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, 10/24/17). Your assignment has three parts:
First, read the section titled “Mathematics and Culture” (pages 3-5) in Lockhart’s essay, (click here). Â 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”. Â What do you think he means? Â 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 10/24 (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 10/24 (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.”
Paul Lockhartâs, âA Mathematicianâs Lament,â was an insightful critique of the way educational systems approach teaching mathematics. Lockhart argues that by directly giving students formulas to memorize – instead of creating a space that fosters creative thinking – results in a mundane mathematic experience. He states, âBy 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.â (p. 5) In other words, the lack of ingenuity in math classes often leads to students being disinterested or unengaged, which can be a great disservice to their overall education. In fact, Lockhart describes the way mathematics is being taught in schools today as âheartbreakingâ; I believe he is referring to the way school systems neglect the creativity of children to enforce a curriculum that is based on rigor. Children are imaginative and creative by nature but the structure of schools and obligatory curriculums do not allow children to explore these aspects of themselves thus, subjects such as mathematics become monotonous.
As a student that has gone through the public education system, I am inclined to agree with Lockhart. The subject of mathematics is often taught in a rigorous way that can be intimidating to students like myself. The idea of math being a creative process filled with imagination, is a foreign concept to me, as I have never before been introduced to mathematical subjects in such a way. While I have never viewed learning math in this way, I now wonder how my perspective on education may have been impacted if such principles were instilled from a younger age. Additionally, I believe that by giving students a problem in which the answer is not obvious, aides in their ability to learn skills that can be applicable to math and the real world such as the ability to rationalize, problem solve, and create. This is especially important because it will encourage students to develop their own individual ways of thinking.
I enjoyed reading Lockhart’s Lament. I agree with most of what it says. I myself like mathematics, but I can understand, having gone through the education system, why most people hate math. It is just presented in a dry way, as just a bunch of formulas and algorithms to memorize. That is not what real mathematics is about. In graduate school, we learn how to do proofs. Proving things is what math is really about. The quote I choose is “Mathematics is the purest of the arts, as well as the most misunderstood.” I do not actually agree with this quote, because I do not think math is either an art or a science. It is in its own category. Nevertheless, there are both scientific and artistic aspects to math, and I agree with Lockhart that it would be more fun if students were exposed to math in an artistic way. I do think the state of mathematics education is heartbreaking, because most people come away from the experience traumatized and hating math. I honestly don’t know if it will ever improve. I am too pessimistic to think anything will ever change. I don’t even know what the right way of teaching math is, a way that will make more people like math. Math is just too difficult a subject to have more than a tiny specialized minority to appreciate it.
One thing that I agree with is the lack of mathematics in todayâs math class because math is more than the knowledge one has in problem solving, itâs about the process of trail and error getting to the solution by testing the endless limitations of ones own creativity. Something I didnât like was nothing…I enjoyed every thing. A quote I liked wasâItâs not at all like science. Thereâs no experiment I can do with test tubes and equipment and whatnot that will tell me the truth about a figment of my imagination. The only way to get at the truth about our imaginations is to use our imaginations, and that is hard workâ because everything is both solvable and unsolvable at the same time. Only the best dreamers can win and I am ok with that. Math is an art because of one example that comes to mind âEuclideaâ. Play it and find out why because I will not explain it. Schools today are heartbreaking but there is nothing we can do because of a limits time frame to learn and standards that students must meet. We can treated like machines. The ones that are quick and smart are good but the slow are trash. Math can be fun and so important to the development of unlimited potential in imagination but due to restraints places on learning the schoolâs children will never know and that is a true tragedy. If I learned that was he was talking about than the math in general would come easy because I why I am doing in and not only how to do it.
One thing Paul Lockhart said that I agree with is that mathematicians enjoy thinking about the simplest possible things. I find this statement to be very true. There is beauty in simplicity. Being able to see the basics of a concept and build it up to an abstract concept. Personally, I enjoy seeing things at its simplest before I tackle more difficult problems. It gives me a sense of confidence to keep achieving.
Lockhart describes mathematics in schools today as âheartbreakingâ. By this Lockhart points out that the beauty of creating critical thinking in school in the subject of math is being taken away. For example, he gave an example of the formula for the area of a triangle. Most teachers, if not all, give their students formulas to remember without making their students’ brains work hard to derive where the formula comes from (in this case formed from a triangle inside a rectangle). The formulas are just drilled into their head which is fine for test-taking but it gives students a lack of true comprehension. During my experiences in school, most of my teachers just gave me the formula at the corner of the whiteboard for reference and made me work a bunch of problems using that formula so I get used to it. The approach worked well for me to learn how to use the formula, but thatâs as far as that went. In my sophomore year of high school, I had this one Bulgarian teacher named Ms. Lekic who took a different approach. She made us manipulate prior algebraic equations to form new formulas to apply to our problems. Because of her, without any prior studying and just using what I learned in my geometry class through her, I passed the Geometry regents with an 89. If I actually reviewed I wouldâve definitely scored higher.
One thing I agree with Paul Lockhart is math is wondering, playing, amusing yourself with our imagination. Because this is like the math thinking process; we ask questions on ourselves before we work on the problem, then we work on it, and finally we review the steps and check it. A quote I chose is ” I was able to create a profound simple beauty out of nothing and change myself in the process .”. I think this is how mathematics was to create and are the processes of working on the math problem. People’s brains contain a lot of thoughts. When they have a new idea, they will use their methods to prove that this idea is feasible. This is very similar to mathematics; people will use formulas, steps, and graphics to illustrate. I had an experience of mathematics as art is worked on the golden ratio. I forgot how we worked on it, but the result we got is the picture of the drawing. On page 5, Lockhart describes mathematics in schools today as ” heartbreaking”, I think he means students don’t understand what the formulas or solving steps mean. Teachers just provide formulas for the board, use it to solve the following example and tell students to work on questions. It is my experience. If somebody asks me ” What is the formula for the area of a rectangle? How do you know that? “, I will say ” the area of a rectangle is width times length, I know it because my teacher told me that”. Teachers should use methods to explain the knowledge; better than just write it down on the board.