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 Friday, 10/14/16). 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 Friday 10/14 (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 your game, and bring it with you to class on Friday 10/14 (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:
Conjecture: The person who goes first always wins.
Question: Is the corner the best move, or the center?
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. Ismail Akram

    Lockhart’s statements ‘the first thing to understand is that mathematics is an art’, and that ‘mathematics in schools today is “heartbreaking”‘ are reminiscent of my higher level maths experience. One of the big divides between ordinary and higher level maths was approach.

    In ordinary level, students were given formulas and and methods to simply regurgitate in order to “do well”. Perhaps I had too much time to ponder about these formulas, but I couldn’t help but ponder the quintessential question of why? Lockhart’s triangle encased within a rectangle was reflective of my higher level teacher’s methodology, I’d never see anything of the sort in ordinary level, students in that class were simply not trusted to be able to think beyond “1/2*base*height”.

    However, I felt like I was given that trust in higher level, my intellect was respected. I was allowed time to make mistakes and ponder “why/why not?”. There’d be may moments of exclaiming eureka in my head upon finally clicking everything into place. My teacher’s explanation of pi is still embedded within my mind to this day.

    While I do agree with Lockhart’s approach to teaching previous archaic formulaic concepts in a more engaging light (drawing a broken line down the triangle to explain its area being half that of a rectangle encasing it), his statement about student’s not learning mathematics today raises contention. This “real maths” he claims to teach is also taught in other places (Ireland, and I’d presume some parts of America). To simply write off that “they’re not learning anything now!” is a bit of an overstatement.

    The following quote stood out to me. “Why don’ we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard?” The dividing bridge between ordinary and higher level maths curriculum screams yes to this question.

    As for experiencing mathematics as an art, I think I’ve lost count at how many times I found myself in awe. While the term art can be subjective, I’d rather describe zero as powerful. For one mere “nothing” to shake the foundations of early maths and wreak havoc is artful in it’s eloquent destruction of previous structures. Charles Seife in his book “Zero The Biography of a Dangerous Idea” discusses “the number zero from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe and its apotheosis as the mystery of a black hole.”

    • Jonas Reitz

      I appreciate that you introduced an international perspective into this discussion – while America is not alone in its problems with math education, there are certainly excellent examples out there (and, in pockets, here in the US as well). The question of how to change deeply engrained habits about math teaching and learning remains (extremely!) difficult, but looking to successful models is a promising direction. Great!

  2. Jose De Leon

    Mathematics and Culture
    During my educational experience beginning from kindergarten throughout college, my processing of understanding how to approach math was memorizing formulas in geometry, table of multiplication, so on. Those days I was questioned me how am I learning to get passing with good grades believing I’m smart and capable to be good at math. I decided to get an Associates in Electronic Engineering technology looking for curiosity about applied content related to technology in daily life. I discovered sometimes knowledge is to create products for society and uses to not evaluate in essence. The invaluable invention of such theory bases on physics makes those procedures make mathematics irrelevant and only focus on revenues and benefits for wealthy people.
    According to Lockhart, mathematics in school is described it as “heartbreaking” means mathematics supposed to be simpler and beautiful patterns using drawings visually such as music or painting for a practice in considers of arts in form of expression. In schools, content is taught to follow recipes to apply for real world problems without thinking in abstract that no contain substance involved in it. The purpose to know basics skills to be able to survive that society dictates what it is applicable.
    Culturally, math is shown as difficult to figure out not explaining in details by teachers using curriculum bases on standardized common core knowledge in education. Those method of teaching can condemn to inspire and motivate using creativity as driving force for joyful moment to understand the beauty behind math as “art of explanation “and “simplest as imaginary “I disagree of lack of mathematics in mathematicians using formal proofs and technical language of symbols as representation of ideas. Education systems must follow instruction based on orders in function relate to employment, level of education to be able to compete. But not thinking ideally in abstract concept at all not thinking in real application that produces limited results to really appreciate math.

    • Jonas Reitz

      Thanks! I’m also very interested in the relationship between applied math and pure math – and I think applied math can be beautiful and inspiring, too! But there does seem to be a difference between math taught for understanding, and math taught simply to find the solution to a specific type of problem, and it might be in this area that we (math educators!) have some real opportunities to make change.

  3. Armando

    Have you ever had an ah-ha moment? Have you ever thought of something you really believe in and you just can’t explain it? Some topic, some idea, some statement that you just can’t put in your own words? Well this has happened to me for years and you know what, this man took all the words I’ve been trying to say and has put it in a beautiful three page essay. I could never agree with anyone so much more!!! Mathematics is so much more than memorizations and theorem and proofs, there really is something beautiful about drawing and adapting a picture and messing around and discovering a conclusion on your own. I just had a situation like this when a fellow classmate and me were messing around with similar triangles. When you reach that conclusion, it is such a great feeling. This quote from the text helps describe my feeling “ I was able to create a profound simple beauty out of nothing, and change myself in the process”. I mean no disrespect by my next comments to any artists, but that’s what I see is art. Art is at times NOTHING! A splatter, lines, dots or even a blank sheet can be beautiful as long as your open to see what is meant by the word beautiful. In Math, we may consider nothing being a line, a point, a ray but it really adds so much to the picture. I agree with this person that people have a misconception with what mathematics really is, and that may be the result of bad experiences in schools due to the idea that our teachers may be restricted to teaching formulas and what I like to call “cheat sheets”. I don’t remember if this was in the 3 pages I read (because I read up to page 8) but he does mention that some teachers have a lack of understanding and appreciation for mathematics and to an extent I agree with that. I feel you must be passionate for what ever you do for a living. Some teachers are teaching for paychecks while other teach to educate and inspire others and that to me is the root of the problem.

    By The Way, If anyone is reading this, is anyone else curious on why Prof. Reitz has this under #project ???? O_o

    • Jonas Reitz

      I like what you’ve said, Armando – and your final points raise a good question. How do we encourage more math teachers to feel the inspiration in math, and share it with their students? How do we attract motivated, interested people into teaching math? These are essential questions for us as a society, I think.

  4. Hanan

    In the section I read titled “Mathematics and Culture” from the essay “A Mathematician’s Lament” I do agree with the way Paul Lockhart criticizes how mathematics is being taught in America today. I do agree with Lockhart for calling mathematics an art that is not seen as one. One quote from the excerpt that stood out to me was on the second paragraph of page 3, “There is no question that if the world had to be divided into the ‘poetic dreamers` and the ‘rational thinkers` most people would place mathematicians in the latter category.” I chose this quote because it explains that most people would not call mathematics an art because most people don’t understand what mathematicians do. He goes on to call math something that is mind blowing and something that allows more freedom of expression that poetry, art, or music.
    Lockhart gives an example of how he sees math as an art with a question about how much space an imaginary triangle is taking up inside of a rectangle. He then finds out how much space the triangle is taking up by placing an imaginary line in the rectangle. The line shows that each piece is cut diagonally in half by the sides of the triangle proving that the triangle takes up exactly one half of the rectangle. He uses this example to show that math is about using your imagination just like an artist to answer questions. He shows that in order to solve this question he did not need to follow a formula. He had to use inspiration, experience, and trial and error.
    On page 5, Lockhart describes mathematics in schools today as heartbreaking. What I think Lockhart means by this is that math today is all about memorizing formulas and procedures and is no longer about logic and creativity. I do agree with this statement. I think that math is no longer taught with creativity and logic and instead taught with memorizing formulas and procedures.

    • Jonas Reitz

      Great! Now, what are we going to do about it? (I’m not sure of the answer, either, but I want to keep trying).

  5. Jpitt

    While reading Lockharts essay I came upon some key points that I can firmly say I agreed to. Lockhart said that Math is also a form of being an artist since it requires creativity. This makes a lot of sense, since with math you may come upon problems, where you would have to think about everything that you’ve learned to be able to come up with a solution. That is certainly one of the fun things about math. Also we can take simple shapes and break them apart, to be able to create proofs, that weren’t originally presented. It requires an imaginative mind to be able to actually understand math and the content that is within.
    There was a certain idea from Lockharts essay that I certainly did not agree with. Lockhart wanted the students today to be able to come up with solutions, without given rules, which would make math extremely complicated for even the brightest students today. If we, the students, had to find rules to figure out math problems without any help, then the world would be going backwards. The proofs that have already been made by past mathematicians, help to make it easier for the students today, while we can also build upon what is already given. By having these proofs, we can make the future easier because we might be able to create new proofs that would help us, the human race, advance and become even more capable of doing things that we never thought were possible. Past knowledge helps to increase the intellect of all, so, without it we would be going backwards instead of forward.
    “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”. I chose this quote because it certainly describes the value of mathematicians. Mathematicians are artist because they have to create patterns which eventually become proofs that would become permanent in the world. Someone might hear these proofs as they are taught in class but they would of never thought to themselves that the mathematician was a artist. They would of just thought, they were a very gifted person with the ability to manipulate what is given, to create more. This requires a mind that can delve into Math to make perspectives or ideas that perpetuate a different reality.
    There have been times where I’ve had to use art to progress in my Math skills. Right now while doing proofs we have to contemplate ideas, while being creative, that would make our given proofs logical. It’s a bit complicated since the wording is the part that gets me. The wording can be complicated, since certain things can be said in different ways, which can lead you to believe that your definitions are off. These proofs require a mind that can be creative with definitions and wording to be able to make a given proposition, a fact.
    I agree with Lockhart that Math today is sort of heartbreaking since today most students just copy definitions and repeat them without really understanding them. It’s pretty much copy and paste, since, everyone today, get the same answers doing the same exact work. This means math wouldn’t be as creative to most unless it becomes a career. Most people today learn these math skills but never actually use them in their everyday life. The only things that we would end up using is our multiplication, addition and subtraction skills but things like knowing the formulas for shapes and quadratic formulas become obsolete. Even if you end up using these things you end up just using formulas that you repeat over and over again which doesn’t require much creativity. This is what is happening to me in school since most of the math skills I’ve picked up probably won’t be used in my life. What might help is when I have to think to solve math problems since sometimes you have to think abstractly. This might help me in the future since training my abstract thoughts may help me in other fields. It’ll help to perfect my thinking, which helps me in every road I end up taking.

    • Jonas Reitz

      Jeron, I especially like the way you disagree with Lockhart – there is something to be said for building on the knowledge that exists already, rather than discovering it all for ourselves! Maybe the question is about how to achieve a balance in our math education, between providing hints and encouraging exploration…

  6. Kevin Truong

    In the section of Lockhart’s essay “Mathematics and Culture” he describes the simplicity of the ideas of math. I agree with his idea of how math can be simple if looked into a deeper perspective. Lockhart said that mathematics in schools today is “heartbreaking”. He says it is heartbreaking because schools today only gives the students the formula and work with it instead of the deep meaning of the formula. Lockhart showed the idea of how the area of the triangle came to be and then showed the formula that schools gives to the students to memorize which then he said “The question has been asked and answered at the same time— there is nothing left for the student to do”. Figuring out the whole idea or concept of the formulas being given is simple and it follows a pattern that can be seen if we put some thought into it.

    I do agree that math is more “rich and fascinating ” when the why and how comes into the equation whether than only the what is being taught. Learning how something came to be can give the person a better understanding of the idea. Lockhart stated that mathematics is the “art of explanation” which means that mathematics is not finding what is the answer, but it is the reason why is that the answer. There is one point where Lockhart said “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” and that caught my attention because he is right that when you’re given the answer there is no more of the problem to be solved, so we just move on without doing anything about it.

    My experience of math is art is not from math, but from physics. The idea is about the same where the real world is the solution and the physics is the art. After taking 3 years of physics I got a better understanding of how the world works in a physical point of view, like how and why gravity works, or how does force play in our everyday lives. Just like math where the formulas are being explained, physics explain where things in our world work.

    • Jonas Reitz

      I also think physics is fascinating, and mathematical knowledge is essential to exploring it deeply – someone once commented “to really do physics, you must know math – because you have to speak the language.”

  7. Gary

    From the reading I agree with how he found the area of the triangle from a rectangle because when taking a math education 3011 course professor Rojas explained and showed an example of how you could find the area of a triangle from a rectangle from using a geoboards on the board. At first when professor Rojas said it I didn’t believe it but then when I read about it I believed it a little more. Something that I didn’t agree with was when he said that mathematics was an art because I really don’t see mathematics as a art now when I’m in college but rather like a algorithm for a computer program. When I was in middle school and high school I could see that mathematics was an art because of the fun activities that teachers would make us do to help us remember the topic better and understand the material for example like defining a math definition and then showing an example of it below the definition and getting posted on the bulletin board inside the classroom. On page 5 when Lockhart describes mathematics as “heartbreaking” because I think the way that teachers in middle school and high school are teaching mathematics from a curriculum that tells them what to teach the students in the class than teaching the students more in depth about the topics. I disagree because I believe that when teachers are teaching the topics from the curriculum because they are teaching the students so they can prepare them for the next grade. When I was middle school and high school mathematics class felt fun with the ways the teachers were reviewing for the test than now with professors just explaining to students anything that they are having trouble with. Teachers and professors are but still teaching from the curriculums but aren’t really teaching from their own knowledge to teach the students about the topics.

    • Jonas Reitz

      Gary, I like your disagreement with Lockhart a lot. What I took away from your writing was that the teachers and their approach have a lot to do with student experience (even if they are just passing on knowledge) – this is a valuable bit of insight for future math teachers!

  8. Violanda

    “Mathematics is an art”-this is how Paul Lockhart starts the chapter on mathematics and culture.I would add also that this is understanable only by mathematicans becausd ordinary people,especially those who don’t like math can not understand this concept.Most of the people who I know are surprised why I chose math as my major.Usually these people don’t know how beautiful the Math is itself.
    However,beside many beautiful names that Lockhart gives to Math,I would like to mention a part where is not so good,but unfortunately it is true the part where he states that it is heartbreaking to see what is being.done to mathematics in school.I agree with this part because many teacher do not teach in the way that math should be taught.They choose a short way how to explain a certain topic.For example,in order to give a deep explanation about how a certain formula can be found from a graph,they only give a formula and most of those formulas are hard to learn.So,that’s why students end up in not seeing Math as it really is.
    My experience learnt me the beauty of math and logic that I had to use in order to solve a certain problem.It is very interesting that I never know what is a sin, cos,tan,or cot, of a certain angle if I do not use the graph.It is more simply for me to use the graph because this is how the professor from hight school taught me.However,there are many formulas which I learnt by heart just to pass an exam or just for a short amount of time.Even though I like math,there are many things that I don’t get,or I get them only for short amountvof time,until I pass the exam.
    However,even though many people dont understand Math still stays an art inside itslef as a song to singer and as a poem to a poet.So bad for some people who don’t understand the beauty of it,but I ylthink that if many math teachers change the way of teaching,many students would change their opinion about Math.

    • Jonas Reitz

      I like your comment “many beautiful names that Lockhart gives to Math” – names are important, and changing the way we talk about math changes the way we think about it. Great!

  9. Tyniqua

    Lockhart’s Lament is a very insightful reading of the world of mathematics from his perspective. The term, “mathematics as art,” is key in his argument and proves to be truthful throughout the history of mathematics. Lockhart uses G.H Hardy’s description of mathematicians, “A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than them, it is because they are made with ideas.” Indeed, it is known that mathematics is an art which is inspired by beauty. I find that there is math in every aspect of life. In order to bring an idea to life we use math. We can make a conjecture in the field of science but we use our knowledge in mathematics in order to come up with useful solutions to put forth an idea. The buildings and landmarks that we are all familiar with is a creation put forth by architects, whose designs are based on a compilation of mathematics, and the end result is someones beautiful imagination coming to life. I understand Lament’s position on the heartbreaking way math is being taught in school, along the lines a memorization of formulas and procedures, without getting a full understanding of the meaning of things such as the area of a triangle. The example he used with a box whose space is shared by a triangle and using the triangle as part of the rectangle is a very useful way to show students why the formula of the area of a triangle works in turn helping them understand that the imagination is what helps you understand the notion. The author states that something as simple as drawing a straight line connecting to points on the rectangle gives a drawing more meaning. In my math ed classes my professor always talk about math in the most whimsical way. It is obvious that she sees the beauty in math and she often mentions to us that we have to use that beauty of shapes, numbers, and patterns to get through to our future students. As I mentioned in class before, there is a reason why NASA has a need for the best scientist, physicist, and mathematicians there are in order to bring to life ideas that allow us to venture to dimensions known and unknown. The combination of those beautiful minds have took us far and beyond in past and will do in the future. Although, I am not the best at math I do see the beauty of it, solving equations is like a fascinating puzzle that you have to use everything you know to solve. One of my favorite puzzles is Suduko where recognizing patterns is crucial in solving and my favorite card game is spades where probability is a key component of winning the game. There is art in everything and the underlying component of such is pure mathematics, I love math!

    • Jonas Reitz

      I love the idea of “The combination of those beautiful minds” – passionate people working together for a common cause! And math is an essential element. Great!

  10. Leonardo

    Lockhart’s essay was very interesting to read. When the phrase, “Mathematics is an art” was introduced, I said to myself, this is indeed a beautiful true statement. Mathematics can be done in many different ways that we, as students, can creat such brilliant ideas in order to come up with conclusions. We could be right or wrong but there’s always another try! The more we try making these ideas, the more adaptable we will become as we can master them in the future. I mean face it we all have to use math every day, even if some does not like the subject very much, but they don’t realize the true beauty. When Lockhart distin

    • Leonardo

      Distinguished* this “triangle in a rectangle” idea, it is definitely showing that mathematics is art! I agree that mathematics can be “heartbroken” because at times students will not get a single clue on what the topic is about in a math class. Maybe is because of the teacher who could not explain very well on the topic or maybe the student is just not interested but it doesn’t hurt to open your eyes and see the inside of mathematics. We can’t get a full understanding on a certain topic that it makes us students feel in a way that we will just see mathematics like its nothing. Me personally, I was very inspired from my math classes in my middle and high school days. I remembered my teachers always give me a smile on their faces just to show that I’m a very confident person and I can accomplish their classes by following their lessons and examples they have shown for to learn. It’s all about the student and teacher following to each other, like if it was a pattern, and that is also known as mathematics at its beauty.

      • Jonas Reitz

        I love your phrase “It’s all about the student and teacher following to each other”!

Leave a Reply

Your email address will not be published. Required fields are marked *