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, 10/18/17). 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 10/18 (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 10/18 (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.”

12 Comments

  1. D. Young

    My Thoughts on Math and its Teaching

    The author of Lockhart’s Lament is obviously disappointed in the lack of teaching mathematics without giving clear reason for the “why” of math. A critical element to forgo. His opinion is colorful and subjective, yet I see his point and in some ways agree. In some ways I disagree. In this position paper, I will try to explain my point of view on mathematics, its impact on the world, both good and bad, and what it means and has meant for thousands of years. The beauty and dangers of math. The healthy and unhealthy ways it has affected us. The abilities that it has given our race to flourish as well as self-destruct. I will attempt to show, as best I can, the power within mathematics.
    The entire physical universe is mathematical. The one true gift that “The Creator of all things” gave us (if you choose to believe that, or whether you believe in a cosmic catalyst that caused the universe) is the gift of math. There isn’t anywhere math is not involved. Physics, chemistry, biology, even the mathematics of language all prove the ubiquity of it. It is a power that, metaphorically speaking, allows us to peek into the mind of God.
    The Pyramids, the Taj Mahal, The Great Wall of China are all examples of the gift math has given mankind. The precision to build great things. Understanding the atom, quantum physics and space travel. It has given us the knowledge of when the planting of crops should happen, early ocean navigation, leading to discovery by knowing the stars positions, these are just a fraction of the ways math has showed mankind the way. Not to mention aiding in the science of sound reasoning along with the calculation of time. The list is endless as to what math has given us.
    It has also given us the ability to kill on a planetary scale. How to land bombs on targets quite precisely. The mathematics of the gun, the drone and so forth. The math of warfare. There is always the flip side of the coin.
    The “why” of math is, of course, paramount. Giving a mind, young and not so young, a “reason” gives the thing being taught meaning and can reveal beauty in it. Be it math or anything else. Without the “why”, it’s just a bunch of meaningless formulas and numbers. Like reading the Constitution of the United States without the history behind it. It just sounds like a mass of “high sounding” words that are ignorable.
    In conclusion, I will say that math must be given the respect of being taught in a way that makes sense. A way that grabs the imagination, as Lockhart alludes to in his writing. It is said that, “With great power comes great responsibility.” With that being said, I charge the math teachers of today with the great responsibility of showing students the greatness of this science/artform, and how the understanding and respect of it and for it, can bring greatness to our lives and our race.

    • Jaroslav E Sykora

      Good piece, Mr Young!

    • Jonas Reitz

      Thanks for being first to respond. I love the perspective and ideas you provide here – what a great piece on “The beauty and dangers of math”! Math teachers of today, take note!

  2. Jaroslav E Sykora

    The provided essay is a cutout of Lockhart’s book of 140 pages with the same title. I read pages 1-18 + “The Conclusion” and don’t feel to be entirely in the picture regarding his novelty and suggestions how Math should be taught. Without reading his solutions for all the problems he so eloquently and provocatively targets, I might easily slide into superficiality and misunderstanding. I like his claim, or idea, of playing with a problem, though, regardless if it should be a Tylor’s approximation exercise or figuring out any product of two three-digit numbers. The playful, not-stressed attitude to Math-solving problems speaks me from heart, although the external conditions (full-time studies, family, employment, friends and hobbies) make it often impossible. However, studying Math for sake of playing and self-entertaining is too idealistic, in my view. The positive aspect of practicality as a motive and a drive is missing in his criticism. I still motivate my curiosity to learn Math with the question what new and else in the real word of materiality and nature can be solved mathematically. I wonder if the Pythagorean theorem, or the value of “Pi” had been discovered accidentally or as a result of an effort to find a way how to part a piece of land, rent it out and earn tax of it. And what about the Euler’s number “e”? I am fascinated by the fact that I learned how to count the compound interest, the total of car plates having three letters and four digits, the area of a curved mirror, using the outcomes of calculus, etc. I completely fell in love with the optimization problems when I took the Cal I class. Those are not abstract schemes, any incidental outcomes of a play. Lockhart claims that Math is a kind of art, and compares it to music and painting. Well, many of his analogies are based on wrong premises. To mention at least two examples: Michelangelo’s ceiling paintings in the Sistine’s Chapel is not a result of a mere play but a work requested by the Pope Julius II. Or, Beethoven’s famous 5th symphony was not a result of a self-entertaining play but was composed on a commission he received from the Sicilian Count, Oppersdorff. And so on. And last: is Math in today’s schools heartbreaking? In his view, it is. Having a wider experience with studying Math, in Europe and the USA, I’d say the content is the same, the methodologies, although quite similar, differ. In Europe the stress put on memorization is stronger than here. Once I had the chance to intern, for about 5o hours, in Pathway Technical HS in Crown Heights, Brooklyn. None of the students passed the Final Exam because they didn’t memorize the formulas. Lockhart claims that the formulas kill the students’ creativity, interest and capacity to advance in Math. He writes that they should be remembered as a byproduct of playful solving exercises. Well, I seriously doubt it.

    • Jonas Reitz

      Great and well-balanced discussion! I think the danger of disregarding math’s practicality is a real one – it’s amazingly practical, and that’s one of its great benefits (and indeed, a big part of the attraction for many people – I also really liked optimization problems, although I think many students do not). I also like the examples you introduce of great art that arose out of commission – artists have to eat, but they can still make amazing, authentic works.

  3. Junior

    For the most part I agree with what the author of Lockhart’s Lament has to say. He is very disappointed in the education’s system on how they teach math. When Lockhart says “By concentrating on what, and leaving out why, mathematics is reduced to an empty shell”. I had come to the realization that I was never really taught why I was doing this or where the formulas came from, throughout my school years. It makes me upset because when I started learning the theories and the proofs of formulas in my college course I found that it was fun and crazy at the same time. To come up with formulas using the simplest concepts and connecting everything together somehow makes math feel like a puzzle, and puzzles are fun. I wish I had learned the why from an earlier age. That being said I agree with some of what Lockhart says the education system is lacking. Although when Lock hart says, “the students. They say, “math class is stupid and boring,” and they are right”. I can’t help but feel that he is wrong in this. I’ve always thought that math was fun and I loved going to my math classes throughout middle and high school. To me the fun was having a tough problem to work out. It felt like a puzzle and once I had finished it I felt like I could solve any problem. Although it is frustrating when you’re are stuck on the problem. That feeling you get once you solve it is very satisfying. I remember in my high school geometry class my teacher made us do an activity on the first day of class. She gave us a picture made up of many different triangles. She gave us three color pencils and said “take these color pencils and color each triangle so that no two of the same colors touch each other.” When I did this activity I had no idea what it was for. My geometry teacher later said to us. “Math is everywhere and that it is all connected with each other from some basic things” After that day I really started appreciating mathematics even more. In conclusion what some of the author says is true. The education system is lacking in some aspects in its ways of teaching, but it is definitely not boring or stupid. The education system has been fair to me from my experiences.

  4. Silvana

    In this reading, Lockhart explains his frustration on how math is being taught to students. I do agree with most of what he said. I really enjoyed reading how Lockhart often compares math to music and art. All do involve mystery, trial and error, inspiration, and imagination. There are problems we can’t solve fast and we must make mistakes to finally get to the answer. Math truly involves a lot of trial and error, as well as much practice. I was introduced to the formula of a triangle on handout. I remember my teacher telling us that we must remember all the formulas on this handout because it would be useful when we would take our regents. I was never explained why I must use these formulas. I never learned how all these formulas came to be. I never did consider to ask why we must use them; but I simply agreed to the fact these formulas work, so I must use them. “ Students are asked to memorize this formula and then “ apply ” it over and over in the “ exercises. ”” I do agree that students are often asked to memorize many formulas without giving any explanation of why. Lockhart’s example of a triangle inside a box is really interesting. He gives an explanation of why and how the area of a triangle works. “ I couldn’t see, and then all of a sudden I could. ” For me this quote relates much to the “ oh wow ” moments we have when we truly understand why and how something works. Many subjects involve math, many professions involve math, a lot of things we do daily involve math. It is important for students to truly understand how and why math works, but to also enjoy it.

  5. jesstopal

    We have to accept mathematics is in our lives, and we live with it. Yes, it is true that many people are unaware that they use mathematics in almost everything, such as many of our most common daily activities and routines: shopping, banking, cooking, making repairs to our homes, etc. Knowing math and why it is vital in so many aspects of our lives is essential.
    In the article “A Mathematician’s Lament,” Lockhart claims that “mathematics is an art.” I agree with that because if you look at the Mona Lisa painting, you will notice that Leonardo Da Vinci used mathematical directions. It is undeniable that numbers have an impact on our lives. Although many see mathematics as consisting of only symbols and sharp rules, it immediately shows that it is an enjoyable area, even though it is a complex one. Even though many people are not aware of the numbers they encounter in many areas of their lives, such as art, music, architecture, and basic sciences, we are in consecutive harmony, and the real fun side of mathematics begins here.
    Lockhart claims that in schools today, mathematics is “heartbreaking.” He means that today, mathematics education in the world is not at the desired level. He claims that many students do not understand math deeply because when they look at the math, it’s like looking at Chinese symbols. Today, a lot of students lack knowledge in mathematics because of that. They have no idea what they are doing because they mostly try to memorize formulas to solve problems. Many students cannot explain many things because education systems are based on memorization. For example, our math teacher wrote “Pi” number on the board. We know that this a is “Pi” number, but we don’t know how to explain if someone asks, “What do you know about this number, and can you give a real-life example?” This number is everywhere. For example, if you are a coffee drinker, you will see it on your coffee cup. Also, Lockhart claims that today’s mathematics education consists of rules, formulas, and definitions to memorize. Students have no connection to the concepts and are only focused on correct answers instead of developing mathematical understanding. He underlined that students need to engage and explore with conceptual ideas, the nature of mathematical concepts, and relationships that build understanding.
    The teacher should try to help students to improve their questions and engage the activity, so students can have discovered their explanations without memorizing. If today’s society realizes that mathematics is imagination and creation and fun, every person would fall in love with math. Our mathematics system makes us remember everything one time and kills our creativity. Many people don’t like math because of that. Today, math education is murdering our new generation’s imagination. Mathematics is imagination, discovery, creativity, and curiosity. For example, if mathematicians weren’t curious in the past, we wouldn’t be able to know what gravity, space, the universe, etc. are. Without mathematics, life is like a dark place without seeing anything. If we don’t see anything, we cannot understand, and we can not make new things or create. Mathematics is like a fun game for us to play in our lives and helps us to see things beautifully with sense meaning.

  6. Samantha.C

    First off I would like to say that the author Lockhart point of view reminds me of Professor Rojas. By that being said I really enjoyed this reading because a lot of what Lockhart stated I understood and I can relate to it. This is not the first time I read or heard something pertaining to the lack of teaching mathematics. His perspective on it is very clear-cut. One thing I really agree with Lockhart about is what is being done to mathematics in school. The beauty of math is lost and the students are lost because they are being asked to memorize formulas and not see where it was created, or how was it created(the proof). Lockhart states “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. It is like saying that Michelangelo created a beautiful sculpture, without letting me see it. ” this quote stood out to me because I feel like this a true statement. Its so much things I’ve learned in elementary school, middle school, and high school that I just don’t understand why I had to use certain formulas for certain problems. I was told to answer problems; I did it and never bother to asks journalistic questions about it. Like, when was the formula generated? Who came up with this formula? How did they come up with this formula? Where’s the proof of the formula? I feel like I never asked those questions before because I just wanted to pass my class and I thought it just was the rules from the teacher. Now that I’m majoring in mathematics and education I question things a lot more to understand and see the bigger picture of it all. I understand things so much better once I see how things are formulated and me seeing the proof. If you asked me today whats the proof of certain formulas, I seriously couldn’t answer you. Thats because everything I learned wasn’t taught in depth. We basically was told what to do. Im actually going through this issue right now as we speak. I feel like my professor don’t really care if we understand whats really going on in the class, as long as he get through the context’s of the subject he’s contempt. It’s very disappointing because I don’t feel like Im learning anything, Im just trying to pass the class. I truly feel that math is such a beautiful work of art and that it takes time and patience from the educator to educate students the right way so they can legitimately understand it all, but also see the beauty in mathematics it self.

  7. Rachel

    I find myself agreeing with Lockhart’s view of math education at the higher levels of mathematics. In the majority of my college classes a teacher will write a theorem on the board and under it a quick proof and ask us to copy it down without much explanation. I feel, as Lockhart does, that the proof should come first. We should try to solve a problem and note a pattern before we are given a formula to memorize. We should give students the chance to figure out a theorem themselves; working through the problem will probably give them a higher level of understanding. Not only that, but the idea of memorization becomes unnecessary. When a student knows why a formula is what it is, they will be able to figure it out again on an exam should they forget it.
    I was fortunate to have some math teachers who also subscribed to Lockhart’s theories on education in this country. His specific example of finding the area of a triangle was explained to me in much the same fashion, and I remember my teacher splitting us up into groups to try and figure out the formula on our own. Looking back, that exercise helped me to get into math in the first place and really changed my perspective on the subject as a whole. It made math fun, like a game. If you can get students interested like that, they’ll want to pay attention and learn the material. Just asking students to remember formulas will not keep them excited, nor will it foster a deeper level of understanding.
    I also agree with Lockhart’s position that mathematics education in schools is “heartbreaking.” It seems like although students may know how to solve a problem, they don’t understand why they are solving it in that way. A student might be able to tell you which formula to use to solve a problem, but if asked why they are only able to point to their notes and say their teacher taught them so. There’s a very shallow level of understanding. It makes sense that Lockhart would be frustrated by this, considering his stance that math is art. Art should be appreciated, so math should be as well. And with the way math is often taught, there’s no room for appreciation of how a problem is solved.

  8. Aleks

    “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“. The quote above I definitely agree with and stood out because based on my own experiences this is what I e observed. Countless times I e heard to “conceptualize, do not memorize” but it always came down to memorization because in the end we need to be assessed with a test. I wish they made a way to be able to look at math in an art form as described in the article because it seemed to be fun to look at math in that perspective. In calculus we studied the three planes and where asked to take three equations and produce a 3D model of there intersections to form the wedge that we wind up finding the volume of by integrating. I liked it because being forced to build out an equations curve helped the image stay with me. I agree with Lockhart that math is heartbreaking. As I write earlier, math has became more about memorization and less creative which makes it less fun in the end.

  9. Franklin Ajisogun

    I really enjoyed “Lockhart’s Lament,” specifically his section on “Mathematics and Culture”. It really speaks to me when Lockhart says ‘mathematics is an art.’ When we look back the ideas and creations that mathematicians have discovered, they have changed the world through the creation of telecom by using phone or computer, what you are using to read my essay. Without mathematics it is not possible to make those ideas and imagination into realities and that is why Lockhart’s statement “Mathematics is the closest thing to art” is correct. For me, as an applied math student and data analyst, I see math as a beauty. The enjoyment of doing math is what motivates me to study the subject. In data analysis, it is interesting how we can use matrices and vectors to represent a dataset which we can use mathematic concept like machine learning which is built of linear algebra to predict what kind of scenario or situation will happen based on the data. I like the statement from the article “we get to play and imagine whatever we want and make patterns and ask question about them.” I can react to this. When I analyze a data set, I have to plot different types of visualizations from the dataset to see the relationship or pattern. From that, I make a hypothesis and research on what I saw.

    Lockhart talks about how schools teach mathematics today. They are “concentrating on what and leaving out why.” I agree with Lockhart’s position that mathematics education in schools is “heartbreaking.” I remembered in high school how my math teacher always gave a math formula to solve the problem without explaining what it is and what it is used for. I remember one of my classmates asked our teacher why the alternate interior angles were equal, assuming that you have 2 parallel lines cut by a transversal. He couldn’t explain it to us. He could only tell us it was true and that we should memorize it. That wasn’t first time that a student stumped him on relatively simple question. He didn’t understand the concepts himself; he just taught a collection of disjointed methods. Sometimes, students need to be taught WHY they are doing what they are doing. This article really speaks to me because a lot of students which I tutor always say to me ‘I am not going to use this stuff again.’ It is surprising how a lot of students see math that way. Mathematics is critical to everyday life by adding and subtracting amounts of money or compound interest for mortgages and car loans. Mathematics is the cornerstone of much of our learning process. Studying math properly opens us up to the patterns that appear throughout our lives. It is a subject that we begin learning very early in life and continue with throughout our academic career because mathematics helps in so many other areas, including most of the sciences, computers, construction and design.

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