art.” My students will tell you that I talk about the beauty or elegance of math at least once per day!
But I think you’ve missed my main point: mathematics education serves two masters and a balance has to be struck.
This is not nearly as true in art classes. If you don’t get to a certain technique, lesson, or assignment, the consequences aren’t really felt. In art classes, you have a bit more freedom to teach what you like.
Like I said in my post, if I just taught whatever I liked in my math class, we would have a lot of fun (we would!!), but we wouldn’t cover all the necessary material. If I’m teaching Calculus and we get caught up having fun with differentiation and spend all year doing this, we may have a lot of fun. But if I never exposed the students to integration, I would have failed as a teacher. The college professor will expect anyone who has had a Calculus course to know certain techniques.
Do you agree that this phenomenon is a LOT more true in math classes than in art classes?
I think that simply incorporating more history and philosophy into the lessons would go a long way, as well as presenting unsolved problems which might be interesting to them. Because it is “practical” math that is being taught, you can give the students a practical problem that really elluded people in history before the solution was found and ask them to try and solve it. Give them a week, and then guide them through the actual solution as it was discovered before you finally lay down the modern and sterile version. Not only will they find it a lot more interesting, but they will be much more likely to remember it and have a better idea of how to apply to real problems that weren’t just made up exercises in a book. That’s my take anyayw.
I generally agree with Lockhart (and love his wit). You raise a very valid counter-point too.
I find it very interesting that us mathematicians can truly appreciate the beauty of our subject, yet virtually all of us went through the traditional system. Therefore, I believe that the best students will reassemble what they have learned and even resurrect the beauty in it.
Peter, I think you’re spot-on.
I think there are some tragedies in math education and Lockhart is correct to point them out. But it’s not clear that his laissez faire approach is the best solution to the problem.
For now, students can thrive in the traditional system if they experience pieces of beauty and coherence along the way, as you suggest. I think that’s not a bad compromise.
The bigger issue–the elephant in the room, perhaps–is getting teachers to love math and know their content deeply. If such teachers are in the classroom, I’m not sure the structure/approach of the curriculum, the state standards, etc, matter that much.
That’s one reason I object to teacher training that focuses so heavily on behavior management, classroom procedures, and innovative activities (group work, experiments, etc). Those things are certainly important, but teachers *knowing* and *enjoying* their content is far more important. I have yet to see teacher training from local school districts that truly encourages teachers to understand their content more deeply.
For most of my life I’ve assumed myself mediocre or lower in intelligence due to my utter lack of understanding or enjoyment in mathematics or science. High school happens and I realize that math is relevant and possibly very interesting, maybe even beautiful, and the same for science. To assume this discovery had anything in anyway to do with school is laughably ludicrous and irritability, rigorously incorrect. This breakthrough was due to the show Numb3rs. It gave me an open mind to what had been the bane of my existence.
I don’t know any history on math, I don’t have a clue what these lines of numbers and symbols I’m being taught mean, and I certainly don’t see anything artistic or creative about it, yet as I look around and find things that echo this mathematician’s descriptions of math, I’ve become curious. What if the torture I’m enduring isn’t actually meant to be torture? It’s an astonishing, barely believable breakthrough.
For years I assumed my future was as a starving artist. For years I questioned the judgments of my parents and rare kind teacher who would insist I was very smart. Now I want to try. I want to learn. I want to be the greatest gosh-darned smart person I can, so I am reaching out for math no matter how long it takes to see it isn’t the monster that’s been beating me, but something maybe pleasant, exciting, and nice to use.
School’s curriculum had me threatening to squander my potential. How many other people are doing the exact same thing?
I feel like it needs to at least lean towards Lockhart’s ideals, because otherwise education in those areas are painful and to be avoided. I held no consideration that I might attempt engineering or physics because the scary monster was involved. Now people are telling me I might be good in those areas. My art teacher loves chemistry and has steady hands that could work surgeries. He’s mentioned being a bit remiss in missing out on the good side of math that he sees now as an adult which he didn’t as a student. He could have gone into some amazing fields. But the curriculum spoiled it for him. As I recall Lockhart mentioning, lots of creative people are being lost from those subjects. It’s true, it’s real, and it’s a tragedy.
Apologies for the backstories and ranting, but I felt it important to bring up that functionality only does so much good for people.
I think a primary condition of Lockhart’s approach is that it be implemented universally, at all levels of math instruction. Placing it here or there among the current system is bound to cause the unresolvable tension you describe.
My understanding is that Lockhart’s students are primary school students. His approach would not be appropriate to secondary school or college math. I would say his point still holds. By the time students reach high school many are traumatized. To extend his analogy with art education you wouldn’t assign the Bargue course to a 3rd grader but one can encourage their creativity in the earliest grade and then as the fine motor skills develop, give more formal instruction in realistic drawing and watercolors etc.