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I did well in high school math. These days, when something involving algebra, trigonometry, geometry etc comes up I feel like I have a good understanding of it

I did well in high school math. These days, when something involving algebra, trigonometry, geometry etc comes up I feel like I have a good understanding of it but my calculus seems weak to non-existent. I'm not sure if it's how I was taught, how I studied it or something else but calculus always seemed like a huge step change in difficulty. That said, I love how this article gives practical hints on how to replicate the insight and solve the question, rather than just the insight itself.

xivzgrev 3y ago
I think it’s a common feeling. Even though they are both “math”, they feel like different skill sets.
mtlguitarist 3y ago
I find personally that my math ability is set to approximately 2-3 levels "below" the highest level math I completed, and I've seen hold for others. I have an applied math bachelors so I've taken analysis, dynamical systems, and other high level math classes, but I find that the stuff that I actually remember at a level to pass undergrad exams is up to linear algebra or maybe a little more advanced. Of course if I were to relearn it'd be much faster, but years of being a software engineer have caused me to forget all that stuff.
- physicles 3y ago
  I was literally about to write this same comment. I did physics so I finished some group theory and complex analysis, but 20 years later all that stuff is gone because I never applied it in other classes. Only the stuff that kept coming up, like calc 1/2 and first order diffeq, really stuck.
  - kaba0 3y ago
    With that said, does anyone know of a good method to relearn math efficiently? I found it to be really hard to self-learn any math topic, most books repeat everything from the basics at the beginning like what a set is, and then suddenly turn into ultra-advanced with “the proof is trivial” all around.
    - wholinator2 3y ago
      I think this experience is typical of self teaching from a math textbook. It's extremely difficult to find a good book that leaves no gaps while simultaneously explaining everything that might be difficult to understand. The key thing when encountering this for me is to expand my horizons and begin looking for videos or other supplementary materials. A teacher would show you the proof, or at least help you along the way, the internet can be used for this as well, up to a point. While it's frustrating and time consuming, self teaching a difficult subject is just like that unless you're a god amongst men (and few of us are). Sometimes you'll want to fight through the strange unproven thing by thinking hard for a couple weeks about it while googling intermittently to find key steps. Sometimes you'll have to give up on it and keep moving. If it's foundational then fighting through can be highly beneficial, but a lot more things are presented as foundational than actually are. I'd recommend finding good books by searching relentlessly on reddit and other forums for opinions, dedicating the time necessary to self teach something difficult (it can take upwards of a year to get through a smaller textbook if you have other things in your life going on), and if you really want it then fight for it. Give it everything you have, really let the problem consume your thoughts because eventually you'll wake up at 3am and know exactly what to do. And finally, move on if you don't want to do that. Try just keeping moving. Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts. Or you could find something you want to know and work backwards through every term that's used until you're at a concept and then attempt to apply it to the larger idea. In general, self teaching math is extremely difficult, and only really works if you're willing to dedicate the time to fight through ideas.
      - nohaydeprobleme 3y ago
        This is a great comment. To add on to a point that really aligns with my experiences: > "Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts." This is a very good approach, and I wish I started doing this earlier. Even in my university math courses, the professors sometimes skipped ahead to have students focus on a few later chapters before coming back, or told the class to skip several pages in the book. I also found that working on later exercises in a textbook would sometimes help me better understand concepts introduced in earlier chapters. Lastly—though this may not be completely relevant to studying mathematics—I've explicitly been taught in various language courses (explicitly for audio courses and implicitly for in-person university courses) that it's okay to move ahead if I know at least 80% of the material. The percentage may be higher for studying math topics, but especially for someone self-learning out of interest or for a specific application, it's much more preferable to move forward and revisit earlier exercises as needed, instead of quit the book. If you find yourself getting lost in later chapters, there is no problem with revisiting earlier chapters. You'd also likely be no worse off (possibly even better) than many undergraduates studying the textbook for a course for the first time. The most important thing is just to not quit the habit of consistent study. Perfectionism in understanding is a pitfall for self-directed studies, which consistency in studying beats every time.
kaba0 3y ago
One of my professors used to say that “even a horse can do derivatives. Integration is the real deal”, another one said that you integrate by “look at it, deeply, deeply, deeply; and then solve it”. The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with equations — once you have it down, you can basically solve it, even if it is harder than the “notice this and that” elegant solution. Most integration problems require this intuition-based solution which has a certain elegance to it. It was especially humbling to me that Wolfram alpha fails most of the interesting calculus problems I encountered during my analysis classes, but after a while I managed to solve most of them. But it unfortunately does disappear after not using it for a time..
- mjd 3y ago
  > “look at it, deeply, deeply, deeply; and then solve it” That's the Feynman method: write down the question, think really hard, then write down the answer. Only three simple steps! Unfortunately, some of us are not Feynman.
  - mrguyorama 3y ago
    I actually hit this personally, because right up UNTIL calculus, math was the Feynman method for me. Everything always just clicked, made perfect sense, and I saw the beauty in it, and it was great fun. Then for calculus, we learned concepts, like what a derivative is, and I understood that, and understood conceptually (as in, what everything "means" and what it tells you) but I could never take that concept knowledge to the practice problems with me. I could follow along as the professor walked us through a problem, showing us what heuristics helped and what patterns to follow and how to manipulate the functions to get to something that followed one of the patterns to pull an answer out of your ass, but I could never commute those heuristics and patterns to novel examples. It's weird because I was great at doing the exact same thing for physics: Taking a novel and purposely opaque problem and finding which pattern it corresponds to.
- gizmo686 3y ago
mjd 3y ago
I think this is really important for good teaching. It's not enough to show the student how to solve the problem. One needs to also show the student the patterns of thought that could have led them to the solution. And it's not enough to show how _someone_ could have been led to the solution, one has to show how _this particular_ student could have figured it out, knowing what they know and being who they are. I have a blog article about this in progress.
- justeleblanc 3y ago
  I'm a professor in a big university in western Europe. Students don't want to be led to thinking of the solution. They want the same exercises that they did during the tutorial, and they want to know in advance how to solve all the exercises. Any attempt to digress from this is met with vitreous eyes.
gcanyon 3y ago
I'm in pretty much the same boat re: calculus, but I think a lot of it has to do with problems just like this. For me, early in my experience with calculus I always looked for the "graph it out"/non-calculus solution. So problems like water leaking out of a bucket, rocket acceleration, and other integrals where the underlying process is in some way linear always fell to non-calculus-based analysis. And thus when I got to problems where actual calculus was required, my non-grounding in the basics pushed me toward rote memorization which (of course) didn't stick over years of non-utilization.

xivzgrev 3y ago

I think it’s a common feeling. Even though they are both “math”, they feel like different skill sets.

mtlguitarist 3y ago

I find personally that my math ability is set to approximately 2-3 levels "below" the highest level math I completed, and I've seen hold for others. I have an applied math bachelors so I've taken analysis, dynamical systems, and other high level math classes, but I find that the stuff that I actually remember at a level to pass undergrad exams is up to linear algebra or maybe a little more advanced. Of course if I were to relearn it'd be much faster, but years of being a software engineer have caused me to forget all that stuff.

physicles 3y ago
I was literally about to write this same comment. I did physics so I finished some group theory and complex analysis, but 20 years later all that stuff is gone because I never applied it in other classes. Only the stuff that kept coming up, like calc 1/2 and first order diffeq, really stuck.
- kaba0 3y ago
  With that said, does anyone know of a good method to relearn math efficiently? I found it to be really hard to self-learn any math topic, most books repeat everything from the basics at the beginning like what a set is, and then suddenly turn into ultra-advanced with “the proof is trivial” all around.
  - wholinator2 3y ago
    I think this experience is typical of self teaching from a math textbook. It's extremely difficult to find a good book that leaves no gaps while simultaneously explaining everything that might be difficult to understand. The key thing when encountering this for me is to expand my horizons and begin looking for videos or other supplementary materials. A teacher would show you the proof, or at least help you along the way, the internet can be used for this as well, up to a point. While it's frustrating and time consuming, self teaching a difficult subject is just like that unless you're a god amongst men (and few of us are). Sometimes you'll want to fight through the strange unproven thing by thinking hard for a couple weeks about it while googling intermittently to find key steps. Sometimes you'll have to give up on it and keep moving. If it's foundational then fighting through can be highly beneficial, but a lot more things are presented as foundational than actually are. I'd recommend finding good books by searching relentlessly on reddit and other forums for opinions, dedicating the time necessary to self teach something difficult (it can take upwards of a year to get through a smaller textbook if you have other things in your life going on), and if you really want it then fight for it. Give it everything you have, really let the problem consume your thoughts because eventually you'll wake up at 3am and know exactly what to do. And finally, move on if you don't want to do that. Try just keeping moving. Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts. Or you could find something you want to know and work backwards through every term that's used until you're at a concept and then attempt to apply it to the larger idea. In general, self teaching math is extremely difficult, and only really works if you're willing to dedicate the time to fight through ideas.
    - nohaydeprobleme 3y ago
      This is a great comment. To add on to a point that really aligns with my experiences: > "Review from time to time but don't let a hard first couple chapters prevent you from ever learning the concepts." This is a very good approach, and I wish I started doing this earlier. Even in my university math courses, the professors sometimes skipped ahead to have students focus on a few later chapters before coming back, or told the class to skip several pages in the book. I also found that working on later exercises in a textbook would sometimes help me better understand concepts introduced in earlier chapters. Lastly—though this may not be completely relevant to studying mathematics—I've explicitly been taught in various language courses (explicitly for audio courses and implicitly for in-person university courses) that it's okay to move ahead if I know at least 80% of the material. The percentage may be higher for studying math topics, but especially for someone self-learning out of interest or for a specific application, it's much more preferable to move forward and revisit earlier exercises as needed, instead of quit the book. If you find yourself getting lost in later chapters, there is no problem with revisiting earlier chapters. You'd also likely be no worse off (possibly even better) than many undergraduates studying the textbook for a course for the first time. The most important thing is just to not quit the habit of consistent study. Perfectionism in understanding is a pitfall for self-directed studies, which consistency in studying beats every time.

kaba0 3y ago

One of my professors used to say that “even a horse can do derivatives. Integration is the real deal”, another one said that you integrate by “look at it, deeply, deeply, deeply; and then solve it”. The point is, many part of high school math is actually really “algorithmic”. I was one of the few in my class who absolutely loved coordinate geometry over “normal” geometry, because I simply felt really comfortable with equations — once you have it down, you can basically solve it, even if it is harder than the “notice this and that” elegant solution. Most integration problems require this intuition-based solution which has a certain elegance to it. It was especially humbling to me that Wolfram alpha fails most of the interesting calculus problems I encountered during my analysis classes, but after a while I managed to solve most of them. But it unfortunately does disappear after not using it for a time..

mjd 3y ago
> “look at it, deeply, deeply, deeply; and then solve it” That's the Feynman method: write down the question, think really hard, then write down the answer. Only three simple steps! Unfortunately, some of us are not Feynman.
- mrguyorama 3y ago
  I actually hit this personally, because right up UNTIL calculus, math was the Feynman method for me. Everything always just clicked, made perfect sense, and I saw the beauty in it, and it was great fun. Then for calculus, we learned concepts, like what a derivative is, and I understood that, and understood conceptually (as in, what everything "means" and what it tells you) but I could never take that concept knowledge to the practice problems with me. I could follow along as the professor walked us through a problem, showing us what heuristics helped and what patterns to follow and how to manipulate the functions to get to something that followed one of the patterns to pull an answer out of your ass, but I could never commute those heuristics and patterns to novel examples. It's weird because I was great at doing the exact same thing for physics: Taking a novel and purposely opaque problem and finding which pattern it corresponds to.
gizmo686 3y ago

mjd 3y ago

I think this is really important for good teaching. It's not enough to show the student how to solve the problem. One needs to also show the student the patterns of thought that could have led them to the solution. And it's not enough to show how _someone_ could have been led to the solution, one has to show how _this particular_ student could have figured it out, knowing what they know and being who they are. I have a blog article about this in progress.

justeleblanc 3y ago
I'm a professor in a big university in western Europe. Students don't want to be led to thinking of the solution. They want the same exercises that they did during the tutorial, and they want to know in advance how to solve all the exercises. Any attempt to digress from this is met with vitreous eyes.

gcanyon 3y ago

I'm in pretty much the same boat re: calculus, but I think a lot of it has to do with problems just like this. For me, early in my experience with calculus I always looked for the "graph it out"/non-calculus solution. So problems like water leaking out of a bucket, rocket acceleration, and other integrals where the underlying process is in some way linear always fell to non-calculus-based analysis. And thus when I got to problems where actual calculus was required, my non-grounding in the basics pushed me toward rote memorization which (of course) didn't stick over years of non-utilization.