In A Mathematician’s Lament, Paul Lockhart analyzes the current state of mathematics education in primary and secondary schools in the US. This page is my response to his lament, some of it sympathetic, a lot of it highly critical. Read his essay first; otherwise my points are out of context.
Let me start with the admission that I went to the “right kind” of secondary school. By the time I graduated, I hated art and music, and I loved mathematics. I hated art and music because “paint by numbers” and “read those notes” was the only thing we did for 9 years. And I loved mathematics because we had to figure out that the area of a triangle was half of the area of a bounding rectangle. My mathematics teacher was my physics teacher; calculus was theory of movement and change. When I asked him whether he covered functions on the complex numbers (fun!), he did; and in physics he threw in Einstein’s special theory of relativity (I didn’t get the physics then).
Along the way, I also encounter the following joke:
Student: “Teacher, I am wondering whether I should become a theoretical physicist or a fantasy author. What do you think?”
Teacher, immediately: “A fantasy author, obviously.”
Student, taken aback by the response: “How come?” “Why obviously?”
Teacher: “For a theoretical physicist you have way too little imagination.”
So I think I understand where the author is coming from when he speaks of the joy of mathematics, its boundless opportunities for creativity. Yes, I see this spirit broken when I watch my kids’ mathematics education. (What’s really sad that the existing mathematics education builds up so much peer pressure that a busy parent can’t overcome it with supplementary lessons.)
Let me also add a deep expression of sympathy from a computer scientist. Mathematicians suffer from a lack of mathematics in primary and secondary schools. Computer scientists suffer from that, and a lack of proper principles at the college level, too. For three decades now, most instructors of introductory courses have chased the current fashion in notation: Fortran, Pascal, C, Eiffel, C++, Java, C#, you name it. Throw in some Scheme “because MIT did it” and you have the whole range. Eight or so languages (at least) over 30 years and no end in sight. And if these instructors are not bending over backwards for fashionable notations, they are chasing other trends: objects first, objects last, objects in the middle; robotics; games; and many more. This is not to deny that borrowing from some of these notations and concepts isn’t justified. The problem is, however, that almost nobody has tried to distill all of this mess to “Newton’s laws of programming and computing”, i.e., the principles of systematic program design and the interpretation of programs as computational processes. So computer “scientists” just keep on hitting their head against the wall, hoping that the headache will go away. For the kids, there is little computational thinking left in our courses until they get to the upper level of a decent Computing curriculum. See Why Computer Science Doesn’t Matter for my lament on that.
There are far fewer of those than there are good mathematics curricula.
So, I feel your pain. And still, I can’t take this lament; I must respond.
My most serious criticism concerns the nature of the essay. It reads like the recording of one long whining session. The author’s point is clear after a couple of pages Why is he wasting the reader’s time? Let’s assume his intended audience includes not just “real mathematicians” but also some of his teacher colleagues in the trenches. He would like them to read and to understand and to appreciate what an abomination they are teaching. Even then, I just don’t think that this essay really achieves its purpose.
It lacks a really positive element. Sure it states a “thesis” or actually two – teaching mathematics should not be required and the elective mathematics courses should lead the students on a magical discovery tour – but there are no arguments. It’s a utopian vision, right up there with Marxism and other such silly stuff.
At a minimum, a good paper should demonstrate that the author has constructed a serious alternative to the current sad state of affairs. Stating the (rather obvious) idea that students should come up with their own conjectures, based on their own interests, and that they should use the standard “fail, revise, try again” cycle of discovery doesn’t do justice to my request. The word “serious” implies a samples for the first year or for the first few years; it doesn’t have to be for the entire 12-year cycle and it doesn’t have to be a yearly/monthly/daily “plan.” It should include suggestions, though, on how to tease out what kind of mathematical subjects the students could be interested in at various stages and how this “interest” could be leveraged to get somewhere. Of course, the author also owes us a demonstration how “random” discussions can be pruned. I am not asking for a lesson plan; the paper correctly explains their weaknesses. I am just asking for suggestions for teachers to get “started” and for how to “prune” fruitless directions that the class may take.
Beyond the minimum, a positive contribution would also explain how to get from the current state of affairs to the desired one. Let’s again restrict the focus. There is no need to explain how to launch a campaign to get a new slate of education officials elected in some state. I would like to hear, however, how a single teacher can convince the headmaster of a (probably private) 12-year school to throw out the mathematics requirement and how to implement the new one. Should a single teacher accompany a generation from first grade to 12th grade? How many teachers do we need for how many students? How are the kids going to fit into their future college? Can a change come only “all at once?”
Just imagine how to convince the teacher unions – the unions exist, but who are the owners of the schools? – to terminate all mathematics teachers and to replace them with well-trained, real mathematicians who learn to teach “on the fly.”