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, 10/15/14). 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 or question about the Bridges and Walking Tours game, and bring it with you to class on Tuesday 10/20 (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 onTuesday 10/20 (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.”
The reading by Lockheart was by far has to be one of the most logical and intuitive readings I have read. It’s one of those things I alwasy knew but never really new how to put together, hence lockhart executed the idea of math as a art form beautifully. More over, I agree to many of his theories or ideas that he presented and discussed in this short chapter reading. But the one I most importantly agree on and feel extemley passionate about is the one of mathematics aspects being deprived from math classes. Now what I mean by this is that, thinking back to my younger self, i would sit in math classe a and just briefed on about a formula then taught it, memorized and applied. All this numerous times, and if I was wrong I was penalized as in points off the question. But no teacher ever allowed, atleast that I remember, to actually create the answer. In other words, the creative aspect was gone. I don’t remember ever sitting in class, giving a problem and really been able to evaluate to it on my own through trials and errors. I was just given a problem and told to use the formula. But how would I know the formula was correct? Well I just had to belive it. So if teachers were to allow the student to actually move mathematical thoughts around in the question, It would be a more permanent and fun experience rather than a tedious or boring one. This brings me to the next part where I discuss the quote from his sayings that I agree too the most. The quote I chose from Lockhearts reading is “Mathematics is an art”. Now what this means to me and why I agree to it is well, it is an artistical aspect. There are many ways to approach a question and get the right answer. Probably not infinite many, but definitely quite a few. For example, and an example he also uses, an artist! Now one can draw an apple many ways, and not have to follow a cookie cutter way. I mean artists can’t draw a apple from the inside out, the outside in, or the bottom-up and vise-versa. In other words, that I can obtain the image of a apple (which in math would be my answer), not depending on where I started or how I finish (formula aspect of math). Like I’ve said, in math, or like an apple, there aren’t infinite many ways to come up the answer or image but if allowed my creativity, I can some how derive the answer by just taking a few routes. Eventually establish the best route, and learning on the way why the other routes or methods don’t work, or not as effective. Like said, if this approach was allowed to students in math rather than the tideous and repetitive way, I belive one would realize that students would enjoy math and less revisions would be needed for the student to understand the concept of the problem. In conclusion, the students would approach the problems with different math aspects, some of prior knowledge and some of recent, but through the students trial and error, he/she would be more effective in learning the concept. Adding on, the artistically aspect of the students have mathematical free will, will really encourage the student to see math not as just computational heavy but as artistic and imaginative.
Thanks for being first to post on Lockhart’s Lament. I really like the connections you make between what you read and your own experience – and I encourage you to try to incorporate those insights into your own teaching, whenever you can (not always easy, since we work within a structured system with many demands, but there is still room to give your students opportunities to play and have fun with math!).
-Prof. Reitz
As a vocalist, (speciality classic jazz) where learning old tunes, revising them into music appreciated today, this article regarding the art of mathematics is music to the ears. In reading his article, the following appreciation, understanding and conversation will follow me this week when socializing with peers who listen to jazz every day.
I will also note that my partner is an artist/ (realist) painter that paints elephants that look real enough to to touch their skin and think it’s real. His paintings create envy from trained artists because he never studied painting(he took a course in color theory at the age of seven). I disagree that spending time on the rudiments of painting will not (without putting paint to canvas) result in a talented artist.
Lockhart’s Lament is an interesting testimonial in absolute truth in his examples of art and music in his first two pages and the latter conversation regarding mathematics. The current American public education system in general fails students, more now than they did 2-3 decades ago. Music, Art, English but especially mathematics have become nothing but rudimentary rhetorical memorization (students in my day used another term). The work assigned (but not required) has erased the “fun” variable from the equation of “learning.”
Today’s society and the label of art and music in American culture specifically would be better addressed with more creativity if “the art of math” was still a part of the language of teaching. Now dead American musicians: John Coltrane, Thelonious Monk, Charlie Parker, Dizzy Gillespie, Duke Ellington, Sarah Vaughn, Ella Fitzgerald, Shirley Horn, Billie Holiday and of course Jimi Hendrix and so many others will never share any comparison in individual genius until math is returned to art.
As an example consider a basic nursery rhyme in my younger day of the basic tune, a tisket, a tasket; by Ella Fitzgerald. The “nursery rhyme” melody is her basic equation; a tisket a tasket a red and yellow basket, etc… commonly mimicked to most toddlers. Her mastery of improvisation in the way she expands on the basic rhyme by creating her own vocabulary using the basic melodious song for as many choruses as the band plays, evokes an art of mathematics missing from the basics of learning. She takes a basic quarter note; she radicalizes, fractures, expounds, compresses, screeches, moans all in the key signature of seven notes and the chords created using those seven notes into chromatic scaled melodies, creating a universal language. (Supposedly she never read notes; perhaps this is why her art was so above approach. She can only be copied.) Even today, when experts argue, Ella always reigns top as the queen of Jazz. The jazz tunes of Ella is dying a slow death. Nevertheless, when introduced to it, her music is still understood, appreciated and hummed by people all over the world, it is the universal language.
Today’s musicians or artists sample old music instead of creating music themselves. If they understood mathematical art which results in comprehension of music theory, increased genius could change this I believe. People who grew up in the culture under the “art of math” speak how they hear songs, who composed it originally, and how it’s better to listen to the old tune. Classical music, classic jazz, classic rock (as it’s called), 60’s, 70’s and 80’s music was more creative and the instrumentalists who mastered their art of music learned the rudiments of the art in mathematics. Listening to this music, making effort to play, improvise on it, learning and watching master musicians play an upright bass, piano, guitar, drums, when asking, the all relate to the art of mathematics as it relates to their instrument.
They break down a tune; they speak notes, tones, tri-tones or chords, major fifth, seventh, ninth, twelfth chords, minor fifths, seventh, ninth, twelfth chords, major and major scale, chromatic scales, jazz scale blues scale and it goes on and on; when the basics are mastered, they all conclude; “it comes to the basic of mathematics.” They each will say, but I’m not good in math, however Wynton Marsalis will say, practice so many hours a day and it boils down to breaking up notes, adding beats to notes, shortening notes, extending notes where rhythms, styles, phrasing, lyrical enunciation within the many scales and it all comes back to mathematics.
As with jazz, so has the math of my mother and father in the 50 and early 60’s slowly seems to have drifted away. Before the public education system changed it’s teaching process, my mother in primary and secondary school was taught mathematics with a structure where she excelled. Even though it was not her favorite subject, she always was able to help with homework, do the problems and explain reasoning behind learning some formula, why you may be able to use it and how it is an art. I envy not being able to remember like she did because her teacher gave her a history, showed her how to derive formulas and ultimately taught her the art of mathematics.
Coming back to school in the middle of a career, needing mathematics for Mechanical Engineering, the theory taught back when I first went to college is missing from my classes. Theorems were always introduced when working before working through math examples. Once the theorem was explained, understand how something was derived made it easier to work through difficult math problems in Calculus. I think it was something that allowed my take away from college to allow me to have some very interesting jobs throughout the years. The math seems harder to understand until I find an old book where the theorem of some equation is explained, followed by some examples. If I go the theory route, followed by examples, I score better on tests.
A discussion with my Mechanical Engineering Professor this evening, we’ve concluded that when given difficult jobs of breaking down scopes of the projects like 2nd Avenue Subway design project (now under construction), the critical thinking ability of finding quickly is missing today. It’s making life much harder for the students to succeed and compete in the market place. I now understand the critical thinking ability I attained was due to the theory in the art of mathematics. Critical thinking must be mastered in order to create. The “art of math” must be reintroduced to the system of education. Without it, students will not be able to compete with those who have access to it.
Steve Jobs, Bill Gates, Mark Zuckerman all dropped out of school to build their empires. As they mastered and realize the “creativity of success outside of the class room because they mastered the “art of math?”
Students in other countries, specifically Asia know and understand the art of Mathematics and they do better on tests, to them math is music like to the musician, music is math.
Great comment, Deborah, for many reasons – I like the connections you make to music, and your description of Ella Fitzgerald’s “A Tisket, A Tasket” inspired me to look it up. Beautiful description, and beautiful song!
Mathematics provide us support in our everyday life. It exist with us but sometimes we do not recognize it. A mathematician, Paul Lockhart said in his article, A Mathematicianâs Lament, âPart of the problem is that nobody has the faintest idea what it is that mathematicians do.â I agree with this statement because mathematics is part our journey and we are connect with it every moment. Everywhere we go, we can see math. The road we are walking on is build with the measurement of length and width which is math. As you all know that the world is circle. It has radius and circumference, the acceleration, the gravity 9.81 which all based on math. they are figure out by the mathematicians. We just do not recognize it.
When Paul Lockhart compared mathematics with an art, I disagree with him because mathematics is not just an art. It has included logics, ideas and proofs which build a structure that has more knowledge than just an art. When you look at an art, you just enjoy it physically or mentally by creating it but when you look at the mathematics, you look at the facts of numbers, learn from it, play with it, think in depth.
âMathematicians enjoy thinking about the simplest possible things, and the simplest possible things are imaginary.â This part of made me interest to read because the author compared the math with imagination which is true. As I said earlier math is everywhere but we do not recognize or imagine it. We picture the math in different possible way to solve a problem. When I took Geometry class in High School, I tried to picture a rotation or translation of a triangle and how does it rotate and translate.
By saying, âToday as schools heartbreaking,â Paul Lockhart tried to explain that now in schools students memorize the math instead of practical examples. He tried to say that showing the image can help students to understand the concept of math. From my experience, when I was in middle school in my country, the math teachers would make us to memorize the formulas in order to solve the problems. It did not help me to visualize the math.
It is clear that math is involved in our daily life but we do not picture it.
I really appreciate your point that math is not *just* art – you are so right, it also happens to be an enormously effective tool for describing our world, and accomplishing concrete, practical results. Maybe we need to think about how to balance these two faces of mathematics?
As I look back over the directions, connecting my statement to part of the essay; the vivid passage he uses G. H. Hardy most explains my experience.
A mathematician like a painter or poet, (in my case with Ella Fitzgerald, a musician and jazz vocalist in her scatting in “A Tisket, A Tasket”) is a maker of patterns. If his (her) patterns are more permanent than theirs, it is because they are made with ideas.
Thanks for your comment and I agree; math is clearly involved in our daily lives in so many ways.
Mathematics have always been a favorite subject of my life. It’s simplicity and creativity just knows no bounds. I share Lockheart’s discontent with the way this subject is being taught in schools. Teachers present formulas as a separate entity that students just have to memorize and apply. It deprives the students’ experience of actually understanding the concepts and how it came to be. It’s similar to giving a student a textbook to memorize instead of giving them an activity to play around with.
Math is a subject of discovery and experimentation. I found that Math classes were the only classes where I had the metaphorical lightbulb lit up in my head. It’s the only time when I had those “A-ha” moments. The struggle to fit the pieces in just the right way with a variety of methods at your disposal and when you finally get the answer you’re looking for, it’s very satisfying. Additionally, the tools and methods are things that you’ve already learned in previous Math classes. You build upon the foundation of Math and stack higher levels of cognition and understanding. This is why I disagree with the spoon feeding of Math formulas and facts. Your’re essentially giving the student a tool he has little understanding of. How can he/she build upon that? Without a solid understanding and foundation, stacking higher levels of Math will just leave a crumbling mess of confusion.
Math as an art? Ironic but very fitting. In my mind, the idea of art is merely an expression of one’s own thoughts in a some sort of medium (statue, canvas, video, etc.). You wouldn’t normally compare the logical reasoning of Mathematics and the expressive and creative aspects of art but I find some similarities. For starters, I don’t believe that Math is a complex and broad topic. In fact, I would argue the opposite. I’ve noticed that in any level of Mathematics, we’ve don’t really stray too far from the common core of Mathematics which are Algebra and number operations. From a different perspective, you can say that most, if not all, complex formulas and theorems are based upon these simple building blocks. Is it not artistic? To take a simple canvas and paints in order to express a complex emotion or idea is like taking the simple ideas of Algebra and operations to build complex reasoning and formulas.
When I think of Mathematics, the pyramids always come to mind. It is laid block after block, foundation after foundation, to create a geometrically beautiful work of art.
The article starts with discussion on music and painting which are two major two forms of art. Then, it continues to talk about mathematics which is another shape of art. I agree that math is also art since we could apply a lot of artistic concepts in math field. Or, we are able to find some math in music and painting. For instance, in music, when composers write a song, they need to draw the notes on the scales. They need to define which note is going to be used and to test it if it works properly and beautifully. Also, in painting, it is important for painters to have a sense of proportion or ratio when they need to draw down the picture of a bridge, a house, or a whole city.
I disagree that math is just simple especially after I encountered so many challenging math problems in math classes. Math is not just about algebra or numbers related concepts; it is more than that. People need to learn math from different perspectives to understand the true meaning and significance of it. As Lockhart showed the example of triangle inside the rectangle, without drawing the dash line, people could have a lot of imagination on the graph. People could propose a question based on the observation. They can also add lines to make it more complicated for others to see what is going to happen. Or, people can simply fill in colors as painters usually do when they have a picture in front of them. Just for a simple graph, people can come up so many ideas, so it is improper to say that math is simple. It depends on how people look at it and from what aspect.
I like the statement âMathematics is the art of explanation.â In other words, mathematics is not going to work without explaining. For example, a teacher is having a lesson on variable. By just giving out the definition of a variable is not enough. Students can look this online which makes no sense of the existence of educators if they only provide the formal definitions without extra work on explanations. That is also why teachers need to spend so much time in lesson plans because they need to be prepared for studentsâ doubts. Teachers also need to help students engaged in the class by different approaches to make them participate and understood. Therefore, in order to fully understand mathematics, we need explanation to make it happen.
I learned about paper folding in elementary math class. I think it is a form of art because we could apply symmetry into artistic work. Paper folding is also considered as a game; students can have different ways to get what they want. This also shows the mathematical idea that math is what peopleâs beliefs on it.
I think the reason that the author mentioned âheartbreakingâ is because, in some institutions, the teacher does not care the course of learning and replaces it by memorizing. I agree with his ideas as I went through. Many of my math teachers always start with definition without any outside sources to make the goal more interesting. I think it is going to be very boring if it continues forever. Apply visual art to math can make the materials easier for students to absorb and express it by using their own words.
1.In the essay â A Mathematicianâs Lamentâ by Paul Lockhart, I almost agree with everything the author said in his essay. The thing that I agree with the most is that he categorizes mathematics as an art, but not a pure science. As the author emphasized, mathematicians should be viewed as makers of patterns of ideas, and the procedure of making such an idea is joyful and beautiful because of the nature of the imagination, therefore, mathematics no longer should be misunderstood as a so -called pure science.
2.The one thing that is do not agree with is the lack of understanding the importance of math. The author said â Mathematics is viewed by the culture as some sort of tool for science and technology. Everyone knows that poetry and music are for pure enjoyment and for uplifting and ennobling the human spirit â. I disagree with this statement because it is clear that the culture view about poetry and music as enjoyment and ennobling the human spirit but they think that math is only a tool for science and technology with is not right because math is a very important subject that require people to think a lot, which is what the society need people to think is better that memorizing facts that might not be useful for them in the long run.
3.The quote that i think stands out in the reading is â Math is not about following directions, itâs about making new directionsâ. I really liked this quote and i chose it because i think that math is not about following formulas and memorizing steps as it is , math is about creating your own ideas and your own formulas. Math is about thinking of many ways and creating new ideas not following one idea all the time because thatâs the way you learned it.
4. I don’t have any experience that relate to the mathematics as art.
5.Lockhart describes mathematics in schools today as âheartbreakingâ because he thinks that the rich adventure of imagination is been reduced , and now it is all facts and steps that we have to memorize and follow , it is not about thinking and creating new ideas , its more about memorizing . I totally agree with him because of my experience of learning in schools. We only had to memorize steps and follow it throughout the whole semester. No one had the chance to think of any other way to solve things or create our own ideas, because we had to follow the steps that were given in class or otherwise it will consider to be wrong. which means we had lack of imagination.
Mathematics has been reduced to shadow of its former self. In Paul Lockhartâs essay âA Mathematicianâs Lamentâ, he mention âThis rich and fascinating adventure of the imagination has been reduced to a sterile set of âfactsâ to be memorized and procedures to be followed.â Paul Lockhart focus about how rich mathematics was, if everyone are to experiment with it to discover concepts about mathematics, but this adventure has been reduced to students memorizing facts and procedures in school, thus mathematics has been reduced to shadow of its former self. I agree with Paul Lockhart, it heart-breaker to watch but I understand students memorizing facts and procedures. When I first enter college, majority of college peers I encountered were memorizing formulas and procedures to solve problems, not learning background of where formulas originated. I donât think is wrong to memorize procedures considering college student are paying for their classes, so there need to memorize to process to get better results in their classes. I had memorizes all formula for my physic 1101 calculus base without prior knowledge of calculus, so I had memorizes formulas it to make up for the lack of knowledge. It not wrong to memorizes procedures but you donât learn anything, you forget it in couple weeks after your final exams.
As I started my Math Education major, my professor told me âIn middle school students are given recipes to solve this to solve thatâ, it harsh true. I remember I had teachers wrote formulas on board, then explains to student âwhat is for?â and âhow use itâ. And students would just copy and do as the teacher told them when encounter the same type problem. Then next year, students donât understand the concepts being told by another math teacher because all they learned before was formulas the year before. I was shocked, I felt all math teacher I ever had was good except null curriculum situations. But as I learn concepts with materials in mathematics to help students to understand, I came love experiment what approach can I take to support my students in learning process, so this what a type art for me in mathematics.
First and foremost, I read this entire 25-page essay. I was hooked since the first paragraph. I agree with all of Lockhartâs statements in the reading. I find it mind blowing that ever since I decided I wanted to become a math teacher, my ideas were similar in that of Lockhart. The idea of letting the students choose or question how to approach any problem or puzzle and not just simply giving them the answer. He points this out repeatedly in the essay and also the idea of a fixed system. âlecture, test, repeat.â I would say 95% of math classes specifically are like this. Teachers give you formulas and do not explain or let you question where they came from. And this is the pivotal point of his argument, students simply become bored and flunk. There is no advancing or exercising oneâs brain. Itâs crazy how deeply I feel about Lockhartâs argument I could go on forever.
I do not agree with everything he hasnât said. I agree with everything he has said in his essay.
Choosing just one quote is hard.
âHow many students taking literature classes will one day be writers? That is not why we teach literature, nor why students take it. We teach to enlighten everyone, not to train only the future professionals. In any case, the most valuable skill for a scientist or engineer is being able to think creatively and independently. The last thing anyone needs is to be trained.â
Not only is he enforcing that no one has to be trained but also comparing to another subject and a profession. School isnât for training, and it shouldnât be thought as a place to be trained in a subject. School or education in itself should be thought as a place of empowering the mind, exercising, forming ideas and reaching conclusions or better yet taking it to the next level. Many classes, and not just mathematics but every subject, has a structure that the education department created where it seems like itâs to âtrainâ people. Maybe I shouldnât say train more like constantly repeating an idea or formula in order for us (students) to memorize. But I canât really argue this because there are very few teachers who teaches in their own language in a way where students understand the principles of a concept and not just memorize the concept.
I have experienced math in the form of art for the first time in my junior year of high school when I was learning trigonometry. My teacher, Mr. Schniederman not only made it fun, but allowed us to question a problem and see what we can come up with or how two things are related, etc. He never really presented us with formulas, the whole class participated in these activities where we figured out the formula ourselves. So not only did we learn where they were derived from but we taught ourselves, it was a different experience. And I am currently experiencing math in the form of art again currently in MAT 2071 ï.
I agree with his statement of school being heartbreaking because the way I see it, school is a place of questioning and learning and exercising the mind. Itâs heartbreaking now because there isnât much of that going on. School as of now is a place of memorizing everything in order to do well on exams so that one can go to a good college and get a good job. When youâre an adult, guess what YOU FORGET ALMOST EVERYTHING! Itâs the pressure of getting accepted to a good college is what breaks the education system. Most math classes compare to the description.
âIf you deny students the opportunity to engage in this activityâ to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofsâ you deny them mathematics itself.â I completely agree with Lockhart. When I was in school and even still today, teachers would never tell you were a formula originated from. They simply showed it to you and expected you to apply it. This is what I feel that Lockhart means by âheartbreakingâ, the fact that the students are being restricted and forced to relinquish their creativity and the art of mathematics. It was not until college in my math education courses that I learned some of the amazing conceptions of were these formulas originated from and fully understood the meaning behind them. There are many students who struggle with math because of their non-understanding of the concepts. If we allowed the students to creatively conjure the formulas instead of automatically presenting it to them then this may allow them to explore and understand the concepts further and find a greater interest in what they are learning. Before reading this essay I did not believe math to be an art. I knew I had an attraction and found an amusement in the numbers and concepts but until reading this essay, I realize that math is a beautiful art and it allows us to express things in different ways and be creative in analyzing features of shapes and aspects. I am currently majoring in math education and would love to see my students as excited about math as I am and would want them to come up with some of their own conjectures and really get into their full potential and engage it in their learning.
”This is a major theme in mathematics; things are what you want them to be. you have endless choices; there is no reality to get in your way”. I totally agree with Lockhart , most of mathematics topics have endless choices, for instance numbers: there is infinite numbers, we can count as much as we want. sets, there is infinite subsets in a set of real numbers for example. function, there is infinite function; in the universe every shape has a function. even dimensions mathematician now believe that there is more than 3 dimensions like we studied in matrices.
I don’t agree with Lockhart by saying all mathematics is art, in my opinion I would say that geometry is more of art but algebra and everything else is not, why is that? well geometry we get to draw shapes that represents objects in real life and we can modify it depends on your creativity but in algebra we are solving a problem using specific pattern and logic and some problems have one way to solve them and get the answer. for example 3+x= 7 there is one x=4 that solves the equation using the logic x=7-3 witch is 4. we don’t create the answer here witch is 4 but instead we follow the logic and proof of addition and subtraction.
”so we get to play and imagine what ever we want and make patterns and ask questions about them”. This is probably the foundation of mathematics , all the theories that has been proved in math were imagined by mathematicians, they imagine and suppose things and ask questions then come up with theories. this is what we learned in class when we got introduced to proofs its is the definition of conjecture and theory .
Am I experienced math as art ? yes but in my early years in school, I remember in my 5th grade a teacher told as to draw a geometry shape in paper and cut it and come up with an object like an airplane for example but math has always been a case where I get teach and remember formulas and apply them in homework.
I think Lockhart means by saying math in schools is ”Heartbreaking” that teachers are not teaching math the way it suppose to be in schools , he probably based his answer on the small percentage of students that are doing good in math in todays education. I don’t agree with him math teachers are doing a good job in schools all the math teachers I had in my schools years were good, they inspired me to be a teacher.