A Mathematician’s Lament (2002) [pdf]

This is how people think of Math class:<p>&quot;Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.&quot;

I actually disagree pretty strongly with this essay. I think Lockhart is advocating for an educational approach that would cater strongly (and exclusively) to a particular learning style, specifically, those learners who thrive when going from abstract to specific (deductive learning).<p>For those of us who do best with <i>inductive</i> learning, the type of education he proposes would bring even greater misery to our grade school education. It was not until I took statistics and probability (for science and engineer majors) in college that I truly began to enjoy math once again. I was able to start with concrete ideas and applications, and then work my way back to the theory behind them.<p>I&#x27;m now reading &quot;Concrete Mathematics&quot; and really enjoying it. Knuth&#x27;s ideas on math education are pretty diametrically opposed to those of Lockhart, as far as I can tell, and they give rise to something very close to my ideal learning environment for math.<p>Why not allow children to follow which ever of the two math paths that is best suited to them, instead of forcing concrete thinkers into an abstract world, and abstract thinkers into a concrete world?<p>Edit: I should probably add that I agree with Lockhart that there&#x27;s a problem in the way math is taught, but I disagree with him on the solution.

I loved mathematics throughout school. I don&#x27;t know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.<p>That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I&#x27;ve learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I&#x27;ve studied makes it sound like classical music does to me.<p>With math, I don&#x27;t really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don&#x27;t know if that&#x27;s irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.

A beautifully written and tragic essay.<p>(Note: What follows is US-centric.)<p>After nine years of teaching mathematics courses (one semester as an undergraduate, 4.5 years as a graduate student, and 4 years as an assistant professor) and navigating university politics, I&#x27;m convinced that this is, at its heart, a cultural issue.<p>There&#x27;s a hatred of mathematics in mainstream American culture that runs very, very deep. And it will probably take generations to change that (if changing it is even possible at this point).

Such a wonderful essay. It&#x27;s very applicable to the teaching of computer science&#x2F;software engineering as well. So much of the problem is the misunderstanding people have about the field. It&#x27;s a creative, constructive discipline, and so much of the instruction is consumption, mimicry, and repetition.<p>Solving well defined problems is relatively easy. Our real problem is that real problems are not well defined.

FWIW one of our professors here at uwaterloo taught a first year abstract algebra &#x2F; number theory class in a very Lockhart-esque way (Math 145; he even quoted Lockhart on one of the assignments). I learned a lot of math and enjoyed myself, but the main problem I observed was figuring out how to fairly grade students, and the fact that the homework took a lot more time than a class taught normally.

I have a lot of sympathy for his point of view. I loved Math growing up. High school drove the interest out of me, and I didn&#x27;t get it back until senior Calculus, when I started doing well again. Then I learned to appreciate CS theory, economic theory, etc. Trying to figure out how to break the cycle for my kids: Stats for practical work, and math for curiosity.

A very similar piece by V.I. Arnold[1]: <a href="http://pauli.uni-muenster.de/~munsteg/arnold.html" rel="nofollow">http:&#x2F;&#x2F;pauli.uni-muenster.de&#x2F;~munsteg&#x2F;arnold.html</a><p>[1]: <a href="http://en.wikipedia.org/wiki/Vladimir_Arnold" rel="nofollow">http:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Vladimir_Arnold</a>

As a Physicist, I feel obliged to mention that black holes were first hypothesised by Physicists (contrary to the essay), albeit through Mathematical enquiry.

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.<p>However I don&#x27;t see why they couldn&#x27;t start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.<p>Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.<p>Geometry, if I recall correctly, was one of the exceptions in early math where you are allowed to veer off the path a bit. Is everything else algorithmic until college?

<a href="https://news.ycombinator.com/item?id=6187014" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=6187014</a><p>With that being said, I&#x27;ve always loved this essay. As of recently, I&#x27;ve viewed it as relevant to the recent argument that programming should be a requirement in American public schools, either as a tool in math and science classes or a free-standing course. This kind of mathematical reform might actually be a prerequisite for programming and computer science, given that it would develop mathematical maturity much more effectively than the current system does.

I think for most people, math is best learned in the context of some application that they care about (the last four words are very important). Few people appreciate the beauty of the abstract game itself.<p>For example, most people who play poker online quickly learn about expected value, probability, and variance.

HN is big on curated lists lately. Is there one for resources to aid in teaching mathematics the way Lockhart would prefer?<p>I&#x27;ve seen this posted so many places so many times that surely there&#x27;s a market for materials and support for it. Where are they?

Lockhart has a book, &quot;Measurement&quot;, that elaborates in the same vein. Quite nice.

I loved mathematics throughout school. I don&#x27;t know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.<p>That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I&#x27;ve learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I&#x27;ve studied makes it sound like classical music does to me.<p>With math, I don&#x27;t really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don&#x27;t know if that&#x27;s irrational reasoning, but I know that understanding math correctly is hard, and writing buggy programs is easy.

I loved mathematics throughout school. I don&#x27;t know about anyone else, but I can see quite clearly how mathematics has shaped the way I think and form concepts, ideas, and understandings of my perception of the world. I would say that mathematics is foundational to my perception, if there exists anything that serves as the base way I interpret information.<p>That said, I paint and I hated painting classes. I hated almost every art class I took. I paint okay, but painting is more about getting rid of negative emotions for me, than anything. I never really liked piano lessons either, I prefer to gain a small ability and spend years perfecting it with a combination of the few I&#x27;ve learned and perfected, into various impromptu permutations. I guess some people call this jazz, but all the stuff I&#x27;ve studied makes it sound like classical music does to me.<p>With math, I don&#x27;t really care about creating it. I just want all of the math in my head, with the right understanding of it, because I think that makes me a better computer scientist and software developer. I don&#x27;t know if that&#x27;s irrational reasoning, but I know that understanding math correctly is hard, and writing code is easy.

Love the bit about the misconception that Mathematics is mainly about utility. I remember reading something about G.H. Hardy (which I can no longer find) in which he said he would get a little bit disappointed if he found that one of his results ended up finding a practical use.

Vital context: <i>The Underground History of American Education</i>, free online: <a href="http://mhkeehn.tripod.com/ughoae.pdf" rel="nofollow">http:&#x2F;&#x2F;mhkeehn.tripod.com&#x2F;ughoae.pdf</a>

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.<p>However I don&#x27;t see why they couldn&#x27;t start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.<p>Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

I often hear the excuse that mainstream courses like Algebra and Calculus are taught first and in a boring way because you have to learn mechanics before getting to the good stuff.<p>However I don&#x27;t see why they couldn&#x27;t start a Calc course with one of those cool documentaries on Newton. For me it was incredibly motivating to hear the questions that drove the theory.<p>Beyond that it seems certain classes like discrete math or combinatorics might allow more creativity and experimentation in secondary school without requiring a ton of foundation.

If the musician woke up in the first sentence, how is he still dreaming in the next one?

My favorite part is &quot;The Standard School Mathematics Curriculum&quot;.

Always a good read.