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I hit a wall in college where math just stopped being something I intuitively "got". I'm sure that given enough time and motivation, I could have continued bein

by Smudge 13y ago

I hit a wall in college where math just stopped being something I intuitively "got". I'm sure that given enough time and motivation, I could have continued being "good at math" even at the higher levels, but these were luxuries I did not have, given everything else on my plate at the time. My biggest problem with math, especially once I got into academia, was how it was taught. So many professors would scribble what seemed like nonsense on the board (symbols that change from professor to professor, or even from lecture to lecture) and then go on to say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!" Finally, I'd find a tutor to explain to me what it was I was missing, and it really WAS obvious. If only it had been taught that way in the first place! Admittedly, not everyone has the same learning style, but the classes I took seemed really tailored towards the students who already had the intuition that I lacked.

mahyarm 13y ago
I agree, college mathematics education is usually horrible, especially in the start. I remember back when I was attempting to read my calculus 1 textbook and not understanding the thing at all. I realized that about 3 or 4 classes later in discrete math the reason why is because it was using fairly foundational mathematical terms and concepts such as sets, proof by induction and so on that it was small wonder why my calculus textbook was incomprehensible to my grade 12 mathematical education. I confronted a professor about why they have it backwards and don't teach discrete math course or similar foundational course FIRST so people can actually read their textbooks, or at least let people take that path. They basically said that since it's not relevant to many majors and it's harder for most people since they don't have the 'mathematical mind' they do it in that backwards way. The professors being Math PhDs, don't adjust their any of their classes enough for the lack of foundational knowledge. It frustrated me very much. I don't think it's a big mystery why you probably hit that college wall when most college curriculums are set up that way.
- short_circut 13y ago
  I think this is due to the fact that many many years ago logic used to be taught in highschool and all of those concepts would have seemed significantly less foreign.
SamBoogie 13y ago
This resonates with me- particularly the 'it obviously follows', (huge jump in complexity and then --->) 'so we already know that x, so obviously- y'. I think a lot of teachers aren't cognizant of the fact that what is obvious to then isn't automatically obvious to students. I had what I would say is at best a mediocre HS math teacher and completely tuned out. It was my fault in the end, but the teacher didn't help. I'm now trying to learn a bunch of things that require math and mathematic theory when you get to higher levels. So - learning math is what I have to do. It's kind of fun, and yes it is hard work. Like, brain-hurting hard work.
aethertap 13y ago
> and I'd sit there wanting to shout "NO, NO it's not obvious!" Toward the end of my degree program, I became the annoying guy in class who would do exactly that. I remember one time in particular (I think the topic was something on wavelets, which I barely remember now anyway) when I stopped the professor and said "Can you explain all of that over again, from the beginning?" Worked great for me, but I'm not sure what the rest of the class thought of it. At the time I just assumed they were as lost as me and would appreciate it, but that may well have not been the case.
- csmuk 13y ago
  Good for you. Didn't work for me at university. We had one lecturer who was great and would adjust his material on his OHP transparencies if questions were asked and explain in detail. Unfortunately questioning others or asking for clarification would result in either being summonsed by your tutor and getting a bollocking or being asked to leave instantly for "not reading the material". I quit after the first year. Best thing I ever did.
- mattdotc 13y ago
  I bet a lot of them were lost. If I were in your class, I probably would not have been bothered by your question. The questions that bother me are the type that stroke the ego of the person asking because they already know the answer.
a1a 13y ago
>> wanting to shout "NO, NO it's not obvious!" Usually "obviously", "trivial" and such are used to point out that: "this should be obvious/trivial by now", if not: you are getting behind/need to study more/be better prepared before class/....
- Smudge 13y ago
  That is how I read these signals, and sometimes it was true. I'd study a bit more, meet with the class's TA or a university-provided tutor to go over the material, and I'd be fine. But even then, in many cases, the professor was simply expecting us to have made an intuitive leap. And those of us who hadn't made it were left behind, with no explanation as to why or what it was we needed to understand.
- munin 13y ago
  you know you've stepped into the woods when you stop hearing "this is obvious" in math classes...
jrs99 13y ago
i got As in almost all my classes in math without really understanding anything. It was like a train I couldn't stop. I couldn't ever slow down for a month and say "I'm going to work on trigonometry this whole month so I can know what I'm doing when I do integration." I just had to memorize the steps to get the right answer, get my A, forget it all, and wait for whatever was coming next. Years later I go straight back to the beginning and figure everything out, starting with Serge Lang's Basic Mathematics. I didn't even know where the Pythagorean Theorem came from, and when I learned it the second time around, it was damn beautiful. my advice is to go back all the way to the beginning and get a book written by a real mathematician. I.M. Gelfand's Algebra and Trigonometry were truly enlightening.
- phasetransition 13y ago
  Thanks for the book suggestion. I too, would like to go through and replace the "method" regions with "conceptual" understanding
- selimthegrim 13y ago
  In that vein, may I also suggest Ordinary Differential Equations by V.I. Arnol'd.
pshin45 13y ago
> So many professors would [...] say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!" Agreed. What makes a great teacher is not the depth of their knowledge of the subject matter, but how well they are able to put themselves in their students' shoes and overcome the "Curse of Knowledge"[1]. Unfortunately for students, university professors are usually hired for the former (knowledge & research) and not the latter (real teaching ability). [1] http://en.wikipedia.org/wiki/Curse_of_knowledge http://en.wikipedia.org/wiki/Curse_of_knowledge
paganel 13y ago
I know that my personal experience won't be of much of help to lots of HN-ers who are passed the highschool - early college-age, but I can say that in my case my love and understanding for Maths was inspired by two great teachers that I had, my highschool Maths teacher and my Calculus prof in my first year in college. Maths is not about arithmetic computations or getting the "exact" answer, is realizing that things like convergent series or Real numbers are extraordinary things, almost magical, as in you somehow get the sense they all come from a different Universe. Sometimes one is lucky enough to have these things revealed to him, like it happened for me.
trentmb 13y ago
I usually just raise my hand and ask the professor to elaborate. If it's genuinely time intensive, they'd ask me to stop by during office hours.
ChuckMcM 13y ago
You have hit upon my biggest gripe with mathematicians, their love of 'notation', or more precisely their love of writing in symbology that makes mathematics seem more arcane than it actually is. When ever I've run up against "impenetrable" math I often ask "So how would you use this?" and connecting it to the real world helped tremendously.
- wdewind 13y ago
  Yeah I totally agree. I took one of the car programming classes online and found that while the instructor was (obviously) really smart, he sucked at programming. He tried to write out mathematical equations in python rather than structure the code in a way that simply described what was going on. At the end of each mini lesson I'd refactor his code so that it made sense (mostly doing small stuff like changing variable names from their corresponding mathematical symbols to the word of what they actually were, or factoring blocks into methods). Eventually I gave up not because I couldn't understand the domain of self driving cars (he was great at explaining that stuff), but because I couldn't keep up with the mathematical syntax (and the online course kept wiping my code and resetting with his, which was extremely frustrating).
  - nemesisrobot 13y ago
    Are you talking about Sebastian Thrun's AI class on Udacity [1]? I haven't yet taken it, but I have on my todo list. [1] https://www.udacity.com/course/cs373 https://www.udacity.com/course/cs373
    - chattoraj 13y ago
      Yes, he is.
  - nightski 13y ago
    If you do not learn the mathematical structure it is going to be hard to do anything further in the field after the course. Why not just bite the bullet and properly learn the pre-reqs? With a basic course in linear algebra (such as Gilbert Strang's on MIT OpenCourseWare) and potentially some intro calculus you should fly through that course.
agumonkey 13y ago
I really felt college level math (abstract algebra, groups, monoids and such) was abstract painting in hieroglyphics, until I made a full turn into programming where you start to speak about abstract patterns that makes absolutely no sense (iterators, monads) at first, then I felt that this way of thinking seemed to be about finding your own solutions by being "mathematical" in the way you model your problem. Then abstract algebra started to feel like 'math + generics' and I went into re-reading my college textbooks. I still don't understand more than 15%, but I feel I have a chance. I agree about the way its taught, teachers either forgot their own learning process, or they're all very advanced brains aiming at younger advanced brains that can unfold the possible application behind the abstractions.
vinbreau 13y ago
I was an art major in a mostly engineering school. I always had trouble with math but loved learning it. I also hit a wall in college like you. The first day of class the teacher asked how many engineering students there were in the class. Just about everyone raised their hand. This was her sign that she could teach fast. I knew I was screwed immediately. I wish she had asked how many art students there were so we could go slower. I remember the math stopped making sense. The teacher would do exactly as you described, saying things like "...it obviously follows that...", etc. A girl who sat next to me would try to explain but was no less clear than the teacher. All the engineering students just got it. My grades started high and then rapidly fell each week until I hit a string of zeros for a month. I was too proud to ask for a drop but eventually did but only after skipping a month of classes. The teacher was kind enough to understand that I was trying but my effort was for naught. She gave me the drop. I've never pursued math any further, having felt defeated.
- doorhammer 13y ago
  I think that's part of what the author is trying to approach in the article. I could be wrong, but the engineering students were probably immersed in math way more than you were, so concepts that seemed natural to them were only because they'd banged their head against it more often. I run into this a lot when people talk to me about programming and I get something faster than they do. I've spent a lot of time reading book after book, listening to podcasts, learning new languages, and studying new concepts, so it can be really easy for me to fit new information or ideas into some context and get them. I don't think that I'm necessarily smarter for it. I think I've just had a passion for it, so I never get tired of reading the articles and absorbing the material. Now I'm personally running into the place where there are a lot of things I've wanted to learn for awhile, but my weak background in math hinders me (machine learning, more advanced algorithm analysis, signal processing, machine vision, etc). I'm working my way through a calc book. I don't think I could have done it in your position, either though. I'd have psyched myself out. I've got to learn it on my on, with my own rhythm. Kind of rambly, but you should find something related to what you like and maybe jump back in. Just find a learning mechanism that's suited to your background :)
skierscott 13y ago
> I hit a wall in college where math just stopped being something I intuitively "got". That's the college experience. Everyone that goes to that school is smart. Since the professors go to this level, all that remains is hard work. They understand that everyone there is smart.

by Smudge 13y ago

mahyarm 13y ago

I agree, college mathematics education is usually horrible, especially in the start. I remember back when I was attempting to read my calculus 1 textbook and not understanding the thing at all. I realized that about 3 or 4 classes later in discrete math the reason why is because it was using fairly foundational mathematical terms and concepts such as sets, proof by induction and so on that it was small wonder why my calculus textbook was incomprehensible to my grade 12 mathematical education. I confronted a professor about why they have it backwards and don't teach discrete math course or similar foundational course FIRST so people can actually read their textbooks, or at least let people take that path. They basically said that since it's not relevant to many majors and it's harder for most people since they don't have the 'mathematical mind' they do it in that backwards way. The professors being Math PhDs, don't adjust their any of their classes enough for the lack of foundational knowledge. It frustrated me very much. I don't think it's a big mystery why you probably hit that college wall when most college curriculums are set up that way.

short_circut 13y ago
I think this is due to the fact that many many years ago logic used to be taught in highschool and all of those concepts would have seemed significantly less foreign.

SamBoogie 13y ago

This resonates with me- particularly the 'it obviously follows', (huge jump in complexity and then --->) 'so we already know that x, so obviously- y'. I think a lot of teachers aren't cognizant of the fact that what is obvious to then isn't automatically obvious to students. I had what I would say is at best a mediocre HS math teacher and completely tuned out. It was my fault in the end, but the teacher didn't help. I'm now trying to learn a bunch of things that require math and mathematic theory when you get to higher levels. So - learning math is what I have to do. It's kind of fun, and yes it is hard work. Like, brain-hurting hard work.

aethertap 13y ago

> and I'd sit there wanting to shout "NO, NO it's not obvious!" Toward the end of my degree program, I became the annoying guy in class who would do exactly that. I remember one time in particular (I think the topic was something on wavelets, which I barely remember now anyway) when I stopped the professor and said "Can you explain all of that over again, from the beginning?" Worked great for me, but I'm not sure what the rest of the class thought of it. At the time I just assumed they were as lost as me and would appreciate it, but that may well have not been the case.

csmuk 13y ago
Good for you. Didn't work for me at university. We had one lecturer who was great and would adjust his material on his OHP transparencies if questions were asked and explain in detail. Unfortunately questioning others or asking for clarification would result in either being summonsed by your tutor and getting a bollocking or being asked to leave instantly for "not reading the material". I quit after the first year. Best thing I ever did.
mattdotc 13y ago
I bet a lot of them were lost. If I were in your class, I probably would not have been bothered by your question. The questions that bother me are the type that stroke the ego of the person asking because they already know the answer.

a1a 13y ago

>> wanting to shout "NO, NO it's not obvious!" Usually "obviously", "trivial" and such are used to point out that: "this should be obvious/trivial by now", if not: you are getting behind/need to study more/be better prepared before class/....

Smudge 13y ago
That is how I read these signals, and sometimes it was true. I'd study a bit more, meet with the class's TA or a university-provided tutor to go over the material, and I'd be fine. But even then, in many cases, the professor was simply expecting us to have made an intuitive leap. And those of us who hadn't made it were left behind, with no explanation as to why or what it was we needed to understand.
munin 13y ago
you know you've stepped into the woods when you stop hearing "this is obvious" in math classes...

jrs99 13y ago

i got As in almost all my classes in math without really understanding anything. It was like a train I couldn't stop. I couldn't ever slow down for a month and say "I'm going to work on trigonometry this whole month so I can know what I'm doing when I do integration." I just had to memorize the steps to get the right answer, get my A, forget it all, and wait for whatever was coming next. Years later I go straight back to the beginning and figure everything out, starting with Serge Lang's Basic Mathematics. I didn't even know where the Pythagorean Theorem came from, and when I learned it the second time around, it was damn beautiful. my advice is to go back all the way to the beginning and get a book written by a real mathematician. I.M. Gelfand's Algebra and Trigonometry were truly enlightening.

phasetransition 13y ago
Thanks for the book suggestion. I too, would like to go through and replace the "method" regions with "conceptual" understanding
selimthegrim 13y ago
In that vein, may I also suggest Ordinary Differential Equations by V.I. Arnol'd.

pshin45 13y ago

> So many professors would [...] say things like "...and the proof is trivial" or "...it obviously follows that...", and I'd sit there wanting to shout "NO, NO it's not obvious!" Agreed. What makes a great teacher is not the depth of their knowledge of the subject matter, but how well they are able to put themselves in their students' shoes and overcome the "Curse of Knowledge"[1]. Unfortunately for students, university professors are usually hired for the former (knowledge & research) and not the latter (real teaching ability). [1] http://en.wikipedia.org/wiki/Curse_of_knowledge http://en.wikipedia.org/wiki/Curse_of_knowledge

paganel 13y ago

I know that my personal experience won't be of much of help to lots of HN-ers who are passed the highschool - early college-age, but I can say that in my case my love and understanding for Maths was inspired by two great teachers that I had, my highschool Maths teacher and my Calculus prof in my first year in college. Maths is not about arithmetic computations or getting the "exact" answer, is realizing that things like convergent series or Real numbers are extraordinary things, almost magical, as in you somehow get the sense they all come from a different Universe. Sometimes one is lucky enough to have these things revealed to him, like it happened for me.

trentmb 13y ago

I usually just raise my hand and ask the professor to elaborate. If it's genuinely time intensive, they'd ask me to stop by during office hours.

ChuckMcM 13y ago

You have hit upon my biggest gripe with mathematicians, their love of 'notation', or more precisely their love of writing in symbology that makes mathematics seem more arcane than it actually is. When ever I've run up against "impenetrable" math I often ask "So how would you use this?" and connecting it to the real world helped tremendously.

wdewind 13y ago
Yeah I totally agree. I took one of the car programming classes online and found that while the instructor was (obviously) really smart, he sucked at programming. He tried to write out mathematical equations in python rather than structure the code in a way that simply described what was going on. At the end of each mini lesson I'd refactor his code so that it made sense (mostly doing small stuff like changing variable names from their corresponding mathematical symbols to the word of what they actually were, or factoring blocks into methods). Eventually I gave up not because I couldn't understand the domain of self driving cars (he was great at explaining that stuff), but because I couldn't keep up with the mathematical syntax (and the online course kept wiping my code and resetting with his, which was extremely frustrating).
- nemesisrobot 13y ago
  Are you talking about Sebastian Thrun's AI class on Udacity [1]? I haven't yet taken it, but I have on my todo list. [1] https://www.udacity.com/course/cs373 https://www.udacity.com/course/cs373
  - chattoraj 13y ago
    Yes, he is.
- nightski 13y ago
  If you do not learn the mathematical structure it is going to be hard to do anything further in the field after the course. Why not just bite the bullet and properly learn the pre-reqs? With a basic course in linear algebra (such as Gilbert Strang's on MIT OpenCourseWare) and potentially some intro calculus you should fly through that course.

agumonkey 13y ago

I really felt college level math (abstract algebra, groups, monoids and such) was abstract painting in hieroglyphics, until I made a full turn into programming where you start to speak about abstract patterns that makes absolutely no sense (iterators, monads) at first, then I felt that this way of thinking seemed to be about finding your own solutions by being "mathematical" in the way you model your problem. Then abstract algebra started to feel like 'math + generics' and I went into re-reading my college textbooks. I still don't understand more than 15%, but I feel I have a chance. I agree about the way its taught, teachers either forgot their own learning process, or they're all very advanced brains aiming at younger advanced brains that can unfold the possible application behind the abstractions.

vinbreau 13y ago

I was an art major in a mostly engineering school. I always had trouble with math but loved learning it. I also hit a wall in college like you. The first day of class the teacher asked how many engineering students there were in the class. Just about everyone raised their hand. This was her sign that she could teach fast. I knew I was screwed immediately. I wish she had asked how many art students there were so we could go slower. I remember the math stopped making sense. The teacher would do exactly as you described, saying things like "...it obviously follows that...", etc. A girl who sat next to me would try to explain but was no less clear than the teacher. All the engineering students just got it. My grades started high and then rapidly fell each week until I hit a string of zeros for a month. I was too proud to ask for a drop but eventually did but only after skipping a month of classes. The teacher was kind enough to understand that I was trying but my effort was for naught. She gave me the drop. I've never pursued math any further, having felt defeated.

doorhammer 13y ago
I think that's part of what the author is trying to approach in the article. I could be wrong, but the engineering students were probably immersed in math way more than you were, so concepts that seemed natural to them were only because they'd banged their head against it more often. I run into this a lot when people talk to me about programming and I get something faster than they do. I've spent a lot of time reading book after book, listening to podcasts, learning new languages, and studying new concepts, so it can be really easy for me to fit new information or ideas into some context and get them. I don't think that I'm necessarily smarter for it. I think I've just had a passion for it, so I never get tired of reading the articles and absorbing the material. Now I'm personally running into the place where there are a lot of things I've wanted to learn for awhile, but my weak background in math hinders me (machine learning, more advanced algorithm analysis, signal processing, machine vision, etc). I'm working my way through a calc book. I don't think I could have done it in your position, either though. I'd have psyched myself out. I've got to learn it on my on, with my own rhythm. Kind of rambly, but you should find something related to what you like and maybe jump back in. Just find a learning mechanism that's suited to your background :)

skierscott 13y ago

> I hit a wall in college where math just stopped being something I intuitively "got". That's the college experience. Everyone that goes to that school is smart. Since the professors go to this level, all that remains is hard work. They understand that everyone there is smart.