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.

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## 7 comments:

I didn't read the article yet, so I don't know if this is relavant, but my mom works in a frum school, and a couple of years ago the school spent a buttload of money on some computer algebra class thing that was supposed to help the students "understand" the math (i.e. not just rote memorization). She says that it was a total failure and the students who used that program ended up not knowing any algebra by the end of the year.

I do think it would be nice if students got a deeper understanding of the math they learn, but I'm not sure if that's something everyone can learn easily, and you really handicap them later if they can't do the basic stuff. My personal experience is that it's often easier for me to more deeply understand some mathematical tool once I have gotten used to using it for a while.

I can see this guy's point of view. I have a completely warped and fearful view of mathematics. I hated it in school, and I still hate it now. This guy might be on to something, though, but I can also see your argument in how students need to learn the basic drills of mathematics. I knew a girl who simply used a calculator all through elementary and high school to the point where we were taking a basic algebra class in college and she didn't know her times tables. Crazy huh?

Professor Ethan Akin of CCNY argues that rote memorization is essential for learning mathematics.

http://math.sci.ccny.cuny.edu/docs?name=MindlessRote.pdf

I didn't read the entire paper, but the few pages I did read really affected the way I think about learning math.

OK, that Akin article, obviously he's right about needing to develop facility with computations, and the way you do that is by doing them, but some of the other stuff in there just leaves me just going "is he serious"?

In particular, commutativity of multiplication. My parents introduced multiplication to me as repeated addition - and, IINM, they didn't mention its commutativity at that time - or if they did, I didn't remember it and certainly didn't understand it. Anyway, so my parents teach me, 15 divided by 3 means 15 divided into groups of 3, while at school they teach me, 15 divided by 3 means 15 divided into 3 groups. How is that the same?! This left me very confused. Sure, the numbers matched up, but I couldn't see any reason why. So I asked my mom about this and then she introduces me to commutativity of multiplication. 5 groups of 3 is 3 groups of 5. Really?! I was pretty incredulous. And then she showed me the rectangle. I tell you, when I was 6 years old, that rectangle was mind-blowing.

I totally fail to understand why we would want kids to think commutativity of multiplication is "obvious" without thinking about it. Anyone who actually understands what it means, but is, you know, only 6 years old and doesn't have much experience with numbers yet, won't find it obvious at all. That rectangle makes for a great demonstration. Why the hell would we want to get people in the habit of people things are "obvious" when they're not? That's a horrible habit to get into, and one that we have to spend a lot of time training them *out* of later!

I would think the ideal would be to teach thinking, leading to not-thinking: Get people to understand, apply the understanding to computations, do them enough so you don't have to think about it. I guess I pretty much agree with Akin there? Timed tests are useful for exactly the reason he describes. (Though I hate flash cards. Yuck.)

But commutativity of multiplication, man... why would he say that?

(Also, after having been surprised by multiplication's commutativity, I'm pretty sure I *was* again surprised to learn that exponentiation wasn't. These are good lessons that should be pointed out, not hidden away from people!)

BTW, Josh, you left out the "http://" on that first link, it doesn't work as a result.

Harry, thanks for the pointer about the link. Fixed now. I agree about commutativity of multiplication. To my mind it seems almost like glossing over unique prime factorization: There's something going on there, and not paying attention to it really isn't helpful.

Hello Josh,

I just thought you might be interested to know that Paul Lockhart was actually one of my teachers in high school. I took a class with when I was a senior and it was great. I was the ideal candidate to get a lot out of his class, but he also taught classes to the people who weren't good and math and didn't particularly enjoy it, and it was my impression they got a lot out of it too.

Toby

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