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A Mathematician’s Lament (2002) [pdf] (maa.org)
159 points by Tomte on May 14, 2017 | hide | past | web | favorite | 27 comments



Oddly enough, I went to high school with Paul Lockhart and I too became fascinated with math despite rather than because of our terrible high school math program.

I think what originally got interested in math was the infamous "new math" program [1] that existed in the 60s.

And the thing with New Math is that was abolished through a backlash of parent and teachers because ... the parents and teachers couldn't understand the new concepts.

And this is the thing. Math is both an abstract enterprise and an enterprise of bookkeepers adding numbers on ledgers, parents adding up grocery bills and so-forth. While New Math might have had some flaws in execution, any effort to produce curriculum suited to students learning abstract mathematics is going to run-up against the "readin', writin' and arithmetic" crowd in a similar fashion to New Math.

[1] https://en.wikipedia.org/wiki/New_Math


This is a good point, and a relevant one when discussion Common Core with people. At the surface level, I don't see anything in the new mathematics curriculum that I would consider negative, but the biggest complaint I hear is that parents can't understand it, and some teachers barely do either. But that hints at a deeper problem.


I think this is indeed part of the problem. One of the problems I've encountered is they sometimes use unusual or unfamiliar terms for what are actually fairly simple concepts, and if you're helping your kids with their homework, it may not be obvious what they're supposed to be doing.

One simple example (which I'm probably mixing up) is "making tens"; when doing addition or subtraction, rather than blindly memorizing a bunch of facts like "3+8=11", they learn to break the numbers up, so you take 3 and 8, "make a ten", and have one left over.

This is actually a great thing to help with understanding of what's going on with math, and after my kids showed me a few examples, I totally got it. But for those who are less open to new ways of teaching, it may just seem like change for the sake of change. And it's not just math; they're also teaching other subjects (such as writing) in newer ways, which I honestly think are fantastic - having kids spend time writing every day is great.

There are also quite a few bad questions in the workbooks; this has probably always been true, but the combination of bad question plus unfamiliar (to parents) concepts causes a strong reaction. Plus the whole "the gubmint is trying to brainwash mah kids!" contingent overreacting about everything....


I think the reaction against New Math was a part of the general reaction against all of what many viewed as the "weirdness" of the 60s, an anti-intellectual attitude that was often used by powers-that-be that also felt threatened by these upheavals.

My, I'd be happy if we had military dictatorship which imposed a new order including the metric system, radians instead of degrees and algebra from 1st grade onward. But I don't think that program would have a sufficient constituency.


Wrong. It was a reaction against the utterly predictable failure of a contentless approach to instruction which started and ended with the rhetoric necessary to convince pseudo-intellectuals that supporting it demonstrated political discernment and moral virtue.


Can you please elaborate on what you mean by "contentless"? I just checked out the wiki page on New Math and it seemed to have covered (with the exception of inequalities) a few topics that usually get covered in the first couple of chapters of an intro to abstract algebra book.


I was actually coming back to delete the original comment as it wasn't very well thought through. I mean, sure, of course there was "content". There were chapter headers as so on and so forth. It would be better to criticise the absense of thought put into having a theory of instruction beyond "Experts do this, therefore if we teach this, we'll create experts"


The gradient between the student and rest of society was too high. Teachers didn't know how to teach it, parents didn't know why it was important. I think now it could fly, just put computer in front of it.


Out of interest (I think this forum is not short of people who are in a position to answer): what is the general opinion among US mathematicians about that "new math" experiment?

I know that it has a terrible reputation, but was it a plain bad idea, or a good idea implemented poorly, or a good idea implemented well but unfairly maligned?


I don't think mathematicians are the people to ask. "New Math" wasn't new math at all - it was standard abstract concepts known to all mathematicians.

The controversy was whether these concepts are useful or helpful to non-mathematicians, and mathematicians may be the least qualified people to answer that.


They may be, but I'm still interested to know what they think. I already know what non-mathematicians-in-general think.


Why would they have an opinion at all, unless they were educators, students, or parents at the time?


I'm a mathematician, and was a student at the time, and am an educator (college professor) and parent now. Not an expert on the New Math, though, but I do teach a lot of people who will be math teachers, both of elementary and high schools, and so will so the Common Core.

My sense is that a lot of the reaction to the New Math was the same as the reaction to dictionaries that came out at the time and included words like "groovy" or said that it is not the crime of the century to write "which" when not preceded by a comma. That is, it was more of a political effect than pedagogical.

That said, there were a lot of people who thought on a lower or more local level, that math teaching that asked students to notice the Associative Law, or that asked students to think about sets, was not directed to immediate gain. That made it wrong in these folks's view, or at least puzzling. They had not learned those things and one thing about Math is that it is unchanging so ... .

I am excited by the Common Core, myself, but I hear a lot of rejection that seems to me to be just general opposition to doing it differently than when the speaker was in school. There was a lot of that in the 60's, for sure.


I'm a parent with a child that's currently being subjected to Common Core (second grade). I'm mathematically inclined, when I took my SAT I scored in the top 1% in math.

My evaluation of Common Core so far is that it's terrible. Students are forced to learn multiple ways of doing basic arithmetic, which is highly counterproductive and confusing if the child mastered the topic (and concept) with the first approach. It also lacks any aspect of what this essay stresses - helping children see the beauty and creativity of math. The emphasis is on learning processes by rote.

We'll see how things progress in the next grade, but I'm not optimistic. Private school is looking very attractive.


I observe that mathematicians in my country often have informed (and strong) opinions on mathematical education. As does the author of "A Mathematician's Lament".

Maybe I have a mistaken impression of how significant "new math" was, but I'd expect people to have thought about it. I mean they've all heard the Tom Lehrer song, right?


I used to tutor some 16 and 17-year olds in math. It was always heartbreaking to see the school curriculum requiring students to essentially know the "name of a thing" e.g. commutativity, without really ever understanding it or using it. Reminded me of Feynman's account of learning from his father, and the difference between really learning something, and just learning the name of a thing: https://haveabit.com/feynman/knowing-the-name-of-something/


I came across this recently. I had always wondered why I managed to find math so fascinating when most of my peers hated it. I think I was just lucky enough to see through the poor state of education into some of the magic underneath. For a (partial) solution to the math education problem, Paul Lockhart has also written a book called "Measurement"[1] which is a very entertaining read.

[1] https://www.amazon.com/Measurement-Paul-Lockhart/dp/06742843...


It seems to me that the musician's nightmare is actually a lived reality for a lot of people, it definitely was like this for me. As a child I took pleasure in inventing melodies on our family's piano so that one observant family member signed me up for piano lessons. I quickly came to hate my lessons so thoroughly that I didn't touch a piano until I was in my mid-twenties, when I rediscovered the joy of making music...


It's basically the difference between authoritarian education and exploratory play-led education.

Extremes of either seem to be destructive, but hitting the sweet spot where the balance is just right is incredibly hard, and also incredibly demanding of time and resources.


Yes, current society and academia have it all wrong. Math is both celebrated and scorned. It's somehow the most applicable thing in the world and yet largely unemployable. People can't imagine a mathematician can do anything, but know you need it to do most everything.


This is an astute observation.

The world leaders during WW2 recognized this - basically, when critical work needed to be done, they knew whom they needed and roped in a large number of the leading "dreamers/thinkers" of the time to work on the Manhattan Project.

The thing that saddens me is that it is often conflict and animosity that awaken this recognition - other examples include the Cold War, artillery calculations in WW1, etc.


To be fair most of the scientists on the Manhattan project were physicists.


> largely unemployable

FWIW, the math major graduates at the college where I teach, to my knowledge, have never had serious trouble getting employment.


This issue is recognized by many professional mathematicians.

An example of a good effort to combat these tendencies is a course by Prof. Bhargava at Princeton, where he introduces good mathematics through card tricks and games: https://www.princeton.edu/main/news/archive/S36/37/98S70/ind...


https://hn.algolia.com/?query=A%20Mathematician%E2%80%99s%20...

... although I don't think we'd heard from a high-school classmate of the author before.


> all without the advice or participation of a single working musician or composer.

Some of the nest mathematicians are the worst teachers. They focus too much on small details, as if they are carrying out their own usual role; and don't intuitively explain the basics concepts that are now too ingrained for them to easily recall the original difficulties they had with learning them.

Music and mathematics are not similar. I also imagine music is more subjective / less standardized. You might be better off with the majority of your time spend with a music teacher than Elton John, for example.


Are there any books with the same kinds of tasks that the author proposed to his students? Would really like to improve my proofing skills.




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