Are you an engineer or an engineern't
April 21, 2017 1:08 PM   Subscribe

In other words, why am I good at abstract & theoretical math and so horrible at applied/computational math?

I am a PhD student in computer science and this has plagued me all my days. I'm in a challenging program and I've always done well in my classwork and contributed to novel research (though I'm not helming my own yet). I learn quickly (as a TA, I frequently need to learn things and teach them simultaneously, while on the spot, and I would say I'm pretty good at it). I love complicated, challenging shit, and have always been drawn to systems design and programming. However, for reasons beyond my comprehension, I've always been somewhat garbage at "engineering" math, or computational math.

My hypotheses are:

1) I did not go to a good high school and did not care much about math in high school, so I never really solidified the fundamentals, and I always end up going back to the well to remember things that other people seem to know like they memorized them when they were 16 and never looked back. It's like if I didn't know my multiplication tables or something-- it doesn't mean I wouldn't be good at math deep down, but in everyday life I'd have to keep looking up multiplication tables, which would make me slow and make me feel stupid.

2) I don't care about applied math as much as theoretical math-- I'm drawn to elegance more than utility. Which to some degree, would explain why I perked up when learning about entropy and information theory in machine learning, but steadfastly did not give a shit about most of the gory details. I tend to also enjoy learning algorithms (because often they are fun, like riddles), whereas I don't necessarily find grinding out computations particularly fun (I might if I were better at it). So I learn more about the things which appeal to me, and don't really care about the rest. Since I do not respond well to extrinsic motivation, it gets left behind.

3) There is something weird about my brain-- dyslexia (dyscalculia?) or something. I think if this were the case, I would have heard of it by now. I'm not sure I've ever met another person who said that theory and algorithms was easy and fun to them, but working with real numbers was torture. I used to know a professor who loved pure math, and said he got straight A's in Calculus because by the time he took the course, the real numbers were trivial to him (lol). I also got A's in Calculus, because I did the work, but I wouldn't say that the real numbers are something I really have an excellent grasp on. At the level of data analysis/machine learning I'm at now, there is a lot of interpretation and judgment calls going on, and I don't have the baseline confidence to really feel facility with this stuff (though it's not wholly mysterious to me either).

Is this a thing? Are other people like this? Is it maybe a combo of gaps in my education and the fact that I genuinely like/do well in math, making kind of a weird mosaic of my own skills? I'm willing to put in the time to shore up my abilities but sometimes I think it might be a waste of time because my brain will just never work the right way. And, of course, as a woman, I feel my own internalized sexism telling me I'm just not good with numbers... but, like, is that even a thing?? (I don't really believe it is? I don't know.)

I remember in high school having one male friend who ostensibly had the same shitty math education as me but seemed to do better in math competitions we attended (as I mentioned I wasn't really that into math so I mostly fucked around at the time). But he would also make lots of mistakes and blah blah, it wasn't like he was a genius compared to me. So maybe we had the same shifty education, but he had more interest in math as an actual discipline, whereas for me it was just another homework assignment. I never really felt like I appreciated math as it's own thing until taking intro math/CS courses when I thought I wanted to be a chemist in freshman year of college. So maybe I just didn't have the curiosity at the time when I was learning the fundamentals, whereas once I got to college, I did and therefore absorbed a lot more. And conveniently, learning how to write proofs and understand deep concepts meant starting from first principles, so my spotty background didn't interfere until the point where we started tying in things it was assumed everybody knew.

Anyway, please help me make sense of this, and take next steps so I don't have secret PhD math shame? Anecdotal tales of people with similar issues (yourself or others, success stories or just stories) would also be so useful, because I feel weirdly alone in this!
posted by stoneandstar to Science & Nature (18 answers total) 6 users marked this as a favorite
Maybe you can give examples of things you can't do that you think you should be able to? I get what you're getting at but there's not much data to make any sort of interpretation.
posted by GuyZero at 1:11 PM on April 21

And, of course, as a woman, I feel my own internalized sexism telling me I'm just not good with numbers... but, like, is that even a thing?? (I don't really believe it is? I don't know.)

Not sure which "thing" you're asking about, but is internalized sexism a thing? Yep. Stereotype threat is apparently the academic term people use to describe the issue.
posted by GuyZero at 1:15 PM on April 21 [6 favorites]

It's hard for me to describe exactly, but definitely things like... concepts in precalculus and what they "mean" (when someone says "oh we'll take the logarithm here," until recently I was always thinking, ok, why) as well as not having great facility with precalculus-level computations (which is somewhat familiarity/memorization but I think that goes hand in hand). Recently in a class a professor described a function and said "what makes the use of this function here problematic," and a student said, "oh, it's not differentiable," which was the case, and if you had asked me "is this function differentiable" I would have said No. But I'm not sure I would have made that deduction in context, like, ever, unless lead there. Reasoning about real numbers with geometry or trigonometry is essentially never illuminating to me. Understanding the significance of various loss functions goes over my head. Etc.

If something is being presented as totally new (as if often the case in algorithms courses), I pick up on it. It's this grey area where I think I'm supposed to know things, and when I try to compensate by learning on my own, I don't know, either it's hard for me or I don't start in a good place, or something.

A lot of this I just need to look up and study, and I have a hunch a lot of other students in my program come from a more conventional background and have just heard of these things before. But I am a little worried about these things not sticking with me.
posted by stoneandstar at 1:22 PM on April 21

Anecdotally, I asked a very smart PhD in mathematics friend of mine who had this really firm grasp on calculus (though that wasn't even close to his focus), what I was missing. Why did these concepts in calculus make the smallest amount of sense to me (enough to get As) but also seem so just out of my grasp that I was some sort of fraud for getting As when I didn't feel like I totally got it. What am I missing? When is it going to click?

His response was "dude I didn't get it until I had to teach it [in grad school]"
posted by czytm at 1:27 PM on April 21 [2 favorites]

Side note: said friend is also adamant that the way we teach these core mathematics classes (trig/precalculus/calc) is wrong and has VERY STRONG OPINIONS about how they should be taught for deeper understanding (the way they work vs how to "do" it). Probably incorrectly paraphrasing his feelings: those core math classes are usually taught with a "this is how you do it" attitude rather than a "this is how it works" attitude.
posted by czytm at 1:38 PM on April 21 [4 favorites]

1) no
2) maybe
3) no

You're spending a huge amount of time and effort on a particular class of problems, a different kind of problem is different and not necessarily easier or harder but different and needs a corresponding amount of effort. Probably not the kind of effort that will help with your dissertation but may be needed if you don't go direct tenure track, that is needing to make a companies software actually work for a paycheck. You'll manage if you need to.
posted by sammyo at 1:40 PM on April 21

I'm not sure I've ever met another person who said that theory and algorithms was easy and fun to them, but working with real numbers was torture.

Yeah, this is actually kind of a half-joking but also half-serious stereotype of some professional mathematicians, particularly those with a focus on pure rather than applied topics: e.g.: very good at extremely complex abstract reasoning but also take way too long to compute the tip on a bar tab. As it turns out, there's all different kinds of mathematical skills and there's no reason to expect that aptitude at one of them would guarantee aptitude at all of them. Some things will likely feel easier and some other things harder.
posted by mhum at 2:15 PM on April 21 [12 favorites]

I think maybe grade school math education tends to lead us to expect that math is a linear progression through levels where everyone that has reach level N has necessarily mastered the same stuff.

In fact it's a huge field that people take different paths for. It's completely normal that at the PhD level you're going to find you don't know stuff that other people consider elementary.

So there's often a transition around grad school where instead of learning things in a linear order, you instead find yourself in a situation where you're trying to do something and suddenly realize you're missing some prerequisite knowledge and have to figure out what exactly the gap is and where to fill it in: that could mean just stopping right there and putting your hand up, or asking someone later, or signing up for a class, or tracking down the right book or paper, or sometimes even figuring out something from first principles, or (usually) some combination of all of those.

"If something is being presented as totally new (as if often the case in algorithms courses), I pick up on it. It's this grey area where I think I'm supposed to know things, and when I try to compensate by learning on my own, I don't know, either it's hard for me or I don't start in a good place, or something."

So, yes, you may have to fill in some of those gaps on your own, and it's completely normal to find that harder than learning something in a course where everything's laid out nicely for you in order.

Some random ideas:

- Don't worry quite as much about what you're "supposed to know". You either know it or you don't, and if you identify something you don't know, try to take it as an opportunity not a failing and don't be embarrassed to talk about it.
- remember that you're a successful student who's already learned a lot and is capable of learning much more, and don't let your sense of shame at a some gap prevent you from asking a stupid question. You'll be afraid of looking bad, and maybe you will sometimes, though sometimes actually it'll be the reverse ("wow, stoneandstar picked that up fast"). As long as you're not just sucking up disproportionate amounts of class time or something, don't be afraid to ask questions.
- Don't forget that if you miss the chance to ask a question in the moment, you can still take notes and go back and ask someone later.
- "Is there a course I need to take"/"book I need to read" is also a good question.
- But maybe be careful about over-generalizing prematurely about your problems. I doubt you're as bad at "applied/computational math" in general as you think, and to the extent you are it's more likely to be a matter of some missing background that you can easily pick up rather than some fundamental personality flaw. Try to focus on the *specific* problems you're having.
posted by floppyroofing at 2:36 PM on April 21 [4 favorites]

...theory and algorithms was easy and fun to them, but working with real numbers was torture

This is totally normal. More than a few of my math-PhD friends count on their fingers and despair at things like estimating tips or mentally multiplying small numbers. It doesn't matter; do what you like. If you really need to calculate things, you'll be able to figure it out.

To some degree, rote memorization can help. Like, seriously, memorizing the multiplication table and integral table and quadratic formula. There are many reasons that this could be dumb and a waste of time, but I'd think about it more like a party trick. Anyway, there's a point at which "why does it work" becomes more of a philosophical question, and just knowing "these two symbols are connected with this other symbol in my head" is enough. Little by little I'll eventually absorb the "real" message.

I often feel as though I have two minds about math: one that verbally understands all of these complex patterns, and another (almost disjoint) mind that can actually calculate things. There's a big gap between them: any time I actually have to come up with a number, there's a significant delay before the calculation-brain wakes up and starts spinning.

I like books like The Art of Problem Solving and things like Youtube's Numberphile -- even/especially now as a STEM PhD. There's a lot of fun stuff you can just mess around with in your head, like why do multiples of three always have digits that add up to a multiple of three? Little mundane problems like that helped me get over the "calculation vs. deep abstract math" dichotomy. Now I tend to think about numbers like a sort of sandbox with some cool patterns buried inside: a toy I can carry around in my head. And if the formality ever gets annoying, usually you can just draw out the patterns instead of the numbers themselves...
posted by miniraptor at 2:41 PM on April 21 [3 favorites]

This is a thing, actually I know a number people with math PhDs that dislike/say they are bad at statistics, computational/applied math etc.

When I was in a CS program, I found both the engineering/architecture and applied math type classes much much harder and more overwhelming (and also just harder to get started in, so I would get behind right away) than the theory classes, which I found not easy exactly but fun and entirely manageable. Actually this is partly why I quit and didn't do well in grad school. Because I couldn't motivate myself to buckle down and get through some of the stuff I felt I was bad at/didn't like.

In addition to your point about the theory-oriented material being easier because it is presented as new, I think there also just tends to be less material. The difficulty is in thinking very abstractly and internalizing some key definitions and concepts, and proving things of course, whereas in the more applied/engineering classes, you are responsible for a ton of content and methods in my experience. So it's just a very different kind of cognitive burden I think.

So my general point is that you are not alone, and that the difficulty you're having is because it's genuinely hard in my opinion (as someone who has always been 'good at math'). My specific suggestion, if you feel like you are lacking in the background/intuition for things like differentiability of functions, and you have time to, you might want to work through some materials for a real analysis course. As a math major in college I was never in love with calculus either but found real analysis more fun/interesting because of the proofs aspect and because you do start from the beginning (formal definition of a limit, etc.) And it will help you internalize all the calculus/real numbers stuff that you don't feel confident about right now.
posted by day late at 2:52 PM on April 21 [1 favorite]

This is a thing for me too - I have a degree in mathematics and near the end of another in computing and statistics, so pure maths, logic, statistical analysis are all fine for me, but my particular struggle is with mechanics - I struggled enough in my late teens to put me off pursuing a career as an engineer, though in hindsight I was much better at it than I realised.
I think 1) definitely - I missed a lot of school in my early to mid teens, and now in my 40s still discover things I missed (or perhaps was never taught at my okay but not especially academic secondary school).
2) is possibly true, but I think might be partly a self-fulfilling prophecy i.e. you find something harder so you assume that rather than it being harder, it must be that you are not good at it, so you put less effort into it, which in turn makes it harder...
3) I think at least part of this might be contextual understanding of the mathematics and how often you get to use particular skills?

I would also add, as a woman who is good at maths, definitely don't discount the internalised sexism thing - I was much better at maths than my peers at school, but was completely put off for some years by a number of things:
1) it became harder as I studied, and I took that to mean not that the work was more challenging, but that I must not be as good at it as I had thought
2) as a girl who was good at maths, I found myself frequently challenged by many of my male peers to prove I was as good or better than them as they couldn't bear the thought a girl was beating them at maths. This seemed to be true even of boys who weren't especially great at maths. This usually took the form of trying to get me to perform mental arithmetic quickly (not one of my skills!) and really undermined my confidence.

There are also a number of studies about how men and women differ in how they perceive their own ability in mathematics relative to the other gender, so definitely don't dismiss the idea that you are downrating your own ability due to being a woman.
posted by kumonoi at 3:03 PM on April 21 [2 favorites]

To me, a lot of this sounds like you haven't *practiced* enough. Like, you're just reading the books and taking the tests, but you're not getting into the nitty gritty details and working through problems and just playing around with what you're learning.
Recently in a class a professor described a function and said "what makes the use of this function here problematic," and a student said, "oh, it's not differentiable," which was the case, and if you had asked me "is this function differentiable" I would have said No. But I'm not sure I would have made that deduction in context, like, ever, unless lead there.
This specifically sounds more like pattern recognition than anything else. It's the kind of intuition you have because you've run into that problem before. You're not going to get that by learning theory on its own. You get that through practice.

I'm a professional programmer, and I love reading about theory and datastructures and algorithms, but until I've actually *used* them for something, there's no way I could contribute intelligently to any conversation about them at work, and I have experiences like that in meetings all the time, where someone says something that is clearly true in retrospect, but that I never would have considered in a million years, and the reason why is that they're speaking from experience.
posted by empath at 3:11 PM on April 21 [2 favorites]

I was going to say something, but empath basically said exactly what I was going to. I pretty much don't deeply understand things until I've had a chance to apply them usefully and that feels like what's going on here to me.
posted by Aleyn at 3:36 PM on April 21

Skipping to the bottom to say this was me. I didn't do math all through high school because, while I was good at it, I was told early that I wasn't REALLY good at it.

Turns out that while I will regularly be like oh 2 x 3? That's 5, I was pretty fucking good at proving things I'd never heard of before. Which I didn't find out until I took my first theoretical math class second semester sophomore year at a top 5 school for math.

And then fluency was the biggest thing. My deficit in early math education was REALLY obvious for anything that involved taking this strategy from this thing and applying it in this new way to a new thing, because I would have to reinvent that strategy, having never seen it before. But any time there was something novel that no one in my Putnam winning international Olympiad type classes had seen before? I would hold my own or kick ass.

(Sadly there was still too much ground for me to make up. Also I got sick. But I used to love this shit, and I want to go back to it.)

Anyway, this is just by way of saying that yes, my brain was like this too. The only way I can describe it is that I just don't care for unimportant details? Which sounds arrogant as hell, but I do it in life, too. The big ideas are what's interesting, and I just can't make myself care about things I can either look up or get a computer to do.

Mmmm. Math.
posted by schadenfrau at 3:51 PM on April 21 [2 favorites]

So, watching a group of mathematicians try to settle up a bill and compute the an experience. It doesn't go well.

I am a mathematician. Employed as one, even. I am not good at computation. I regularly use my fingers---in front of class---to help me get basic arithmetic correct. Arithmetic !=mathematics.
posted by leahwrenn at 4:28 PM on April 21 [4 favorites]

Have you heard of the spherical cow?

I once worked for a company whose niche was translating the results of PhDs into instructions for Navy sonar operators. The PhDs often didn't recognize how far from the real world they wandered, or how far the theoretical result was from a useful instruction. They approached the applied problem as an abstract problem. I once saw a paper that began "Consider the ocean as a Banach Space..." So it sounds to me like you have an ordinary case of what I thought of as the "PhD disease."

Only a mathematician can be elated to prove that something exists without finding the something exactly, or "calculate" a value as the sum of other abstract, unknowable objects and think he'd done a good day's work.

Here is a personal anecdote on the difference between abstract knowledge and a visceral understanding of how it applies to the real world. It's so embarrassing simple that I don't usually tell it. Shortly after my graduation of college with a degree in Math, I was on my uncle's small sail boat in fog and rain. Visibility was 1/8 mile or less. We had a chart showing course and distance from one navigation buoy to another. Going from one buoy to another, my uncle would make a computation of the sort 1 nautical mile divided by 5 knots = 12 minutes. And in 12 minutes, we would arrive at the next mark. I was familiar with velocity as first derivative, and acceleration as a second derivative, etc., but the experience of seeing the buoys appear out fog on time gave me a visceral understanding of what it meant.

So, I think that like any good PhD student, you are solving "applied" problems in a theoretical, abstract way. Maybe you should do more examples by hand.

BTW, memory is essential for greatness in mathematics. I could never have been a great mathematical, but when I reached my limit, it was largely because my memory, which is at least average, was not good enough, and I had to go back to first principles too often.
posted by SemiSalt at 6:00 AM on April 22 [1 favorite]

Applied computations are ugly and annoying unless you actually care about the application. So I would recommend avoiding generalizations, and just assuming that the applications you've encountered in coursework so far are toy models you don't personally connect with.
posted by yarntheory at 7:53 PM on April 22

I'd like to second the suggestion to look at some real analysis if you want a better conceptual understanding of calculus. As a bonus, you get to think about the Cantor set and other intuition-breaking objects.
posted by yarntheory at 7:55 PM on April 22

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