Lockhart’s Lament: A Cogent Criticism of Math Education

I’ve received multiple requests to blog about Paul Lockhart’s “A Mathematician’s Lament.” Lockhart is a math teacher who is fed up with elementary and high school math education. I haven’t blogged about Lockhart's piece primarily because I agree with most of what he has to say and also a lot of people have already talked about it. (See for example, Scott Aaronson’s insightful commentary.)

Lockhart’s thesis is that much of mathematics education is simply wrong. According to Lockhart, the vast majority of our math education before college is rote learning that does not convey what mathematics is about. Lockhart argues that much of what children do in high school would be the equivalent of painting by numbers if we translated it into art. Mathematics is far more about exploration and understanding than it is about rote memorization. Lockhart argues that, by failing to let children understand and explore, we are not even teaching them mathematics. Lockhart further argues against rote math education based on practicality i.e. that these are techniques children will need when they are older.

Lockhart makes many good points and I recommend that people read his piece. As someone who has worked for many summers with the PROMYS program which uses a method similar to that outlined by Lockhart, I have much sympathy for his viewpoint. However, there are three problems with his thesis.

First, Lockhart overemphasizes the willingness of students to do exploratory mathematics. Exploration is intrinsically difficult. Moreover, it is difficult to get people to do math exploration if they don’t want to. If one tries to get youngsters to explore and they can’t do it effectively , the result is that parents will do the “exploration” for them. I’m sure there are readers of this blog who remember their parents “helping” with art projects back in elementary school.
Second, Lockhart underemphasizes the actual importance of rote learning and drills in picking up basic mathematics. Students need to be able to add, subtract, divide and multiply. They need to be able to do these things quickly in real life. Moreover, they need to do them enough times that they develop an intuition for orders of magnitude and when answers look right or wrong. That requires drilling in arithmetic from a young age. Lockhart addresses this issue briefly, but his response is unsatisfactory.

Third, Lockhart’s choice of focus on specific aspects of the high school and elementary curriculum is poor. He is correct in his criticism of the large amounts of trig memorization that occur. But he is incorrect in his example of the quadratic formula. In order to have an intuitive understanding of parabolas and other curves of degree 2, you need to know the quadratic formula. Moreover, the formula comes up frequently enough in later math classes that not knowing it would be a serious barrier. Finally, there are some items that educated people just need to know. Understanding the quadratic formula is one of those things that educated people just need to know in the same way that you can’t be an educated citizen of the United States and not know who Abraham Lincoln was.

Despite these criticisms, Lockhart is essentially correct. There are many serious problems with how we teach math and Lockhart correctly identifies many of them. While the massive overhaul that he outlines may not be necessary, it would substantially help matters if children were exposed at a much earlier age to what mathematics actually is, a subtle and beautiful art.

Alan Turing, Apologies, and Cthulhu

The British Government has finally apologized for its treatment of Alan Turing. Turing was one of the greatest mathematicians of the twentieth century. He was responsible for founding computer science and he lead the effort to crack the Enigma encryption used by the Germans during World War II. This work saved many Allied lives and according to some historians proved crucial to the victory over the Axis forces. Without Turing's work, our world would look very different. However, Turing was gay. In 1952, Turing was convicted for engaging in homosexual acts. He was forced to undergo hormone therapy which lead to weight gain and other problems. Turing's security clearance was revoked. At the time, homosexuals were considered a security risk because of the potential of blackmail. The fact that the entire risk of blackmail was because they were considered a security risk apparently did not matter. Nor did it matter that since Turing was publicly gay, there was no possible risk of blackmail. Turing's ongoing consulting work with the government was terminated. Turing's life took a steady downhill side. In 1954, he committed suicide.

I am ambivalent about this apology. On the one hand, it is good to acknowledge how horribly Britain treated one of the saviors of civilization. On the other hand, apologies to the long dead always strike me as hollow. The living always face more than enough issues that are of far more practical importance than assuaging the feelings of the long-deceased.

Rather than discuss the pros and cons of such apologies, I am instead going to suggest three pieces of further reading.

First, Wikipedia has an excellent biography of Turing which explains his accomplishments and his mistreatment in far more detail than one can easily do in a short blog entry.

Second, Greg Egan, an excellent science fiction writer, has written a short story imagining a world in which Turing's life went slightly differently. In this case, "slightly differently" means had the assistance of a time-traveling robot. The story is more serious than one might think from that summary. The story looks at Turing's interactions with C.S. Lewis. I'm not sure the story is completely fair to Lewis overall, but it is very well-written and is an amusing what-if. Like most of Egan's writing, there's just enough plausibly correct mathematics to make it interesting.

Third, Charles Stross has written an amusing novel The Atrocity Archives in which Turing figures in the background. The essential premise is that Turing did not commit suicide but was assassinated by the British government to cover up far scarier discoveries he made (so presumably the Brits still owe Turing an apology in that universe). In that novel, mathematics is deeply connected to magic and thinking about certain theorems can accidentally lead to summonings of Cthulhu and other eldritch horrors. Turing was killed for discovering a series of powerful theorems including a proof that P=NP which if invoked improperly could destroy our universe. Unlike the Egan story, this is not a story I can claim has much in the way of serious merit. But it is very fun. By most accounts, Turing was a man with a sense of humor about things. I'd like to think that he'd smile to know that fifty years after he was dead, Great Britain would be apologizing to him at the same time that people were reading novels which linked him to Lovecraftian horrors.

Health Care, Death Panels and GPFM

Recently, Sarah Palin and other prominent Republicans have repeated claims that the proposed health care legislation would result in the creation of so called "death panels."This claim was quickly determined to be an egregious falsehood. However, the damage was already done and all elements of the proposed legislation that dealt with end-of-life issues were removed. My father has a piece up at the Oxford University Press Blog arguing that in fact death panels aren't such a bad idea. Essentially, the point is that one of the major reasons our health care costs are so high in the United States is because we go through tremendous effort to extend life during the last few months of life for many elderly. It is an issue very much worth discussing.

However, there are two related matters I'd like to mention. First, we should not lose track of how incredibly mendacious Palin and her compatriots have been about this matter. The fact that Bush's earlier proposed health care legislation had nearly identical end-of-life provisions is just one of the many ways in which Palin has simply become divorced from anything resembling reality. At this point, one must wonder if reality has taken out a restraining order on Palin.

Second, as someone who grew up with a very Jewish ethic about end-of-life issues it is emotionally very hard to agree to actively deciding that we will no longer provide health care to certain people. In classical Jewish thought, every effort must be made to continue to sustain life, regardless of cost. However, this view was constructed when medical technology was very different. Two-thousand years ago we didn't have the option to add a few painful months of life to someone at the cost of millions of dollars. We do now. This different situation may require serious reexamining of this sort of belief. This probably isn't an issue for most people in the US or at least the form it will take will be very different. Catholicism for example has for a long time tried to make a distinction between ordinary and extraordinary standards of medical care. In event, as medical technology becomes even more advanced, more and more value systems are going to need to serious think about these issues. Go read his piece here.

Jack Chick’s New Tract

Jack Chick has a new tract, “Some Like it Hot.” I’ve discussed Chick’s tracts before. This one, however, takes us to a new perspective: The entire theological system as explained by the Devil. Like all good Jack Chick tracts, even the title of the tract provides amusement. The title is an apparent attempt at something resembling wordplay referring to the 1959 movie of that title starring Marylin Monroe and Jack Lemmon. Apparently, Chick’s idea of a clever pop cultural reference is to a film that came out fifty years ago. It seems that Chick’s understanding of pop culture stopped sometime in the 1960s. This is consistent with other data, such as his continuing bashing of rock music as satanic while not addressing more modern forms of devil music such as rap.

The premise of this new tract is that a grandfather and his grandson find themselves both in Hell. While wandering around the landscape, they wonder why Henry, the intervening parent, is not there and then realize that he accepted Jesus as his personal lord and savior. The Devil then decides that he wants to talk to the two of them. The Devil gives a long rambling speech, outlining the basic theology. Jack Chick’s Devil seems to be not only incompetent, but doesn’t understand basic time management. As with any Jack Chick tract, we need some bashing of other religions. Thus, the Devil declares:

I alone control every major religion in the world! Isn’t that a pleasant surprise. We’re overrun with religious leaders down here… and all their followers! We’re “blessed” with popes galore and “holy men” like Buddha and Muhammad… all of them got here by trusting their “good works.”

This paragraph is vintage Chick. We’ve got the required Catholic bashing and we’ve got the deep misunderstanding of other religions. Memo to Jack: There’s nothing in Buddhism remotely resembling a notion of “good works.” We understand that you can only think in terms of your religion and its own theological disputes, but that doesn’t mean that everyone else thinks in the same way as you.

The Devil’s rant is accompanied by pictures of various people burning in Hell. One of them appears to be dressed as a Pope, another has peyos and a black hat, while another is wearing a turban. Apparently good Christians never wear turbans.

The Devil continues his rant, explaining that he really hates Henry because Henry saved so many souls. Since the Devil cannot take out his wrath on Henry who is in Heaven, he decides to take it out on these two and thus throws them into a bottomless pit.

This raises a number of issues. First, even Jack Chick seems to think that there’s some element of justice regarding people who go to Hell. So what theological justification allows the Devil to torture people who happen to be related to people he doesn’t like? Second, is this tract really a great argument for accepting Jesus if the most likely result is that one’s family is stuck in a burning hot wasteland where one can actually find and talk to people you know? It may be that Chick is just getting soft in his old age and so we don’t have the default setting of eternal all consuming flame while surrounded in darkness. However, if I had accepted Jesus as my personal lord and savior and had relatives who had not, I’d be tempted not to witness to people. Maybe, I’m just a horrible human being, but I’d have a lot of trouble trying to save strangers if it meant my friends and loved ones would suffer more in Hell. I don’t think that’s what Jack Chick is trying to accomplish.