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Book Review: A Modern Day Jeremiah Laments Math Instruction

6/10/2013

 
If you read up on math pedagogy long enough you will see a reference to Paul Lockhart’s Mathematician’s Lament. It even has it’s own Wikipedia entry.

It is a marvelous little book of 140 pages, that makes a simple, 3 part argument about how to improve mathematics education the US.

Unfortunately, 89 pages of the book remains unwritten and it contains the third, decisive part of the argument.

The argument looks like this:

1)      Math as it is taught in the US is  boring

2)      Math doesn’t need to be boring. In fact, math is interesting and beautiful

3)      We can teach children the beauty and fascination of math in US schools by doing X.

Lockhart devotes 90 pages of the book to the first proposition, about 88 pages more, I estimate, than is necessary.

Lockhart suggests that the root of the problem lies in how teachers conceive or mathematics. Actually, it’s how everyone (save mathematicians) conceive of mathematics. We see it as a rigid, rule-based, practical. Lockhart offers mathematics as aesthetically pleasing problem solving. No more, no less. 

Mathematical ideas are inherently interesting, charming, fun. If you’re not interested, you’re doing math wrong—or being taught math wrong.
Picture
So why is math taught wrong?

Lockhart suggests that teaching math requires intense personal relationships with students, “choosing engaging and natural problems suitable to their tastes, personalities, and levels of experience,” and being flexible and open to the students’ shifts in curiosity. Lockhart doubts most teachers are interested in this sort of thing, and he suggests that most teachers see it as too much work (p. 44).

Later (p. 82) bureaucrats are blamed; they won’t allow individual teachers to follow their instincts. Later, we’re told more bluntly that schools ruin not just math, but all subjects.

But what of the third part of the argument in which Lockhart ought to tell us how make things better?

His vision of “better” is that teachers will pose interesting problems to students—he offers many compelling examples—and students will work on these problems under the guidance (ideally, the minimal guidance) of the teacher. The best learning, he avers, is where the student is doing math (i.e., creating arguments, finding patterns) rather than executing formulae described by others.

There is an irony here. Lockhart describes mathematical arguments as two-headed. One head is relentlessly logical, and rigidly insists that an argument be airtight. The other head has aesthetic criteria, and seeks an argument that is elegant, lovely, and that sheds light as it proves.

Lockharts prescription is all lovely and not enough logic.

For someone who excoriates a system that would allow people who don’t know the history of math to teach it, Lockhart is surprisingly quick to write about educational practice in the absence of any knowledge of its history.

Giving students interesting problems and aiding their efforts to solve them as the workhorse method of classroom learning—that idea has been mainstream for about 100 years. Further, surveys of teachers consistently show that they believe this method (and closely aligned methods) to be not only effective but desirable.

Teachers don’t fail to use these methods because they are lazy or because bureaucrats won’t let them.

These methods are really hard to pull off. Your knowledge of math needs to be very deep because the problem may pivot in an unexpected direction. Your classroom management needs to be flawless because you are expecting the students to work more independently.  And both knowledge of math and classroom management will be tapped further by the fact that you must make many decisions in the moment, as the classroom situation is very fluid.

Picture
The unfortunate thing about Mathematicians Lament is that Lockhart has put his finger on a real problem but is so caught up in righteous indignation that he loses the chance of doing any good. Simple scolding won’t do it. Jeremiah had compassion for the benighted in the Book of Lamentations.

A much more effective approach has been adopted by Hung-Hsi Wu, a mathematician at Berkeley. Wu has argued that a key problem is that today’s math teachers—products of the American system themselves—don’t get math. The solution is to teach them some math. (I once listened to a 30 minute explanation by Wu of why our system of whole numbers works the way it does. Quite literally, stuff that first graders can and should know. I was spellbound.)

In contrast to Lockhart, Wu has some faith in teachers. If they understand mathematics, they will teach it. He is also less dogmatic than Lockhart, who unthinkingly assumes that they only way to learn a topic is to practice it the way experts practice it. Indeed, some important elements that Lockhart wants to see—especially discovery—are present in some quite traditional approaches, especially the Japanese approach to teaching, as described by Jim Stigler.

Righteous indignation should be an occasional guilty pleasure, not a blueprint for math education.

David Wees link
6/10/2013 06:04:00 am

Of note: the book is an extension of the original essay available here on Keith Devlin's site: http://www.maa.org/devlin/lockhartslament.pdf

If the discussion about how to change our current system looks like an after-thought, that is because it certainly is. Surely it is possible that people can produce works which expose problems without always having a solution? The problem is beautifully captured and easily presented to others, which in itself is exceedingly challenging. If you asked 10 math and math education experts what they thought were the biggest challenges facing math education, I'm pretty sure you would get 10 different answers...

Paul Lockhart has also produced another book called Measurement in which he describes a history of the ideas around measurement in mathematics.

Karst
6/10/2013 10:43:21 am

Lockhart's book can be thought of as providing his personal way of going about accomplishing part 3 ("We can teach children the beauty and fascination of math in US schools by doing X".)---at least, at the high school level.

Further discussion of Lockhart and his book can be found at Jason Rosenhouse's evolutionblog at National Geographics' scienceblogs.com:

http://scienceblogs.com/evolutionblog/?s=Lockhart

Rosenhouse is a mathematician who has published on the Monty Hall problem, and on Sudoku.

Roger Sweeny
6/11/2013 07:15:02 am

I am getting increasingly tired of jeremiads like Lockhart‘s. If only we teach right, we will make our students see how wonderful our subject is. They will be interested and excited, willing to work hard, and by the end of the course, they will understand and enjoy.

Bull bleep. Math is no more “inherently interesting, charming, fun” than soccer is or basketball is or “World of Warcraft” is. The best teaching in the world is not going to inspire most high school students to love math, any more than it would inspire me to love any of the latter three.

When I read “Mathematician’s Lament,” I hear religion--a wonderful faith that the universe has been made so that math is “inherently interesting, charming, fun,” and that people have been made so they can see that. Those who understand math have attained an elevated status and have an obligation to pass on its great truths to those who are as yet unenlightened.

So it is fine to force young people to take math courses that they are not interested in and will probably never use. It will make them better people.

Except that it won’t. Most will feel some mixture of boredom, resentment, and failure. Like requiring me to watch the entire regular season of the Denver Nuggets and pass a number of tests and quizzes about it.

No doubt much math teaching is bad. It can and should be better. But for most students, the major problem isn’t bad preaching. It’s that they are forced to attend chapel in the first place.

EB
6/11/2013 07:45:45 am

I got pretty far in math, for a girl in the 1960's -- College Algebra/Trigonometry. Liked some parts better than others. Used some parts later, other not so much. BUT, I did not need to (and did not want to) do math the way a mathematician does it. What I was aiming for was to see the math I was learning as a functional tool where all the parts fit together. I did not need to explore and find my own solutions except when I was working on a geometry proof. I did not need to develop alternative algorithms., That might have been nice, but would have slowed the whole process down greatly. A good math teacher, at any level from K-12 except when working with very gifted math students, needs to combine instruction on how to do math with instruction on how to figure out what math procedure to use in solving the different types of problems you might run across. Good teachers always made sure that the applications were addressed creatively; they didn't just make you memorize facts and formulas.

Philip McIntosh link
6/14/2013 10:16:57 am

Not sure we read the same book. I think it should be required reading for all math teachers, principals, superintendents, school board members and parents of kids who are enrolled in public school math classes. Lockhart hit the nail on the head when he suggests early school math should consist mostly of puzzles and logic conundrums that encourage thought and not rote learning of facts.

EB
6/16/2013 02:50:32 am

Philip, I do believe that puzzles and logic conundrums should be part of the curriculum (and not just in the early years). But are you claiming that they should replace learning the basic arithmetic operations entirely, just because you have to memorize some facts in order to do the operations? I found that being able to do the operations strengthened my ability to do some kinds of puzzles, and certainly bolstered my sense of logic. Why would you postpone at least beginning children on the stepwise process of learning the use of mathematics as a tool for everyday (and academic) life?


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