OpenLab #5: 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, 10/12/17). 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 10/12 (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. If you could have one question answered about your game, what would it be? If you wanted to be a master of your game, and be able to solve any challenge that was given to you, what would you need to know? Write down a conjecture or question about the bridges and walking tours game, and bring it with you to class on Thursday 10/12 (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 corner 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. Kelly

    Mathematics may be beautiful and require creativity, but that does not make it art. If that were true then every field of study would be considered an art. History, sciences, and accounting all require or possess these two traits. The concept of art would be made irrelevant because everything is art. It seems that Lockhart has confused the concept of creativity and imagination with art.
    All of life requires creativity and imagination, whether we are aware of it or not. Every time we read a menu we must imagine what the food will look, taste, and smell like. Is that then considered art? Furthermore, an artist has no limits as to medium, topic choice, or format whereas a mathematician encounters limits at every turn. 1 + 1 = 0 must be true. These sorts of limitations would make math an extremely limited form of art if it were, in fact, an art form. Other arts may be limited by requirements of the genre, but those rules can, and often are, broken. More so, artist are often celebrated for breaking those rules. Would a mathematician who created a new and complicated proof be celebrated if s/he broke the rules of math? What if s/he said that –(P^Q) does not equal (-P)v(-Q)? Do we still accept the proof or do we reject it? Art has no limits, whereas math encounters limits around every corner. It is these limits that help to prevent math from being considered a form of art.
    However, I do agree with Lockhart’s assessment about the “heartbreaking” state of math in schools today. Math has been taught by rote for decades. “Learn these equations and do these problems” has been par for the course in public school systems. The idea of creativity is rarely mentioned in math courses. Sure, many math teachers will try to use problems that involve real world examples, but for the most part math is presented in a dry and methodical way which eschews creativity for rote knowledge. There certainly is room for creativity in math and math education. Often that creativity comes in the form of the teachers attempt to draw the students in. How the teacher engages the subject and the student is where much of the creativity will come into play. It’s this interaction that will help to draw the students in, as much as the course material, and help to stem the heartbreaking trends we have seen in the past few decades.

  2. Evelin

    After reading the article, I completely agree that mathematics has become like an empty shell since students are concentrating more on the “what” portion rather than the “why” portion. Students are just fed all types of formulas and are instructed to memorize them instead of being taught exactly why or how that formula came to be or when to apply them. That doesn’t guarantee that the student fully understood the concept more like that the student just understood it at that moment, but it is not likely that the student will know when to apply the concept in real-world situations.

    I think what the author means when he described mathematics in schools as “heartbreaking” is that it’s sad or upsetting to see that students are not giving the freedom to explore and be creative when it comes to mathematics. I agree with him, it is upsetting to see students not being given the opportunity to be able to “play” with different ideas. In my experience with mathematics, I wasn’t able to explore the concepts. I didn’t even know you could explore the concept, I always thought that formulas were just facts and that was that. In my geometry class, where there’s a huge field of exploration, was just reduced to memorizing all of the theorems. Which obviously led to me forgetting everything in geometry. It wasn’t until when I took my MEDU courses, when I was given the opportunity to explore and fully understand why. My professor had us come up with our own conjectures by using different hands-on objects. I was so used to doing the “what” portion that building up the concept from scratch was so confusing. It was even a bit frustrating to not know whether I was doing it correct or not. These courses really put me out of my comfort zone, but it definitely helped my understanding.

  3. Stephanie Cuate

    One quote that I think stands out in the reading is on page 6, “Math is not about following directions, it’s about making new directions.” I chose this quote because math is much more than just knowing your formulas or knowing what steps has to be follow in order to get the right answer. Math, to me, is about having the chance to be creative and finding your own ways to solve a problem. Which is a reason why I enjoy math because I’m able to be creative at times, since in my mind I could picture so many different ways to solve a problem that could or could not work. And creating these new methods or ideas, will help see math differently or even better, it’s a way to learn something in math or maybe all. Students shouldn’t be told that that’s the only way to solve this problem, instead teachers should inspire them be more creative and find new ways.

    On page 5, Lockhart describes mathematics in schools today as “heartbreaking.” What I think he means by this is that math is not being taught where students are able to learn deep about a topic, instead teachers are just giving students “facts” to remember. Teachers just want their students to memorize formulas and procedures that have to be followed, without exploring deep in mathematics. Which to me seems like students aren’t really learning anything. I agree with Lockhart because I have had teachers before that expect you to know what to do just because they have given you a formula which you just have to plug in numbers into the formula to get the answer. But doing this won’t help the students see the idea of math, that may or may not lead students into being creative or inspire in math. Students are not understanding the main reason why we need to use math in life. Teachers are just throwing information to students without motivation them, they simply have to memorize and apply them. This could be a reason why many students end up disliking math because they feel like they aren’t good at it, since they aren’t good at memorizing all these formulas. This shouldn’t be the case that math is becoming “heartbreaking” in school, we should somehow chance this where students are able to have time to be creative and to have time to explore in mathematics.

  4. Miralia

    In the article,” A Mathematician’s Lament by Paul Lockhart.” He describes the state of math education, focusing on school mathematics, how math should be more inspired. Lockhart states, ”Mathematics is the purest of the arts, as well as the most misunderstood.” I agree with that statement because mathematics should be something that students can be used to develop and express their own imagination and also a way to be aware of their own capacities as they are moving forward in their sturdy. However sometimes it is not the case as Lockhard mentions on his article because students learned in a way that prevent them to express their own ideas. Since teaching math seem to be more about taking notes and do as the teachers told. This type of issue make the students find math boring sometimes. I remember when I was in high school I never considered math as an art until I read this article. I didn’t have the chance to bring my own ideas since my school years was more about taking note do what the teachers told me to do. In the article Lockhart describes mathematics in school today as a “heartbreaking.” I agree in some how because the way some professors or teachers are teaching nowadays is totally confused and also boring. students need to develop their own imagination and challenges themselves in order to be able to come up sometimes with their own ideas which is something that could really help them. However, it is not the case sometimes for them because some teachers require them to do what they give them and that can also prevent some students to develop their own knowledge.

  5. Yasmine

    When Lockhart describes mathematics in school today as “heartbreaking”, I think he means many teachers these days do not teach mathematics the way they are supposed to. They tend to build the boundaries that prevent students from exploring and discovering the reality of mathematics by throwing formulas without engaging students to do the real mathematics. I agree with him and I feel really sad to hear many students complaining about mathematics more than any other subject. That’s because they do not get a chance to explore the joy of mathematics. On the other hand, there exist teachers who really make math fun but they are few.

    I had good experience with good math teachers out of The United States, but there are still a lot that need to be learned and clarified. There are many concepts that I thought I learned them in high school, but when I started college I realized I did not quietly learn them. Let consider which I thought I learned it few years ago and it is about 3.14. But I did not quite learn it till I started this class and another math education class where I found it easier to understand and to apply.

    I agree with him when he says that giving students formulas and making them applying these formulas over and over again there is no problem left to be solved. As I learned from my math education class, when we ask students to use formulas directly to solve problems we do not really ask them to think cognitively because everything is given. These kind of questions are considered to be lower level of cognitive demand because they required students to use what they memorized instead of thinking and creating their own way of solving in order to achieve higher level of cognition.

  6. Josvenia

    Reading this was as though this man took many of my thoughts and put it into paper. I love that he saw math as an art because I have always seen math as a puzzle. I big puzzle you put together to get this beautiful picture. Which is why I agree with him in math being the art of explanation. It’s the art of explanation because you can explain this puzzle or this art piece in so many ways and be correct.
    The one thing I disagree with is the explanation of the triangle. I guess I didn’t really understand how students would get that putting a line would mean that the triangle takes up half the box. You can let your students have a creative mind while teaching them at the same time for example. You make an exercise where the students have to figure out how much of the triangle is taking up the space of the box. And from there you teach them the formula. Somethings in math you can’t just throw it out there and expect students to understand or know it, especially when they have been taught for years that is the point of math to do computations.
    Picking a quote is rather difficult because I feel as though I relate to all of it. It makes sense and he is right in many of the things he is saying. “This rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be
    memorized and procedures to be followed.” I truly believe that this is why students hate math. Because it’s not fun, it’s not creative, it’s not their own it’s something already given to them and ask they have to do is memorize. Although I am not a fan of college math for this reason, we can see that in calc 2 and 3 you enter a whole new world of math full of pictures and puzzles that need to be put together to be able to understand what’s in front of you and when you finally understand it you love it. This is what math should be not facts but your creative thinking brought together with facts to build a beautiful picture. Growing up I was always good at math and I loved it but I learn to love it more when I took trigonometry. The teacher would always tell us that this was a puzzle and when we would put all the pieces together we would be able to see the beautiful picture. And she was right every time I’m able to understand and learn something new in math I feel excited because I am able to add it to my list of beautiful puzzle pieces. Also seeing things as a whole when we finish the chapter or thinking of how it looks when I am done also makes me see it as a beautiful painting because this little section when put together looks beautiful.
    I agree with him up to a certain point. I honestly think that the best thing that happened to our education system and out schools is that we got a different curriculum. Everybody hates the new curriculum because it’s not something that is understandable but it teaches students to look at math in a different way, to be able to explain math, and also to try different ways to solve math. But at the same time I agree with him. My experience and many people in this country is a terrible experience with math. They need to memorize all these things and they can never get the computation right. Understanding it doesn’t matter as long as you can do the work and you know the equation. Still today teachers treat math like this and don’t care about the students trying to enjoy what they are learning but are more worried about just getting the material through your head and giving a lecture.

  7. Sonam

    Mathematic is one of the most important subjects in worldwide. It is a subject that we starting to learn since kindergarten and it always will be in our school program until we graduate from high school, math also required for most the majors in college too. But according to many studies many students think that math is one of the most boring subjects in the school. But what makes math boring than other subject like science, english, music, paint or history? Well, I think the answer is already gave in the Lockhart’s essay “Lockhart’s Lament”. In the assay, Lockhart stated that: “The first thing to understand is that mathematics is an art. The difference between math and the other arts. Such as music and painting, is that our culture does not recognize it as such” I totally agree with this quote. I like the idea that mathematics is an art. I believe a piece of work can be considered as an art, because it has a soul and expresses the creator or author’s spirit. Such as paint or music. They are the tools for a one to express their thought or idea. and mathematician express their thought through mathematics. But not many people understand them because our culture does not recognize it.
    But, why our couture does not recognize it? We have to go back to beginning of our discussion. we start getting familiar with math through the school. But the system in the school failed to introduce math in a correct way to the students. I believe that is reason why Lockhart describes in article mathematics in schools today as “heartbreaking”. In my own personal experience in school, many teachers teach student mathematics by show them solve problems by just use formulas and example problems, especially during the High school. Teacher want their students to pass the regents exams, so, many times they do not have time to show the student the concept behind the formulas or make the lesson more interesting and colorful, That is the place many student start getting lost, feeling math is boring, it gives students a sense that math is memorization instead reasoning and understanding. But that is not teacher’s fault, because that is how the system is works.

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