The subject of this post is an essay commonly referred to as “Lockhart’s Lament” (actually “A Mathematician’s Lament”) about mathematics education, and some of my observations and opinions relating those ideas to homeschooling and volunteering.
Before going any further, if you have not read the essay, or the articles about the essay in Keith Devlin’s MAA column, then stop reading here, and start reading there. There are better words there, including an interesting follow-up article with a response from Lockhart to reader comments. Even if you do not agree with all of his ideas, “Lockhart’s Lament” is a must read for anyone with a stake in mathematics education, whether you are a teacher, student, parent, or even administrator (!).
Lockhart’s essay rings very true to me. My focus here is on two ideas in particular. First, as I have said before, mathematics is not a vocational skill. It seems fashionable today to try to motivate students with applications of mathematics that they will be able to use in the “real world.” Compute the tax and tip on a restaurant bill; compute the interest earned on a savings account. Such applications typically end up consisting of problems that, in the real world, people do not solve with pencil and paper. They solve them with calculators, lookup tables, handbooks, or web sites.
It is this requirement of application that bothers me. Who cares if there is an application? I will admit that at times an application might be useful as a means of making a problem less abstract and more concrete, but why can’t the application be fun? What kid cares about the tax on a restaurant bill? As Lockhart suggests:
Play games! Teach them Chess and Go, Hex and Backgammon, Sprouts and Nim, whatever. Make up a game. Do puzzles. Expose them to situations where deductive reasoning is necessary. Don’t worry about notation and technique, help them to become active and creative mathematical thinkers.
How can we possibly get away with something like this? This sounds like a mathematical environment where there is no fixed curriculum, there is no pre-planned road ahead, but instead the students have at least some influence on where the road leads, with that influence stemming from the directions in which their interest takes them.
This brings me to my second and more important point: maybe we can’t have such an environment… at least in a school system. As one reader responded, “educational systems almost inevitably entail measuring results, an activity from which Lockhart clearly recoils.” I completely agree here… because it is that anticipation of measurement that causes the problem. When it is known that a school’s effectiveness will be evaluated based on student performance on specific tests of specific “skills,” the curricula naturally adapt to teach to the test, and student interest no longer has any influence on the direction of study. It is an inadequate metaphor, but mathematics education is rather quantum mechanical; the very act of measuring disturbs that which is being measured.
(It is not just my own observation that standardized tests measure the school much more than they measure the student. I was amused to find that, at least here in Maryland, the description of the High School Assessment program seems to acknowledge this.)
All of this suggests to me that environments with more freedom of direction have a lot of potential for engaging students. It is a lot easier to teach when no one is watching. Homeschooling seems like an attractive example of this… but one of which I am suspicious for several reasons. Homeschooling is attractive because of its simple efficiency. Whether using a fixed curricula or not, the student:teacher ratio is so much lower that there is little to hold the student back. But accuracy is another matter. The parent/teacher has to be able to react knowledgeably and creatively to the student’s evolving interest; as Lockhart points out:
Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher. If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it?
How many parents are equipped to think on their mathematical feet in this way? While we’re at it, how many school teachers are equipped to do so? This is where I think there is great potential for something like a compromise: mathematical professionals volunteering, working with students in public schools in an independent study environment. It works— and I speak from experience, both as a student and as a wannabe teacher. Mathematics can indeed be beautiful and useful, pure and applied, rigorous and recreational, all at the same time.
Possibly Wrong 
Nice post.
It’s a good point you make about homeschooling efficiency. The student-teacher ratio is fantastic and there is no need for anybody to slow down (or speed up) to move with the herd. That counts for a lot.
Also a valid point – how many parents are skilled enough in [pick a subject] to do the job? I’d say that “almost surely” there is some subject for which a set of parents does not have the necessary skill set to do a good job. I think homeschoolers have recognized this too, often forming co-ops to pool skill sets and teach each others’ kids.
More interestingly, to me (as an ardent defender of homeschooling), is the point about school teachers not having the necessary capacity for various subjects (mathematics, in this case). I would say that this and general inefficiency of the public education system is what has driven a significant rise in homeschooling. We’d love to send the kids off to school during the day – but we just can’t stomach the massive waste of time it incurs. Many homeschoolers only have a few hours of instruction during the day – and still get just as much educating done.
I point this out simply to note that I believe the education system is fundamentally broken (or significantly damaged). If it were fixed, then I suspect you would see a drop in homeschooling. The homeschooling parents are out to defend the best interests of their children – and will gladly send them off to school once it becomes a palatable alternative.
“If it were fixed, then I suspect you would see a drop in homeschooling.”
It is worth emphasizing Lockhart’s comment in his sequel, that his essay was “a lament, not a proposal.” It is not clear to me that the system can be fixed, since by its nature publicly funded support of education requires accountability– and thus evaluation– of the schools providing that education. And that evaluation leads to “teaching to the test.” Both homeschooling and volunteering by professionals have the advantage of not requiring that evaluation… but the smaller numbers that makes either option so attractive also implies that such an education almost certainly cannot be provided to all children. It’s Most Children Left Behind.
Lest I seem too cozy with homeschooling, I should clarify that although I like the idea, I tend to not trust the implementation. That is, parents may homeschool because they feel they can teach subject X more efficiently, where in this post X = mathematics. In that case I am simply suspicious of the individual subject matter expertise. But parents may also homeschool because they want their children to learn Y instead of Z, where typically Y = creation and Z = evolution. This also does not seem useful; learning pseudoscience quickly does not sound more attractive to me than learning science more slowly.