He doesn’t have to give proofs. Just explaining the intuition behind each formula doesn’t take that long and will help the students understand how and when to use them. Giving intuitions really isn’t esoteric trivia for advanced students, it’s something that will make solving problems easier for everyone relative to if they just memorized each individual case where each formula applies.
I suspect this is typical mind fallacy at work. There are many students who either can’t, or don’t want to, learn mathematical intuitions or explanations. They prefer to learn a few formulas and rules by rote, the same way they do in every other class.
There are many students who either can’t, or don’t want to, learn mathematical intuitions or explanations. They prefer to learn a few formulas and rules by rote, the same way they do in every other class.
Former teacher confirming this. Some students are willing to spend a lot of energy to avoid understanding a topic. They actively demand memorization without understanding… sometimes they even bring their parents as a support; and I have seen some of the parents complaining in the newspapers (where the complaints become very unspecific, that the education is “too difficult” and “inefficient”, or something like this).
Which is completely puzzling for the first time you see this, as a teacher, because in every internet discussion about education, teachers are criticized for allegedly insisting on memorization without understanding, and every layman seems to propose new ideas about education with less facts and more “critical thinking”. So, you get the impression that there is a popular demand for understanding instead of memorization… and you go to classroom believing you will fix the system… and there is almost a revolution against you, outraged kids refusing to hear any explanations and insisting you just tell them the facts they need to memorize for the exams, and skip the superfluous stuff. (Then you go back to internet, read more complaints about how teachers are insisting that kids memorize the stuff instead of undestanding, and you just give up any hope of a sane discussion.)
My first explanation was that understanding is the best way, but memorization can be more efficient in short term, especially if you expect to forget the stuff and never use it again after the exam. Some subjects probably are like this, but math famously is not. Which is why math is the most hated subject.
Another explanation was that the students probably never actually had an experience of understanding something, at least not in the school, so they literally don’t understand what I was trying to do. Which is a horrible idea, if true, but… that wouldn’t make it less true, right? Still makes me think: Didn’t those kids at least have an experience of something being explained by a book, or by a popular science movie? Probably most of them just don’t read such books or watch those movies. -- I wonder what would happen if I just showed the kids some TED videos; would they be interested, or would they hate it?
By the way, this seems not related to whether the topic is difficult. Even explaining how easy things work can be met by resistance. This time not because it is “too difficult”, but because “we should just skip the boring simple stuff”. (Of course, skipping the boring simple stuff is the best recipe to later find the more advanced stuff too difficult.) I wonder how much impact here has the internet-induced attention deficit.
Speaking as a student: I sympathize with Benito, have myself had his sort of frustration, and far prefer understanding to memorization… yet I must speak up for the side of the students in your experience. Why?
Because the incentives in the education system encourage memorization, and discourage understanding.
Say I’m in a class, learning some difficult topic. I know there will be a test, and the test will make up a big chunk of my grade (maybe all the tests together are most of my grade). I know the test will be such that passing it is easiest if I memorize — because that’s how tests are. What do I do?
True understanding in complex topics requires contemplation, experimentation, exploration; “playing around” with the material, trying things out for myself, taking time to think about it, going out and reading other things about the topic, discussing the topic with knowledgeable people. I’d love to do all of that...
… but I have three other classes, and they all expect me to read absurd amounts of material in time for next week’s lecture, and work on a major project apiece, and I have no time for any of those wonderful things I listed, and I have had four hours of sleep (and god forbid I have a job in addition to all of that) and I am in no state to deeply understand anything. Memorizing is faster and doesn’t require such expenditures of cognitive effort.
So what do I do? Do I try to understand, and not be able to understand enough, in time for the test on Monday, and thus fail the class? Or do I just memorize, and pass? And what good do your understanding-based teaching techniques do me, if you’re still going to give me tests and base my grade on them, and if the educational system is not going to allow me the conditions to make my own way to true understanding of the material?
Ah. I think this is why I’m finding physics and maths so difficult, even though my teachers said I’d find it easy. It’s not just that the teachers have no incentive to make me understand, it’s that because teachers aren’t trained to teach understanding, when I keep asking for it, they don’t know how to give it… This explains a lot of their behaviour.
Even when I’ve sat down one-on-one with a teacher and asked for the explanation of a piece of physics I totally haven’t understood, they guy just spoke at me for five/ten minutes, without stopping to ask me if I followed that step, or even just to repeat what he’d said, and then considered the matter settled at the end without questions about how I’d followed it. The problem with my understanding was at the beginning as well, and when he stopped, he finished as if delivering the end of a speech, as though it were final. It would’ve been a little awkward for me to ask him to re-explain the first bit… I thought he was a bad teacher, but he’s just never been incentivised to continually stop and check for understanding, after deriving the requisite equations.
And that’s why my maths teacher can never answer questions that go under the surface of what he teaches… I think he’d be perfectly able to understand it on the level to give me an explanation, as when I push him he does, but otherwise…
His catchphrase in our classroom is “In twenty years of questioning, nobody’s ever asked me that before.” He then re-assures us that it’s okay for us to have asked it, as he assumes we think that having asked a new question is a bad thing...
If you’re really curious, have you considered a private maths tutor? I wouldn’t go anywhere near the sort of people who help people cram for exams, but if there’s a local university you might find a maths student (even an undergrad would be fine) who’d actually enjoy talking about this sort of thing and might be really grateful for a few pounds an hour.
Hell, if you find someone who really likes the subject and can talk about it you may only have to buy them a coffee and you’ll have trouble getting them to shut up!
Thanks for the tip, and no, I hadn’t considered going out and looking for maths students. I mainly spend my time reading good textbooks (i.e. Art of Problem Solving). I had a maths tutor once, although I didn’t get out of it what I wanted.
Oops, I didn’t mean to sound quite so arrogant, and I merely meant in the top bit of the class. If you do want to know my actual reasons for thinking so, off the top of my head I’d mention teachers saying so generally, teachers saying so specifically, performance in maths competitions, a small year group such that I know everyone in the class fairly well and can see their abilities, observation of marks (grades) over the past six years, and I get paid to tutor maths to students in lower years.
Word of advice: don’t put too much attention into your “potential”. That’s an unfalsifiable hypothesis that you can use to inflate your ego without actually, you know, being good. Look at your actual results, and only those.
I schlepped through physics degree without understanding much of anything, and then turned to philosophy to solve the problem...the rest is ancient history.
Some of them probably did, but most didn’t. The “no homework and no additional study at home” part was meant only for computer science, which I taught.
Say I’m in a class, learning some difficult topic. I know there will be a test, and the test will make up a big chunk of my grade (maybe all the tests together are most of my grade). I know the test will be such that passing it is easiest if I memorize — because that’s how tests are. What do I do?
This is not usually true in the context of physics. I recently taught a physics course, the final was 3 questions, the time limit was 3 hours. Getting full credit on a single question was enough for an A. Memorization fails if you’ve never seen a question type before.
Say I’m in a class, learning some difficult topic. I know there will be a test, and the test will make up a big chunk of my grade (maybe all the tests together are most of my grade). I know the test will be such that passing it is easiest if I memorize — because that’s how tests are.
Not all tests are like that. I had plenty of tests in math that did require understanding to get a top mark. Memorization can get you enough points to pass the test but not all points.
There are also times where the problem isn’t necessarily memorization, but just lapse of insight that makes it hard to realize that a problem as presented matches one of your pre-canned equations, even though it can be solved with one of them. Panic sets in, etc.
In situations like that, particularly in those years when you have calculus and various transforms in your toolkit (even if they aren’t strictly /expected/), you can solve the problem with those power tools instead, and having understood and being able to derive solutions to closely related problems from basic principles ought to be fairly predictive of you being able to generate a correct answer in those situations.
My first explanation was that understanding is the best way, but memorization can be more efficient in short term, especially if you expect to forget the stuff and never use it again after the exam. Some subjects probably are like this, but math famously is not. Which is why math is the most hated subject.
Another explanation was that the students probably never actually had an experience of understanding something, at least not in the school, so they literally don’t understand what I was trying to do.
What do you think about these other possible explanations?
Some of these students really can’t learn to prove mathematical theorems. If exams required real understanding of math, then no matter how much these students and their teachers tried, with all the pedagogical techniques we know today, they would fail the exams.
These students really have very unpleasant subjective experiences when they try to understand math, a kind of mental suffering. They are bad at math because people are generally bad at doing very unpleasant things: they only do the absolute minimum they can get away with, so they don’t get enough practice to become better, and they also have trouble concentrating at practice because the experience is a bad one. Even if they can improve with practice, this would mean they’ll never practice enough to improve. (You may think that understanding something should be more fun than rote learning, and this may be true for some of them, but they never get to actually understand enough to realize this for themselves.)
The students are just time-discounting. They care more about not studying now, then about passing the exam later. Or, they are procrastinating, planning to study just before the exam. An effort to understand something takes more time in the short term than just memorizing it; it only pays off once you’ve understood enough things.
The students, as a social group, perceive themselves as opposed to and resisting the authority of teachers. They can’t usually resist mandatory things: attending classes, doing homework, having to pass exams; and they resent this. Whenever a teacher tries to introduce a study activity that isn’t mandatory (other teachers aren’t doing it), students will push back. Any students who speak up in class and say “actually I’m enjoying this extra material/alternative approach, please keep teaching it” would be betraying their peers. This is a matter of politics, and even if a teacher introduces non-mandatory or alternative techniques that are really objectively fun and efficient, students may not perceive them as such because they’re seeing them as “extra study” or “extra oppression”, not “a teacher trying to help us”.
It could be different explanations for different people. This said, options 1 and 2 seem to contradict with my experience that students object even against explaining relatively simple non-mathy things. My experience comes mostly from high school where I taught everything during the lessons, no homeword, no home study; this seems to rule out option 3.
Option 4 seems plausible, I just feel it is not the full explanation, it’s more like a collective cooperation against something that most students already dislike individually.
I’m closer to the typical mind than most people here with regard to math. I deeply loved humanities and thought of math and mathy fields as completely sterile and lifeless up until late high school, when I first realized that there was more to math than memorizing formulas. And then boom it became fun and also dramatically easier. Before that I didn’t reject the idea of learning using mathematical intuitions, I just had no idea that mathematical intuitions were a thing that could exist.
I suspect that most people learn school-things by rote simply because they don’t realize that school-things can be learned another way. This is evidenced by how people don’t choose to learn things they actually find interesting or useful by rote. There are quite a few people out there who think “book smarts” and “street smarts” are completely separate things and they just don’t have book smarts because they aren’t good at memorizing disjointed lists of facts.
This is hard to test. What we need here are studies that test different methods of teaching math on randomly selected people.
Of course people self-selecting to participate in the study would ruin it, and most people hate math after the experience and wouldn’t participate unless paid large sums.
On the other hand, a study of highschool students who are forced to participate also isn’t very useful because the fact of forcing students to study may well be the major reason why they find it a not fun experience and don’t study well.
If they get a few formulas and rules by rote, but can’t figure out when to apply them because they lack understanding, what does that actually get them?
It’s not a waste of time to give them a chance of getting something out of it, even if they’re almost certainly doomed in this regard.
I’m not saying it’s a bad thing in itself, but there’s usually not enough time in class to do it; it comes at the expense of the rote learning which these students need to pass the exams.
He doesn’t have to give proofs. Just explaining the intuition behind each formula doesn’t take that long and will help the students understand how and when to use them. Giving intuitions really isn’t esoteric trivia for advanced students, it’s something that will make solving problems easier for everyone relative to if they just memorized each individual case where each formula applies.
I suspect this is typical mind fallacy at work. There are many students who either can’t, or don’t want to, learn mathematical intuitions or explanations. They prefer to learn a few formulas and rules by rote, the same way they do in every other class.
Former teacher confirming this. Some students are willing to spend a lot of energy to avoid understanding a topic. They actively demand memorization without understanding… sometimes they even bring their parents as a support; and I have seen some of the parents complaining in the newspapers (where the complaints become very unspecific, that the education is “too difficult” and “inefficient”, or something like this).
Which is completely puzzling for the first time you see this, as a teacher, because in every internet discussion about education, teachers are criticized for allegedly insisting on memorization without understanding, and every layman seems to propose new ideas about education with less facts and more “critical thinking”. So, you get the impression that there is a popular demand for understanding instead of memorization… and you go to classroom believing you will fix the system… and there is almost a revolution against you, outraged kids refusing to hear any explanations and insisting you just tell them the facts they need to memorize for the exams, and skip the superfluous stuff. (Then you go back to internet, read more complaints about how teachers are insisting that kids memorize the stuff instead of undestanding, and you just give up any hope of a sane discussion.)
My first explanation was that understanding is the best way, but memorization can be more efficient in short term, especially if you expect to forget the stuff and never use it again after the exam. Some subjects probably are like this, but math famously is not. Which is why math is the most hated subject.
Another explanation was that the students probably never actually had an experience of understanding something, at least not in the school, so they literally don’t understand what I was trying to do. Which is a horrible idea, if true, but… that wouldn’t make it less true, right? Still makes me think: Didn’t those kids at least have an experience of something being explained by a book, or by a popular science movie? Probably most of them just don’t read such books or watch those movies. -- I wonder what would happen if I just showed the kids some TED videos; would they be interested, or would they hate it?
By the way, this seems not related to whether the topic is difficult. Even explaining how easy things work can be met by resistance. This time not because it is “too difficult”, but because “we should just skip the boring simple stuff”. (Of course, skipping the boring simple stuff is the best recipe to later find the more advanced stuff too difficult.) I wonder how much impact here has the internet-induced attention deficit.
Speaking as a student: I sympathize with Benito, have myself had his sort of frustration, and far prefer understanding to memorization… yet I must speak up for the side of the students in your experience. Why?
Because the incentives in the education system encourage memorization, and discourage understanding.
Say I’m in a class, learning some difficult topic. I know there will be a test, and the test will make up a big chunk of my grade (maybe all the tests together are most of my grade). I know the test will be such that passing it is easiest if I memorize — because that’s how tests are. What do I do?
True understanding in complex topics requires contemplation, experimentation, exploration; “playing around” with the material, trying things out for myself, taking time to think about it, going out and reading other things about the topic, discussing the topic with knowledgeable people. I’d love to do all of that...
… but I have three other classes, and they all expect me to read absurd amounts of material in time for next week’s lecture, and work on a major project apiece, and I have no time for any of those wonderful things I listed, and I have had four hours of sleep (and god forbid I have a job in addition to all of that) and I am in no state to deeply understand anything. Memorizing is faster and doesn’t require such expenditures of cognitive effort.
So what do I do? Do I try to understand, and not be able to understand enough, in time for the test on Monday, and thus fail the class? Or do I just memorize, and pass? And what good do your understanding-based teaching techniques do me, if you’re still going to give me tests and base my grade on them, and if the educational system is not going to allow me the conditions to make my own way to true understanding of the material?
None. No good at all.
Ah. I think this is why I’m finding physics and maths so difficult, even though my teachers said I’d find it easy. It’s not just that the teachers have no incentive to make me understand, it’s that because teachers aren’t trained to teach understanding, when I keep asking for it, they don’t know how to give it… This explains a lot of their behaviour.
Even when I’ve sat down one-on-one with a teacher and asked for the explanation of a piece of physics I totally haven’t understood, they guy just spoke at me for five/ten minutes, without stopping to ask me if I followed that step, or even just to repeat what he’d said, and then considered the matter settled at the end without questions about how I’d followed it. The problem with my understanding was at the beginning as well, and when he stopped, he finished as if delivering the end of a speech, as though it were final. It would’ve been a little awkward for me to ask him to re-explain the first bit… I thought he was a bad teacher, but he’s just never been incentivised to continually stop and check for understanding, after deriving the requisite equations.
And that’s why my maths teacher can never answer questions that go under the surface of what he teaches… I think he’d be perfectly able to understand it on the level to give me an explanation, as when I push him he does, but otherwise…
His catchphrase in our classroom is “In twenty years of questioning, nobody’s ever asked me that before.” He then re-assures us that it’s okay for us to have asked it, as he assumes we think that having asked a new question is a bad thing...
Edit: Originally said something arrogant.
If you’re really curious, have you considered a private maths tutor? I wouldn’t go anywhere near the sort of people who help people cram for exams, but if there’s a local university you might find a maths student (even an undergrad would be fine) who’d actually enjoy talking about this sort of thing and might be really grateful for a few pounds an hour.
Hell, if you find someone who really likes the subject and can talk about it you may only have to buy them a coffee and you’ll have trouble getting them to shut up!
Thanks for the tip, and no, I hadn’t considered going out and looking for maths students. I mainly spend my time reading good textbooks (i.e. Art of Problem Solving). I had a maths tutor once, although I didn’t get out of it what I wanted.
Why do you think that?
Oops, I didn’t mean to sound quite so arrogant, and I merely meant in the top bit of the class. If you do want to know my actual reasons for thinking so, off the top of my head I’d mention teachers saying so generally, teachers saying so specifically, performance in maths competitions, a small year group such that I know everyone in the class fairly well and can see their abilities, observation of marks (grades) over the past six years, and I get paid to tutor maths to students in lower years.
Still, edited.
Word of advice: don’t put too much attention into your “potential”. That’s an unfalsifiable hypothesis that you can use to inflate your ego without actually, you know, being good. Look at your actual results, and only those.
I schlepped through physics degree without understanding much of anything, and then turned to philosophy to solve the problem...the rest is ancient history.
From what I hear, philosophy is mostly ancient history.
It’s mostly mental masturbation where ancient history plays the role of porn...
writes down in list of things people have actually said to me
Kinda like this site. :-)
This site has different preferences in pr0n :-P
I had this experience in a context of high school, with no homework and no additional study at home.
None of the students’ classes assigned any homework?!
Some of them probably did, but most didn’t. The “no homework and no additional study at home” part was meant only for computer science, which I taught.
This is not usually true in the context of physics. I recently taught a physics course, the final was 3 questions, the time limit was 3 hours. Getting full credit on a single question was enough for an A. Memorization fails if you’ve never seen a question type before.
Not all tests are like that. I had plenty of tests in math that did require understanding to get a top mark. Memorization can get you enough points to pass the test but not all points.
It’s more useful than that, even.
There are also times where the problem isn’t necessarily memorization, but just lapse of insight that makes it hard to realize that a problem as presented matches one of your pre-canned equations, even though it can be solved with one of them. Panic sets in, etc.
In situations like that, particularly in those years when you have calculus and various transforms in your toolkit (even if they aren’t strictly /expected/), you can solve the problem with those power tools instead, and having understood and being able to derive solutions to closely related problems from basic principles ought to be fairly predictive of you being able to generate a correct answer in those situations.
What do you think about these other possible explanations?
Some of these students really can’t learn to prove mathematical theorems. If exams required real understanding of math, then no matter how much these students and their teachers tried, with all the pedagogical techniques we know today, they would fail the exams.
These students really have very unpleasant subjective experiences when they try to understand math, a kind of mental suffering. They are bad at math because people are generally bad at doing very unpleasant things: they only do the absolute minimum they can get away with, so they don’t get enough practice to become better, and they also have trouble concentrating at practice because the experience is a bad one. Even if they can improve with practice, this would mean they’ll never practice enough to improve. (You may think that understanding something should be more fun than rote learning, and this may be true for some of them, but they never get to actually understand enough to realize this for themselves.)
The students are just time-discounting. They care more about not studying now, then about passing the exam later. Or, they are procrastinating, planning to study just before the exam. An effort to understand something takes more time in the short term than just memorizing it; it only pays off once you’ve understood enough things.
The students, as a social group, perceive themselves as opposed to and resisting the authority of teachers. They can’t usually resist mandatory things: attending classes, doing homework, having to pass exams; and they resent this. Whenever a teacher tries to introduce a study activity that isn’t mandatory (other teachers aren’t doing it), students will push back. Any students who speak up in class and say “actually I’m enjoying this extra material/alternative approach, please keep teaching it” would be betraying their peers. This is a matter of politics, and even if a teacher introduces non-mandatory or alternative techniques that are really objectively fun and efficient, students may not perceive them as such because they’re seeing them as “extra study” or “extra oppression”, not “a teacher trying to help us”.
It could be different explanations for different people. This said, options 1 and 2 seem to contradict with my experience that students object even against explaining relatively simple non-mathy things. My experience comes mostly from high school where I taught everything during the lessons, no homeword, no home study; this seems to rule out option 3.
Option 4 seems plausible, I just feel it is not the full explanation, it’s more like a collective cooperation against something that most students already dislike individually.
I’m closer to the typical mind than most people here with regard to math. I deeply loved humanities and thought of math and mathy fields as completely sterile and lifeless up until late high school, when I first realized that there was more to math than memorizing formulas. And then boom it became fun and also dramatically easier. Before that I didn’t reject the idea of learning using mathematical intuitions, I just had no idea that mathematical intuitions were a thing that could exist.
I suspect that most people learn school-things by rote simply because they don’t realize that school-things can be learned another way. This is evidenced by how people don’t choose to learn things they actually find interesting or useful by rote. There are quite a few people out there who think “book smarts” and “street smarts” are completely separate things and they just don’t have book smarts because they aren’t good at memorizing disjointed lists of facts.
This is hard to test. What we need here are studies that test different methods of teaching math on randomly selected people.
Of course people self-selecting to participate in the study would ruin it, and most people hate math after the experience and wouldn’t participate unless paid large sums.
On the other hand, a study of highschool students who are forced to participate also isn’t very useful because the fact of forcing students to study may well be the major reason why they find it a not fun experience and don’t study well.
If they get a few formulas and rules by rote, but can’t figure out when to apply them because they lack understanding, what does that actually get them?
It’s not a waste of time to give them a chance of getting something out of it, even if they’re almost certainly doomed in this regard.
I’m not saying it’s a bad thing in itself, but there’s usually not enough time in class to do it; it comes at the expense of the rote learning which these students need to pass the exams.
This is very much true, as I was one of those students myself. I did care about passing exams, not learning math.