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I’ve taught mathematics for over 20 years in higher education and I strongly disagree with your view on the relative ease of learning math on one’s own. Let’s

I’ve taught mathematics for over 20 years in higher education and I strongly disagree with your view on the relative ease of learning math on one’s own. Let’s take a simple example. Here’s the distributive property for rings: a(b + c) = ab + ac It’s easy for me to convince someone that because of this property we get the following: 3(x + 2) = 3x + 6 It’s much harder to convince someone that 3x + 6 = 3(x + 2) It’s hard to convince someone that factoring is just using the distributive property reading right to left instead of left to right. It’s much harder to convince them of this just from reading it in a book. The nuance will be lost. They need examples that one goes over in a classroom to see this and a book can’t contain enough examples without being too many pages or without boring the reader with what appears to them to be minutia. They can’t grasp the nuance from reading. They grasp it from doing problems and being guided on the problems at the time they do them. Books and videos can not replace a teacher. Your experience is not the norm and you should not use it as a guide for what is realistically possible for most people.

rrrrrrrrrrrr2 4y ago
I agree with you except for the very last point, that "students need to be guided on problems at the time they do them". I think this issue is fundamental to the tradeoffs of self studying that I've experienced. I want to point out: the guide doesn't necessarily have to be a human. And textbooks accomplish this by giving examples that walk through how to solve problems similar to the exercises. Unfortunately this is a useful secret about textbook and problem design that is not common knowledge to students. (I.e., the principle of charity, principle of relevance, Chekhov's gun, etc.) I agree with the point of your example, but maybe not the choice of example. For your example, Gallian (which imo is the standard intro algebra book) includes this property in the definition of a ring: "Property 6. a(b + c) = ab + ac and (b + c)a = ba + bc". (Probably in order to disclaim the confusion you're talking about.) Then he goes on to derive 6-7 other properties of groups that will (of course) be useful in the exercises. I definitely found it's really important when self-studying to pick the right textbook. Definitely you have to be an experienced educator to write a good book that anticipates most of these things. But I agree that it's impossible for any textbook to anticipate every place someone reading it might get stuck. Let's say for a given student it happens 5-6 times in a really high quality textbook like Gallian or Rudin PoMA. They have 3 options (1) ask on mathematics.stackexchange.com and probably get an answer because it's a great community, (2) try to figure it out themselves, or (3) pay $600 and invest 3h/wk to take an algebra class that might cover the first 1/2 of the textbook in 4 months. I think where we disagree is what are the constraints of the tradeoff between attending a class vs. reading the book. My experience has been that, if the subject is interesting enough, reading a good quality textbook is cheaper and 2-3x faster than taking a course, but at certain points it can be much more challenging. I think where it depends on which student is how much more challenging, i.e., will they be able to dig themselves out of those holes in a few minutes or a few hours. I was personally in the middle of these two extremes, but I still thought it was worth it to self study after taking 3-4 classes in the math department, then I skipped a bunch (7 semesters) of analysis/algebra/topology and came back and took 2-3 grad courses, where I didn't really understand the main goal of the subjects until I took the classes. And then I went into CS industry and never used any of it again. But I don't regret the experience; it was legitimately interesting to learn about math. I think you have to be legitimately interested in a subject to self study it successfully. But that's true about studying serious math in general though: you have to be unrelentingly into it, or else you're just really misguided and shouldn't be there, given the high competition, poor odds for any future in math, and no practical use for any of the theory. That's my experience with it and why I made the argument that I made.
- syzarian 4y ago
  My example was geared toward those people learning beginning algebra in 9th grade or so or the great many students who place into pre college level math during their first semester of undergrad. I used the word “ring” because as I was writing my comment I didn’t want any readers to think my example only applied to integers, or rationals, or reals. So I said “ring” trying to capture a nuance that very few people would appreciate. I contend you are an outlier in the extreme. I think very few people can read a math book and get anything out of it unless they have had training. For one thing, in lower level math we lie to students all the time. We just can’t be honest about things because students at that level can’t appreciate the insane number of nuances. For example, ask a first year calculus student what “dx” means and their answer will be complete nonsense but they don’t know that. We can’t tell students what “dx” really is because it is quite beyond their level of understanding. Thanks for the discussion. I appreciate your perspective.
  - rrrrrrrrrrrr2 4y ago
    Obligatory: https://www.smbc-comics.com/comics/20141206.png https://www.smbc-comics.com/comics/20141206.png
- pointedset 4y ago
  I think your quote from Gallian doesn't address syzarian's example, actually. The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one possible left-right confusion, but I would argue most weak students believe it holds even when it doesn't, so in a sense they're too permissive in their reasoning. However, syzarian's example is about a different left-right confusion: whether you can read an equality both forwards and backwards. I've seen this confusion in students: they will readily believe that you can distribute a factor over a sum (going forward), but be very skeptical about the act of factoring out (going backward), even though it's justified by the same equation. In this case, the students aren't permissive enough in their reasoning. This is the kind of misconception (that equality has a "direction") that's much easier to suss out with an in-person interaction, whether it's with a teacher or other students.
2Gkashmiri 4y ago
1+1 x 2 this has been doing rounds on twitter how people do not have grasp of basic maths and it points out a good argument for people who have been taught "good basics"... like i recently asked a doctor who says they are bad at maths and they just blurted out 3. i asked why and they responded, "uh.... bodmas?" its such a small thing really, like in your primary and secondary education, of all the classes you attend, this is just a tiny tiny topic that must've been taught but because you were taught its importance and use it as a building block early on, it just feels like second nature. like the tables of 2-9. pretty hard for kids to remember but once they do and they are taught to HOW to use it, it really helps them
- rrrrrrrrrrrr2 4y ago
  I think this is an apples to oranges comparison. You get taught pemdas or bomdas early on and then use it hundreds of times throughout school and the rest of your life. The value of learning it is clear. Whereas ring theory is very much specific to algebra and you're not going to use it programming, doing chemistry, or whatever else outside math. Finding motivation to learn pure math is a bit harder. GP's example illustrates a good point. I agree with their point.
ido 4y ago
I think that OP's fundamental error is talking about school vs university. When I started my bachelor's studies in math the professor covered 3 years of high-school math in the first 2-3 weeks of the first semester. Up to that point I too thought I could just learn by reading the textbook. University (assuming a good one, mine wasn't world famous but has a good reputation nationally & in Europe) level maths is completely different level than high school level math, and having access to the professors makes a big difference (although I was often the only student coming by at their office hours if it wasn't before a test, so I guess not everyone agrees with that).
- rrrrrrrrrrrr2 4y ago
  If you're talking about me (GP not OP); when I said "school" I meant during my undergrad at a university. I'm claiming that I skipped 2-3 years of math at university by self studying, specifically 2 years of analysis, 1 year of topology, and some algebra. You can absolutely do this for university math using textbooks if you're motivated enough. Which I was. Sorry for the colloquialism. "School" means university in the U.S. tech industry, but apparently not in Europe. Edit: Reply to child comment before I log off. Haha I agree. The really valuable things I learned from math professors had nothing to do with the curriculum though. It was more like stories about all the people they saw going crazy trying to get tenure, ultimately failing and then seeking revenge against those who denied them; how they all go to conferences sometimes and drink very heavily afterwards in the nearest bar; and just little colloquialisms. Math academia was a really interesting subculture of very interesting people.
  - ido 4y ago
    No worries! I think for undergraduate studies I’m sure you can. Although motivation levels of “motivated enough” have to be pretty high! My experience was that pretty quickly professors (and TAs) provided insight and guidance I couldn’t (or at least didn’t) get from just reading the textbook. But also that this increases the further you go in your education- in a 200 student auditorium class for linear algebra or calculus the professor isn’t going to be giving you any individual attention but in my school even by the 3rd-4th semester classes were already often small (15-20 students) and we had actual conversations with the professors about the material (and beyond).

rrrrrrrrrrrr2 4y ago

I agree with you except for the very last point, that "students need to be guided on problems at the time they do them". I think this issue is fundamental to the tradeoffs of self studying that I've experienced. I want to point out: the guide doesn't necessarily have to be a human. And textbooks accomplish this by giving examples that walk through how to solve problems similar to the exercises. Unfortunately this is a useful secret about textbook and problem design that is not common knowledge to students. (I.e., the principle of charity, principle of relevance, Chekhov's gun, etc.) I agree with the point of your example, but maybe not the choice of example. For your example, Gallian (which imo is the standard intro algebra book) includes this property in the definition of a ring: "Property 6. a(b + c) = ab + ac and (b + c)a = ba + bc". (Probably in order to disclaim the confusion you're talking about.) Then he goes on to derive 6-7 other properties of groups that will (of course) be useful in the exercises. I definitely found it's really important when self-studying to pick the right textbook. Definitely you have to be an experienced educator to write a good book that anticipates most of these things. But I agree that it's impossible for any textbook to anticipate every place someone reading it might get stuck. Let's say for a given student it happens 5-6 times in a really high quality textbook like Gallian or Rudin PoMA. They have 3 options (1) ask on mathematics.stackexchange.com and probably get an answer because it's a great community, (2) try to figure it out themselves, or (3) pay $600 and invest 3h/wk to take an algebra class that might cover the first 1/2 of the textbook in 4 months. I think where we disagree is what are the constraints of the tradeoff between attending a class vs. reading the book. My experience has been that, if the subject is interesting enough, reading a good quality textbook is cheaper and 2-3x faster than taking a course, but at certain points it can be much more challenging. I think where it depends on which student is how much more challenging, i.e., will they be able to dig themselves out of those holes in a few minutes or a few hours. I was personally in the middle of these two extremes, but I still thought it was worth it to self study after taking 3-4 classes in the math department, then I skipped a bunch (7 semesters) of analysis/algebra/topology and came back and took 2-3 grad courses, where I didn't really understand the main goal of the subjects until I took the classes. And then I went into CS industry and never used any of it again. But I don't regret the experience; it was legitimately interesting to learn about math. I think you have to be legitimately interested in a subject to self study it successfully. But that's true about studying serious math in general though: you have to be unrelentingly into it, or else you're just really misguided and shouldn't be there, given the high competition, poor odds for any future in math, and no practical use for any of the theory. That's my experience with it and why I made the argument that I made.

syzarian 4y ago
My example was geared toward those people learning beginning algebra in 9th grade or so or the great many students who place into pre college level math during their first semester of undergrad. I used the word “ring” because as I was writing my comment I didn’t want any readers to think my example only applied to integers, or rationals, or reals. So I said “ring” trying to capture a nuance that very few people would appreciate. I contend you are an outlier in the extreme. I think very few people can read a math book and get anything out of it unless they have had training. For one thing, in lower level math we lie to students all the time. We just can’t be honest about things because students at that level can’t appreciate the insane number of nuances. For example, ask a first year calculus student what “dx” means and their answer will be complete nonsense but they don’t know that. We can’t tell students what “dx” really is because it is quite beyond their level of understanding. Thanks for the discussion. I appreciate your perspective.
- rrrrrrrrrrrr2 4y ago
  Obligatory: https://www.smbc-comics.com/comics/20141206.png https://www.smbc-comics.com/comics/20141206.png
pointedset 4y ago
I think your quote from Gallian doesn't address syzarian's example, actually. The two properties you quoted are about the fact the distributivity works whether you're multiplying on the left or on the right. That's one possible left-right confusion, but I would argue most weak students believe it holds even when it doesn't, so in a sense they're too permissive in their reasoning. However, syzarian's example is about a different left-right confusion: whether you can read an equality both forwards and backwards. I've seen this confusion in students: they will readily believe that you can distribute a factor over a sum (going forward), but be very skeptical about the act of factoring out (going backward), even though it's justified by the same equation. In this case, the students aren't permissive enough in their reasoning. This is the kind of misconception (that equality has a "direction") that's much easier to suss out with an in-person interaction, whether it's with a teacher or other students.

2Gkashmiri 4y ago

1+1 x 2 this has been doing rounds on twitter how people do not have grasp of basic maths and it points out a good argument for people who have been taught "good basics"... like i recently asked a doctor who says they are bad at maths and they just blurted out 3. i asked why and they responded, "uh.... bodmas?" its such a small thing really, like in your primary and secondary education, of all the classes you attend, this is just a tiny tiny topic that must've been taught but because you were taught its importance and use it as a building block early on, it just feels like second nature. like the tables of 2-9. pretty hard for kids to remember but once they do and they are taught to HOW to use it, it really helps them

rrrrrrrrrrrr2 4y ago
I think this is an apples to oranges comparison. You get taught pemdas or bomdas early on and then use it hundreds of times throughout school and the rest of your life. The value of learning it is clear. Whereas ring theory is very much specific to algebra and you're not going to use it programming, doing chemistry, or whatever else outside math. Finding motivation to learn pure math is a bit harder. GP's example illustrates a good point. I agree with their point.

ido 4y ago

I think that OP's fundamental error is talking about school vs university. When I started my bachelor's studies in math the professor covered 3 years of high-school math in the first 2-3 weeks of the first semester. Up to that point I too thought I could just learn by reading the textbook. University (assuming a good one, mine wasn't world famous but has a good reputation nationally & in Europe) level maths is completely different level than high school level math, and having access to the professors makes a big difference (although I was often the only student coming by at their office hours if it wasn't before a test, so I guess not everyone agrees with that).

rrrrrrrrrrrr2 4y ago
If you're talking about me (GP not OP); when I said "school" I meant during my undergrad at a university. I'm claiming that I skipped 2-3 years of math at university by self studying, specifically 2 years of analysis, 1 year of topology, and some algebra. You can absolutely do this for university math using textbooks if you're motivated enough. Which I was. Sorry for the colloquialism. "School" means university in the U.S. tech industry, but apparently not in Europe. Edit: Reply to child comment before I log off. Haha I agree. The really valuable things I learned from math professors had nothing to do with the curriculum though. It was more like stories about all the people they saw going crazy trying to get tenure, ultimately failing and then seeking revenge against those who denied them; how they all go to conferences sometimes and drink very heavily afterwards in the nearest bar; and just little colloquialisms. Math academia was a really interesting subculture of very interesting people.
- ido 4y ago
  No worries! I think for undergraduate studies I’m sure you can. Although motivation levels of “motivated enough” have to be pretty high! My experience was that pretty quickly professors (and TAs) provided insight and guidance I couldn’t (or at least didn’t) get from just reading the textbook. But also that this increases the further you go in your education- in a 200 student auditorium class for linear algebra or calculus the professor isn’t going to be giving you any individual attention but in my school even by the 3rd-4th semester classes were already often small (15-20 students) and we had actual conversations with the professors about the material (and beyond).