I found this 10+ year old essay by mathematician (and teacher) Paul Lockhart on all the things wrong with education in mathematics education in K-12 and on mathematics as a form of art, entitled “A Mathematician’s Lament”:
He’s got some great stuff in there, highly quotable, such as “PRE-CALCULUS: A senseless bouillabaisse of disconnected topics. Mostly a half-baked attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful.”
Although it makes for great reading and has a lot of good points, I’m not sure I agree 100%. I do remember the transition of going into more advanced mathematics in college and the professor essentially laughing at the formal way we had been taught to do proofs in high school geometry. My first year in university, when we were proving something, he had us write it out in a much more conversational style, with the mathematical notation flowing with the text, as sentences. It totally blew my mind. In the same way, when I was writing my first really scientific paper, I was laughed at by the PI for using passive voice, as we had been taught all the way through in our lab reports: “50 mL of water was pipetted” instead of “we then added 50 mL of water”. The first sounds ridiculous in retrospect, and the latter has exactly the same information but sounds like it was written by a human being.
However, I would stress some of the practical applications of mathematics as being really important and useful, and I am a little skeptical of the ability of most math teachers to really teach the beauty of mathematics in such a free form way, and I am also skeptical that a push in this direction won’t move us away from actual learning of skills. For example, if you just teach poetry and encourage the beauty of literature, you end up with people who don’t know where to put commas in a business email. I know many highly educated people with this problem, presumably because they were taught English by teachers who probably had dreams of being like Robin Williams in Dead Poets Society and awakening their student’s minds instead of teaching them grammar. Paul Lockhart obviously isn’t saying that you don’t teach any technique or any formality, but I think it is worth mentioning this as a potential problem. I think the same is true of teaching some of the fundamentals. The New Math which was a real step backward in mathematics education is the fault of the fictitious Nicolas Bourbaki, a pseudonym for a bunch of French mathematics researchers who were trying to make sure math was more rigorous and beautiful. I’m not always sure that when you plan out what should be part of mathematics curriculum you shouldn’t include just as many physicists and engineers as mathematicians in the discussion.
Instead of learning how calculate, students are encouraged to learn all sorts of abstract concepts instead of learning how to do useful things with mathematics. At the same time, kids are often taught weird algorithmic, formal solutions for problems that aren’t even that relevant. For example, I did some substitute teaching of high school mathematics, and one of the things students were being taught was the Sieve of Eratosthenes. In principle, I can see that it might be considered useful as a way to think about prime numbers, but these kids were having trouble with fractions and arithmetic. It’s not that people won’t have access to calculators, but you should know roughly what the calculations should be like. Learning how to do rough, back of the envelope estimates is an incredibly useful skill and I’d note like to see it be replaced by the art of mathematics for art’s sake.
For example, this response to a mathematics problem from the Common Core was making the rounds with a lot of parents:
Overall, I don’t have any particular solution, but I do like the essay by Dr. Lockhart, and I encourage those interested in mathematics education to read it.
Part of what is beautiful and satisfying about mathematics is that there is a correct answer. There aren’t as many of the same semantic arguments that cause so much discussion in much of human discourse. That’s also what makes mathematics hard but also worthwhile.
Part of my personal lament I suppose is that I don’t think a lot of my mathematics education prior to University was very good. Although I did have some great teachers in my early education, my later years in K-12 were terrible. For example, in 8th grade (middle school), I was given a textbook from the high school and told to sit in the back of the class and teach myself the material, as I was more advanced than the other students (partly because I enjoyed mathematics, partly because I had come from a better school system). I just had to pass the midterm and final exams that came down from the high school. That wasn’t a great method of education, particularly as the textbook sucked. Maybe if I Dr. Lockhart had been my teacher, I would have more happy memories of my early education.
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