Richard Feynmann claimed that he wasn’t exceptionally intelligent, but that he focused all his energies on one thing. Of course he was exceptionally intelligent, but he makes a good point.
I think one way to improve your intelligence is to actually try to understand things in a very fundamental way. Rather than just accepting the kind of trite explanations that most people accept—for instance, that electricity is electrons moving along a wire—try to really find out and understand what is actually happening, and you’ll begin to find that the world is very different from what you have been taught and you’ll be able to make more intelligent observations about it.
The intervening years might have glazed their memories with a euphoric tint, but about 80 percent recall Feynman’s lectures as highlights of their college years. “It was like going to church.” The lectures were “a transformational experience,” “the experience of a lifetime, probably the most important thing I got from Caltech.” “I was a biology major but Feynman’s lectures stand out as a high point in my undergraduate experience … though I must admit I couldn’t do the homework at the time and I hardly turned any of it in.” “I was among the least promising of students in this course, and I never missed a lecture. … I remember and can still feel Feynman’s joy of discovery. … His lectures had an … emotional impact that was probably lost in the printed Lectures.”
Trying to actually understand what equations describe is something I’m always trying to do in school, but I find my teachers positively trained in the art of superficiality and dark-side teaching. Allow me to share two actual conversations with my Maths and Physics teachers from school.:
(Teacher derives an equation, then suddenly makes it into an iterative formula, with no explanation of why)
Me: Woah, why has it suddenly become an iterative formula? What’s that got to do with anything?
Teacher: Well, do you agree with the equation when it’s not an iterative formula?
Me: Yes.
Teacher: And how about if I make it an iterative formula?
Me: But why do you do that?
Friend: Oh, I see.
Me: Do you see why it works?
Friend: Yes. Well, no. But I see it gets the right answer.
Me: But sir, can you explain why it gets the right answer?
Teacher: Ooh Ben, you’re asking one of your tough questions again.
(Physics class)
Me: Can you explain that sir?
Teacher: Look, Ben, sometimesnot understanding things is a good thing.
And yet to most people, I can’t even vent the ridiculousness of a teacher actually saying this; they just think it’s the norm!
Teacher: Look, Ben, something not understanding things is a good thing.
Ahem:
“Headmaster! ” said Professor Quirrell, sounding genuinely shocked. “Mr. Potter has told you that this spell is not spoken of with those who cannot cast it! You do not press a wizard on such matters!”
Amusing, although I’ll point out that there are some subtle difference between a physics classroom and the MOR!universe. Or at least, I think there are...
I will only say that when I was a physics major, there were negative course numbers in some copies of the course catalog. And the students who, it was rumored, attended those classes were… somewhat off, ever after.
And concerning how I got my math PhD, and the price I paid for it, and the reason I left the world of pure math research afterwards, I will say not one word.
A visit to wikipedia suggests that “secondary school” can refer to either what we in the U.S. call “middle school / junior high school”, or what we call “high school”. That’s a fairly wide range of grade levels. In which year of pre-university education are you?
I didn’t upvote you but I would have if you hadn’t mentioned it; it would have been because I appreciate people answering questions and finishing comment threads rather than leaving them hanging forever unresolved.
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.
I haven’t seen them mentioned in this thread, so thought I’d add them, since they’re probably valid and worth thinking about:
the utility of a math understanding, combined with the skills required for doing things such as mathematical proofs (or having a deep understanding of physics) is low for most humans. much lower than rote memorization of some simple mathematical and algebraic rules. consider, especially, the level of education that most will attain, and that the amount of abstract math and physics exposure in that time is very small. teaching such things in average classrooms may on average be both inefficient and unfair to the majority of students. you’re looking for knowledge and understanding in all the wrong places.
the vast majority of public education systems are, pragmatically speaking, tools purpose built and designed to produce model citizens, with intelligence and knowledge gains seen as beneficial but not necessary side effects. ie, as long as the kids are off the streets—if they’re going to get good jobs as a side effect, that’s a bonus. you’re using the wrong tools, for the job (either use better tools, or misuse the tools you have to get the job you want done, right)
I’ve noticed that one of the biggest thing holding me back in math/physics is an aversion to thinking too hard/long about math and physics problems. It seems to me that if I was able to overcome this aversion and math was as fun as playing video games I’d be a lot better at it.
Amtal (or Amtal Rule) – “Common rule on primitive worlds under which something is tested to determine its limits or defects. Commonly: testing to destruction.”[3] “To know a thing well, know its limits. Only when pushed beyond its tolerances will true nature be seen. – The Amtal Rule.”[6]
From here. Or as I just think of it, if you don’t at least have a hard time sometimes, if not fail sometimes, you’re not shooting high enough.
Almost, but not quite. “If you never get a game over, you’re playing games that are too easy” would indeed be a Umeshism, but this is a complaint about easy games rather than a suggestion that I should be playing harder ones.
Not in my experience, unless you’re talking about trouble teaching them. It’s very possible to run out of classes before you hit anything truly difficult (in my country there are no more classes after Masters level, a PhD student is expected to be doing research—the american notion of “all but dissertation” provokes endless amusement, here you’re “all but dissertation” from day 1).
A system where a non-genius math student never faces a challenging math class would probably “provoke endless amusement” from an American grad student, since to them it means that the program is too weak to be considered serious.
If you literally never had trouble in math class, you are a rare mind of the Newton/Gauss calibre, and you should go get your Field’s medal before you are 40 :).
I had trouble in my Masters (a combination of course choice and bad luck) and so didn’t do a PhD. But we’re talking about the top university in at least the country, and by some accounts the hardest non-research course in the world. I’m pretty sure that going a different route I could’ve got to the point of starting a PhD before hitting anything difficult.
I do sometimes think I should’ve chased the Fields medal, but I’m ultimately happier the way things turned out. I worked my ass off the whole time in school/university; nowadays I earn a good living doing fun things, but my evenings and weekends are my own, and I’ve got a much better social life.
And at least some math instructors effectively teach that if you aren’t already finding (their presentations of) math fascinating, that you must just not be a Math Person.
Math is a bit like liftening weights. Sitting in front of a heavy mathematical problem is challenging. The job of a good teacher isn’t to remove the challenge. Math is about abstract thinking and a teacher who tries to spare his students from doing abstract thinking isn’t doing it right.
Deliberate practice is mentally taxing.
The difficult thing as a teacher is to motivate the student to face the challenge whether the challenge is lifting weights or doing complicated math.
The job of a good teacher isn’t to remove the challenge.
The job of a good teacher is to find a slightly less challenging problem, and to give you that problem first. Ideally, to find a sequence of problems very smoothly increasing in difficulty.
Just like a computer game doesn’t start with the boss fight, although some determined players would win that, too.
The job of a good teacher is to find a slightly less challenging problem, and to give you that problem first. Ideally, to find a sequence of problems very smoothly increasing in difficulty.
No. Being good at math is about being able to keep your attention on a complicated proof even if it’s very challenging and your head seems like it’s going to burst.
If you want to build muscles you don’t slowly increase the amount of weight and keep it at a level where it’s effortless. You train to exhaustion of given muscles.
Building mental stamina to tackle very complicated abstract problems that aren’t solvable in five minutes is part of a good math education.
Deliberate practice is supposed to feel hard. A computer game is supposed to feel fun. You can play a computer game for 12 hours. A few hours of delibrate practice are on the other usually enough to get someone to the rand of exhaustion.
If you only face problems in your education that are smooth like a computer game, you aren’t well prepared for facing hard problems in reality. A good math education teaches you the mindset that’s required to stick with a tough abstract problem and tackle it head on even if you can’t fully grasp it after looking 30 minutes at it.
You might not use calculus at your job, but if your math education teaches you the ability to stay focused on hard abstract problems than it fulfilled it’s purpose.
You can teach calculus by giving the student concrete real world examples but that defeats the point of the exercise. If we are honest most students won’t need the calculus at their job. It’s not the point of math education. At least in the mindset in which I got taught math at school in Germany.
If you want to build muscles you don’t slowly increase the amount of weight and keep it at a level where it’s effortless.
You don’t put on so much weight than you couldn’t possibly lift it, either (nor so much weight that you could only lift it with atrocious form and risk of injury, the analogue of which would be memorising a proof as though it was a prayer in a dead language and only having a faulty understanding of what the words mean).
Yes, memorizing proof isn’t the point. You want to derive proofs. I think it’s perfectly fine to sit 1 hours in front of a complicated proof and not be able to solve the proof.
A ten year old might not have that mental stamia, but a good math education should teach it, so it’s there by the end of school.
This kind of philosophy sounds like it’s going to make a few people very good at tackling hard problems, while causing everyone else to become demotivated and hate math.
Motivation has a lot to do with knowing why you are engaging in an action. If you think things should be easy and they aren’t you get demotivated. If you expect difficulty and manage to face it then that doesn’t destroy motivation.
I don’t think getting philosophy right is easy. Once things that my school teachers got very wrong was believing in talents instead of believing in a growth mindset.
I did identify myself as smart so I didn’t learn the value of putting in time to practice. I tried to get by with the minimum of effort.
I think Cal Newport wrote a lot of interesting things about how a good philosophy of learning would look like.
There a certain education philosophy that you have standardized tests, than you do gamified education to have children score on those tests. Student have pens with multiple colors and are encouraged to draw mind maps. Afterwards the students go to follow their passions and live the American dream. It fits all the boxes of ideas that come out of California.
I’m not really opposed to someone building some gamified system to teach calculus but at the same time it’s important to understand the trade offs. We don’t want to end up with a system where the attention span that students who come out of it is limited to playing games.
I think that the way good games teach things is basically being engaging by constantly presenting content that’s in the learner’s zone of proximal development, offering any guidance needed for mastering that, and then gradually increasing the level of difficulty so as to constantly keep things in the ZPD. The player is kept constantly challenged and working at the edge of their ability, but because the challenge never becomes too high, the challenge also remains motivating all the time, with the end result being continual improvement.
For example, in a game where your character may eventually have access to 50 different powers, throwing them at the player all at once would be overwhelming when the player’s still learning to master the basic controls. So instead the first level just involves mastering the basic controls and you have just a single power that you need to use in order to beat the level, then when you’ve indicated that you’ve learned that (by beating the level), you get access to more powers, and so on. When they reach the final level, they’re also likely to be confident about their abilities even when it becomes difficult, because they know that they’ve tackled these kinds of problems plenty of times before and have always eventually been successful in the past, even if it required several tries.
The “math education is all about teaching people how to stay focused on hard abstract problems” philosophy sounds to me like the equivalent of throwing people at a level where they had to combine all 50 powers in order to survive, right from the very beginning. If you intend on becoming a research mathematician who has to tackle previously unencountered problems that nobody has any clue of how to solve, it may be a good way of preparing you for it. But forcing a student to confront needlessly difficult problems, when you could instead offer a smoothly increasing difficulty, doesn’t seem like a very good way to learn in general.
When our university began taking the principles of something like cognitive apprenticeship—which basically does exactly the thing that Viliam Bur mentioned, presenting problems in a smoothly increasing difficulty as well as offering extensive coaching and assistance—and applying it to math (more papers), the end result was high student satisfaction even while the workload was significantly increased and the problems were made more challenging.
If you intend on becoming a research mathematician who has to tackle previously unencountered problems that nobody has any clue of how to solve, it may be a good way of preparing you for it.
Not only research mathematicians but basically anyone who’s supposed to research previously unencountered problems. That’s the ability that universities are traditionally supposed to teach.
If that’s not what you want to teach, why teach calculus in the first place? If I need an integral I can ask a computer to calculate the integral for me. Why teach someone who wants to be a software engineer calculus?
There a certain idea of egalitarianism according to which everyone should have an university education. That wasn’t the point why we have universities. We have universities to teach people to tackle previously unencountered problems.
If you want to be a carpenter you don’t go to university but be an apprentice with an existing carpenter. Universities are not structured to be good at teaching trades like carpenting.
Not only research mathematicians but basically anyone who’s supposed to research previously unencountered problems.
Isn’t that rather “problems that can’t be solved using currently existing mathematics”? If it’s just a previously unencountered problem, but can be solved using the tools from an existing branch of math, then what you actually need is experience from working with those tools so that you can recognize it as a problem that can be tackled with those tools. As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on. The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
But I wasn’t thinking that much in the context of university education, more in the context of primary/secondary school. Math offers plenty of general-purpose contexts that may greatly enhance one’s ability to think in precise terms: to the extent that we can make the whole general population learn and enjoy those concepts, it might help raise the sanity waterline.
I agree that calculus probably isn’t very useful for that purpose, though. A thorough understanding of basic statistics and probability would seem much more important.
There an interesting paper about how doing science is basically about coping with feeling stupid.
No matter whether you do research in math or whether you do research in biology, you have to come to terms with tackling problems that aren’t easily solved.
One of the huge problems with Reddit style New Atheists is that they don’t like to feel stupid. They want their science education to be in easily digestible form.
As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
I agree, that’s an important skill and probably undertaught.
The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
Nobody understands all math. For practical purposes it’s often more important to know which mathematical tools exist and having an ability to learn to use those tools.
I don’t need to be able to solve integrals. It’s enough to know that integrals exists and that Wolfram Alpha will solve them for me.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on.
I’m not saying that one shouldn’t spend any time on easy exercises. Spending a third of the time on problems that are really hard might be a ratio that’s okay.
A thorough understanding of basic statistics and probability would seem much more important.
Statistics are important, but it’s not clear that math statistics classes help. Students that take them often think that real world problems follow a normal distribution.
If that’s not what you want to teach, why teach calculus in the first place? If I need an integral I can ask a computer to calculate the integral for me. Why teach someone who wants to be a software engineer calculus?
Calculus isn’t as important to software engineering as some other branches of math, but it can still be handy to know. I’ve mostly encountered it in the context of physical simulation: optics stuff for graphics rendering, simplified Navier-Stokes for weather simulation, and orbital mechanics, to name three. Sometimes you can look up the exact equation you need, but copying out of the back of a textbook won’t equip you to handle special cases, or to optimize your code if the general solution is too computationally expensive.
Even that is sort of missing the point, though. The reason a lot of math classes are in a traditional CS curriculum isn’t because the exact skills they teach will come up in industry; it’s because they develop abstract thinking skills in a way that classes on more technical aspects of software engineering don’t. And a well-developed sense of abstraction is very important in software, at least once you get beyond the most basic codemonkey tasks.
The reason a lot of math classes are in a traditional CS curriculum isn’t because the exact skills they teach will come up in industry
To that extend the CS curriculum shouldn’t be evaluated by how well people do calculus but how well they do teach abstract thinking.
I do think that the kind of abstract thinking where you don’t know how to tackle a problem because the problem is new is valuable to software developers.
This is a very strong set of assertions which I find deeply counter intuitive. Of course that doesn’t mean it isn’t true. Do you have any evidence for any of it?
Which one’s do you find counter intuitive? It’s a mix of referencing a few very modern ideas with more traditional ideas of education while staying away from the no-child-left-behind philosophy of education.
I can make any of the points in more depths but the post was already long, and I’m sort of afraid that people don’t read my post on LW if they get too long ;) Which ones do you find particularly interesting?
Of course bad instructors can say this as easily as good ones.
But isn’t it true to say that if you have reasonably wide experience with different presentations of math, and you don’t find any of them fascinating, then you’re probably not a Math Person? Or do Math People not exist as a natural category?
Or do Math People not exist as a natural category?
I’d be ever so interested in the answer to this question. It seems really obvious that some people are good at maths and some people aren’t.
But it’s also really obvious that some people like sprouts. And it turns out as far as I’m aware that it’s possible to like sprouts for both genetic and environmental reasons. I’d love to know the causes of mathematical ability. Especially since it seems to be possible to be both ‘clever’ and ‘bad at maths’. Does anyone know what the latest thinking on it is?
My recent experiences trying to design IQ tests tell me that that’s both innate and very trainable. In fact I’d now trust the sort of test that asks you how to spell or define randomly chosen words much more than the Raven’s type tests. It’s really hard to fake good speling, whereas the pattern tests are probably just telling you whether you once spent half an hour looking closely at the wallpaper. Which is exactly the reverse of the belief that I started with.
Related: some people believe that programming talent is very innate and people can be sharply separated into those who can and cannot learn to write code. Previously on LW here, and I think there was an earlier more substantive post but I can’t find it now. See also this. Gwern collected some further evidence and counterevidence.
It was probably mentioned in the earlier discussions, but I believe the “two humps” pattern can easily be explained by bad teaching. If it hapens in the whole profession, maybe no one has yet discovered a good way to teach it, because most of the people who understand the topic were autodidacts.
As a model, imagine that a programming ability is a number. You come to school with some value between 0 and 10. A teacher can give you +20 bonus. Problem is, the teacher cannot explain the most simple stuff which you need to get to level 5; maybe because it is so obvious to the teacher that they can’t understand how specifically someone else would not already understand it. So the kids with starting values between 0 and 4 can’t follow the lessons and don’t learn anything, while the kids with starting values 5 to 10 get the +20 bonus. At the end, you get the “two humps”; one group with values 0 to 4, another group with values 25 to 30. -- And the worst part is that this belief creates a spiral, because when everyone observed the “two humps” at the adult people, then if some student with starting value 4 does not understand the lesson, we don’t feel a need to fix this; obviously they were just not meant to understand programming.
What are those starting concepts that some people get and some people don’t? Probably things like “the computer is just a mechanical thing which follows some mechanical rules; it has no mind, and it doesn’t really understand anything”, but you need to feel it in the gut level. (Maybe aspies have a natural advantage here, because they don’t expect the computer to have a mind.) It could probably help to play with some simple mechanical machines first, where the kids could observe the moving parts. In other words, maybe we don’t only need specialized educational software, but also hardware. A computer in a form of a black box is already too big piece of magic, prone to be anthropomorphized. You should probably start with a mechanical typewriter and a mechanical calculator.
If it hapens in the whole profession, maybe no one has yet discovered a good way to teach it, because most of the people who understand the topic were autodidacts.
A lot of effort has gone into trying to invent ways of teaching programming to complete newbies. If really no-one has succeeded at all, then maybe it’s time to seriously consider that some people can’t be taught.
A claim that someone cannot be taught by any possible intervention would be a very strong claim indeed, and almost certainly false. But a claim that no-one knows how to teach this even though a lot of people have tried and failed for a long time now, makes predictions pretty similar to the theory that some people simply can’t be taught.
As a model, imagine that a programming ability is a number. You come to school with some value between 0 and 10. A teacher can give you +20 bonus.
This model matches the known facts, but it doesn’t tell us what we really want to know. What determines what value people start out with? Does everyone start out with 0 and some people increase their value in unknown, perhaps spontaneous ways? Or are some people just born with high values and they’ll arrive at 5 or 10 no matter what they do, while others will stay at 0 no matter what?
I don’t know if educators have tried teaching the concepts you suggest explicitly.
A lot of effort has gone into trying to invent ways of teaching programming to complete newbies. If really no-one has succeeded at all, then maybe it’s time to seriously consider that some people can’t be taught.
The researcher didn’t distinguish the conjectured cause (bimodal differences in students’ ability to form models of computation) from other possible causes (just to name one — some students are more confident, and computing classes reward confidence).
Clearly further research is needed. It should probably not assume that programmers are magic special people, no matter how appealing that notion is to many programmers.
Once upon a time, it would have been a radical proposition to suggest that even 25% of the population might one day be able to read and write. Reading and writing were the province of magic special people like scribes and priests. Today, we count on almost every adult being able to read traffic signs, recipes, bills, emails, and so on — even the ones who do not do “serious reading”.
A problem with programming education is that it is frequently unclear what the point of it is. Is it to identify those students who can learn to get jobs as programmers in industry or research? Is it to improve students’ ability to control the technology that is a greater and greater part of their world? Is it to teach the mathematical concepts of elementary computer science?
We know why we teach kids to read. The wonders of literature aside, we know full well that they cannot get on as competent adults if they are literate. Literacy was not a necessity for most people two thousand years ago; it is a necessity for most people today. Will programming ever become that sort of necessity?
Literacy was not a necessity for most people two thousand years ago; it is a necessity for most people today. Will programming ever become that sort of necessity?
That was the thinking at the dawn of personal computing, back in the 80s.
You think the general population the future will hacking code into text editors? That isn’t even ubiquitous in the industry, since you can call yourself a developer if you only know how to us graphical tools. They’ll be doing something, but it will be analogous to electronic music production as opposed .tk p.suing an instrument.
My bet would be on childhood experience. For example the kinds of toys used. I would predict a positive effect of various construction sets. It’s like “Reductionism for Kindergarten”. :D
The silent pre-programming knowledge could be things like: “this toy is interacted with by placing its pieces and observing what they do (or modelling in one’s mind what they would do), instead of e.g. talking to the toy and pretending the toy understands”.
An anecdatum. The only construction set I had as a boy was lego, and my little sister played with it too. As far as I know, there was no feeling that it was my toy only. We’re five years apart so all my stuff got passed down or shared.
My sister’s very clever. We both did degrees in the same place, mine maths and hers archaeology.
She’s never shown the slightest interest in programming or maths, whereas I remember the thunderbolt-strike of seeing my first computer program at ten years old, long before I’d ever actually seen a computer. I nagged my parents obsessively for one until they gave in, and maths and programming have been my hobby and my profession ever since.
I distinctly remember trying to show Liz how to use my computer, and she just wasn’t interested.
My parents are entirely non-mathematical. They’re both educated people, but artsy. Mum must have some natural talent, because she showed me how to do fractions before I went to school, but I think she dropped maths at sixteen. I think it’s fair to say that Dad hates and fears it. Neither of them knew the first thing about computers when I was little. They just weren’t a thing that people had in the 70s, any more than hovercraft were.
Every attempt my school made to teach programming was utterly pointless for me, I either already knew what they were trying to teach or got it in a few seconds.
The only attempts to teach programming that have ever held my attention or shown me anything interesting are SICP, and the algorithms and automata courses on Coursera, all of which I passed with near-perfect scores, and did for fun.
So from personal experience I believe in ‘natural talent’ in programming. And I don’t believe it’s got anything to do with upbringing, except that our house was quiet and educated.
You’d have had to work quite hard to stop me becoming a programmer. And I don’t think anything in my background was in favour of me becoming one. And anything that was should have favoured my sister too.
I’ve got two friends who are talented maths graduates, and somehow both of them had managed to get through their first degrees without ever writing programs. Both of them asked me to teach them.
The first one I’ve made several attempts with. He sort-of gets it, but he doesn’t see why you’d want to. A couple of times he’s said ‘Oh yes, I get it, sort of like experimental mathematics’. But any time he gets a problem about numbers he tries to solve it with pen and paper, even when it looks obvious to me that a computer will be a profitable attack.
The second, I spent about two hours showing him how to get to “hello world” in python and how to fetch a web page. Five days later he shows me a program he’s written to screen-scrape betfair and place trades automatically when it spots arbitrage opportunities. I was literally speechless.
So I reckon that whatever-makes-you-a-mathematician and whatever-makes-you-a-programmer might be different things too. Which is actually a bit weird. They feel the same to me.
A lot of effort has gone into trying to invent ways of teaching programming to complete newbies. If really no-one has succeeded at all,
That seems like rather a strong claim. Everyone who can program now was a complete newbie at some point. Presumably they did not learn by a bolt of divine inspiration out of the blue sky.
The sources linked above claim that some can be taught, and some (probably most of the population) can’t, no matter what you do. And of those who can learn, many become autodidacts in a suitable environment.
Of course they don’t reinvent programming themselves, they do learn it from others, but the same could be said of any skill or knowledge. And yet there are skills which clearly have very strong inborn dispositions. It’s being claimed that programming is such a skill, and an extreme one at that, with a sharply bimodal distribution.
It was probably mentioned in the earlier discussions, but I believe the “two humps” pattern can easily be explained by bad teaching. If it hapens in the whole profession, maybe no one has yet discovered a good way to teach it, because most of the people who understand the topic were autodidacts.
Bad teaching? There’s an even simpler explanation (at least regarding programming): autodidacts with previous experience versus regular students without previous experience. The fact that the teaching is often geared towards the students with previous experience and suffers from a major tone of “Why don’t you know this already?” throughout the first year or two of undergrad doesn’t help a bit.
“I can teach you this only if you already know it” seems like bad teaching to me.
Yes, that is the definition of bad teaching. My assertion is that CS departments have gotten so damn complacent about receiving a steady stream of autodidact programmers as their undergrad entrants that they’ve stopped bothering with actually teaching low-level courses. They assign work, they expect to receive finished work, they grade the finished work, but it all relies on the clandestine assumption that the “good students” could already do the work when they entered the classroom.
Only a small fraction of math has practical applications, the majority of math exists for no reason other than thinking about it is fun. Even things with applications had sometimes been invented before those applications were known. So in a sense most math is designed to be fun. Of course it’s not fun for everyone, just for a special class of people who are into this kind of thing. That makes it different from Angry Birds. But there are many games which are also only enjoyed by a specific audience, so maybe the difference is not that fundamental. A large part of the reason the average person doesn’t enjoy math is that unlike Angry Birds math requires some effort, which is the same reason the average person doesn’t enjoy League Of Evil III.
Spot on. Pure, fun math does benefit society directly in at least one way, however, in that the opportunity to engage in it can be used to lure very smart people into otherwise unpalatable teaching jobs.
In fact, that seems to be the main point of “research” in most less-than-productive fields (i.e. the humanities).
Pure, fun math does benefit society directly in at least one way, however, in that the opportunity to engage in it can be used to lure very smart people into otherwise unpalatable teaching jobs.
Is it clear that this is in the best interests of society? It would seem to me the end result is bad teaching. Back when I was in undergrad, the best researchers were the worst teachers (for obvious reasons- they were focused on their research and didn’t at all care about teaching).
When I was in grad school in physics, the professor widely considered the strongest teacher was denied tenure (cited AGAINST him in the decision was that he had written a widely used textbook),etc.
Also, the desire for tenured track profs to dodge teaching is why the majority of math classes at many research institutions were taught by grad students.
In graduate school, for special topics class there were usually only 1 or 2 professors that COULD teach a certain class (and only 3 or 4 students interested in taking it)- so when you are talking cutting edge research topics, its a necessity to have a researcher because no one else will be familiar enough with whats going on in the field.
Outside of that, not really. Good teaching takes work, so if you put someone in front of the class whose career advancement requires spending all their time on research, then the teaching is just a potentially career destroying distraction. Also, at the intro level, subject-pedagogy experts tend to do better (i.e. the physics education group were measurably more effective at teaching physics than other physics groups. So much so that I think now they exclusively teach the large physics courses for engineers)
I mean, it’s easier to get research positions with those professors, and those are learning experiences, but the students generally get very little out of it during the actual class.
He parted with and published nothing except under the extreme pressure of friends. Until the second phase of his life, he was a wrapt, consecrated solitary, pursuing his studies by intense introspection with a mental endurance perhaps never equalled. I believe that the clue to his mind is to be found in his unusual powers of continuous concentrated introspection. A case can be made out, as it also can with Descartes, for regarding him as an accomplished experimentalist. Nothing can be more charming than the tales of his mechanical contrivances when he was a boy. There are his telescopes and his optical experiments, These were essential accomplishments, part of his unequalled all-round technique, but not, I am sure, his peculiar gift, especially amongst his contemporaries. His peculiar gift was the power of holding continuously in his mind a purely mental problem until he had seen straight through it. I fancy his pre-eminence is due to his muscles of intuition being the strongest and most enduring with which a man has ever been gifted. Anyone who has ever attempted pure scientific or philosophical thought knows how one can hold a problem momentarily in one’s mind and apply all one’s powers of concentration to piercing through it, and how it will dissolve and escape and you find that what you are surveying is a blank. I believe that Newton could hold a problem in his mind for hours and days and weeks until it surrendered to him its secret. Then being a supreme mathematical technician he could dress it up, how you will, for purposes of exposition, but it was his intuition which was pre-eminently extraordinary - ‘so happy in his conjectures’, said De Morgan, ‘as to seem to know more than he could possibly have any means of proving’. The proofs, for what they are worth, were, as I have said, dressed up afterwards—they were not the instrument of discovery.
...the appointee’s wife was granted a divorce from him because
of appointee’s constantly working calculus problems in his head
as soon as awake, while driving car, sitting in living room, and
so forth, and that his one hobby was playing his African
drums. His ex-wife reportedly testified that on several occasions
when she unwittingly disturbed either his calculus or his drums
he flew into a violent rage, during which time he attacked her,
threw pieces of bric-a-brac about and smashed the furniture.
Indeed, terse “explanations” that handwave more than explain are a pet peeve of mine. They can be outright confusing and cause more harm than good IMO. See this question on phrasing explanations in physics for some examples.
http://www.reddit.com/r/askscience/comments/e3yjg/is_there_any_way_to_improve_intelligence_or_are/c153p8w
reddit user jjbcn on trying to improve your intelligence
If you’re not a student of physics, The Feynman Lectures on Physics is probably really useful for this purpose. It’s free for download!
http://www.feynmanlectures.caltech.edu/
It seems like the Feynman lectures were a bit like the Sequences for those Caltech students:
Trying to actually understand what equations describe is something I’m always trying to do in school, but I find my teachers positively trained in the art of superficiality and dark-side teaching. Allow me to share two actual conversations with my Maths and Physics teachers from school.:
(Physics class)
And yet to most people, I can’t even vent the ridiculousness of a teacher actually saying this; they just think it’s the norm!
Ahem:
For every EY quote, there exists an equal and opposite EY PC Hodgell quote:
(That was P.C. Hodgell, not EY.)
Good point, I’ll correct it.
Amusing, although I’ll point out that there are some subtle difference between a physics classroom and the MOR!universe. Or at least, I think there are...
I will only say that when I was a physics major, there were negative course numbers in some copies of the course catalog. And the students who, it was rumored, attended those classes were… somewhat off, ever after.
And concerning how I got my math PhD, and the price I paid for it, and the reason I left the world of pure math research afterwards, I will say not one word.
Were there tentacles involved? Strange ethereal piping? Anything rugose or cyclopean in character?
I think we can safely say there were non-Euclidean geometries involved.
Were there also course numbers with a non-zero complex part?
PEOPLE NEED TO STOP QUOTING PROFESSOR QUIRRELL LIKE HE IS ELIEZER YUDKOWSKY
HE IS NOT
THERE ARE SOME IMPORTANT DIFFERENCES AND THEY ARE VISIBLE AND THEY MATTER
THANK YOU
What level of school?
Secondary school.
A visit to wikipedia suggests that “secondary school” can refer to either what we in the U.S. call “middle school / junior high school”, or what we call “high school”. That’s a fairly wide range of grade levels. In which year of pre-university education are you?
Oh, okay. After I finish this year, I’ll study at school for one final year, and then go to university.
Edit: I am confused that this got five up votes, and would be interested in hearing an explanation from someone who up voted it.
I didn’t upvote you but I would have if you hadn’t mentioned it; it would have been because I appreciate people answering questions and finishing comment threads rather than leaving them hanging forever unresolved.
Cheers.
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.
I haven’t seen them mentioned in this thread, so thought I’d add them, since they’re probably valid and worth thinking about:
the utility of a math understanding, combined with the skills required for doing things such as mathematical proofs (or having a deep understanding of physics) is low for most humans. much lower than rote memorization of some simple mathematical and algebraic rules. consider, especially, the level of education that most will attain, and that the amount of abstract math and physics exposure in that time is very small. teaching such things in average classrooms may on average be both inefficient and unfair to the majority of students. you’re looking for knowledge and understanding in all the wrong places.
the vast majority of public education systems are, pragmatically speaking, tools purpose built and designed to produce model citizens, with intelligence and knowledge gains seen as beneficial but not necessary side effects. ie, as long as the kids are off the streets—if they’re going to get good jobs as a side effect, that’s a bonus. you’re using the wrong tools, for the job (either use better tools, or misuse the tools you have to get the job you want done, right)
I’ve noticed that one of the biggest thing holding me back in math/physics is an aversion to thinking too hard/long about math and physics problems. It seems to me that if I was able to overcome this aversion and math was as fun as playing video games I’d be a lot better at it.
You have to want to be a wizard.
Plenty of us took the Wizard’s Oath as kids and still have a hard time in math classes sometimes.
I think everyone has trouble in math class, eventually.
From here. Or as I just think of it, if you don’t at least have a hard time sometimes, if not fail sometimes, you’re not shooting high enough.
If I don’t get a game over at least once, the game is too easy.
Is that an Umeshism?
Almost, but not quite. “If you never get a game over, you’re playing games that are too easy” would indeed be a Umeshism, but this is a complaint about easy games rather than a suggestion that I should be playing harder ones.
Not in my experience, unless you’re talking about trouble teaching them. It’s very possible to run out of classes before you hit anything truly difficult (in my country there are no more classes after Masters level, a PhD student is expected to be doing research—the american notion of “all but dissertation” provokes endless amusement, here you’re “all but dissertation” from day 1).
A system where a non-genius math student never faces a challenging math class would probably “provoke endless amusement” from an American grad student, since to them it means that the program is too weak to be considered serious.
If you literally never had trouble in math class, you are a rare mind of the Newton/Gauss calibre, and you should go get your Field’s medal before you are 40 :).
I had trouble in my Masters (a combination of course choice and bad luck) and so didn’t do a PhD. But we’re talking about the top university in at least the country, and by some accounts the hardest non-research course in the world. I’m pretty sure that going a different route I could’ve got to the point of starting a PhD before hitting anything difficult.
I do sometimes think I should’ve chased the Fields medal, but I’m ultimately happier the way things turned out. I worked my ass off the whole time in school/university; nowadays I earn a good living doing fun things, but my evenings and weekends are my own, and I’ve got a much better social life.
Huh. Yes, I guess that in retrospect I wouldn’t be the only one.
This is your secret?
You have to want to learn how to be a wizard.
You have to like to learn how to be a wizard.
Good video games are designed to be fun, that is their purpose. Math, um, not so much.
And at least some math instructors effectively teach that if you aren’t already finding (their presentations of) math fascinating, that you must just not be a Math Person.
Math is a bit like liftening weights. Sitting in front of a heavy mathematical problem is challenging. The job of a good teacher isn’t to remove the challenge. Math is about abstract thinking and a teacher who tries to spare his students from doing abstract thinking isn’t doing it right.
Deliberate practice is mentally taxing.
The difficult thing as a teacher is to motivate the student to face the challenge whether the challenge is lifting weights or doing complicated math.
The job of a good teacher is to find a slightly less challenging problem, and to give you that problem first. Ideally, to find a sequence of problems very smoothly increasing in difficulty.
Just like a computer game doesn’t start with the boss fight, although some determined players would win that, too.
No. Being good at math is about being able to keep your attention on a complicated proof even if it’s very challenging and your head seems like it’s going to burst.
If you want to build muscles you don’t slowly increase the amount of weight and keep it at a level where it’s effortless. You train to exhaustion of given muscles.
Building mental stamina to tackle very complicated abstract problems that aren’t solvable in five minutes is part of a good math education.
Deliberate practice is supposed to feel hard. A computer game is supposed to feel fun. You can play a computer game for 12 hours. A few hours of delibrate practice are on the other usually enough to get someone to the rand of exhaustion.
If you only face problems in your education that are smooth like a computer game, you aren’t well prepared for facing hard problems in reality. A good math education teaches you the mindset that’s required to stick with a tough abstract problem and tackle it head on even if you can’t fully grasp it after looking 30 minutes at it.
You might not use calculus at your job, but if your math education teaches you the ability to stay focused on hard abstract problems than it fulfilled it’s purpose.
You can teach calculus by giving the student concrete real world examples but that defeats the point of the exercise. If we are honest most students won’t need the calculus at their job. It’s not the point of math education. At least in the mindset in which I got taught math at school in Germany.
You don’t put on so much weight than you couldn’t possibly lift it, either (nor so much weight that you could only lift it with atrocious form and risk of injury, the analogue of which would be memorising a proof as though it was a prayer in a dead language and only having a faulty understanding of what the words mean).
Yes, memorizing proof isn’t the point. You want to derive proofs. I think it’s perfectly fine to sit 1 hours in front of a complicated proof and not be able to solve the proof.
A ten year old might not have that mental stamia, but a good math education should teach it, so it’s there by the end of school.
This kind of philosophy sounds like it’s going to make a few people very good at tackling hard problems, while causing everyone else to become demotivated and hate math.
Motivation has a lot to do with knowing why you are engaging in an action. If you think things should be easy and they aren’t you get demotivated. If you expect difficulty and manage to face it then that doesn’t destroy motivation.
I don’t think getting philosophy right is easy. Once things that my school teachers got very wrong was believing in talents instead of believing in a growth mindset.
I did identify myself as smart so I didn’t learn the value of putting in time to practice. I tried to get by with the minimum of effort.
I think Cal Newport wrote a lot of interesting things about how a good philosophy of learning would look like.
There a certain education philosophy that you have standardized tests, than you do gamified education to have children score on those tests. Student have pens with multiple colors and are encouraged to draw mind maps. Afterwards the students go to follow their passions and live the American dream. It fits all the boxes of ideas that come out of California.
I’m not really opposed to someone building some gamified system to teach calculus but at the same time it’s important to understand the trade offs. We don’t want to end up with a system where the attention span that students who come out of it is limited to playing games.
I think that the way good games teach things is basically being engaging by constantly presenting content that’s in the learner’s zone of proximal development, offering any guidance needed for mastering that, and then gradually increasing the level of difficulty so as to constantly keep things in the ZPD. The player is kept constantly challenged and working at the edge of their ability, but because the challenge never becomes too high, the challenge also remains motivating all the time, with the end result being continual improvement.
For example, in a game where your character may eventually have access to 50 different powers, throwing them at the player all at once would be overwhelming when the player’s still learning to master the basic controls. So instead the first level just involves mastering the basic controls and you have just a single power that you need to use in order to beat the level, then when you’ve indicated that you’ve learned that (by beating the level), you get access to more powers, and so on. When they reach the final level, they’re also likely to be confident about their abilities even when it becomes difficult, because they know that they’ve tackled these kinds of problems plenty of times before and have always eventually been successful in the past, even if it required several tries.
The “math education is all about teaching people how to stay focused on hard abstract problems” philosophy sounds to me like the equivalent of throwing people at a level where they had to combine all 50 powers in order to survive, right from the very beginning. If you intend on becoming a research mathematician who has to tackle previously unencountered problems that nobody has any clue of how to solve, it may be a good way of preparing you for it. But forcing a student to confront needlessly difficult problems, when you could instead offer a smoothly increasing difficulty, doesn’t seem like a very good way to learn in general.
When our university began taking the principles of something like cognitive apprenticeship—which basically does exactly the thing that Viliam Bur mentioned, presenting problems in a smoothly increasing difficulty as well as offering extensive coaching and assistance—and applying it to math (more papers), the end result was high student satisfaction even while the workload was significantly increased and the problems were made more challenging.
Not only research mathematicians but basically anyone who’s supposed to research previously unencountered problems. That’s the ability that universities are traditionally supposed to teach.
If that’s not what you want to teach, why teach calculus in the first place? If I need an integral I can ask a computer to calculate the integral for me. Why teach someone who wants to be a software engineer calculus?
There a certain idea of egalitarianism according to which everyone should have an university education. That wasn’t the point why we have universities. We have universities to teach people to tackle previously unencountered problems.
If you want to be a carpenter you don’t go to university but be an apprentice with an existing carpenter. Universities are not structured to be good at teaching trades like carpenting.
Isn’t that rather “problems that can’t be solved using currently existing mathematics”? If it’s just a previously unencountered problem, but can be solved using the tools from an existing branch of math, then what you actually need is experience from working with those tools so that you can recognize it as a problem that can be tackled with those tools. As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on. The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
But I wasn’t thinking that much in the context of university education, more in the context of primary/secondary school. Math offers plenty of general-purpose contexts that may greatly enhance one’s ability to think in precise terms: to the extent that we can make the whole general population learn and enjoy those concepts, it might help raise the sanity waterline.
I agree that calculus probably isn’t very useful for that purpose, though. A thorough understanding of basic statistics and probability would seem much more important.
There an interesting paper about how doing science is basically about coping with feeling stupid.
No matter whether you do research in math or whether you do research in biology, you have to come to terms with tackling problems that aren’t easily solved.
One of the huge problems with Reddit style New Atheists is that they don’t like to feel stupid. They want their science education to be in easily digestible form.
I agree, that’s an important skill and probably undertaught.
Nobody understands all math. For practical purposes it’s often more important to know which mathematical tools exist and having an ability to learn to use those tools.
I don’t need to be able to solve integrals. It’s enough to know that integrals exists and that Wolfram Alpha will solve them for me.
I’m not saying that one shouldn’t spend any time on easy exercises. Spending a third of the time on problems that are really hard might be a ratio that’s okay.
Statistics are important, but it’s not clear that math statistics classes help. Students that take them often think that real world problems follow a normal distribution.
Calculus isn’t as important to software engineering as some other branches of math, but it can still be handy to know. I’ve mostly encountered it in the context of physical simulation: optics stuff for graphics rendering, simplified Navier-Stokes for weather simulation, and orbital mechanics, to name three. Sometimes you can look up the exact equation you need, but copying out of the back of a textbook won’t equip you to handle special cases, or to optimize your code if the general solution is too computationally expensive.
Even that is sort of missing the point, though. The reason a lot of math classes are in a traditional CS curriculum isn’t because the exact skills they teach will come up in industry; it’s because they develop abstract thinking skills in a way that classes on more technical aspects of software engineering don’t. And a well-developed sense of abstraction is very important in software, at least once you get beyond the most basic codemonkey tasks.
To that extend the CS curriculum shouldn’t be evaluated by how well people do calculus but how well they do teach abstract thinking. I do think that the kind of abstract thinking where you don’t know how to tackle a problem because the problem is new is valuable to software developers.
This is a very strong set of assertions which I find deeply counter intuitive. Of course that doesn’t mean it isn’t true. Do you have any evidence for any of it?
Which one’s do you find counter intuitive? It’s a mix of referencing a few very modern ideas with more traditional ideas of education while staying away from the no-child-left-behind philosophy of education.
I can make any of the points in more depths but the post was already long, and I’m sort of afraid that people don’t read my post on LW if they get too long ;) Which ones do you find particularly interesting?
Of course bad instructors can say this as easily as good ones.
But isn’t it true to say that if you have reasonably wide experience with different presentations of math, and you don’t find any of them fascinating, then you’re probably not a Math Person? Or do Math People not exist as a natural category?
I’d be ever so interested in the answer to this question. It seems really obvious that some people are good at maths and some people aren’t.
But it’s also really obvious that some people like sprouts. And it turns out as far as I’m aware that it’s possible to like sprouts for both genetic and environmental reasons. I’d love to know the causes of mathematical ability. Especially since it seems to be possible to be both ‘clever’ and ‘bad at maths’. Does anyone know what the latest thinking on it is?
My recent experiences trying to design IQ tests tell me that that’s both innate and very trainable. In fact I’d now trust the sort of test that asks you how to spell or define randomly chosen words much more than the Raven’s type tests. It’s really hard to fake good speling, whereas the pattern tests are probably just telling you whether you once spent half an hour looking closely at the wallpaper. Which is exactly the reverse of the belief that I started with.
Related: some people believe that programming talent is very innate and people can be sharply separated into those who can and cannot learn to write code. Previously on LW here, and I think there was an earlier more substantive post but I can’t find it now. See also this. Gwern collected some further evidence and counterevidence.
It was probably mentioned in the earlier discussions, but I believe the “two humps” pattern can easily be explained by bad teaching. If it hapens in the whole profession, maybe no one has yet discovered a good way to teach it, because most of the people who understand the topic were autodidacts.
As a model, imagine that a programming ability is a number. You come to school with some value between 0 and 10. A teacher can give you +20 bonus. Problem is, the teacher cannot explain the most simple stuff which you need to get to level 5; maybe because it is so obvious to the teacher that they can’t understand how specifically someone else would not already understand it. So the kids with starting values between 0 and 4 can’t follow the lessons and don’t learn anything, while the kids with starting values 5 to 10 get the +20 bonus. At the end, you get the “two humps”; one group with values 0 to 4, another group with values 25 to 30. -- And the worst part is that this belief creates a spiral, because when everyone observed the “two humps” at the adult people, then if some student with starting value 4 does not understand the lesson, we don’t feel a need to fix this; obviously they were just not meant to understand programming.
What are those starting concepts that some people get and some people don’t? Probably things like “the computer is just a mechanical thing which follows some mechanical rules; it has no mind, and it doesn’t really understand anything”, but you need to feel it in the gut level. (Maybe aspies have a natural advantage here, because they don’t expect the computer to have a mind.) It could probably help to play with some simple mechanical machines first, where the kids could observe the moving parts. In other words, maybe we don’t only need specialized educational software, but also hardware. A computer in a form of a black box is already too big piece of magic, prone to be anthropomorphized. You should probably start with a mechanical typewriter and a mechanical calculator.
A lot of effort has gone into trying to invent ways of teaching programming to complete newbies. If really no-one has succeeded at all, then maybe it’s time to seriously consider that some people can’t be taught.
A claim that someone cannot be taught by any possible intervention would be a very strong claim indeed, and almost certainly false. But a claim that no-one knows how to teach this even though a lot of people have tried and failed for a long time now, makes predictions pretty similar to the theory that some people simply can’t be taught.
This model matches the known facts, but it doesn’t tell us what we really want to know. What determines what value people start out with? Does everyone start out with 0 and some people increase their value in unknown, perhaps spontaneous ways? Or are some people just born with high values and they’ll arrive at 5 or 10 no matter what they do, while others will stay at 0 no matter what?
I don’t know if educators have tried teaching the concepts you suggest explicitly.
http://www.eis.mdx.ac.uk/research/PhDArea/saeed/
The researcher didn’t distinguish the conjectured cause (bimodal differences in students’ ability to form models of computation) from other possible causes (just to name one — some students are more confident, and computing classes reward confidence).
And the researcher’s advisor later described his enthusiasm for the study as “prescription-drug induced over-hyping” of the results …
Clearly further research is needed. It should probably not assume that programmers are magic special people, no matter how appealing that notion is to many programmers.
Once upon a time, it would have been a radical proposition to suggest that even 25% of the population might one day be able to read and write. Reading and writing were the province of magic special people like scribes and priests. Today, we count on almost every adult being able to read traffic signs, recipes, bills, emails, and so on — even the ones who do not do “serious reading”.
A problem with programming education is that it is frequently unclear what the point of it is. Is it to identify those students who can learn to get jobs as programmers in industry or research? Is it to improve students’ ability to control the technology that is a greater and greater part of their world? Is it to teach the mathematical concepts of elementary computer science?
We know why we teach kids to read. The wonders of literature aside, we know full well that they cannot get on as competent adults if they are literate. Literacy was not a necessity for most people two thousand years ago; it is a necessity for most people today. Will programming ever become that sort of necessity?
That was the thinking at the dawn of personal computing, back in the 80s.
Turns out the answer is “no”.
“Not yet.”
You think the general population the future will hacking code into text editors? That isn’t even ubiquitous in the industry, since you can call yourself a developer if you only know how to us graphical tools. They’ll be doing something, but it will be analogous to electronic music production as opposed .tk p.suing an instrument.
Computing hasn’t even existed for a century yet. Give it time.
There will come a day when ordinary educated folks quicksort their playing cards when they want to put them in order. :)
I insertion sort. :P
Doesn’t almost everyone? I’ve always heard that as the inspiration for insertion sorting.
No way, I pigeonhole sort.
My bet would be on childhood experience. For example the kinds of toys used. I would predict a positive effect of various construction sets. It’s like “Reductionism for Kindergarten”. :D
The silent pre-programming knowledge could be things like: “this toy is interacted with by placing its pieces and observing what they do (or modelling in one’s mind what they would do), instead of e.g. talking to the toy and pretending the toy understands”.
An anecdatum. The only construction set I had as a boy was lego, and my little sister played with it too. As far as I know, there was no feeling that it was my toy only. We’re five years apart so all my stuff got passed down or shared.
My sister’s very clever. We both did degrees in the same place, mine maths and hers archaeology.
She’s never shown the slightest interest in programming or maths, whereas I remember the thunderbolt-strike of seeing my first computer program at ten years old, long before I’d ever actually seen a computer. I nagged my parents obsessively for one until they gave in, and maths and programming have been my hobby and my profession ever since.
I distinctly remember trying to show Liz how to use my computer, and she just wasn’t interested.
My parents are entirely non-mathematical. They’re both educated people, but artsy. Mum must have some natural talent, because she showed me how to do fractions before I went to school, but I think she dropped maths at sixteen. I think it’s fair to say that Dad hates and fears it. Neither of them knew the first thing about computers when I was little. They just weren’t a thing that people had in the 70s, any more than hovercraft were.
Every attempt my school made to teach programming was utterly pointless for me, I either already knew what they were trying to teach or got it in a few seconds.
The only attempts to teach programming that have ever held my attention or shown me anything interesting are SICP, and the algorithms and automata courses on Coursera, all of which I passed with near-perfect scores, and did for fun.
So from personal experience I believe in ‘natural talent’ in programming. And I don’t believe it’s got anything to do with upbringing, except that our house was quiet and educated.
You’d have had to work quite hard to stop me becoming a programmer. And I don’t think anything in my background was in favour of me becoming one. And anything that was should have favoured my sister too.
And another anecdote:
I’ve got two friends who are talented maths graduates, and somehow both of them had managed to get through their first degrees without ever writing programs. Both of them asked me to teach them.
The first one I’ve made several attempts with. He sort-of gets it, but he doesn’t see why you’d want to. A couple of times he’s said ‘Oh yes, I get it, sort of like experimental mathematics’. But any time he gets a problem about numbers he tries to solve it with pen and paper, even when it looks obvious to me that a computer will be a profitable attack.
The second, I spent about two hours showing him how to get to “hello world” in python and how to fetch a web page. Five days later he shows me a program he’s written to screen-scrape betfair and place trades automatically when it spots arbitrage opportunities. I was literally speechless.
So I reckon that whatever-makes-you-a-mathematician and whatever-makes-you-a-programmer might be different things too. Which is actually a bit weird. They feel the same to me.
That seems like rather a strong claim. Everyone who can program now was a complete newbie at some point. Presumably they did not learn by a bolt of divine inspiration out of the blue sky.
The sources linked above claim that some can be taught, and some (probably most of the population) can’t, no matter what you do. And of those who can learn, many become autodidacts in a suitable environment.
Of course they don’t reinvent programming themselves, they do learn it from others, but the same could be said of any skill or knowledge. And yet there are skills which clearly have very strong inborn dispositions. It’s being claimed that programming is such a skill, and an extreme one at that, with a sharply bimodal distribution.
Bad teaching? There’s an even simpler explanation (at least regarding programming): autodidacts with previous experience versus regular students without previous experience. The fact that the teaching is often geared towards the students with previous experience and suffers from a major tone of “Why don’t you know this already?” throughout the first year or two of undergrad doesn’t help a bit.
“I can teach you this only if you already know it” seems like bad teaching to me. Not sure if we are not just debating definitions here.
I don’t think we’re even debating.
Yes, that is the definition of bad teaching. My assertion is that CS departments have gotten so damn complacent about receiving a steady stream of autodidact programmers as their undergrad entrants that they’ve stopped bothering with actually teaching low-level courses. They assign work, they expect to receive finished work, they grade the finished work, but it all relies on the clandestine assumption that the “good students” could already do the work when they entered the classroom.
Exactly.
Only a small fraction of math has practical applications, the majority of math exists for no reason other than thinking about it is fun. Even things with applications had sometimes been invented before those applications were known. So in a sense most math is designed to be fun. Of course it’s not fun for everyone, just for a special class of people who are into this kind of thing. That makes it different from Angry Birds. But there are many games which are also only enjoyed by a specific audience, so maybe the difference is not that fundamental. A large part of the reason the average person doesn’t enjoy math is that unlike Angry Birds math requires some effort, which is the same reason the average person doesn’t enjoy League Of Evil III.
Spot on. Pure, fun math does benefit society directly in at least one way, however, in that the opportunity to engage in it can be used to lure very smart people into otherwise unpalatable teaching jobs.
In fact, that seems to be the main point of “research” in most less-than-productive fields (i.e. the humanities).
Is it clear that this is in the best interests of society? It would seem to me the end result is bad teaching. Back when I was in undergrad, the best researchers were the worst teachers (for obvious reasons- they were focused on their research and didn’t at all care about teaching).
When I was in grad school in physics, the professor widely considered the strongest teacher was denied tenure (cited AGAINST him in the decision was that he had written a widely used textbook),etc.
Also, the desire for tenured track profs to dodge teaching is why the majority of math classes at many research institutions were taught by grad students.
Interesting. Did there seem to be any pedagogical benefit to having relatively easy access to research-level experts, though?
In graduate school, for special topics class there were usually only 1 or 2 professors that COULD teach a certain class (and only 3 or 4 students interested in taking it)- so when you are talking cutting edge research topics, its a necessity to have a researcher because no one else will be familiar enough with whats going on in the field.
Outside of that, not really. Good teaching takes work, so if you put someone in front of the class whose career advancement requires spending all their time on research, then the teaching is just a potentially career destroying distraction. Also, at the intro level, subject-pedagogy experts tend to do better (i.e. the physics education group were measurably more effective at teaching physics than other physics groups. So much so that I think now they exclusively teach the large physics courses for engineers)
I mean, it’s easier to get research positions with those professors, and those are learning experiences, but the students generally get very little out of it during the actual class.
Thinking for a long time is one of the classic descriptions of Newton; from John Maynard Keynes’s “Newton, the Man”:
Paul Graham also mentions focus in this article.
I think math is more fun than playing video games. But I guess it’s subjective.
Lucky you.
He brags shamelessly about his wide variety of interests: Drumming, lockpicking, PUA, biology, Tana Tuva, etc.
The Feynman divorce:
You’re right.
Indeed, terse “explanations” that handwave more than explain are a pet peeve of mine. They can be outright confusing and cause more harm than good IMO. See this question on phrasing explanations in physics for some examples.