OpenLab #8: 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, 11/6/14). 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 or question about your game, and bring it with you to class on Thursday 11/6 (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.  Think about the game you worked on last week (the MIU game, the bridges and walking tours game, or the mutilated checkerboards game).  Each of these games 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 Thursday 11/6 (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:
Question: Is the corner the best move, or the center?
Conjecture: The person who goes first always wins.
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.”

32 thoughts on “OpenLab #8: Lockhart’s Lament

  1. Reading mathematics and culture was really interesting. I definitely agree that math is a pure art and I do agree that math in the most misunderstood but then again what isn’t? I also believe that math is the core that brings everything together such as science, English, music, etc. It is the center piece that brings the world together. I agree that Mathematicians enjoy thinking about the simplest possible things because for example mathematicians hate writing so they uses symbols to write out there expression or equation. Although for a mathematician writing out in symbols can be easy for them to read but for everyone one else, no-one will understand what was written without a mathematician explaining it to them. I also agree that math is about wondering and amusing yourself with your imagination because I myself like to imagine numbers and think about them. I had a dream once that I created my own formula, I don’t know if it’s because I was taking number theory and it was making my brain go crazy or I was thinking about that class to hard, but I do have my own imaginations with numbers and shapes and having my own ideas.
    The quotes that I believe stands out most to me are:
    (1) What matters is the beautiful idea of chopping it with the line, and how that might inspire other beautiful ideas and lead to creative breakthroughs in other problems— something a mere statement of fact can never give you.
    (2) That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations, and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!
    (3) 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. Mathematics is an art of explanation.
    I chose these three quotes because I like how it depicts the meaning of why math is an art and what makes it an art. I also like the fact that he uses the words ideas, imaginary, patterns, and creations because I believe these are one of the few words that can depict the meaning of math because in math we have our ideas, where we can create using imaginary patterns.
    I think I had many experience as math as an art in my dreams, or when I’m randomly thinking about shapes and numbers. I also experience this when I’m explaining a certain problem that I know to one of my classmates because not only do I do more research but I also try to find the root of why that concept I the way it is and why the solution works that way. Explaining more helps me realize that I use math as an art because when I explain I like to go into detail and I like to think and create my own problems and discoveries.
    Lockhart says that schools today are heart breaking because the adventure of creating and imagining mathematical patterns has been reduced to facts, meaning that students are unable to create, imagine, and discover in a math class. All a teacher can do is give us facts or formulas where certain things have to be memorized and procedures have to be followed by certain steps. I agree with Lockhart because there is no freedom in math class where students are able to develop their own ideas, we are given out information that we have to learn and we use this to solve equations and problems. The only discovery we can make is by asking questions, however when we ask questions to a teacher we are obviously given a fact that has been proven throughout our times however we have not really made a discovery of our own, we just have to go by what the book or what the teacher says and it’s pretty depressing.
    There is no joy when solving a problem or applying a formula because the question has been answered either the same way, or has been answered many times. There is no new discovery in this, which is why after answering a math question students are left to do nothing else. Of course learning it is something good because you become open minded and more knowledgeable then before, however what matters is that inspiration of wanting to continue creating new ideas and imagining/discovering new things, it’s something a fact can’t give you. He also states that by removing the creative process and leaving with a result process you are guaranteed that no one will have any real engagement with the subject. I agree with this because students will not get inspired, why you think students are drained after math class or find math boring, it’s because there is a loss of discovery/imagination in a math class.

    1. I like your phrase “that inspiration of wanting to continue creating new ideas and imagining/discovering new things, it’s something a fact can’t give you” – I encourage you to be on the lookout for ways to encourage that inspiration, within the structure imposed by our school system. And good luck!

    2. I really like you idea that you think that math is the core that brings everything together such as science and as any subject and i also think that it is a very important subject because it involves a lot of thinking which the society need, thinking is more important and better than just memorizing facts . I also like your experience in creating your own formula and i’m sure it required you a lot of thinking in order to create this formula. I really like your way of thinking about creating new ideas and formulas.

  2. 1 Basically, I 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. If you ask me, what do I not agree with with the author? There is a minor disagreement with regard to whether or not mathematics should be eliminated from the public school system. According to the author’s belief, poetry and music are not part of public school curriculum because “they are pure enjoyment and for uplifting and ennobling the human spirit. Mathematics still exists as a mandatory subject in all the public school systems because we make it to be important, but ignore its natural attributes such as what poetry and music have. I do not have any solid evidence to oppose his claim, but I think there are some object reasons that music cannot be imposed as a mandatory subject such as budget and space because of the necessity of instruments.
    3. The quote I would choose to summarize the main idea of this article is
    “Why do colleges care if you can fill in numbered regions with the corresponding color?”
    “Oh, well, you know, it shows clear-headed logical thinking.
    Even though it is a dialogue from anonymous, it reflects the reality of why teachers and parents make students engage in certain subjects and expect them to reach certain goal regardless of the enjoyment of learning it. Similarly, we often admire someone’s intelligence if he/she is good at math. There is no correlation between scoring high at math and being a mathematicians. The distorted purpose of learning mathematics produces the lament that the author resonates with.
    4. I have never had any experience of mathematics as art. However, I heard that models bodies coincide with the golden ratio and designers use the symbol of ∏ to have their designs stand out. If that counts, I think it is an experience of mathematics as art.
    5 Lockhart thinks of mathematics in schools today as” heartbreaking” because “the rich and fascinating adventure of the imagination has been reduced to a sterile set of “facts” to be memorized and procedures to be followed”. I 100% agree with him because if I recall the experience of learning math no matter in what kind of school , the only things that pop out in my head are formulas , symbols, abstract axioms , postulates and infinite practice and tests. I never have gotten a chance to see its beauty and the leisurely nature of mathematics.

    1. Your discussion of “the distorted purpose of learning mathematics” inspires an excellent general question to consider when approaching all teaching – “what is the purpose of this?” When I ask myself this question, I find it is complicated – there are many reasons to teach a subject, some imposed by external factors (the school, department, or culture) and some arising from within – and not all of them are as bleak as Lockhart describes. But thinking intentionally about the ultimate purpose of the material can help shape the way you teach, and sharing your thoughts with your students can help them to see the point of what you are asking them to do. Keep it up!

    2. I agree with you about your experience that the only thing comes in your mind is formulas and steps. I also have the same experience because this is the way they taught us math . We only got to think of math as memorizing formulas and following steps, but we did not have or get the chance to experience the beauty of math by creating new ideas.

  3. 1. I really enjoyed reading mathematics and culture. I totally agree with everything the author said. The one thing that I agree the most is ” it is every bit as mind blowing as cosmology or physics and allows more freedom of expression than poetry, art, or music . Mathematics is the purest of the arts, as well as the most misunderstood”. I agree with this statement because i think that math is really about mind blowing. Math involve a lot of thinking in order to arrive to your solution. A lot of people like to try problems without following formulas , they like to create their own formulas and try out many ways to get it right which require a lot of thinking. I also agree that math is the purest of art as well as it is the most misunderstood because sometimes as well as people see it a piece of art that is very easy and creative , it can become very hard and understandable.
    2. The one thing that is do not agree with is the lack of understanding the importance of math. The author said ” 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 “. 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 ” Math is not about following directions, it’s about making new directions”. 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 that’s the way you learned it.
    4. I really do not have any experience of math as art, but in my point of view i think that dealing with drawing in math is consider art . I think that drawing circles, squares and triangles is considered art for me. I think that math is more about dealing with numbers than looking at shapes , so for me any kind of shape is considered to be math art.
    5. Lockhart describes mathematics in schools today as “heartbreaking” 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.

    1. I also think math is mind blowing! But I’m afraid many students in math classes in America today do not share this perspective… Part of the job of math teachers, I think, is to find ways to blow students’ minds – although you will not find it anywhere in the job description. Good luck!

  4. 1- One thing that I agree about the reading is the mathematics is an art. I believe this because like the author said math is a way of creating and discovering new ideas. As the author stated math is like any other art on is inspire to create and discover.
    I don’t agree with the fact that the author said that we should let students create their own problems meaning that students shouldn’t create their own rules or formulas in mathematics. The formulas that are in mathematics exist for a reason; many mathematicians’ years ago have discovered and proved their logic behind their formulas.

    2- “Mathematicians enjoy thinking about the simplest possible thing, and the simplest possible things are imaginary.”
    I chose this quote because I think the quote is true, mathematicians don’t think about heavy mathematical equations as most people think, mathematicians think about simple thing that one can imagine meaning that it can be pictured nice and friendly. Pictures like the one he gave as an example. Math shouldn’t be pictured as a scary image.

    3- I think that math is an art in every way because everything revolves around mathematics, the person who made or thought of a structure used the math art tools to make it possible. I believe math is the art tool for everything because the structures are so precise thanks to the tools provide by math.

    4- I think he means that students are just told to memorize facts without having the choice of exploring as to why it works. I think he means that we are like a computer and are being programed to know certain facts and when the moment is right we recall those facts and put it to use.
    I neither agree nor disagree because this meaning goes both ways for me. I sometimes think the way the author thinks other time I don’t because not everyone is interested in math.

    5- My own experience goes both ways because sometimes I want to understand and discover other times I would rather memorize something and recall it when necessary

    1. I like your point about mathematicians and simplicity. Even complicated-looking ideas are sometimes simple at their core. Every once in a while in my own math education I will be struggling with a complicated subject, and eventually something will just ‘click’ and it will suddenly seem very simple – as though I saw a deeper pattern, which makes all the surface complication make sense in it didn’t before. These moments don’t happen too often for me, but they are wonderful!

  5. According to Mr. Lockhart, math is art. I agree to an extent to this subjective idea, and I would suspect he would agree to the idea that art is not the sole classification that fits math. To pigeonhole math into such a classification does a great disservice to math itself. Math is science also; we can observe the nature of the beast and derive hypotheses and theories then apply them to create technology or derive new techniques to further our study. Math is also a universal language that everyone understands to an extent, regardless of what cultural limitations exist. It is the tool we use to create the art that math is. It is logical reasoning. It is a cycle… I could go on and on. It is more than all of that combined; it is MATH.
    It is no secret that math education in the United States has been less than the ideals that many of us swear by. Lockhart says “nobody has the faintest idea what mathematicians do.” This is said in spite of the fact that throughout everyone’s academic life, they will come into contact with math and learn as we did up to a point. And for the most part, he would be correct in saying so.
    Much of it is the result of this epidemic and cultural misconception (as I would term them) that it is a “necessary evil” or “soulless rationalism” (terms I have heard attributed to math). I have heard advertisements including testimonials where people have not used any of the math they learned in high school. In the most recent decade or so, we have seen elections where politicians have decried “fuzzy math” with little to no alternate math as a rebuttal. In our media, we see school-based sitcoms in which the subject that vexes students the most is some level of high-school math. Is there any doubt that there is anti-math sentiment at nearly all demographics? As future math educators, we can expect this cultural stigma to torment us in the form of this one question we will be condemned to answer for the rest of our careers, “why do we need math?”
    Only recently have I thought of math as an art (it took some “scientific” observation to do so). I am unfortunately one of those who were taught math in the method Lockhart decries. It took years of accidental discoveries and self-imposed study in order to come to this conclusion myself. I had observed and read the math and science behind some Renaissance art and have worked with mathematical tessellations. I have seen complex architecture whose designs were made possible by complex math beyond my comprehension. I have seen math dedicated to origami to be used in various sciences and nanotechnology. Math as the basis for art makes it art.
    Lockhart declares the state of math education as of 2002 to be “heartbreaking”. Even then, statistics showed that the quality of math education in our country had been slipping among developed nations (at that time, like 14th or something and now 28th or so). As a product of said education pre-2002, I could see that. As I have said before, I spent most of my math education waiting for the teacher to tell me what to do and doing as told. It had deprived me of a lot of conceptual understanding; I could tell others I know the material, but I could not explain the intricacies as some of my classmates could. I have seen other classmates being sent to lower academic tracks or left back because they could not achieve similar academic success in a system that likely favored a learner such as myself. Of course, you could sense even back then that socioeconomic status had a lot to do with who received a decent math education and who did not.

  6. I agree with Lockhart that mathematics in the classroom today does not allow the student to think and discover for themselves. Mathematics really has become all about memorizing formulas and procedures without really understand why or how they work. Mathematics in classrooms today has been reduced to giving students the formulas and then letting them plug the appropriate numbers in and if they did it correctly then they understand the math and the teacher beleives that they have taught them something of value. I believe that this is also the reason that Lockhart refers to math in today’s school as “heartbreaking” and I agree. We are taking the creativity away from children, when this is the time we should be promoting creative thinking. Based on my personal experience with math in my school days, I have never experience math as an art form. I think if I had I would appreciate it more that I do now. As a teacher I want my students to be excited about learning to think outside the box and come up with their own ideas. Then I will know that I have been my job.

    1. Great! And of course the challenging thing is doing this within a school system that is very focussed on fixed goals (like test scores), and doesn’t care about this kind of creative skill. Meeting this challenge requires creativity of another sort – best of luck!

  7. 1.Usually when people talk about math the very first things that come to their mind is a bunch of equations and formulas and it doesn’t seem to interest many when it is thought that way. The one thing that I agree is that mathematics is an art and not just science as many view it as. What I also like about is how he relates mathematics to imaginations and that it should be thought of something as beautiful and simple. His concern about how mathematics is taught in schools today is true because the beauty behind a mathematical concept is rarely highlighted.
    2.I don’t have any major disagreements however apart from how mathematics can be fun and enjoyable I don’t seem to agree when he says that “the only way to get the truth out of the imaginations is to use them, which is hard work” and yet his principle or perspective of mathematics is “simple and beautiful.”
    3.“By concentrating on what, and leaving out why, mathematics is reduced to an empty shell.” I like this quote because the author encourages that math is not just restricted to accept the way it is for example theories and formulas but you’re also given the opportunity to ask why it should or shouldn’t be accepted until you are convinced. I think asking questions when learning mathematics is what makes it interesting as a subject. I learn a lot by asking questions as well and sometimes I get others involved too making it seem really important. Often times one might find math really frustrating especially when is neither a yes nor a no answer but that’s what make math really different.
    4.In my MEDU 2010 class I learn about tessellations using geometry sketchpad and using certain polygons that can create perfect tessellations, so far this is the only thing I remember experiencing math as an art.
    5.I think he means that the quality of education in teaching math to the students nowadays is not up to standard. In other words math is not taught the way it should be taught and I totally agree with him that it’s “heart breaking.” To be honest I was never taught to find the area of a triangle in a creative and a very imaginative way as he did, instead I remember being spoon fed with the formula and that’s how I learnt to find the area of a triangles and a list of other shapes and till now I’m not really convinced how it got there.

    1. I’m glad you mention the importance of asking questions – it’s true, this simple act can have a huge affect on your understanding (and also on your experience in a class). But creating an environment in which students feel comfortable asking questions is a challenge!

  8. 1. He said “What matters is the beautiful idea of chopping it with the line, and
    how that might inspire other beautiful ideas and lead to creative breakthroughs in other
    Problems” I totally agree that the beautiful idea is the key. Beautiful idea helps mathematicians seek for patterns, and the important is to use them to formulate new conjectures.

    2. First, He said “The first thing to understand is that mathematics is an art. “
    Second, he said “The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation”
    Lockhart adore Mathematic. Creative is necessary in art and Mathematic. Is mathematics an art work? Mathematic need to be proof, but do we know how to proof of art work? Is there right or wrong in the art work? Mathematics is the truth. If some equation is correct today, it will be correct on tomorrow with no argue. If Mathematic was an art, equation would be the prefect art work, and Mathematicians would be the hardest artist.

    3. He said” If I chop the rectangle into two pieces like this, I can see that each piece is cut diagonally in half by the sides of the triangle. “I agree with it because it is a fact, and it will never change. There is always a better way to explain Mathematic, and that is how we improved, especially abstract concepts.

    4. Mathematics have example as art work, for example, Tessellation. Tessellations can be generalized to beautiful picture and higher dimensions. Tessellation is a prefect art work.

    5. Lockhart describes mathematics in schools today as “heartbreaking” because teacher taught their student to memorize and procedure the formula only. I agree to use different methods to explain mathematics. However, I doubt student know how to solve the problem in creative way like Lockhart. According to bloom’s taxonomy, creating is the hardest level to compete. It is easier to have an equation for student to remember, understand, and apply in the beginning. And then, student can analyze, evaluate, and create the problem

    1. I love that you mention Bloom’s taxonomy – and you’re right, this is an advanced mode of engagement. But it’s also the kind of work that is most rewarding to the person doing it – so if you are able to get people creating, they are motivated to continue because the act of creating is a joy!

  9. 1) I do agree that mathematics is an art because in math, one can play around with shapes and numbers and mold it to anything they please if it helps them prove a certain point. The process of thinking, making mistakes, and discovering new things are the sketches that a mathematician takes to create the art (theory). I also agree that “by removing the creative process and only leaving the results of the process, you virtually guarantee that no one will have any real engagement with the subject” because when people are in middle school and high school learning about math they are just taught formulas and how to apply them to problems without really understanding where the formula came from and how it can be applied to the real world. That’s why I believe that most people don’t consider math a real art because growing up they just memorized things and from there they just viewed math as a bunch of complicated formulas and steps you have to memorize. And if one views something like that it’s obviously boring.
    2) I don’ really disagree with anything.
    3) “If you deny students the opportunity to engage in this activity- to pose their own problems, and make their own conjectures… to be creatively frustrated, to have inspiration… [then] you deny them mathematics itself.”
    I chose this quote because I agree with it. The funnest part in math is trying to figure stuff out through thinking, error, and etc. If one takes that away than math is just a dull subject.
    4)Yes I have art created though the use of mathematics. Its all around us. All the building structures, the bridges, the cars, and planes; all of them required math to create.
    5) Math is now really heartbreaking. Like I said earlier, all we really do is memorize stuff. To be honest I really do believe that it’s a little necessary to teach math like that because math is really a language and if you don’t understand certain concepts or how to express them then you won’t understand thing and you won’t know how to express certain ideas. The only thing that they should change is give a little background information on how certain formulas help us in the real world just to keep their interest (the history/ the way something came to be is always interesting).

    1. I like your inclusive perspective on art and mathematics – and especially the suggestion that art is all around us (not just something that is locked up in museums). Having this perspective makes the world a much more interesting place.

  10. In the essay “ A Mathematician’s Lament” by Paul Lockhart, I agree with the author that mathematics is an art and our cultures does not recognize it as mathematics is not as expressive as any other type of art. I also agree that people have the common opinion about mathematics that it is connected with science, because most of my friends who doesn’t like math they seemed to have same belief about mathematics.
    Honestly I do not see anything in this essay that I cannot agree less.
    The quotes that I like are “The art is not in the “truth” but in the explanation, the argument.” , “Mathematics is the art of explanation”, “That little narrative is an example of the mathematician’s art: asking simple and elegant questions about our imaginary creations,and crafting satisfying and beautiful explanations. There is really nothing else quite like this realm of pure idea; it’s fascinating, it’s fun, and it’s free!”. The reason why i have chosen this quotes because it simplify in short that mathematics is an art and it is full of ideas , imaginations and patterns.
    No I did not have an experience of mathematics as art.
    Lockhart describes mathematics in schools today as “heartbreaking” because mathematics in school should be creative, fun and imaginary but it is rather boring and full of memorization of facts that has a series of procedure that is needed to follow. Yes I agree with Lockhart because in school students are enforced to memorize the formulas, facts and procedure to do the math, which actually reduces the imaginations of a student to be creative in math.

  11. I agree with paul Lockhart that the way mathematics is presented today to high school or student or pupils in the elementary schools yields less room to creativity. everything is made easy in then way that students do not even need to think or shape their creativity in order to solve a math problem. I agree with him something needs to be done in the math edu system. this problem is pertaining not only in the high school or elementary school, but we have the same problem in colleges and universities. definitely this is general situation and they really no one we can blame.
    we cannot blame either innocent students or our deer professors and teachers. their duties is not as easy as many peoples think. Paul Lockhart has to realize that we are living in the 20s and when one generation switch another, some values are gained while others values are lost. the actual computerized and busy generation of students cannot be educated as M.Lockhart had been teach in 1950 or whatever. even the vision that people have for teaching profession has dramatically changed through the last 40 decades. teachers and professors used to be classified among honorable personalities in any society; governments used to make sure that teachers have all the power in their hand in order to lead that honorable profession. we cannot blame our teachers today saying they do not help student to be more creative, they do their best with the pressure they have to face from their departments , from the student parents and from the students themselves. most of teachers do not even have the freedom to make their own syllabus, professors are not trusted and being watched by the government and their departments; they are also rated by their students and being a good teacher today does not mean being creative, but having “A” grade students in the class.
    Paul Lockhart said “faculty do not have enough time to be creative”. time is what every body is missing in the 20s. teachers as students do not have time. many high school or college students have a job while they are studying because the cost of life is higher compared to the 1950s or 1960 and most of parents can afford their student’s children. this implies that there are less time left to these students to think about their lessons and to be creative. this a social fact and we cannot fight against that.
    reforms need to be done not only for math education but the education whole education system needs to be reviewed. teachers needs more power , more freedom and dignity in order to fulfill their jobs. parents need to get involved in their child education and this will really help teachers. when I was in the elementary school and up to the high school, my parents used to make sure my homework are done every class day and not letting the after class program to take care of my homework.

    1. Unlike Lockhart, who spends time pointing out that there is a problem with math education but doesn’t offer many solutions, you’ve outlined for us a number of key areas that must be reformed in order to fix these problems. Great!

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