A Mathematician's Proposal

I just finished reading a tiny tome called A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form by Paul Lockhart. It directly relates to the MLCS course Heather and I are developing (more on that later) so it had me at hello, shall we say.

Some excerpts:


“A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.


Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.


As for the primary and secondary schools, their mission is to train students to use this language—to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely.  One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”


In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”


In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. “It’s a lot for them to learn, but later in college when they finally get to hear all this stuff, they’ll really appreciate all the work they did in high school.” Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one. “To tell you the truth, most students just aren’t very good at music.  They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn’t care less about how important music is in today’s world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable— every note in the right place, perfect calligraphy, sharps, flats, just beautiful.  She’s going to make one hell of a musician someday.”

Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. “Of course!” he reassures himself, “No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!”

A few more short nuggets of gold from this book…


“I can understand the idea of training students to master certain techniques-I do that too. But not as an end itself. Technique in mathematics, as in any art, should be learned in context. The great problems, their history, the creative process-that is the proper setting. Give your students a good problem, let them struggle and get frustrated. See what they come up with. Wait until they are dying for an idea, then give them some technique. But not too much.”

“So how do we teach our students to do mathematics? By choosing engaging and natural problems suitable to their tastes, personalities, and levels of experience.”


“Mathematics is not a language, it’s an adventure.”


“Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple.”


“Efficiency and economy do not make good pedagogy.”


“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?”


“First of all, forget the symbols-they don’t matter. Names never matter. …The only thing that matters in mathematics is what things are, and more important, how they act.


[Referencing proof] “So how do we it? Nobody really knows. You just try and fail and get frustrated and hope for inspiration. For me it’s an adventure, a journey. I usually know more or less where I want to go, I just don’t know how to get there. The only thing I do know is that I’m not going to get there without a lot of pain and frustration and crumpled-up paper.”


To read the entire book, all 140 small pages, head to amazon. To read the 25 page essay that led to the book, here you go.

His original essay was posted on maa.org and unleashed a firestorm. Many commented and soon after, Paul wrote a follow-up response.


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I don’t agree with everything in the book or his original essay but so much resonated with me. I’ve been saying for a while to my students and colleagues that we’re focusing on the wrong things, that “x” isn’t the point. The notation and language don’t matter nearly as much as we claim they do. We’ve gotten mired down with the minutiae and lost sight of our original goal: solving problems that in some way involve numbers. Not algebra, not Please Excuse My Dear Aunt Sally, not which side of the fence you’re on about graphing calculators. None are the point.


I could go on all day but Paul did a much better job and was concise to boot. Tis the nature of academics to form committees, commiserate, and change little. I love to talk as much as the next chatty Kathy but action is far more important to me and this field. So, here is a mathematician’s proposal:


New Life and Mathway, of which I’m a member, has spent the last year developing new goals and outcomes, new course descriptions, and a new vision for what developmental mathematics should be. My colleague, Heather Foes, and I are taking said list of outcomes and bringing them to life.


This course is fundamentally like nothing we teach now at the dev. level, which is exciting and daunting all at the same time. Acquiring skills is not the goal of the course; acquiring the ability to think, solve problems, and see the world with a mathematical slant is.


An example: we may give them a situation (one that’s interesting and relevant to them) where interpreting a graph is needed. Some students will have no difficulty but some may not be able to get past the 2 dimensional aspect of it and need more help in that. Having tools, some individual and online, that assist them to improve their skills can be very beneficial. But the end goal is not the skill or proving mastery of it. The end goal is that original situation that needed to be understood and just happened to have a graph in it.


Sure we’ll have to do some direct instruction but that will not be the whole class. Students will ask for help and tools when they need them. Tell when they ask but not before then and not too much.


I want to get away from students "working the system" or just "playing the game" like I see so much in our current classes. Even some of my most successful students in the traditional courses like algebra, trig, calculus really don't grasp mathematically what we're doing, with good reason. That's not our focus and hasn't been in a long time. But in our statistics courses, Math for Elementary Education courses, and some other "non-traditional" courses, they have a much better grasp at what the point is. Those courses aren't perfect but the goals, methods, and outcomes are not fully focused on a set of skills but instead of way of thinking. I don't think it's coincidental that the pass rates are higher in those courses and students often enjoy them more.

The graph example I mentioned above is just a tiny view into our vision; it's certainly not the course or structure.  We envision starting with interesting, messy problems and letting them play some.  They're going to ask for help and need tools.  When they do, we have something to offer but not in the ways we're doing things now.  Open ended problems are frightening for students and teachers (by darn, we want an answer!) so we'll use some unique solution problems as well.  It's all about balance.


Talk is cheap so Heather and I are putting our time and energy where our mouths are, creating the course and everything an instructor and school would need to be successful. It’s not a book or an online homework system. Yes, there will be materials and tools for the both the instructor and student to journey into mathematics in a completely new way and be successful at it. But we envision so much more. I’ve seen through our school’s redesign and rollout of our new program that one facet of change or one resource will never be enough, that you’ve got to accommodate for instructors just as we do for students. They need many resources that work in ways that make sense to them. Some will want online, some want print, some want face-to-face workshops. Give them what they ask for so that they will feel prepared to take the course and materials and make change a reality.


This is a vague description at best; more will be forthcoming.  But until then, as Paul encouraged, it's pretty exciting to just play in the sandbox called mathematics.