OpenLab Assignment: Lockhart’s Lament

In 2002, a mathematician named Paul Lockhart wrote an essay called “A Mathematician’s Lament,” a passionate criticism of mathematics education in America.  It has become widely known among mathematicians and mathematics educators – not everyone agrees with everything he says (though many do), but everyone seems to have something to say about “Lockhart’s Lament,” as it is called.  For this week’s assignment, you will read a short excerpt (three pages) from his essay and respond to the prompts below.

Assignment (Due Tuesday, 10/20). Your assignment has three parts:

First, read the section titled “Mathematics and Culture” (pages 3-5) in Lockhart’s essay, (click here).  If you’re interested, I encourage you to read more, starting at the beginning – but this is not required.

Second, write a response to what you read and post it in the comments below.  Your response should be at least 300 words. Your response should represent your own thoughts and opinions on what you read, and can include responses to any or all of the following:

  • What is one thing that you agree with in the reading? Explain why.
  • What is one thing that you do not agree with? Explain.
  • Choose one quote that you think stands out in the reading.  Give the quote, and explain why you chose it.
  • Have you ever had an experience of mathematics as art?
  • On page 5, Lockhart describes mathematics in schools today as “heartbreaking”.  What do you think he means?  Do you agree? How do your own math experiences in school compare to his description?

Third, and most important, I want you to write down a conjecture about the Bridges and Walking Tours game, and bring it with you to class on Tuesday 10/20 (do NOT post it here).  Consider Lockhart’s example of a triangle drawn inside a rectangle.  He described the process of playing around with this picture, until he arrives at the basic idea for calculating the area of a triangle.  He contrasts this with a traditional math class, in which the formula is given to students without providing them any opportunity to explore the problem on their own.  The bridges and walking tours game is a little like the triangle-rectangle picture – it’s fun to play around with, but you may not be sure what the point is.  You’ve had a chance to play with it a bit, and try some different challenges.  Now what?  Your job is write down a conjecture (a guess!) or a question about your game.  Write down a conjecture or question about the bridges and walking tours game, and bring it with you to class on Thursday 10/20 (do NOT post it here).

Here is an example: Let’s imagine that you have just been introduced to the game Tic-Tac-Toe.  After playing it for a while, you might come up with one of the following conjectures about Tic-Tac-Toe:
Conjecture: The person who goes first always wins.
Conjecture: The center is the best move.
Conjecture: It’s impossible to win, no matter who goes first.

ps.  Paul Lockhart retired from being a first-rate research mathematician in order to teach math at a private elementary school here in Brooklyn, Saint Ann’s School, where he says “I have happily been subversively teaching mathematics (the real thing) since 2000.”


  1. Allison Kirshner

    In Lockhart’s Lament, Paul Lockhart poses the thesis that “mathematics is an art.” This largely conflicts with my experiences in mathematics, and as a result, my opinions will be shaped by these experiences. One of the key attractions that I find in learning math is that it provides a level of certainty. Unlike most other subjects taught in school, math is largely non-subjective. It is this certitude that I find most appealing. Two plus two equals four, and no matter what anyone says, this is a pure fact, true and demonstrable. It is in this reliability of result that makes math so different from art. The merits of a painting, for example, are abstract. Opinions can and will differ; some may say a work of art is beautiful, others find it ugly, some may see a specific object, while others see an abstract concept.

    While I agree with Lockhart that schools would be better served in posing questions for students to explore, I disagree with his statement that, “by removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject.” Human beings and children especially are naturally curious. By providing students with the facts on which mathematics is based, schools offer the building blocks of knowledge. With these facts, an individual is free to rearrange these “blocks” at will. In the twenty-first century, most students in the United States have the time and resources to explore topics that are interesting to them. Curiosity is a natural phenomenon and doesn’t need to be taught.

    Lockhart offers the example of teaching students to naturally discover the formula for the area of a triangle. I find, and I think many students would agree, that this is insincere, contrived and artificial. If I worked on finding the formula for the area of a triangle for a week in math class and then discovered that a ten second search on Google would have gotten the solution, I would have felt betrayed. Provide students with the facts and the tools to feed their curiosity, and nature will provide the will and drive to discover and expand the limits of their understanding.

  2. Ihn Lee

    I am in the second year at city tech majoring in math education. I thought how to teach math was simply done by explaining concepts and solve problems. However, Lockhart’s essay made me think about how I learn and teach math.

    I agree with his argument ‘By concentrating on what, and leaving out why mathematics is reduced to an empty shell.’ Doing the math to me is using the formula to solve the problems. You can’t possibly think there is an art in math that K-12 and/or even college math classes. All we need is to practice many problems to learn application skills and memorize appropriate formulas. I think what we are learning is an empty shell without the inside part which is art. I don’t ever feel poetic when I do the math while I enjoy doing it. I guess I am satisfied with the empty shell of the math.

    The way I learn math was exactly how he described in his essay. When I was 10 years old, I enjoyed doing arithmetic math even with the numbers from cars’ plates. I remember that I was excited and trying to solve made up problems quicker and quicker. While I was enjoying math, I was able to submit the exam first and receive the best grade in the class. However, learning calculus and the pressure to have a good grade in SAT in highschool made me overwhelmed and questioned myself why I had to study all these. There was no art in it, but everybody including teachers, students, and their parents were heading to the one destination, a good grade of SAT.

    My major was mechanical major because I was accepted with my SAT score in South Korea. However, one of the calculus professors at city tech made change my mind. The professor was so passionate and that made me motivated. I wanted to teach students in the way the professor taught us. So I changed my major to math education. In the class, I was good at calculus. I have been avoiding higher math because I was scared to fail again. But this time was different. I didn’t have to take the SAT test. I didn’t have to memorize and get used to all the equations and problems from geometry, vector, calculus, and probability, and statistics at once. Without the pressure from the test, I was able to breathe and feel the calculus itself. It took years to shake off the fear of calculus because of the SAT test. But how do I teach students the art of math?

  3. Jack Berner

    As a student pursuing a degree in mathematics education, I found Lockhart’s Lament a very interesting read that will certainly influence my approach to teaching in the future. My understanding of mathematics as a discipline and its classification, however, is unchanged.

    Lockhart complains about “the lack of mathematics in our mathematics classes,” in that actual mathematics is a process of experimentation, discovery, and argumentation. Speaking from personal experience, I can attest to the absence of true mathematics in American public schooling. My memories of specific classroom experiences are now hazy, but a general impression remains, made memorable by the repetition of a specific kind of interaction —

    Student: “Why is [blank] true?”
    Teacher: “Because it is.” Or worse, “because I’m telling you it is.”

    There were of course exceptions to this unfortunate pattern, and I wouldn’t have the love for and awe of mathematics that I do today if it were not for the one teacher who encouraged my curiosity and skepticism. But, as they say, ‘the exception proves the rule.’ Many readers of Lockhart’s Lament would likely retort that children and their developing brains are not capable of coming up with and proving mathematical theories. There is evidence this isn’t true though, and that, in fact, fostering such an understanding of mathematics has a positive impact on students’ performance on the kinds of aptitude tests that many believe demand only rote memorization. For example, in Japan mathematics is taught with an emphasis on the kind of understanding that Lockhart encourages. Instead of the classroom model of ‘I, we, you’ that is so pervasive in American classrooms, Japanese math teachers start their lessons off with ‘you’, forcing students to search for and try to explain the potential reasoning in a problem presented to them. Only at the end of a lesson does the teacher reveal something like a formula, or a description of whatever official consensus now exists on the topic. In one study, having taken tests designed to evaluate mathematical aptitude, ten year old Japanese students scored higher than their American counterparts 92 percent of the time. Ironically (and somewhat depressingly), Japanese educators reformed how mathematics is taught in their schools based on the work of American scholars.

    So, I strongly agree with Lockhart’s argument that math education, as it most often occurs in our schools, is destructive or “heartbreaking.” I also see how the cultural stereotype of mathematicians as uncreative drones is a mischaracterization, which perpetuates this problem. However, I don’t think we should squeeze mathematics under this umbrella of the arts in order to fix its public image. I believe that would be trading one mischaracterization for another. I am not denying the creativity behind the process of real mathematics, but the end results – unshakable theorems with meanings so absolute and singular – bear little if any resemblance to the products of artistic expression like music, poetry, etc. If we want to make more people appreciate the value and beauty of mathematics, then we should advertise it in respect to its own merits, of which, in my opinion, there are many.

  4. Denyese Gonsalves

    Like Lockhart, I have always viewed Mathematics as an overlooked art form. In my mathematical journey my own experience of math is the arithmetical relationship of how opposites are related to equivalences. For me to process multiplication and subtraction I see the beauty in knowing that division and addition will help me decipher and imagine the entire operation. However, at times in thinking out of the box and conveying this to a teacher I would be shut down as the teacher may say to me, “What does that have to do with 3 * 6 = 18, why are you telling me that 18 can be divided by 6 multiplied 3 times?” I embarrassingly shy away whilst saying in my mind, 3×6=18 because 18 can be divided by 3 or 6 and 6 is just 2*3. This is exactly why Lockhart has used a dramatic term such as “lament” to describe his anguish. “By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument…Mathematics is the art of explanation…to cobble together their own explanations and proofs, you deny them mathematics itself (Lockhart 5). I concur that this is why I may have issues with computerized courses such as WebAssign? I feel as if I am in a box with WebAssign. You put the answer in, and wrong or right pops up with no explanation. I am not a robot and as such robotic learning does not appeal to me. Not being able to explain my thought process to mathematical computerized programs really irks me. “Now where did this idea of mine come from? How did I know to draw that line (Lockhart 4). Not to bash WebAssign more but I felt that in trying to complete most of the homework, my thought process was reduced to procedure, follow method and type in answer. “This rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” (Lockhart 5).

  5. Jared Cruz

    I couldn’t agree any more with the following statement, “By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.” (5) That pretty much sums it up. I know from firsthand experience, all the way back to the elementary school days we were told by our math teacher, “This is going to be on the test!” That translates to, “We want you to regurgitate everything you have been taught.” Luckily, I was able to pull out the formulas my…head pretty quickly. I considered myself a smarty-pants. Anyway, I wondered I can do all this arithmetic and computational work, but what about it? How is this going to help me in the future? How am I going to remember all of this? I thought.

    The huge issue as worded by Lockhart, “If you deny students the opportunity to engage in activity — to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself.” (5) Basically, I never got a true opportunity to “play” with mathematics nor did I ever truly get to use my “imagination”. An opportunity many students don’t get!

    The fact is that students are being plunged with meaningless facts then being asked to memorize certain formulas and apply them to exercises. (5) That is a huge problem because not only does it slap a boring label on math, but leads to other issues with comprehension. If students regurgitate only what they are given in lectures it leaves many open gaps. There isn’t a concrete fundamental understanding which leads to struggles with higher-level math courses. With that being said, I hope that in my teaching career I will not have to feed my students facts, but have their imaginations ignited and have them genuinely learn something.

  6. Luis lora

    “The mathematics curriculum doesn’t need to be reformed, it needs to be scrapped”.
    I agree with this quote absolutely. The curriculum isn’t working, it’s not working here in New York, it’s not working somewhere else in Wyoming. Mathematics courses are missing so much of what makes a subject interesting. It’s missing the whos, what, wheres, when and why . If it weren’t for my own curiosity my education would consist of procedural displays of monkey see monkey do. This is why passing exams are just regurgitation of material and do not express an individual ability to reason and express their ideas. I can’t tell you how many exams i failed not caring to try because of the unrelenting repetition of it all. Why cant grades be proved using other methods. Can you imagine taking the same formatted test with the numbers and wording changed but everything else is the same?
    “Teaching is not about information. It’s about having an honest intellectual relationship with your students. It requires no method, no tools, and no training. Just the ability to be real. And if you can’t be real, then you have no right to inflict yourself upon innocent children.”
    This really stood out to me because I agree that a teacher needs to be somebody you can have a conversation with . Math can be relatable to pretty much anything but when it’s fused to correspond to your students’ lives then that is a conversation worth having. Yes its tricky to relate but that’s where having an open creative class helps. That’s where having conversations not just about math helps. Students don’t need an Einstein but more of a Feymann in the classroom surely I’m not joking .
    I never experienced math as art in any schooling.Although when i’m at home i love looking at fractals and any geometrical image. I think this is where math becomes art. Not only in its shapes but its configuration. I used to be a carpenter so one could say building a house using all the knowledge from trig, geo and algebra is creating art or not.
    I think Lockhart was correct in this interpretation of a heartbreaking system.”The main problem with school mathematics is that there are no problems. “Here is a type of problem. Here is how to solve it. Yes it will be on the test. Do exercises 1-35 odd for homework.” What a sad way to learn mathematics: to be a trained chimpanzee”. I think Lockharts understands there is a problem with the education system and it involves this criteria. A lot of classrooms are run this way and students tend to suffer the worst from them. This description is fairly accurate as I look back on my own learning. Was this really learning and why was this allowed ?

  7. Bless

    The passage is about a comparison being made by a mathematician, Paul Lockhartment, between mathematic being categorized as art and those that are considered as art. Furthermore, he mentioned some negativities of how math being taught in schools.
    From the beginning of the passage he stated that “mathematics is an art and furthermore he said that mathematics is the purest of the art.” I disagree with that statement. In my perspective, mathematics is not an art, it worth more than that. I consider mathematics to be the foundation of everything because everything we do has a little bit of mathematics. For example, the artist must know how many quarts of paint needed to color a drawing. The poet must know how many lines in each stanza. Imagine living in a world where mathematics is considered as art, how would that be?
    One thing among all those that I agree with in the reading is the following “There is really nothing else quite like this realm of pure idea, it is fascinating, it’s fun and it’s free.” I find this quote to be extremely interested because it is somehow defining the beauty of mathematics. In my perspective, I think mathematics is more than a realm of pure idea because it is kind of grasp almost everything. Nowadays, the development of technology makes mathematics more valuable than before. Though, there is no technology without mathematics because it is used to develop the modern sophisticate technology. Moreover, the same way watching a paint or listening to music is fascinating and fun; it is the same when solving a math exercise. To me, what makes more excited is because mathematicians always start with an imaginary idea and conclude it to be real.
    In my understanding, I think it is a way for Lockhart to express his emotional disturbance of the way mathematics is being taught in schools. I sort of agree with him because some teachers, especially in high school, are not qualified to teach mathematics. Instead of helping students to understand the concept, they encourage students to memorize their thoughts. Meanwhile, comparing my own experience to his description, I think gradually mathematics starts losing its true value. When I was in high school, mathematics was taught in a way that student could develop their own understanding and make their own discovery. However, it was not good enough, they were negatively criticized the strategy being used to teach mathematics in schools. I think if nothing being done, mathematics will be taken over by technology.
    Moreover, I have had ever an experience of math as art but I will not be surprised to see that it’s being categorized as art. Overall, the idea of that short passage is to not only prove mathematics as an art just like paint and music and etc., but also to warn the concerned people of not losing the quality of how mathematics should be taught.

  8. Matthew Schwartz

    One thing that I agree with in the reading is the quote cited by GH Hardy, that mathematical patterns are more permanent than painters or poets. I agree with this because I believe fundamentally that everything at it’s core is in fact mathematics, so therefore the patterns of painters and poets are simply imperfect human combinations of deeper mathematical patterns.

    I do not agree sentimentally of what Lockhart is trying to get at here. I do believe that there is an art to the discovery of mathematics, but that is not because mathematics itself is necessarily an art, but because of our human constraints everything first has to move through our emotionally driven animal brains. A mathematical pattern either exists or it does not exist, regardless of whether or not that pattern describes anything in our universe. As Max Tegmark says, “in the multiverse, all mathematical structures are realized” , meaning that we can develop maths in this universe that has no contact with reality, but somewhere, it does, at the very least on paper. And yes, that is beautiful, it is one of the most beautiful things I can concieve of, and yes, it requires much creativity and abstract thinking to come up with these concepts- I myself find myself to get along well with many artists, sometimes more so than people working in other STEM fields, but I would argue that what they are doing is just a version of mathematics, as opposed to we are doing something more like art. Everything is math as far as I’m concerned.

    “Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical,
    subversive, and psychedelic, as mathematics.” – I chose this quote because this is something that I feel completely and perfectly describes my experience every time I learn something in class that I then get to graph in a python script.

    I consider math to be the most beautiful and purest form of art (in a certain way) when I discovered that the mandelbrot set can describe the logistic map, which can describe population modeling in the real world- I see a kind of art that was created by nature herself.

    I think that he way maths is taught (at least pre college, and some college classes) is completely heartbreaking. Personally I had professors that drove me away from maths. It is heartbreaking because all many children are taught is to memorize formulas, and this makes children hate maths. I recently watched “the code” on netflix with my girlfriend, who absolutely hates math, but after seeing the connections it has between music, socitey, culture and nature, she said to me “if he was my math professor as a kid I could have been a mathematician right now”.

  9. Irina Chernyavskiy

    I have worked in a few different areas of education for quite some time so I definitely have seen time and time again students wanting to only learn enough to pass a course or not really knowing (or caring a lot of the time let’s face it) why the subject of math is useful or relevant.There is no real curiosity or interest there it’s just application and memorization of facts and formulas. The essence has all but erased for everyone but a select few as Lockhart stated “By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject.” as it stands now mathematics exists as a gateway separating students based on how well they adapt and apply knowledge.

    I disagree with Lockhart on considering math an art. While creative problem solving is involved, math has been used as a tool. I certainly consider it as such and one that is important in our daily lives and survival. For this reason everyone should have a chance to experience what mathematics really is. Students should be introduced to mathematics slowly with an emphasis on basic blocks as a foundation then building upon that knowledge. Everyone can learn even if some students don’t think so they can they just need the opportunity it is truly “heartbreaking” as Lockhart put it that the state of math education in the United States is such a shell.

  10. hanzhang

    I agree with the author that mathematics is art. In our culture, we normally thought of art is music,painting,poets that we learned, but just make math outside of the art ? why ? perhaps people think there is only number that created at the first beginning of universe and it is not so sensitive or virtual crushing ideas that make us feel art. But I agree the mathematics is art that we need natural curiosity and love of pattern making with math. This is also a creation and art feeling.
    I really like the comments of the math as follows: “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas. “The words: “patterns made of ideas” that raise my passion of mathematics. When we did math, I am also feeling myself an architect who want to show flowing works of art. It creates a sense of accomplishment.
    “Mathematics that is not following the direction, it is about make new direction. “I partly agree with it. Because if you want a pupil to create, that is impossible because he has no ideas of mathematics ,how could he make new directions. I thought the students should learn the ideas under the directions. When they get enough “ideas”, they can make the patterns of their own and also create new directions.
    I do not agree with the author is that he thought the temporary mathematics education is hopeless nightmare. As I know, in some school , the math teacher has experiences in art teaching they sometime teach the kids both math and art.because I thought nowadays, the situation is improving.

  11. Christopher Cruz

    Lockhart’s introduction I felt was key to his argument. The idea that a strong foundation is needed for any successful artist. Artist meaning the usual (Painter, Musician, Singer,….) and even the unusual artist( the mathematician)

    Lockhart’s statement that “Mathematics is an art” almost drew a tear to my eye. I am often looked at weird when I say an equation is beautiful or the route to solution was incredible. Like fine art, I find that math requires a certain eye to attain its true beauty. Whether a multistep problem we solve ourselves or seeing a connection form in people we may assist in math.

    When Lockhart described mathematics in schools as being heartbreaking I feel he is mentioning how poorly math can be presented in schools nowadays. While some students may like the subject early on in their academic careers , they often claim it gets too hard in later years and begin to dislike the subject.

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