On Math Teaching: Lockhart's Lament(maa.org)
maa.org
On Math Teaching: Lockhart's Lament
http://www.maa.org/devlin/LockhartsLament.pdf
23 comments
The problem is that schools don't teach logic. Everything that they teach is arbitrary.
English class is essentially potpourri. Random order of books + Shakespeare. No teaching as to what makes writing a joy to read. Only emphasis on the relevant social issues.
History class is horrible, no explanations needed. It stinks because History has so much potential. If you have a good teacher, the sort that teaches you shades of grey, then it's amazing. But that's not the curriculum. Also, there's no emphasis on what MATTERS, which is current events. Rarely do you learn anything 80s onward. That means that most high school graduates are functionally retarded in terms of modern politics and news.
The science classes vary between schools, but it's making certain types of science MANDATORY that just lack logic. Bio and chem? I get physics, but even there: most people just don't need to know, and NOBODY learns in a way that makes them like it.
Electives are universally horrid. Teachers regularly have the attitude of "I didn't specialize in it, so standards don't matter."
...and there's math, which is explained properly here.
I've speculated before on what I think a logical school would be like, and it would be something like the following:
-Math and physics form the core learning at a younger age, because with them you can do anything.
-English courses focus not on movements, not even necessarily on "we think these guys are great," but emphasize different attitudes on how to write. Aesthetic, deconstructionist... teaching students to actually look at how writing works. Also, pop literature needs to be examined as well. Not teaching it ignores a field of literature that never gets brought up.
-Philosophy would be a great starter high school course.
-What we call "health/fitness" would focus less on STDs in 7th grade and focus MUCH, MUCH, MUCH more on understanding other people. It took me until about sixteen to learn that people aren't solely irrational beings, and the fact that it took me that long is that school is heavy on mysticism (like, assuming that older ALWAYS means better). That should be fixed.
-Emphasis on current affairs and technologies. Learning about the past should be focused on events rather than sweeping ideologies, and more comprehensive on a few key areas.
-All classes should emphasize writing and conveying ideas clearly. But that means no grading things based on length, either: people good at writing short should.
-No grading at all, actually. Grading has done nobody any good, least of all colleges that assume a high GPA necessarily means a working mind.
English class is essentially potpourri. Random order of books + Shakespeare. No teaching as to what makes writing a joy to read. Only emphasis on the relevant social issues.
History class is horrible, no explanations needed. It stinks because History has so much potential. If you have a good teacher, the sort that teaches you shades of grey, then it's amazing. But that's not the curriculum. Also, there's no emphasis on what MATTERS, which is current events. Rarely do you learn anything 80s onward. That means that most high school graduates are functionally retarded in terms of modern politics and news.
The science classes vary between schools, but it's making certain types of science MANDATORY that just lack logic. Bio and chem? I get physics, but even there: most people just don't need to know, and NOBODY learns in a way that makes them like it.
Electives are universally horrid. Teachers regularly have the attitude of "I didn't specialize in it, so standards don't matter."
...and there's math, which is explained properly here.
I've speculated before on what I think a logical school would be like, and it would be something like the following:
-Math and physics form the core learning at a younger age, because with them you can do anything.
-English courses focus not on movements, not even necessarily on "we think these guys are great," but emphasize different attitudes on how to write. Aesthetic, deconstructionist... teaching students to actually look at how writing works. Also, pop literature needs to be examined as well. Not teaching it ignores a field of literature that never gets brought up.
-Philosophy would be a great starter high school course.
-What we call "health/fitness" would focus less on STDs in 7th grade and focus MUCH, MUCH, MUCH more on understanding other people. It took me until about sixteen to learn that people aren't solely irrational beings, and the fact that it took me that long is that school is heavy on mysticism (like, assuming that older ALWAYS means better). That should be fixed.
-Emphasis on current affairs and technologies. Learning about the past should be focused on events rather than sweeping ideologies, and more comprehensive on a few key areas.
-All classes should emphasize writing and conveying ideas clearly. But that means no grading things based on length, either: people good at writing short should.
-No grading at all, actually. Grading has done nobody any good, least of all colleges that assume a high GPA necessarily means a working mind.
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I agree with all, but: grading is currently a horrible means of evaluation, but still some sort of evaluation. Without evaluation --read, accountability-- no forced studies can acomplish anything. And currently school is nothing else than forced studies. Demeaning, even.
On the other hand, if you make school optional and create a culture of liking or perceiving as essential to attend school for one's well-being, then motivation is way different, and grading is then really an impediment.
On the other hand, if you make school optional and create a culture of liking or perceiving as essential to attend school for one's well-being, then motivation is way different, and grading is then really an impediment.
I think - this is just me, mind you - that interviews are the best way of weeding out really bright kids from pretenders. It means you can't hide behind any words and have to rely on who you actually are. Although, granted, that's not quite as efficient.
"It's amazing how many people don't know the very basics of it like, for instance, how multiplication is just a compact way of expressing a repetitive sum."
You mean to say, of course, that many people usually think this is so, while it being obviously wrong... right?
Otherwise, my friend, I think you should forget all about what your elementary school teacher said and re-examine say multiplication of fractions for starters.
You mean to say, of course, that many people usually think this is so, while it being obviously wrong... right?
Otherwise, my friend, I think you should forget all about what your elementary school teacher said and re-examine say multiplication of fractions for starters.
This lament applies to ALL the subjects in school, not just math. The problem is you just can't teach the joy of each one of these subjects to each of the students.
Some students will enjoy certain subjects and just 'get it'. But most won't, and I do not think there is any to teach it. You just can't.
So you teach what you can: at least they can read, even if they don't understand literature. Same for math: at least they can manipulate the symbols.
And the proof of the problem is the "new math". It tries very hard to do what he is saying and it's been a failure. At the end of it students don't know how to do math.
Except for those students who excel at math - but they would excel at it no matter how it was taught.
Some students will enjoy certain subjects and just 'get it'. But most won't, and I do not think there is any to teach it. You just can't.
So you teach what you can: at least they can read, even if they don't understand literature. Same for math: at least they can manipulate the symbols.
And the proof of the problem is the "new math". It tries very hard to do what he is saying and it's been a failure. At the end of it students don't know how to do math.
Except for those students who excel at math - but they would excel at it no matter how it was taught.
There is a stunning gap in basic capability between the best worst students, but even within subjects there are large ranges. I found Calculus II in collage a lot easer than mesmerizing multiplication tables in 2nd grade.
Many people say they hate math but love logic puzzles like SUDOKU which is math but not the type of thing they learn in school. The important thing is to not confuse students that learning the basic tools like addition and spelling have anything to do with the type of work you do as an engineer or writer. The way the system works you need to have a wide range of basically irrelevant skills before they are willing to teach real math the gray areas of history. Grouping writing poetry in the same subject as grammar is probably harmful because teachers spend so much time on grammar they the skip poetry because it's to "advanced" and it's not on the tests etc. Yet, they don't build off of each other and many of the greatest poets lived before grammar or spelling rules where written down.
PS: I expect most students would pay more attention to the basics if I they had some use for them.
Many people say they hate math but love logic puzzles like SUDOKU which is math but not the type of thing they learn in school. The important thing is to not confuse students that learning the basic tools like addition and spelling have anything to do with the type of work you do as an engineer or writer. The way the system works you need to have a wide range of basically irrelevant skills before they are willing to teach real math the gray areas of history. Grouping writing poetry in the same subject as grammar is probably harmful because teachers spend so much time on grammar they the skip poetry because it's to "advanced" and it's not on the tests etc. Yet, they don't build off of each other and many of the greatest poets lived before grammar or spelling rules where written down.
PS: I expect most students would pay more attention to the basics if I they had some use for them.
I think that 'finding some use for them' is an ultimately doomed approach, if only because each student is going to have different ways in which they want to use the mathematical tools with which they are presented.
My father, who is an 'educator', seems to think that the best way to learn trigonometry is to use it to teach carpentry, but I can't think of anything that would be more dull, for several reasons. One, because I imagine that well less than half of the class would have a genuine interest in carpentry; and Two, because building things with wood requires very little trigonometric calculation, so nearly all of the examples would be highly contrived.
My father, who is an 'educator', seems to think that the best way to learn trigonometry is to use it to teach carpentry, but I can't think of anything that would be more dull, for several reasons. One, because I imagine that well less than half of the class would have a genuine interest in carpentry; and Two, because building things with wood requires very little trigonometric calculation, so nearly all of the examples would be highly contrived.
Trig is useless. There is about 2weeks of useful math and the rest is a waste of time. People like putting trig on tests but the basic 4 year HS math program should be:
Logic + Algebra I, Geometry with a touch of number theory and topology, Grab Bag (1 year including the useful parts of Algebra II, Trig, and pre Calculus), Calculus I.
Logic + Algebra I, Geometry with a touch of number theory and topology, Grab Bag (1 year including the useful parts of Algebra II, Trig, and pre Calculus), Calculus I.
Bingo! School is not really about imparting knowledge.
It's a place to keep kids, teach conformity and a very basic level of text and math comprehension, that's all there is to it.
It's a place to keep kids, teach conformity and a very basic level of text and math comprehension, that's all there is to it.
I agree with this to some extent, but I think the author goes too far. The purpose of math education is not to make theoretical mathematicians. Only a small number of students in math classes will become mathematicians, and the rest are acquiring a set of tools that they'll use to solve problems in the real world. Proofs and conjectures and abstract reasoning, interesting as they may be, are irrelevant to most people's needs.
Here's a better analogy: a mathematician is a composer, and people who use math are singers and instrumentalists. There will always be a few people who love composing, and their work is important. But the majority of people have neither the ability nor the motivation to write original compositions. They'd rather just be given music to play.
For the composers among us, this can be hard to stomach. Writing music is so exhilarating, and playing is so mundane! How can anyone prefer the latter to the former? Well, the fact is, most people do. The rest of humanity really is different from us, and we need to accept that.
Here's a better analogy: a mathematician is a composer, and people who use math are singers and instrumentalists. There will always be a few people who love composing, and their work is important. But the majority of people have neither the ability nor the motivation to write original compositions. They'd rather just be given music to play.
For the composers among us, this can be hard to stomach. Writing music is so exhilarating, and playing is so mundane! How can anyone prefer the latter to the former? Well, the fact is, most people do. The rest of humanity really is different from us, and we need to accept that.
This is one of the best pieces I have read on mathematical education in quite some time.
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I don't necessarily agree with the article. Math was my main subject at the University, so while I work as a software developer, on paper I am a mathematician.
I don't agree with the premise that math is an art, and that "Better to not have math classes at all than to do what is currently being done." First and foremost, Math is something which learns one to think abstractly in patterns. And this I believe is indispensable and useful for almost everyone - even if the concrete pieces of knowledge will be forgotten later or not directly applicable to one's work or life. Even the so-called "drill" is somewhat useful.
Also, during my work in software companies, I was able to notice when someone didn't receive solid Math at university level - they were lacking in the area of their ability to think more abstractly and thoroughly about more complex problems.
Of course, the education could be improved by having better teachers with passion and ability to explain the ideas, no doubt about that.
I don't agree with the premise that math is an art, and that "Better to not have math classes at all than to do what is currently being done." First and foremost, Math is something which learns one to think abstractly in patterns. And this I believe is indispensable and useful for almost everyone - even if the concrete pieces of knowledge will be forgotten later or not directly applicable to one's work or life. Even the so-called "drill" is somewhat useful.
Also, during my work in software companies, I was able to notice when someone didn't receive solid Math at university level - they were lacking in the area of their ability to think more abstractly and thoroughly about more complex problems.
Of course, the education could be improved by having better teachers with passion and ability to explain the ideas, no doubt about that.
I was sad to see that not once did the author mention the works of Polya. Discovering new ideas in mathematics ultimately boils down to solving an existing problem in a new way or solving a completely new problem.
I believe students should be posed the fundamental questions that forced many of the most beautiful ideas in math to be discovered. The advantage we have now is that if the students get stuck, they don't have to wait 2000 years for someone to discover how to find the area under a curve. We can give them hints. The teacher can compress time with a well thought out question pointing the students in the right direction.
I believe students should be posed the fundamental questions that forced many of the most beautiful ideas in math to be discovered. The advantage we have now is that if the students get stuck, they don't have to wait 2000 years for someone to discover how to find the area under a curve. We can give them hints. The teacher can compress time with a well thought out question pointing the students in the right direction.
Old news but this is something worth duping.
Imagine an ideal math class as Lockhart might envision: one that focuses on the art, joy, and beauty of mathematics, in which students genuinely play with mathematical ideas. Then, ask yourself how the progress of these students could be evaluated via standardized testing. The two mix poorly.
I think the goal of school is supposed to be to teach children how to evaluate which actions will be most beneficial to them.
What school actually teaches is how to be dishonest/illogical in all the correct ways for a given society.
Can anyone link to the original pdf? Scribd always crashes my computer (no other Flash apps seem to).
Click on the title outside of [scribd] for the original PDF.
God, I remember reading this on reddit. Brilliant article.
I didn't read it all so far (but I will), but I think if people were taught mathematics in a more explanatory way there would be a lot more of them interested in it, and therefore interested in areas related to it, or at least we wouldn't have a society so afraid of it.
It's amazing how many people don't know the very basics of it like, for instance, how multiplication is just a compact way of expressing a repetitive sum. And things like that happen because they were trying to remember numbers and formulas instead of understanding the concept.
It's just hard to forget about a concept, which gives the basis to understanding more complex things. I think maths teaching would be more productive if we spent more time elaborating on why things are done in some way (or how to come up with a concept just by reasoning about it, without doing calculations and writing symbols on paper) than trying to teach more complex concepts for someone who doesn't understand the basics, and therefore won't grasp the complex ones.
There's clearly something wrong about teaching maths, and very likely other subjects too, or maybe something wrong about teaching (anything).