pavlukan's review
4.0
One of my Internet friends advised me to read this essay after I had admitted to him that I had little desire to connect my life with mathematics in the future.
It would’ve been OK to answer in such a manner if I didn’t know maths, didn’t study it and my grades sucked, but the reality is different. For the past two years, I’ve been studying this subject for 75 minutes every single day. And those 75 minutes have paid off in terms of grades and knowledge. However, that doesn’t mean that I enjoy mathematics, as many of my classmates tend to believe. I’m right on the border of being indifferent towards it and disliking it, jumping from one place to another depending on the topic or field I’m delving into.
Some also believe that I’m a genius, which is completely untrue. I’ve just learned to excel in the curriculum of my country, not in true maths. This belief was further ingrained into me after reading A Mathematician’s Lament by Paul Lockhart.
Even though Paul Lockhart’s essay concerns itself mainly with the system of education of mathematics in the USA, its critique can be applied to other countries as well. Belarus, Russia, Germany, France, the UK, China, India, Australia, etc. suffer from the same issues as the United States when it comes to maths education.
Although the essay was written in 2002 and published in 2009, nothing has changed since then. Countless students (and teachers) around the globe are still tortured with the abomination that we call mathematics today.
I know from experience and from conversations with my classmates that many parents are worried that their children don’t get good grades on maths tests and that they view the subject as boring. They, of course, like to delude themselves that their children are merely too young to understand the practical applications and the necessity of mathematics today.
The author of the essay argues that the students are actually right, not the parents or teachers, most of whom don’t even know their subject well enough to educate others. However, he also says that we shouldn’t blame them because they’ve also been indoctrinated and enslaved by the broken education system.
There are teachers who are good at mathematics and are extremely passionate about it, but they can’t spread that love and inspire other children’s curiosity because they have to follow the program that the government has set in stone.
One of the most popular quotes from this essay is this one:
“The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.”
Paul Lockhart goes as far as to claim that mathematics is the purest form of art, with which I have to politely disagree because I don’t understand the meaning of “pure” in this context. Is there a metric that can tell us the level of purity of one art versus another? Is it the level of abstraction required for comprehension? Or is it the amount of imagination involved? Without a clear definition, I can as well declare that music is the purest form of art, or painting.
At any rate, I do like the approach he proposes for learning mathematics. It is based on actual problem-solving, the use of intuition, and exploration, not on the knowledge of fancy symbols, the ability to solve boring problems, and extreme rigidity.
When I was only beginning to learn geometry, I remember when I “discovered” two beautiful patterns about the medians of triangles. They always seemed to cross each other in one spot and that spot divided them into two parts, one of which was twice as big as the other one. I formulated a conjecture and tried to prove it. I don’t think any language in the world can explain the amount of awe, concentration and interest I experienced at that moment. That situation is still haunting me to this day. I’ve not felt anything like it in a long time.
Although Paul Lockhart’s approach revolves around fun, exploration, and actual mathematics, this doesn’t mean that studying it should always be pleasant. The actual comprehension of beauty can only arrive when you are hitting your head against a problem and then a solution comes into your mind.
This also doesn’t mean that you will never have to use or remember different formulae. Sometimes, a student is very close to getting over a conundrum, but all they lack is the knowledge of one theorem or formula.
And I want to take a brief moment to talk about another art form that resides in the field of mathematics itself: the art of proof.
For Paul, every proof should satisfy two primary criteria:
It should be logically sound.
Obvious, isn’t it? If your proof isn’t logically sound, then it’s not really a proof.
It should be elegant and simple.
This one is harder to deal with because almost everybody on this planet can come to objective conclusions, but not everybody can do it in a simple, elegant and creative manner.
I disagree with the second point, though.
Yes, it will be very great if our argument isn’t only logical, but stunning to look at and very easy to understand. Sadly, this is a lofty goal that not many people manage to achieve consistently. Striving to make your proofs as elegant and as simple as humanly possible is going to lead to a great deal of unnecessary disappointment and frustration.
It’s like forcing a beginner coder to write extremely clean code.
Such things come with practice. Just do your best, forget the rest. You will always have time to come back and refine your proofs.
I think every student who views the subject as tedious and boring should read this essay. And I think every maths teacher must read it as well. Adults have a habit of forgetting that they were once students too. It will freshen up their memories and probably hurt their ego, but that’s not an entirely bad thing.
It would’ve been OK to answer in such a manner if I didn’t know maths, didn’t study it and my grades sucked, but the reality is different. For the past two years, I’ve been studying this subject for 75 minutes every single day. And those 75 minutes have paid off in terms of grades and knowledge. However, that doesn’t mean that I enjoy mathematics, as many of my classmates tend to believe. I’m right on the border of being indifferent towards it and disliking it, jumping from one place to another depending on the topic or field I’m delving into.
Some also believe that I’m a genius, which is completely untrue. I’ve just learned to excel in the curriculum of my country, not in true maths. This belief was further ingrained into me after reading A Mathematician’s Lament by Paul Lockhart.
Even though Paul Lockhart’s essay concerns itself mainly with the system of education of mathematics in the USA, its critique can be applied to other countries as well. Belarus, Russia, Germany, France, the UK, China, India, Australia, etc. suffer from the same issues as the United States when it comes to maths education.
Although the essay was written in 2002 and published in 2009, nothing has changed since then. Countless students (and teachers) around the globe are still tortured with the abomination that we call mathematics today.
I know from experience and from conversations with my classmates that many parents are worried that their children don’t get good grades on maths tests and that they view the subject as boring. They, of course, like to delude themselves that their children are merely too young to understand the practical applications and the necessity of mathematics today.
The author of the essay argues that the students are actually right, not the parents or teachers, most of whom don’t even know their subject well enough to educate others. However, he also says that we shouldn’t blame them because they’ve also been indoctrinated and enslaved by the broken education system.
There are teachers who are good at mathematics and are extremely passionate about it, but they can’t spread that love and inspire other children’s curiosity because they have to follow the program that the government has set in stone.
One of the most popular quotes from this essay is this one:
“The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such.”
Paul Lockhart goes as far as to claim that mathematics is the purest form of art, with which I have to politely disagree because I don’t understand the meaning of “pure” in this context. Is there a metric that can tell us the level of purity of one art versus another? Is it the level of abstraction required for comprehension? Or is it the amount of imagination involved? Without a clear definition, I can as well declare that music is the purest form of art, or painting.
At any rate, I do like the approach he proposes for learning mathematics. It is based on actual problem-solving, the use of intuition, and exploration, not on the knowledge of fancy symbols, the ability to solve boring problems, and extreme rigidity.
When I was only beginning to learn geometry, I remember when I “discovered” two beautiful patterns about the medians of triangles. They always seemed to cross each other in one spot and that spot divided them into two parts, one of which was twice as big as the other one. I formulated a conjecture and tried to prove it. I don’t think any language in the world can explain the amount of awe, concentration and interest I experienced at that moment. That situation is still haunting me to this day. I’ve not felt anything like it in a long time.
Although Paul Lockhart’s approach revolves around fun, exploration, and actual mathematics, this doesn’t mean that studying it should always be pleasant. The actual comprehension of beauty can only arrive when you are hitting your head against a problem and then a solution comes into your mind.
This also doesn’t mean that you will never have to use or remember different formulae. Sometimes, a student is very close to getting over a conundrum, but all they lack is the knowledge of one theorem or formula.
And I want to take a brief moment to talk about another art form that resides in the field of mathematics itself: the art of proof.
For Paul, every proof should satisfy two primary criteria:
It should be logically sound.
Obvious, isn’t it? If your proof isn’t logically sound, then it’s not really a proof.
It should be elegant and simple.
This one is harder to deal with because almost everybody on this planet can come to objective conclusions, but not everybody can do it in a simple, elegant and creative manner.
I disagree with the second point, though.
Yes, it will be very great if our argument isn’t only logical, but stunning to look at and very easy to understand. Sadly, this is a lofty goal that not many people manage to achieve consistently. Striving to make your proofs as elegant and as simple as humanly possible is going to lead to a great deal of unnecessary disappointment and frustration.
It’s like forcing a beginner coder to write extremely clean code.
Such things come with practice. Just do your best, forget the rest. You will always have time to come back and refine your proofs.
I think every student who views the subject as tedious and boring should read this essay. And I think every maths teacher must read it as well. Adults have a habit of forgetting that they were once students too. It will freshen up their memories and probably hurt their ego, but that’s not an entirely bad thing.
prynne31's review against another edition
challenging
informative
reflective
medium-paced
4.5
Definitely makes you think! Educational concepts apply to a lotore than just math education.
dashtaisen's review
adventurous
challenging
emotional
funny
hopeful
inspiring
mysterious
reflective
fast-paced
4.0
I might have minor quibbles with some of the more extreme statements, but this was a validating and inspiring read
This is more of a rant about the way things are than a plan for how to fix them — but I mean the title makes it pretty clear that’s what we’re in for, so this isn’t a criticism!
This is more of a rant about the way things are than a plan for how to fix them — but I mean the title makes it pretty clear that’s what we’re in for, so this isn’t a criticism!
keytinker's review
2.75
Perhaps overly idealistic but a compelling argument nonetheless. However, the literary construction of the argument and the tone of the author leave much to be desired. Thought-provoking content, but poorly presented.
amihanbooks's review
4.0
As someone who has always placed math as one of my least favorite subjects, this was thought-provoking and life-changing. I regret not growing up experiencing mathematics as an art form rather than rote memorization. I'm no mathematician, but when he said, "to do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration, to be in a state of confusion, not because it makes no sense to you, but because you gave it sense and you still don't understand what your creation is up to; to have a breakthrough idea, to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty." I could definitely relate to this whenever I create something. I'm convinced!
That being said, he did tend to get a bit ~rambly~, oftentimes just plain ranting and making the same point over and over again. But it's okay! Everyone tends to ramble when they're passionate about something, and he has every right to be angry with the system.
However, I may be biased with this, but I'm going to have to disagree with him when he claimed "Mathematics allows more freedom of expression than poetry, art, or music. It is an art form more profound than any poem."
I didn't think there was any need to bash on other art forms in order to prove that mathematics is beautiful in and of itself. He had already made a perfectly sound argument as to why it is an art form, so I found it quite unnecessary to go further and claim math is superior to any other. But then again, he's a mathematician and I'm an art person, so to each their own. I do think all forms of art can be equally beautiful in their own ways though.
All in all, I highly recommend this to EVERYONE, whether you love math or not. Actually, ESPECIALLY if you hated math growing up. Mathematics deserves to be appreciated, and it's important to know that what we've been taught in school is not at all what it's really about. We've actually been taught the complete opposite. Read the book to find out what math is!
That being said, he did tend to get a bit ~rambly~, oftentimes just plain ranting and making the same point over and over again. But it's okay! Everyone tends to ramble when they're passionate about something, and he has every right to be angry with the system.
However, I may be biased with this, but I'm going to have to disagree with him when he claimed "Mathematics allows more freedom of expression than poetry, art, or music. It is an art form more profound than any poem."
I didn't think there was any need to bash on other art forms in order to prove that mathematics is beautiful in and of itself. He had already made a perfectly sound argument as to why it is an art form, so I found it quite unnecessary to go further and claim math is superior to any other. But then again, he's a mathematician and I'm an art person, so to each their own. I do think all forms of art can be equally beautiful in their own ways though.
All in all, I highly recommend this to EVERYONE, whether you love math or not. Actually, ESPECIALLY if you hated math growing up. Mathematics deserves to be appreciated, and it's important to know that what we've been taught in school is not at all what it's really about. We've actually been taught the complete opposite. Read the book to find out what math is!