Me: But sir, can you explain why it gets the right answer?
So you wanted to know not how to derive the solution but how to derive the derivation?
I wouldn’t blame the teacher for not going there. There’s not enough time in class to do something like that. Bringing the students to understand the presented math is hard enough. Describing the process of how this math was found, would take too long. Because especially for harder problems there were probably dozens of mathematicians who studied the problem for centuries in order to find those derivations that your teacher presents to you.
I wouldn’t blame the teacher for not going there. There’s not enough time in class to do something like that.
What’s wrong with saying something to the effect of “There’s a theorem—it’s not really within the scope of this course, but if you’re really interested it’s called the fixed-point theorem, you can look it up on Wikipedia or somewhere”?
Derive the derivation? Huh? And you say that’s different from ‘understanding’ it. No, I just didn’t have the most basic of intuitive ideas as to why he suddenly made an iterated equation, and I didn’t understand why it worked, at any level. It was all just abstract symbol manipulation with no content for me, and that’s not learning.
Furthermore, he does have the time. We have nine hours a week. With a class size of four pupils.
He may actually not know. People who teach maths are often not terribly good at it. Why don’t you post the equation and the thing he turned it into? One of us will probably be able to see what is going on.
In all fairness, at university, being lectured by people whose job was maths research and who were truly world class at it, I remember similar happenings. Although they have subtler ways of telling you to shut up. Figuring out what’s going on between the steps of a proof is half the fun and it tends to make your head explode with joy when you finally get it.
I just gave a couple of terms of first year maths lectures, stuff that I thought I knew well, and the effort of going through and actually understanding everything I was talking about turned what was supposed to be two hours a week into two days a week, so I can quite see why busy people don’t bother. And in the process I found a couple of mistakes in the course notes (that of course get passed down from year to year, not rewritten from scratch with every new lecturer).
Because especially for harder problems there were probably dozens of mathematicians who studied the problem for centuries in order to find those derivations that your teacher presents to you.
In my school math education we had the standard that everything we learn get’s proved. If you are not in the habit of proving math, students are not well prepared for doing real math in university which is about mathematical proofs.
In general the math that’s not understood but memorized gets soon forgotten and is not worth teaching in the first place.
That’s a great rule, but it still has to have limits. Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
In Germany every student has to select two subjects called “Leistungskurse” in which he gets more classes. In my case I selected math and physics which meant we had 5 hours worth of lessons in those subjects per week.
When I went to high school in Israel we had a similar system, but extra math wasn’t an option (at least not at my school).
A big part of an undergrad math (or CS) degree is spent on these subjects. I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
The German school system starts by separating students into 3 different kind of schools based on the academic skill of the student: Hauptschule, Realschule and Gymnasium. The Gymnasium is basically for those who go to university. That separation starts by school year 5 or 7 depending on where in Germany the school is located.
You have more than 3 years of math classes at school. I think proving stuff started at the 8 or 9 school year. At the beginning a lot of it focused on geometry.
At the time I think it was 4 hours of math per week for everyone. I think there were many cases where the students who were good at math had time to prove things while the more math adverse students took more time with the basic math problems.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
Might as well be a description of almost all the non-CS math content in my CS undergrad degree. (The only core subjects missing are probability and statistics). Of course, the depth and breadth and quality of treatment may still be different. But maybe an average high school in Israel is really that much worse than a good high school in Germany.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school. (And they had only 10 years of school in total, ages 7 to 17, while we had 12, ages 6 to 18.) I always thought his stories may have been biased, because he went on to get a graduate degree in applied math and taught undergrad math at a respected Russian university. So I thought maybe he also went to a top high school and/or associated with other students who were good at math and enjoyed it.
But I know there is a wide distribution of math talent and affinity among people. There are definitely enough students for math-oriented schools, or extra math classes or programmes in large enough schools, at that level of teaching. I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly. But there’s no reason these problems should be universal.
In university students often spend time in large lectures in math classes. There’s no real to expect that to be a lot more effective than a 15 person course with a good teacher.
I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly.
In our times the incentives go against teaching like this. in Berlin centralized math testing effectively means that all schools have to teach to the same test and that test doesn’t contain complicated proofs.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school.
Yes, the difference between a math education at bad school with only 3 hours per week at the end and the math education at a good school in Germany with 5 hours per week might be the freshman year of a non-CS math content of a CS undergrad degree.
What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
EDIT: Personally, I think going into computer science would have been easier if in high school I learned logic and set theory my last two years rather than trigonometry and calculus.
What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
The thing that’s wrong is exactly that it would indeed have to be instead of calculus. And then students would not pass the nationally mandated matriculation exams or university entry exams, which test knowledge of calculus. One part of the system can’t change independently from the others. I agree that if you’re going to teach just one field of math, then calculus is not the optimal choice.
I do believe that for every field that’s taught in highschool, the most important theories and results should be taught: evolution, genetics, cell structure and anatomy in biology; Newtonian mechanics, electromagnetism and relativity in physics (QM probably requires too much math for any high-school program); etc.
There won’t be time to prove and fully explain everything that’s being shown, because time is limited, and it’s better that all the people in our society know about classical mechanics and EM and relativity, than that they know about just one of them but have studied and reproduced enough experiments to demonstrate that that one theory is true compared to all alternatives of similar complexity.
And similarly, I think it would be better if everyone knew about the fundamental results of all the important fields of math, than being able to prove a lot of theorems in a couple of fields on highschool exams but not getting to hear a lot of other fields.
As far as possible, we should allow students to learn more and help guide them to the sciences. But scientists are in the end a small minority of the population and some things are important to teach to everyone. I don’t think calculus passes that test, and neither does classic geometry and analytic geometry, which received a lot of time in my school.
Instead I would teach statistics, basic probability theory, programming (if you can sell it as applied math), basic set and number theory (e.g. countable and uncountable infinities, rational and real numbers), basic computer science with some important cryptography results given without proof (e.g. public-key encryption). At least one of these should demonstrate the concept of mathematical proofs and logic (set theory is a good candidate).
Interesting question. I’m a programmer who works in EDA software, including using transistor-level simulations, and I use surprisingly little math. Knowing the idea of a derivative (and how noisy numerical approximations to them can be!) is important—but it is really rare for me to actually compute one. It is reasonably common to run into a piece of code that reverses the transformation done by another pieces of code, but that is about it. The core algorithms of the simulators involves sophisticated math—but that is stable and encapsulated, so it is mostly a black box. As a citizen, statistics are potentially useful, but mostly just at the level of: This article quotes an X% change in something with N patients, does it look like N was large enough that this could possibly be statistically significant? But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
I see computer science as a branch of applied math which is important enough to be treated as a top-level ‘science’ of its own. Another way of putting it is that algorithms and programming are the ‘engineering’ counterpart to the ‘science’ of (the rest of) CS and math.
Programming very often involves math that is unrelated to the problem domain. For instance, using static typing relies on results from type theory. Cryptography (which includes hash functions, which are ubiquitous in software) is math. Functional languages in particular often embody complex mathematical structures that serve as design paradigms. Many data structures and algorithms rely on mathematical proofs. Etc.
But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
That is also a fact that ought to be taught in school :-)
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.
So you wanted to know not how to derive the solution but how to derive the derivation?
I wouldn’t blame the teacher for not going there. There’s not enough time in class to do something like that. Bringing the students to understand the presented math is hard enough. Describing the process of how this math was found, would take too long. Because especially for harder problems there were probably dozens of mathematicians who studied the problem for centuries in order to find those derivations that your teacher presents to you.
What’s wrong with saying something to the effect of “There’s a theorem—it’s not really within the scope of this course, but if you’re really interested it’s called the fixed-point theorem, you can look it up on Wikipedia or somewhere”?
Derive the derivation? Huh? And you say that’s different from ‘understanding’ it. No, I just didn’t have the most basic of intuitive ideas as to why he suddenly made an iterated equation, and I didn’t understand why it worked, at any level. It was all just abstract symbol manipulation with no content for me, and that’s not learning.
Furthermore, he does have the time. We have nine hours a week. With a class size of four pupils.
He may actually not know. People who teach maths are often not terribly good at it. Why don’t you post the equation and the thing he turned it into? One of us will probably be able to see what is going on.
In all fairness, at university, being lectured by people whose job was maths research and who were truly world class at it, I remember similar happenings. Although they have subtler ways of telling you to shut up. Figuring out what’s going on between the steps of a proof is half the fun and it tends to make your head explode with joy when you finally get it.
I just gave a couple of terms of first year maths lectures, stuff that I thought I knew well, and the effort of going through and actually understanding everything I was talking about turned what was supposed to be two hours a week into two days a week, so I can quite see why busy people don’t bother. And in the process I found a couple of mistakes in the course notes (that of course get passed down from year to year, not rewritten from scratch with every new lecturer).
In my school math education we had the standard that everything we learn get’s proved. If you are not in the habit of proving math, students are not well prepared for doing real math in university which is about mathematical proofs.
In general the math that’s not understood but memorized gets soon forgotten and is not worth teaching in the first place.
That’s a great rule, but it still has to have limits. Otherwise you couldn’t teach calculus without teaching number theory and set theory and probably some algebraic structures and mathematical logic too.
We actually did learn number theory, set theory, basic logic and algrebraic structures such as rings, groups and vector spaces.
In Germany every student has to select two subjects called “Leistungskurse” in which he gets more classes. In my case I selected math and physics which meant we had 5 hours worth of lessons in those subjects per week.
When I went to high school in Israel we had a similar system, but extra math wasn’t an option (at least not at my school).
A big part of an undergrad math (or CS) degree is spent on these subjects. I don’t believe the study everything, prove everything you do level is attainable with 5 hours per week for 3 years at the high-school level, even with a very good self-selected student group.
The German school system starts by separating students into 3 different kind of schools based on the academic skill of the student: Hauptschule, Realschule and Gymnasium. The Gymnasium is basically for those who go to university. That separation starts by school year 5 or 7 depending on where in Germany the school is located.
You have more than 3 years of math classes at school. I think proving stuff started at the 8 or 9 school year. At the beginning a lot of it focused on geometry.
At the time I think it was 4 hours of math per week for everyone. I think there were many cases where the students who were good at math had time to prove things while the more math adverse students took more time with the basic math problems.
What did the most advanced students (say, top 15%) study and prove by the end of highschool?
It’s been a while but before introducing calculus we did go through the axioms and theorems of limit of a function.
Peano’s axioms and how you it’s enough to prove things for n=0 and that n->n+1 were basis for proofs.
Your previous comment:
Might as well be a description of almost all the non-CS math content in my CS undergrad degree. (The only core subjects missing are probability and statistics). Of course, the depth and breadth and quality of treatment may still be different. But maybe an average high school in Israel is really that much worse than a good high school in Germany.
I now recall that my father, who went to high school in Kiev in the 70s, used to tell me that the math I learned in the freshman year, they learned in high school. (And they had only 10 years of school in total, ages 7 to 17, while we had 12, ages 6 to 18.) I always thought his stories may have been biased, because he went on to get a graduate degree in applied math and taught undergrad math at a respected Russian university. So I thought maybe he also went to a top high school and/or associated with other students who were good at math and enjoyed it.
But I know there is a wide distribution of math talent and affinity among people. There are definitely enough students for math-oriented schools, or extra math classes or programmes in large enough schools, at that level of teaching. I just assumed based on my own experience that the schools themselves wouldn’t be good enough to support this, or wouldn’t be incentivized correctly. But there’s no reason these problems should be universal.
In university students often spend time in large lectures in math classes. There’s no real to expect that to be a lot more effective than a 15 person course with a good teacher.
In our times the incentives go against teaching like this. in Berlin centralized math testing effectively means that all schools have to teach to the same test and that test doesn’t contain complicated proofs.
Yes, the difference between a math education at bad school with only 3 hours per week at the end and the math education at a good school in Germany with 5 hours per week might be the freshman year of a non-CS math content of a CS undergrad degree.
What is wrong with learning logic, set theory, and number theory before (or in the context of high school, instead of) calculus?
EDIT: Personally, I think going into computer science would have been easier if in high school I learned logic and set theory my last two years rather than trigonometry and calculus.
The thing that’s wrong is exactly that it would indeed have to be instead of calculus. And then students would not pass the nationally mandated matriculation exams or university entry exams, which test knowledge of calculus. One part of the system can’t change independently from the others. I agree that if you’re going to teach just one field of math, then calculus is not the optimal choice.
I do believe that for every field that’s taught in highschool, the most important theories and results should be taught: evolution, genetics, cell structure and anatomy in biology; Newtonian mechanics, electromagnetism and relativity in physics (QM probably requires too much math for any high-school program); etc.
There won’t be time to prove and fully explain everything that’s being shown, because time is limited, and it’s better that all the people in our society know about classical mechanics and EM and relativity, than that they know about just one of them but have studied and reproduced enough experiments to demonstrate that that one theory is true compared to all alternatives of similar complexity.
And similarly, I think it would be better if everyone knew about the fundamental results of all the important fields of math, than being able to prove a lot of theorems in a couple of fields on highschool exams but not getting to hear a lot of other fields.
Really? I think it’s very beautiful and it’s what hooked me. And it’s the bit the scientists use. What would you teach everyone instead?
As far as possible, we should allow students to learn more and help guide them to the sciences. But scientists are in the end a small minority of the population and some things are important to teach to everyone. I don’t think calculus passes that test, and neither does classic geometry and analytic geometry, which received a lot of time in my school.
Instead I would teach statistics, basic probability theory, programming (if you can sell it as applied math), basic set and number theory (e.g. countable and uncountable infinities, rational and real numbers), basic computer science with some important cryptography results given without proof (e.g. public-key encryption). At least one of these should demonstrate the concept of mathematical proofs and logic (set theory is a good candidate).
Interesting question. I’m a programmer who works in EDA software, including using transistor-level simulations, and I use surprisingly little math. Knowing the idea of a derivative (and how noisy numerical approximations to them can be!) is important—but it is really rare for me to actually compute one. It is reasonably common to run into a piece of code that reverses the transformation done by another pieces of code, but that is about it. The core algorithms of the simulators involves sophisticated math—but that is stable and encapsulated, so it is mostly a black box. As a citizen, statistics are potentially useful, but mostly just at the level of: This article quotes an X% change in something with N patients, does it look like N was large enough that this could possibly be statistically significant? But usually the problem with such studies in the the systematic errors, which are essentially impossible for a casual examination to find.
I see computer science as a branch of applied math which is important enough to be treated as a top-level ‘science’ of its own. Another way of putting it is that algorithms and programming are the ‘engineering’ counterpart to the ‘science’ of (the rest of) CS and math.
Programming very often involves math that is unrelated to the problem domain. For instance, using static typing relies on results from type theory. Cryptography (which includes hash functions, which are ubiquitous in software) is math. Functional languages in particular often embody complex mathematical structures that serve as design paradigms. Many data structures and algorithms rely on mathematical proofs. Etc.
That is also a fact that ought to be taught in school :-)
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.