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Background:

I studied Physics and Mathematics (double major) in my Bachelor, and now I'm working in the field of (theoretical) biophysics (I'm a Master's student). I have had a fair amount of research experience so far which gave me a lot of ideas about how research is being done. Most of my contributions to such a project involved wiring a bunch of code, improving an experiment, improving the data collection process etc. but nothing mathematical.

Problem:

I have no problem reading papers, especially theoretical ones; I would describe myself as familiar with a lot of fundamental ideas employed in research. If I want to dig deep into theoretical a paper, I don't have a problem. But the issue is that I feel like my mathematical competence is eroding, both high-level and low-level. I do a lot of errors in algebraic calculations, forget how to integrate certain functions, etc. and I am internally scared from finding out I am bad at mathematics, given that I used to love mathematics.

But unfortunately, right now there is nothing that challenges and pushes my limits. Because, to my experience, physics students only use certain mathematical methods, so once you learn how to do those (all of which can be done via Mathematica), there is nothing that can push your limits, unless you are doing very mathematical research.

So, now whenever I need to calculate something, I go and put it into Mathematica because I have done the same type of calculations over and over again many times, and I don't want to do it again.

I'm afraid that if I start working on a theoretical project which requires mathematical competence, I'll fail because I haven't been practising the art of doing mathematics in a true sense for a while.

Question: Given all these, how to keep up my game? How to push my mathematical skills even when I am not working on a theoretical project?

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    +1 I really sympathise with this feeling. My PhD was very computational rather than theoretical (physics) and nowadays I too make silly/trivial algebra and calculus mistakes. I'm hoping there's a better solution than just revising undergrad algebra/calculus and doing practice problems...
    – astronat
    May 22 at 14:32
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    'physics students only use certain mathematical methods, so once you learn how to do those (all of which can be done via Mathematica)' Caution: I once wasted several months examining the details of what I thought was a novel solution I'd got to a theoretical physics problem, but was actually an artefact of two independent computer algebra systems giving the same wrong answer for the lead coefficient in a Fourier series ansatz for a partial differential equation. Don't get me wrong, I remain a big fan of computer algebra, but one always has to have some human verification. May 22 at 15:49
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    @DanielHatton Thanks for sharing your experience. That is true; human verification is important, but even in that case the mathematics used in most of (bio)physics research is really not advanced. It is just a re-usage same method in different ways over and over again.
    – Our
    May 22 at 15:52
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    this is just to say that I know how it feels and my sympathy is totally with the TS. Whether you gain your skill back or not, I hope you find a way to make your peace with it. Realizing that you were good at something but aren't anymore can be a terrible feeling.
    – sleepy
    May 22 at 17:44

13 Answers 13

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First there’s no magical “being good at mathematics,” so it’s not something you can suddenly lose. If you want to get better you can practice more and get better, and if you practice less you’ll get rustier, but neither state is permanent. Your anxiety around your math skills may be due to this kind of black-and-white thinking about math skill.

In terms of practical advice, one thing that really helps keep you sharp on the basics is teaching. Maybe try signing up to do a little tutoring, or do something with the undergrad math club.

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    Good suggestion re tutoring/teaching!
    – J W
    May 22 at 15:01
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    How to improve: Hop over to MSE, pick an interesting problem or two every day. May 22 at 16:07
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    @Aruralreader: I suggest posting that as an answer.
    – J W
    May 22 at 16:27
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    Re teaching/tutoring - you might want to check out upchieve.org. They provide free math tutoring to low income and minority students through their online portal. You may want to consider volunteering as a tutor with them as a way of brushing-up on your math skills, and helping out some low income and minority students at the same time.
    – mti2935
    May 23 at 10:45
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    I had a professor in college who said the only time people know Calc 2 is when they're teaching it. It just fades out of their minds and they forget how to do some of the more complex integrations despite having taught it 5-10 times. There's nothing wrong with forgetting information, but teaching definitely is a great way of retraining your brain to think in certain ways. May 24 at 19:23
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It's quite normal for knowledge and skills to fade if you don't regularly use them. Since you don't want to do routine but possibly tedious/lengthy calculations by hand and prefer to keep using Mathematica, how about learning some new mathematics? Find a book on something that strikes you as potentially interesting - it doesn't necessarily have to be directly related to your research - and work your way through it. Just be sure to check the prerequisites for the book first so you don't jump into something too difficult for where you currently are and end up frustrated.

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    I agree with the tip to try learning something new. Furthermore, I would not really recommend trying to ``relearn'' things known and forgotten (unless needed as prerequisite for something else), since I suspect the OP will find that rather boring and not stimulating.
    – Blazej
    May 24 at 7:50
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I like to see knowledge as three-fold - there is what you know, what you don't know and what you know that you don't know. As you time goes by, many things of the first category fade out and find themselves in the third category.

I remember when I was preparing my finals and I knew all common integration techniques and some tricky trigonometric substitutions. Now I forgot most of it, but for my research, if I stumble upon such problem, I can easily find those in my old books or online. So practically, this shift of category for my knowledge is transparent for my work.

And what it was traded for, is a kind of mathematical soft-skill but the kind that make someone "great at mathematics" (not that I am particularity great; rather better, this is obviously a spectrum), which is a deeper theoretical insight, a wider mathematical culture (mathematical culture is eminently in the third category!) and some kind of a problem-solving creativity.

So in the end, it's your mathematical ability at large that you may want to enhance or maintain and now the task is way more enjoyable - first such abilities do not fade away so fast and second, to enhance it, like J W suggested, you can just look for a new topic that you have leisure or work interest in and "work your way through it".

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    This is better than the top voted answer, which implies "memorizing integrals" or "miswriting numbers" is some how something that is both important and a prerequisite at being good at math. School makes you memorize those things and take off points for silly mistakes for practical reasons, not because it's actually required to do math. Don't expect yourself to remember those things or be perfect at copying values, humans are not robots, we are not meant to memorize every thing and its impossible for us to avoid mistakes. We don't even treat quantities linearly by default.
    – Krupip
    May 24 at 16:55
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Frame shift: being good at math doesn't mean getting the right answer, but rather knowing how to solve the problem (or knowing where to find how to solve the problem).

Computers are almost infallible; humans are not. So you, and everyone else in the world, are going to make computational mistakes. It is as inevitable as death and taxes. It is not something to fret over. In fact one could argue that if a calculation is complicated enough, it's irresponsible to not check it with a computer.

Your skill as a mathematician is not tied to how many computational mistakes you make. It's much more important to know what to do and how to do it, because once you know these things, you can always program a computer to do the calculation flawlessly for you.

As an aside, ask one of the many mathematicians on this SE how much of their work is done on pen-and-paper.

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    There's an interesting comment on the question by Daniel Hatton about a professional time when a computer algebra system gave him the wrong answer (in fact, two such systems). May 23 at 13:02
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    @DanielR.Collins right, but it implies that recognizing when the computer output is incorrect is also a mathematical skill.
    – Allure
    May 23 at 23:38
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I studied physics. About the second year I started to realize how things were interconnected and it was a revelation for me. Fantastic times. Then I started to teach and it is only then when I fully grasped what I was teaching because I had to understand (and not only learn).

Teaching will help you a lot to keep up.

Then I moved to industry and all my physics and mathematical skills started to erode, to the point that when I once looked at my notes in physics I could barely understand what was there.

I was very disappointed and sad because I loved physics (and to some extent - math) and realized that it will go downhill.

Fast forward many years. I now have children who are starting to learn some more advanced maths (differentials for instance).

This is at this point that I realized how deeply math and physics is engrained in my mind. They had a hard time understanding differentials. The main reason was because they simplycould not see the reason for these strange operations. I explained them what a differential is for, and what it is (in that order, going though some examples from physics). It helped me to overcome the feeling, and I see that it will still be a quite some time before I stumble upon something in their curriculum that will make me go into "well, dad cannot help you on that one" mode.

So try to teach somewhere (I took some volunteering help with more advanced homework for children who are not helped at home) - the higher the level the better.

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This is a tough one, and something I myself wrestle with. It is also one of the current issues I have with academia/advanced degrees as a whole (overly specialized leading to atrophy of your general skill set).

All I can offer is that advanced degrees seem to build your "learning ability" in that they require you to jump into a complicated, foreign area and learn it quickly.

Thus, while you might lose your general skill set, you are gaining the ability to learn complex topics quickly. The answer to your conundrum then becomes "if I forget/lose skill x, I will be able to quickly re-learn it in the future if I ever need to use it again".

As other posters have mentioned, however, it is also not a bad idea to try to maintain your general skill set if you have time. Many of us don't, but if you do have spare time, tutoring or working though a textbook or online course can help keep you fresh.

If you don't use it, you will lose it. If a skill is not being used in your current job, you need to find an extracurricular avenue to continue to exercise that skill set if you want to keep it strong. Otherwise, the only solution I have found is to accept that it is going to fade but to have hope that I will be able to re-learn it quickly if I ever need to use it again.

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Similar personal experience:

No, your abilities are not fading. At least, not as much as you assume at the first glance.

What really happens:

You face challenges of the same domain, but of increasing complexity. The increasing complexity comes from two sources:

  1. The natural development of the scientific knowledge in the field (remember, you are not in the high school anymore and the development happens as you learn)
  2. Your own increasing experience, encouraging you to skip more and more "trivia" as you are thinking.

Both processes promote simple and innocent mistakes and ommissions. They can slow you down while you double-check and cross-check, but they can't fail you.

And there is nothing bad in refreshing some memories and abilities if you have to.

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There are some excellent answers touching on the role of time on task, teaching or tutoring, and solving problems for fun. To this, I would like to add the aspect of mathematical creativity, which some people feel does drop off sharply with age.

In this article and blog, Lila Guterman evaluates the hypotheses that in mathematics in particular, young people are more capable of breakthrough results. There are a lot of citations of practioners that think this is true. A counterargument is that mathematics has expanded rapidly, so there are more young than old mathematicians. They also mention the Fields Medal, which is for mathematicians 40 or younger. Furthermore, there is an argument that older people might have the same capability, but are just not able to focus as much because they already have a larger set of responsibilities.

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@NoahSnyder makes an excellent point about teaching. However, we've recently had a couple of questions on here from people who felt their skills and/or productivity were lower than they would like, and who disclosed that they were working unhealthily long hours. So, just in case, I'll advise: make sure you're taking the time to get an appropriate amount amount of sleep, a healthy diet, and sufficient physical exercise. I think I can safely leave it to you to carry out a quick (and occasionally repeated, since it's still an active research area) literature search to find out how much sleep is appropriate, what constitutes a healthy diet, and how much exercise is sufficient, from the point of view of cognitive function.

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how to keep up my game? How to push my mathematical skills

Karsten mentioned "solving problems for fun"

Some publications have "problem" columns, with problems to solve, and solutions published later based on what was sent in.

See Are there any more mathematical journals or websites with the “problems and solutions”?

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Skills require an application. It's normal to let skills lapse if you don't need them, and to only refresh your memory when you do need them.

Some people solve maths problems for fun. You're probably not one of those people, otherwise you wouldn't be here.

Assuming maths ability is merely a tool for you to do your job and not an end-goal in itself, then you will inherently get good at the bits of maths you use, and let things go which you don't use. That's just how it goes. You will still be good enough to learn or refresh whatever you need in future, though. Either you already know it and just need to remind yourself how to do it, or you have enough basic knowledge to work through the learning process.

Your supervisor should be fine with this. A scientist/engineer won't know all the answers when they're doing new work, because if they already had the answers then it wouldn't be new! The important part is that you know in general terms where you're going with it. You should be able to be up-front with your supervisor about this, and just put the time in to get your skills up the curve. Chances are that your supervisor doesn't know it either, so they should welcome a post-grad expanding the scope of what his team are capable of.

No-one remembers everything forever. You're only at the start of your career here, so you probably didn't learn this more than a few years ago. As you go forwards, you can absolutely expect to see something in 20 years time where you think "hang on, I did something about this as an undergrad" and have to go back and refresh your memory. You aren't expected to have this at your fingertips forever, because humans don't work like that. :)

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If you have the time and inclination, consider contributing to (or at least perusing) Mathematics SE: this is a good (and humbling) way to keep you on your mathematical toes.

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I used to be good at algebraic manipulation and at integration tricks, too. Now, I'm not, but it doesn't matter, because Mathematica can do those things far better than you or I ever could.

The world has changed, so the skills needed to survive and prosper have changed, too.

Many of us used to know how to...

  1. Do long division with a pencil and paper
  2. Add weights expressed in ounces, pounds, and stones
  3. Use a slide rule, tables of logarithms, and a desk calculator

These skills are all obsolete, and I'm not too worried about the fact that they're now pretty rusty. I've learned new things, instead ... how to write code, how to use a calculator, and how to use Mathematica.

Of course, not everything we learned in our mathematical youth is obsolete. I personally still get great value from my knowledge of classical geometry, my intuition about approximation, my ability to draw pictures, and my ability to express ideas clearly.

So, when considering how to "keep up your game", the first decision is which skills are worth preserving, and which should be replaced by new ones. The game has changed.

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