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 Thursday, 10/21/21). 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 Thursday 10/21/21 (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 10/21/21 (do NOT post it here).

Here are some examples: 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. setsandstripes

    From reading Lockhart’s Lament, the quote that resonated with me the most was “things are what you want them to be”. This concept makes me think and I have yet to put those thoughts into words… I appreciated Lockhart’s thoughts because I’ve recently realized through my studies that I view math as an art form, too. With an emphasis on memorizing formulas and timed exams, and a competitiveness to getting to the “right answer”, I’ve found that modern math classes focus more on form than substance with the outcome being that nearly all students forget what they had “learned” shortly after the course ends, even after getting an “A” in the course. I’ve recently thought about that a lot lately, since I graduated undergrad about ten years ago, I couldn’t tell you much of what I learned. I had to retake both pre-calculus and calculus this past year because I couldn’t remember anything from undergrad, and it brought to mind the emphasis on memorization and number crunching rather than the creative and challenging work of seeking to understand not just what we were doing but why we were doing it. Now being a little older and wiser, I was just yesterday thinking about wanting to better understand why we were using certain integration techniques in calculus. I think over the past few decades our society has established a desire for instant gratification and also with this information age the belief that all answers have already been figured out. I also think critical thinking is challenging and it’s likely perceived as easier to not have to expose oneself to the wins and losses of a more artistic and creative approach. In college, I learned a foreign language abroad and found that I benefited more from speaking the language while out in the local community than in the classroom. Despite how frustrating it was at times to struggle in front of native speakers, those mind-stretching moments were what strengthened my ability to truly understand the language, rather than just merely memorizing it. In closing, I too would like to see a greater emphasis on the creative elements of math, so I’m happy to at least know that others are thinking the same.

    • Jonas Reitz

      Great! I think of one of the big challenges as a prospective math teacher is to try to figure out how to give students the experience of “playing and exploring” math rather than just trying to learn the shortest way to a right answer. I’m guessing this is a lifelong struggle (with inevitable successes and failures)… keep at it, and good luck!

  2. Rachel J Dawidowicz

    To start, I find it interesting how he mentions that people who don’t study math don’t see mathematics as an art. That is something I disagree with. Maybe I am biased as an art enthusiast and musician, but if you look at the Mondrian Puzzle, it is a mathematical puzzle that challenges congruency, and the artist Wassily Kandinsky used mathematical concepts in his art. If you’d want to argue music, Bach’s Crab Canon when twisted takes the structure of the Mobius strip. It’s not that people don’t recognize mathematics as an art, it’s that they are unable to see mathematics behind the fine & performing arts.
    I do agree with his statement “That’s what math is— wondering, playing, amusing yourself with your imagination.” Many mathematical (even scientific) theories we take for granted today, a civilization or even a person had to sit down, experiment, and wonder about the properties of mathematics to come to a specific conclusion.
    That’s where I agree that math is an art. Artistic expression requires imagination and experimentation much like mathematics.
    As Lockhart calls mathematics taught in schools ‘heartbreaking’ I understand why. In the contemporary era of education is standardized testing. So the teacher quickly has to provide algorithmic approaches to make math easier to boost test scores. Students only get a surface-level appreciation of mathematics and are unable to see the bigger picture.
    Hear me out here, this is why art education is so crucial to our education system. Yes, the art teacher can’t necessarily explain the mathematical concepts that are in cubism or some abstract art, but if students have that prior art knowledge the math teacher can show Kandinsky, Brock, or Picasso in a geometry class to show where math appears in art.
    In a more advanced math class showing the drafts of Frank Loyd Wright can also spark that curiosity. Even showing how graphic designers use (x,y,z) planes to create animations all show that math is an art (or even an interchangeable application to both).
    The biggest problem is that schools teach math not only at surface level but only use what I’ll call ‘scientific and mathematical applications.’ If a math teacher were to teach ratios, even if students cannot completely understand the mathematics behind it, going to home depot and buying various size tubes can show students where ratios apply in music. (And even students can play with the contraption)
    As a programmer who is still learning Maya, using math to make the ideal surface of some three-dimensional shape, both art, and math. And lastly, I can also say that every day I experience mathematics in art. Sitting in my orchestra as I simultaneously match the time signatures with the notes, or notice that the more keys I press down on my oboe the lower the pitch, the less the higher, or even how the tautness of my lips against the reed affects the pitch of the note. The two are interchangeable, but I realize that math is an art because not only because I love nothing more than a day at the Met, but also I love to explore the deeper realms of mathematics.

    If anyone is interested in the art vs math argument
    Mondrian puzzle

    Crab Cannon

    The art behind Vangough’s ‘the starry night’

    • Jonas Reitz

      Wow, I’m loving the video on Mondrian already! I had always been aware of his interestingly-rectangular artwork, but didn’t realize how much was going on under the surface – and there’s a really nice open problem here, in figuring out the function that gives the “best score” for an nxn grid. I’m looking forward to watching the other videos you recommend.

      Since you’re interested in math, art and music, you might be interested in the book Godel, Escher, Bach by Douglas Hofstadter. This is a sometimes technical, sometimes philosophical, always fascinating look at ideas involving self-reference as they appear in a wide variety of different disciplines (very few people actual finish reading it all the way through – but people find it inspiring nonetheless).

  3. suerissberger

    One quote that stands out in the reading is that mathematics is the art of explanation. Lockhart talks about the importance of giving students opportunities to make their own discoveries and creating spaces to ask questions, be wrong, and try again. I couldn’t agree more with this and think that making mistakes and learning from them helps cultivate better math students and a better understanding of math. In the classes I teach, when a question such as, “Triangle ABC reflected across the y-axis to get Triangle A’B’C’”, we follow up immediately with, “How do you know?” It helps them consider their answer and why they didn’t answer with translation or rotation, and reinforces their own understanding of it when they’re asked to think about how they arrived at a specific answer or conclusion.

    In terms of experiencing mathematics as an art, what I’ve found interesting and heartbreaking is that when students are given flexibility and room to “play”, they often seem to tense up and freeze in their work. I suppose this is because by the time they reach high school, most of their mathematics education has been more rote and procedural, which does have its strengths, but this type of learning creates a student that relies on rules and procedures to “try” at math. When given a discovery task or the opportunity to “play” with something, to create 3D figures with clay, they need a lot of support in order to get started and do the work. So the heartbreaking that Lockhart refers to is the idea that students often receive a series of facts in schools, often devoid of real context (just stating for example, the associative or commutative properties), and absent of any activity or task that leaves space for students to “figure out” math on their own.

    • Jonas Reitz

      I really hear your comment about students not always being ready to dive into “free play” around a math idea. My own experience is that you need to scaffold such play, and carry it out in relatively low stakes assignments with sufficient time to explore (and without pressure of finding the “one right answer”). But this is not always easy, with the pressures of time and a fixed syllabus to cover!
      ps. Our class project on “Bridges and Walking Tours” is an attempt to do just this…

  4. Angie

    While reading one quote that stands out to me was “ By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.” The author is implying that nowadays no one knows the reason or explanation as to why things are the way they are. Which is why some people find math useless and boring, because they don’t go through the thought process nor have a chance to create their own art in math. Honestly, this is something I never paid attention to until now because I always thought math was about just memorizing a formula but it’s more than that. On page 5, Lockhart describes mathematics in schools today as “heartbreaking”. I think he means that math is broken, the generations now will never experience how math was before. I agree because students in the present day are just given a formula sheet with formulas to memorize and don’t get a chance to develop reasons on their own. My own math experiences compare to his description because high school dealt with a whole bunch of formulas, but barely any proof as to why and how it came to be. Which is why I believed in math all you had to do was memorize the formulas and you can solve any problem handed to you. Which I feel is slightly true, but I also agree that math is about using your imagination. Like when we are doing proofs, we have to think about why something is true, and what steps someone has to take to come to that conclusion.

    • Jonas Reitz

      My own math experiences were really mixed – most of them involves rote memorization and blind “plugging-and-chugging”, but i was lucky enough to have a couple teachers who pulled back a corner of the curtain so I could peek at the magic behind. Sometimes one good experience is all it takes!

  5. Jaroslav

    I would start with the quote: “Nevertheless, the fact is there is nothing as dreamy a poetic, nothing as radical, subversive, as psychedelic, as mathematics.” – Dreamy, poetic, radical, subversive, and even psychedelic?
    I had lived for almost all my life with the understanding that math is for rational thinkers only, not dreamy fellows or even psychedelic folks. All my teachers from K-1 to K-12 had made me believe that. “Didn’t you study Newton’s law of motion? So sit down! You fell!”.
    Then the shift in my understanding of math came in a private conversation with my adviser at CityTech, Professor Kennedy (one would say quite late). She said something like (I apologize that it is not an exact quotation): “There are many ways how to reach conclusion in mathematics, not only one. It would bore me to memorize formulas and apply them with a surgical precision. Math is pretty creative endeavor, free in ways of solving the tasks, which requires imaginative ways of thinking and intuition.” It gave me a kick. It was revolutionary!
    This semester I started to learn The History of Mathematics in MAT4080 and also MEDU 1021. I was surprised and astonished to withes how many various approaches there are. How the Ancient Egyptians, Old Babylonians , Chinese, and Indians had been solving multiplication, quadratic equations, and geometrical shapes – it was just fascinating to study. And in the other class, MEDU 1021, we learned about contemporary method that students in Japan are using for visual multiplication, and Laplace grid, and Staircase, Towers, and Castles, and … I realized that Heron’s formula gave me the same answer for obtuse, scalene, equilateral, … triangles as Euclid’s, that the Pythagoream theorem existed in certain form thousands years before Pythagoras, that Ancient Chinese mathematics figured out the enigmatic “pi” before Archimedes, etc. Math is really very creative. I think that once we realize it, it would not be any “heartbreaking” torture of students as it might be now, but an adventurous play with numbers, symbols, and conjectures.

    • Jonas Reitz

      I love your story about my colleague Prof. Kennedy – I find her so inspiring as a math educator! And your comment that math can be “an adventurous play” really resonates with my own experience 🙂

  6. Nina Melendez

    In reading the section on “Mathematics and Culture” in LOCKHART’S LAMENT I felt a wave of anxiety, maybe even a level of discomfort, because I agreed with so much in this section which caused me to be genuinely sad. I agree that in teaching and consequently learning mathematics we focus more on the WHAT; what is the solution?, what are the steps?, what is the formula?. This does lead towards forgetting the WHY and takes away from that discovery of patterns. Mathematics is structured like a machine where things are learned mechanically without understanding why. I do think that making the connection between mathematics and other forms of art serves to explain in which realm mathematics should lie, but I do think this explanation falls short. Looking only at the freedom that is afforded by culture to other arts, while ignoring the respect that is afforded to mathematics, further disjoins the link between these arts. I do agree that the discovery and creativity of the of the “why” in mathematics and consequently in all arts is one of if not the most valuable component cause it was allows us to discover new things. This being said whether it be society, culture, human nature; the “what” sometimes gets more of a spotlight especially when that what is focused on a finished product, which could range from a painting, composition, or solution. Again, reiterating my sentiment expressed at the start, it gives a sense of anxiety. Having a background in visual arts, mathematics, my short lived stage career(being in front of an audience is not for me), and also my 9 year stint with piano; I have always found that mathematics is an art form and it is so obviously so to my that I struggle to explain it to others. As a math teacher & art teacher I have seen many times my students struggle with the fact that I like both “art” and math. To them you are either an art person or a math person and if you are both you are lucky. Explaining and showing them what comes so naturally to me(thanks to the exposure I had to both through my entire life) is a challenge. Students are so used to “these are the steps to solve this” that when given the opportunity to explore it is more a shock than a welcomed challenge.
    I guess the difficult thing to grapple with is what is the solution? and if there is a solution will it be met with resistance.

    • Jonas Reitz

      One of the best ways of breaking down the idea that math and art are separate (and totally incompatible) things is just by being someone interested in both – especially if you’re in a position (like a teacher) to share your passions. And don’t get discouraged – change is slow.
      Keep up the good work!

  7. anik

    In the easy “a mathematician’s Lament”, pail Lockhart considers mathematics as an art and he also tries to justify his claims throughout his piece of writing. I completely agreed with him and I also believe mathematics is just a different type of art event though society doesn’t consider that. I believe any type of art like Music and painting allows us to be more creative because there is no right and wrong and you can create wherever you want based on your vision and creativity. Mathematics is not so different because In math you can not only create problems by yourself but also solve the problems in so many different proper ways. Honestly, I can’t just disagree with any of his claims and opinion because as a student I also feel like most people really don’t understand the real beauty of mathematics and they only care about getting better grades in their transcript, so in the future, they can have better jobs and more money. Nowadays in High school, most of the teachers recommend their students to memorize formulas and blindly apple them to solve problems and I completely agree with the writer. If mathematics was all about memorizing to pass the classes then there wouldn’t be any fundamental differences between math and history. Throughout my student life, I have always tried to understand the problems before I apply the formula and finally, I choose mathematics as my major. One of the quotes I liked from this piece of writing “A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”. I also believe that ideas will stay in our memories for a lifetime and I will be passed to the next generation which makes mathematics much more significant for our development because mathematics is all about ideas that generate from our imagination or deep thinking.

    • Jonas Reitz

      I really like the quote you chose – I love the idea of being a “maker of patterns,” and likening this to other arts. Sometimes (like MC Escher, a great artist and math enthusiast) it is literally true! And ideas, although they are hard to get your hands on, have the potential to last forever. Great!

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