In 2002, Paul Lockhart wrote an article about the state of mathematics education called A Mathematician’s Lament. Lockhart argues passionately that what children are typically taught is neither useful nor interesting. He believes children would get far more from emulating what mathematicians actually do, unstructured mathematical exploration and proofs. One of the problems, Lockhart acknowledges, is that many math teachers don’t understand or appreciate what he calls “real math,” leaving them unable to teach it. While I agree we need to improve our math teaching, and I agree that exposing many more children to “real math” is a good idea, I think Lockhart is at once too optimistic and too pessimistic.

Peter Gerdes, a graduate student in mathematical logic at Berkeley, critiques the over-optimism for me. While some children would greatly prefer “real math,” many students will be indifferent or downright hostile to it regardless. Why? Because math is difficult, much more difficult for some than others, and because there are serious social consequences associated with actual and relative mathematical skills.

But both Lockhart and Gerdes are too pessimistic in that they either deny or ignore the benefits from learning the math schools currently teach. Though more research should be done on evaluating the benefits, I will note that the math skills that are measured on standardized tests are highly predictive of later academic and labor market outcomes. There is even evidence that national economic productivity and growth is affected by the skills measured on such tests. These and other considerations lead me to believe that we could improve mathematics education a lot, even if we didn’t make any of the sort of changes Lockhart recommends.

Most importantly, what Lockhart recommends, and what we’re currently doing, are not mutually exclusive. In fact, though we will have to make tradeoffs, they are both necessary and (on some margins) complementary.

2. Some kids do find a way to learn and love math, despite the way its taught.

3. Applied math can be relevant, useful, beautiful, and fun.

4. People have different learning styles and tastes. Some students actually prefer the the more structured traditional pedagogy with lots of practice doing similar problems.

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I agree with some points in your response, and disagree with others. Seen as a series of actual changes to math education, Lockhart’s vision is wildly over-optimistic in its scope. (As some kind of long-range “what-if” scenario, I think it a lot more to offer, but even then it’s very optimistic)

As regards the pessimism you mention, I have to point out a logical error: correlation is not causation. The results of standardized testing may correlate with future success, but that doesn’t mean that math skills are causing this success- if nothing else, it may be a superset of test-taking skills, which could be more directly challenged by math than other subjects, that influence future success.

There is also a fair amount of narrowness in assuming that the success that these skills correlate with has some special privilege that exempts them from improvement. Just because this set of math skills offers a means to success doesn’t mean that there aren’t other means of bringing about these improvements, nor does it mean that these improvements cannot be brought to a wider audience through different means. Could both approaches be integrated together? Let us hope so.

The main objection I have to Lockhart’s ideas is that they are generally too radical. Like so many theoretically fantastic ideas in education reform, there is no clear way of applying them incrementally, at least not in a way that would be weigh the huge effort needed against the benefits gained. The current education system is, for very understandable reasons, extremely slow to change, and this makes reform something of a Gordian Knot.

Thanks for the comment Ethan. I certainly agree that we should continue scrutinizing the claim that increasing mathematics achievement (of the type measured by standardized tests) would improve economic, or other, outcomes. Correlation is not proof of causation, but it is supportive evidence, especially when you have a theory of the mechanism. This is all I claimed.

I take your third paragraph to be saying that improving math skills isn’t the only way to improve economic outcomes, reduce poverty, etc. I heartily agree with that as well.

I disagree with Ethan’s suggestion that Lockhart’s idea is all-or-nothing. On the contrary, incremental changes involve changing your attitude a bit in the classroom. When given the chance prompt your students:

“Why is this called this?”

“What if we change this hypothesis? think about that for a minute before you respond.”

“Can someone give me a sentence highlighting the idea here?”

I use these kinds of questions with my 7 year old son and I use them when teaching differential equations here at Cornell. Choice of words is not sacred, nor is choice of method of solution, and the pupil that has been encouraged to initiate some of the math on their own will find it more useful and meaningful than the pupil who hasn’t.

One of Lockhart’s points is that “math course” has come to mean “rule following course.” Ultimately, it would be nice if the two were separated into distinct courses, so that those willing to do whatever it takes to please the teacher and those who have ability and interest in actual mathematics were not conflated. No doubt the former is more important to the GDP than the latter, that is beside the point, which is that real math isn’t given a chance.

In Lockhart’s view there are two extremes, one mechanical and suffocating, and the other inspiring and rewarding. Math pedagogy itself could be seen in similar terms. On the one hand you have curriculum designers which want to standardize math nationally, and essentially remove any reliance on the specific teachers (that is, if you can follow the teacher’s edition of the text you’ll be a good teacher according to pedagogy X), and on the other hand you have a lots of radical ideas in the name of progress, which are not easily codified or executed, one of which that teachers should be inspired themselves, and find projects which are interesting to them and the students.

I completely agree with Lockhart’s viewpoint, and as much as I love to dream up radical utopian pedagogy schemes, I see the application as incremental in the smallest ways: buck the system, just a bit, and emphasize the creative process.

Thanks for the comment Andrew. I too, think there is room to learn from Lockhart without radical reforms. In fact I tried to argue that it was possible to be too radical.

I agree with some points in your response, and disagree with others. Seen as a series of actual changes to math education, Lockhart’s vision is wildly over-optimistic in its scope. (As some kind of long-range “what-if” scenario, I think it a lot more to offer, but even then it’s very optimistic)

As regards the pessimism you mention, I have to point out a logical error: correlation is not causation. The results of standardized testing may correlate with future success, but that doesn’t mean that math skills are causing this success- if nothing else, it may be a superset of test-taking skills, which could be more directly challenged by math than other subjects, that influence future success.

There is also a fair amount of narrowness in assuming that the success that these skills correlate with has some special privilege that exempts them from improvement. Just because this set of math skills offers a means to success doesn’t mean that there aren’t other means of bringing about these improvements, nor does it mean that these improvements cannot be brought to a wider audience through different means. Could both approaches be integrated together? Let us hope so.

The main objection I have to Lockhart’s ideas is that they are generally too radical. Like so many theoretically fantastic ideas in education reform, there is no clear way of applying them incrementally, at least not in a way that would be weigh the huge effort needed against the benefits gained. The current education system is, for very understandable reasons, extremely slow to change, and this makes reform something of a Gordian Knot.

Thanks for the comment Ethan. I certainly agree that we should continue scrutinizing the claim that increasing mathematics achievement (of the type measured by standardized tests) would improve economic, or other, outcomes. Correlation is not proof of causation, but it is supportive evidence, especially when you have a theory of the mechanism. This is all I claimed.

I take your third paragraph to be saying that improving math skills isn’t the only way to improve economic outcomes, reduce poverty, etc. I heartily agree with that as well.

I disagree with Ethan’s suggestion that Lockhart’s idea is all-or-nothing. On the contrary, incremental changes involve changing your attitude a bit in the classroom. When given the chance prompt your students:

“Why is this called this?”

“What if we change this hypothesis? think about that for a minute before you respond.”

“Can someone give me a sentence highlighting the idea here?”

I use these kinds of questions with my 7 year old son and I use them when teaching differential equations here at Cornell. Choice of words is not sacred, nor is choice of method of solution, and the pupil that has been encouraged to initiate some of the math on their own will find it more useful and meaningful than the pupil who hasn’t.

One of Lockhart’s points is that “math course” has come to mean “rule following course.” Ultimately, it would be nice if the two were separated into distinct courses, so that those willing to do whatever it takes to please the teacher and those who have ability and interest in actual mathematics were not conflated. No doubt the former is more important to the GDP than the latter, that is beside the point, which is that real math isn’t given a chance.

In Lockhart’s view there are two extremes, one mechanical and suffocating, and the other inspiring and rewarding. Math pedagogy itself could be seen in similar terms. On the one hand you have curriculum designers which want to standardize math nationally, and essentially remove any reliance on the specific teachers (that is, if you can follow the teacher’s edition of the text you’ll be a good teacher according to pedagogy X), and on the other hand you have a lots of radical ideas in the name of progress, which are not easily codified or executed, one of which that teachers should be inspired themselves, and find projects which are interesting to them and the students.

I completely agree with Lockhart’s viewpoint, and as much as I love to dream up radical utopian pedagogy schemes, I see the application as incremental in the smallest ways: buck the system, just a bit, and emphasize the creative process.

Thanks for the comment Andrew. I too, think there is room to learn from Lockhart without radical reforms. In fact I tried to argue that it was possible to be too radical.