Feeling guilty while learning math

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Feeling guilty while learning math
So, I am studying for a Masters degree that involves quite a lot of mathematics. It happens quite frequently that I am learning something and I feel the need to stop and just spend some time doing research to REALLY understand the topic and not just understand the mechanics of the topic. However, one's time is limited and many times I have to just move on or else I won't cover the material necessary for the upcoming exam.
For example, I rarely have the time to understand proofs. We are thrown so much material that I don't think it is realistic for a student to actually understand proofs.
Anyway, I would like to hear your thoughts on this. Is it normal when learning math to learn a subject "gradually" (e.g. intuition, then mechanics, then proper theory)?
Thanks =)
For example, I rarely have the time to understand proofs. We are thrown so much material that I don't think it is realistic for a student to actually understand proofs.
Anyway, I would like to hear your thoughts on this. Is it normal when learning math to learn a subject "gradually" (e.g. intuition, then mechanics, then proper theory)?
Thanks =)
Re: Feeling guilty while learning math
From what I hear, maths is a young person's game. Ageing professors can't understand the work they did when they were young.

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Re: Feeling guilty while learning math
I think I would disagree, I didn’t graduate until I was 49
Re: Feeling guilty while learning math
I’ve been spending a small amount of time going through the signals and systems lectures on YouTube from MIT. I missed some subtle details as a student as I rushed through. I’ll never use this stuff but some of the implications of the mathematical results intrigue me. It makes me wonder why it all seems to hang together.
Doing the same thing off and on with the applied math lectures.
No practical use. Just looking for meaning I guess.
I’m 50 and I’m having a much easier time understanding this stuff than when I rushed through in my 20s while trying to only learn enough to get a good score on my exams.
Doing the same thing off and on with the applied math lectures.
No practical use. Just looking for meaning I guess.
I’m 50 and I’m having a much easier time understanding this stuff than when I rushed through in my 20s while trying to only learn enough to get a good score on my exams.
Re: Feeling guilty while learning math
Fundamentally, mathematics is a matter of comprehension, and cohesion. Maths makes sense from a bottoms up approach. That's why rote memorization does fuck all for your long term understanding, and appreciation, of math.
I don't understand anything, ANYTHING, unless it ties in with other pieces of information. So my time at uni (nanoscience) was fraught with 'slow learning' because we had exams every 10 weeks and that wasn't enough time for me to build knowledge from the ground up. Once a few concepts start to settle I am, however, able to extrapolate very far from those few disjoint pieces of information.
Twice during my first solid state physics course I asked a whathappensif question during lectures that had awarded a Nobel prize or a major breakthrough in the past. One of those was incidentally the behavior of the Xray free electron laser which we used for experiments during my PhD.
Now, slow learning when it comes to math is, in my completely biased opinion, how it's supposed to be. It feels horrible because the educational system appreciates halfassed goodenough work in one third the time, more than it appreciates education. Over time it will feel better, and getting your fundamentals right will serve you better in the long run than the fast track.
I don't understand anything, ANYTHING, unless it ties in with other pieces of information. So my time at uni (nanoscience) was fraught with 'slow learning' because we had exams every 10 weeks and that wasn't enough time for me to build knowledge from the ground up. Once a few concepts start to settle I am, however, able to extrapolate very far from those few disjoint pieces of information.
Twice during my first solid state physics course I asked a whathappensif question during lectures that had awarded a Nobel prize or a major breakthrough in the past. One of those was incidentally the behavior of the Xray free electron laser which we used for experiments during my PhD.
Now, slow learning when it comes to math is, in my completely biased opinion, how it's supposed to be. It feels horrible because the educational system appreciates halfassed goodenough work in one third the time, more than it appreciates education. Over time it will feel better, and getting your fundamentals right will serve you better in the long run than the fast track.
Re: Feeling guilty while learning math
Proofs can be used to play with. There are always assumption in a proof, and one thing some people do is they ignore one of the assumptions and then try to figure out all the places the proof now leaks.LookingInward wrote: ↑Sun Mar 03, 2019 2:52 pm...
For example, I rarely have the time to understand proofs. We are thrown so much material that I don't think it is realistic for a student to actually understand proofs.
...
You can try to think up the strangest, brainfuckiest edge cases to see how it all sticks together and you'll come to appreciate why mathematicians are so fond of proofs. They are an intellectual play ground, and all that play is where you learn a lot of math!
Re: Feeling guilty while learning math
If I were you I would just feel happy to have the ability to memorize applications or algorithms so readily. It's kind of like being good at Burpees competition. Doing proofs is more like being able to figure out the ramifications of social behaviors. My favorite math class ever was based on the book Modern Algebra with Applications. I understood the proofs almost immediately, but I didn't have time to work through the applications problems, because I was breastfeeding a baby who was hospitalized for several weeks with a severe kidney infection that term. The professor gave me an A anyways, because the class was overloaded with Chinese students who diligently did all the work, but clearly didn't comprehend the proofs. OTOH, with Calculus, my comprehension of the proofs came far later than my ability to simply use the tools.
Also, I have found that it is actually quite possible to have interesting conversations about concepts in math with a bright third grader. Because, for example:
https://math.stackexchange.com/question ... ubtraction
Also, I am firmly in the camp that holds that what we can know of math is unique to our species, and ultimately arbitrary. So, it really isn't any better or worse, or unpure vs. pure, to use one part of your brain or body vs. another when you do math. We all start by counting on our fingers.
Also, I have found that it is actually quite possible to have interesting conversations about concepts in math with a bright third grader. Because, for example:
https://math.stackexchange.com/question ... ubtraction
Also, I am firmly in the camp that holds that what we can know of math is unique to our species, and ultimately arbitrary. So, it really isn't any better or worse, or unpure vs. pure, to use one part of your brain or body vs. another when you do math. We all start by counting on our fingers.
Re: Feeling guilty while learning math
It sounds overly simplistic but I had a hard time understanding math until I started looking at it as a language as opposed to a set of formulas that fit together like Legos. Legos fit together in very specific ways  teaching math as a set of legos doesn't help students extrapolate solutions unless the problem sets are structured in specific ways (like the pictures that come inside the lego box that help intuit the build).
But teaching and learning math like a language, helps a student learn the "syntax" which allows them to string formulas together or create new formulas that wouldn't be intuitive under the Legotaught paradigm. I hope this makes sense. I found geometry and physics to be more intuitive to learn while pure math was a struggle for me until I encountered the language metaphor in an open letter written by a professor to high school math teachers; sadly I can't find the letter or I would link it.
And I don't think math is only for the young.
But teaching and learning math like a language, helps a student learn the "syntax" which allows them to string formulas together or create new formulas that wouldn't be intuitive under the Legotaught paradigm. I hope this makes sense. I found geometry and physics to be more intuitive to learn while pure math was a struggle for me until I encountered the language metaphor in an open letter written by a professor to high school math teachers; sadly I can't find the letter or I would link it.
And I don't think math is only for the young.
Re: Feeling guilty while learning math
@Campitor:
+1
I am only kind of good at math BECAUSE I am very good at language. I do not even possess the level of spatial analysis necessary to catch a very slow moving softball in an open field. Logic rules. Graphs suck.
+1
I am only kind of good at math BECAUSE I am very good at language. I do not even possess the level of spatial analysis necessary to catch a very slow moving softball in an open field. Logic rules. Graphs suck.
Re: Feeling guilty while learning math
I'm 15 years removed from college life, but at the time I was studying for my degree, I can totally relate to the comments from the OP. Not specifically with mathematics in particular, but on a whole bunch of topics. With a full schedule of classes, and possibly working parttime concurrently, and keeping some semblance of a social life (to stay sane), it was literally impossible to dive deep enough into every single topic for complete understanding. I often found myself thinking similar things along the lines of "oh well, I'll just understand this part 'good enough' to pass the next midterm." I think the majority of students did similar things. Of course, I undertook a very rigorous program at a well respected engineering university...
Re: Feeling guilty while learning math
The best advice I ever got from my dad was "remember the rules for the test  you will understand the concept later. Trust that it will happen."
So true. It wasn't needed for everything, but when it was, it really was.
It was often years later that I really understood the concepts of a few subjects . I got A's because I could memorize the rules much more quickly  aka in time for the test. There is a reason Einstein took so long on his thoughts... he had to really understand what was going on.
If you are doing a Ph.D., you will understand that narrow little subject area thoroughly and completely. The same is not necessarily true of all the topics on the qualifying exams.
So true. It wasn't needed for everything, but when it was, it really was.
It was often years later that I really understood the concepts of a few subjects . I got A's because I could memorize the rules much more quickly  aka in time for the test. There is a reason Einstein took so long on his thoughts... he had to really understand what was going on.
If you are doing a Ph.D., you will understand that narrow little subject area thoroughly and completely. The same is not necessarily true of all the topics on the qualifying exams.
Re: Feeling guilty while learning math
Ha! I guess I'm more the "graphs rule, proofs suck" kinda guy.
Despite the engineering degree that included an automatic math minor, calculus was never my thing. My biggest problem was connecting the theory to reality, as I quickly figured out I'm a visual thinker and many of the pure equations made no spatial sense. It wasn't until it got past useless academic proofs and rote memorization of derivatives and integrals to things like vectors that it started to click for me, and my "aha" moment with differential equations was when I learned I could visualize them as a system of springs, dampers, and weights. Even simple things like what integrals really are didn't connect until a smart engineering professor taught me how to perform numerical integration via spreadsheet and I could see how each step works.
So personally, I think the biggest challenge in learning math is simply figuring out how you personally process numbers. Fit the knowledge to your preferred mental framework and it gets a lot easier.
Re: Feeling guilty while learning math
@Tyler9000:
Yeah, different brains work different. That's why I usually find it handy to keep an engineer (or two!) in my pocket
Yeah, different brains work different. That's why I usually find it handy to keep an engineer (or two!) in my pocket
 jennypenny
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Re: Feeling guilty while learning math
I've always 'heard' complex math as a kind of musical composition in my head. My brain relies on tones more than charts or graphics to understand math. My shrink during my HS and college years thought it was a sign of mental illness. It wasn't until many years later that I learned through blog posts and essays on the internet that others understood math the same way. Even now, when I'm writing something challenging I can 'hear' the different concepts. Part of the reason I became so focused on language was that I was determined to learn how to articulate what I was hearing.
@OP: You're probably a 'questioner' per Gretchen Rubin. Take an hour sometime to google her/the term to learn some tactics for reigning in that instinct so that it doesn't slow you down too much. It's a good instinct IMO, but it needs boundaries.
@OP: You're probably a 'questioner' per Gretchen Rubin. Take an hour sometime to google her/the term to learn some tactics for reigning in that instinct so that it doesn't slow you down too much. It's a good instinct IMO, but it needs boundaries.

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Re: Feeling guilty while learning math
Could you elaborate on the differential equations and the numerical integrations examples? I'm currently taking a class on differential equations so it might be helpful. I am also a spatial thinker bth.Tyler9000 wrote: ↑Mon Mar 04, 2019 9:51 pmIt wasn't until it got past useless academic proofs and rote memorization of derivatives and integrals to things like vectors that it started to click for me, and my "aha" moment with differential equations was when I learned I could visualize them as a system of springs, dampers, and weights. Even simple things like what integrals really are didn't connect until a smart engineering professor taught me how to perform numerical integration via spreadsheet and I could see how each step works.
As my study continues, I think I've come to the conclusion that my fear of not "getting" things was getting in the way and I am just focusing on learning little by little.
I also had an epiphany that someone who is good at math is not the one who knows how to solve a complicated integral, but the one who knows how to think and solve problems. I spent years afraid of not remembering how to do certain things in math. But now I realize that if I have the need to use a tool I'm good at learning it. And I no longer feel embarrassed of using an online integral calculator for example. It makes me move faster and it actually helps me learn more!
Re: Feeling guilty while learning math
The differential equations topic comes down to the idea of system equivalence. Every equation has variables for flow, effort, compliance (potential energy), inductance (kinetic energy), and resistance (energy dissipation). The cool thing is that very different types of systems all have those same variables and just call them different things. So the effort variable may be Force in a mechanical system or Voltage in an electrical system. And resistance could be a damper in a mechanical system or a valve in a fluid system. By learning how to translate problems to the language of a different system, you may find one that makes more intuitive sense to you. For example, I often would visualize circuit design problems in terms of fluid flow, or translate thermal problems to a mechanical system of springs and dampers. You still have to learn the math to solve anything, but once you understand the system being modeled the math starts to make more sense.LookingInward wrote: ↑Fri Mar 15, 2019 12:09 amCould you elaborate on the differential equations and the numerical integrations examples? I'm currently taking a class on differential equations so it might be helpful. I am also a spatial thinker bth.
I don't remember the specific details on numerical integration with Excel offhand, but there seem to be plenty of examples online. The basic idea is that you set up the spreadsheet to solve for the area of increasingly smaller chunks under the curve, using the iteration feature to converge on a precise answer. For me, the process of solving the problem logically instead of via an esoteric equation transform helped me understand what was happening and how I might apply the concept to real problems instead of just passing the next calculus test.
Last edited by Tyler9000 on Fri Mar 15, 2019 12:55 pm, edited 4 times in total.

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Re: Feeling guilty while learning math
Me too. I wonder if anyone has ever classified different types of mathematical thinking.My biggest problem was connecting the theory to reality, as I quickly figured out I'm a visual thinker and many of the pure equations made no spatial sense.
Re: Feeling guilty while learning math
A good mathematician can solve problems, a great mathematician can solve sets of problems, and a profound mathematician can generate sets of problems for others to solve.LookingInward wrote: ↑Fri Mar 15, 2019 12:09 amI also had an epiphany that someone who is good at math is not the one who knows how to solve a complicated integral, but the one who knows how to think and solve problems.
Disclaimer: I am not a mathematician but it sounded smart.

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Re: Feeling guilty while learning math
Interestingly enough, that's similar to how the software engineering progression goes at both of the two companies I've worked so far.
New grad level: Can complete small tasks within features
Mid level: Can complete features
Senior: Can complete projects; it's typical for seniors to give some features of their projects to more junior engineers
Staff: Can solve broad problems; it's typical for staff to generate the broader goal and give projects/features to more junior engineers
There are levels above staff and the main difference is the scope of the problems you're solving increases.
The same general idea of increasing scope and abstraction of problems being solved applies to this progression and, at the staff level, project/problemtosolve generation is required. In contrast, at the lower levels projects/problemtosolve are typically more known up front even if the software engineer has to scope it and come up with a viable solution.
At my current company, a common refrain is that the move from senior to staff is the hardest because you have to learn the "problem generation" skill for the first time.
New grad level: Can complete small tasks within features
Mid level: Can complete features
Senior: Can complete projects; it's typical for seniors to give some features of their projects to more junior engineers
Staff: Can solve broad problems; it's typical for staff to generate the broader goal and give projects/features to more junior engineers
There are levels above staff and the main difference is the scope of the problems you're solving increases.
The same general idea of increasing scope and abstraction of problems being solved applies to this progression and, at the staff level, project/problemtosolve generation is required. In contrast, at the lower levels projects/problemtosolve are typically more known up front even if the software engineer has to scope it and come up with a viable solution.
At my current company, a common refrain is that the move from senior to staff is the hardest because you have to learn the "problem generation" skill for the first time.