Somewhere in my first year of graduate school, I decided I was behind, and I decided the fix was to work until I wasn’t. I don’t remember making this decision so much as arriving at it, the way you arrive at a habit; a seminar where two sentences in a row didn’t make sense, a hallway conversation where someone used a word I’d have had to ask them to define, and underneath both of those a single, quiet, constant calculation running in the background: everyone else here got a head start, and the only tool I have to close that gap is effort. So I applied the tool. I applied it again in my postdoc, when the fear wasn’t about catching up anymore but about whether any of it would amount to something. I applied it again as an assistant professor, when the fear was about who was watching and what they’d conclude. Three different fears, one response, every time: work harder.
I still believe some version of that response was right. I’m no longer sure it was right for the reasons I thought at the time.
I don’t think this was a private confusion. If you’ve spent any part of a mathematical career afraid of a gap you couldn’t quite name, you’ve probably reached for the same tool, more hours, more drafts, more hard work, without stopping to check whether it was built for the lock actually in front of you.
Here’s what I mean. All three of those fears felt, from the inside, like the same problem, a gap between where I was and where I needed to be, closable with enough hours. The first one was real in a way the seminar made obvious in the moment, a sentence I couldn’t parse, a term I’d have had to ask someone to define, the kind of gap that has a visible shape while you’re inside it. But they weren’t the same problem. The first one, not knowing enough yet, actually was a gap that effort closes; that’s what a skill gap is. The second and third weren’t. No amount of work was going to tell me in advance whether a result would land, or change what a senior colleague concluded about me over a hallway conversation. Working harder didn’t touch either of those. What it did, I think, was give me something to do with the feeling, which is not nothing, but it isn’t the same thing as solving the problem the feeling was about.
This has a name, or at least half of it does. Psychologists distinguish between a “fixed mindset,” the belief that ability is a basically stable quantity you either have or don’t; and a “growth mindset,” the belief that ability develops through effort, strategy, and feedback. The claim, made famous by Carol Dweck’s research, is that people who hold the growth belief seek out challenge, treat difficulty as information rather than as a verdict, and keep going where a fixed-mindset person would quietly stop. It’s a genuinely attractive idea, and it’s worth being honest about how well it holds up, the way we tried to be honest about Dunning-Kruger a few essays back. The evidence is real but smaller and messier than its popularity suggests: two major meta-analyses examined the same set of intervention studies and reached different conclusions depending on how they handled the statistics, one finding effects close to zero once study quality was accounted for, the other finding small but genuine effects concentrated in specific groups and delivery conditions. So: not a myth, not a magic key either. A real psychological tendency with a population-level effect that’s modest and contested, which leaves most of the actual weight, for something like a mathematical career, on how it plays out in one particular life rather than on what a randomized trial of middle schoolers can tell you.
This was, more or less, the belief running underneath my own response, even if I couldn’t have named it at the time: if ability develops with effort, then working harder wasn’t just something to do with the fear, it was a plan, one I thought was actually capable of closing the distance I felt.
The fixed-mindset alternative isn’t just a less optimistic belief, it produces specific, costly behavior. If ability is fixed, then a struggle isn’t information, it’s a verdict, so the safer move is to avoid situations that might return one: the hard course not taken, the visiting speaker’s question not asked for fear it reveals what you don’t know, the subfield not entered because starting over as a beginner again, at this stage, would be too legible a sign. And effort itself gets reinterpreted as evidence against you rather than for you, if you were actually good at this, wouldn’t it come easier? That’s a genuinely strange inversion once you say it plainly: the belief punishes exactly the behavior, trying, visibly, in public, that would let you improve. You’ve probably felt the inversion too, effort read backward, as evidence against you instead of for you, the moment where trying harder starts to feel like a confession instead of a plan. I’ve watched it in students more clearly than I ever saw it in myself: the ones who stop raising their hand not because they’ve stopped being curious, but because the cost of being visibly wrong has started to outweigh the value of finding out. It’s worth adding, too, that growth mindset isn’t a license for effort of any kind, even Dweck’s later work pushed back on treating more hours, undirected, as the same thing as improvement.
Where it plays out cleanly, I now think, is exactly the skill-gap case, and only that case. Technique, the tools of a subfield, giving a talk, writing clearly, advising a student: these are trainable, and “I can improve this with deliberate effort” is simply the correct belief to hold about them, without much nuance required. Mathematics may need this reminder more than most fields, because it carries an unusually strong mythology of fixed gift, the prodigy, the person who either has it or doesn’t, that can make an ordinary, learnable skill gap feel like a permanent verdict on your standing.
But a research career runs into a second kind of difficulty that isn’t a skill gap at all, and conflating the two is where the trouble starts. Sometimes a project is hard because the path is long and technical, you know roughly what you’re doing, and more time and effort will get you there. Sometimes a project is hard because the approach itself might be wrong, and no amount of pushing on a broken approach gets you anywhere except further from where you’d have needed to turn around. A useful test, though not always a decisive one: is this taking longer than expected, or is this not working the way I expected? The first calls for persistence. The second calls for something closer to a cold look at the evidence. And sometimes, this is the part I find hardest to sit with, the test doesn’t resolve at all, because the information you’d need to answer it doesn’t exist anywhere yet, for anyone. That’s not a personal failure of nerve or skill. It’s just the actual state of a field at its edges, and mistaking it for a referendum on your own ability is its own trap, maybe a worse one than picking the wrong strategy.
If any of this is useful beyond one career, it’s probably this: before reaching for effort as the answer, it’s worth asking plainly what kind of gap is actually in front of you.
In practice, that question is worth asking directly, not just holding in the abstract:
- If it’s a skill gap, you don’t yet know the technique, the literature, the tools, push. This is close to the one place “just work harder” is uncomplicated advice, and doubting it is usually fear dressed up as caution.
- If it’s outcome uncertainty, will this result hold, will this paper land, is this problem tractable, don’t reach for more hours as the fix. Ask the test question instead: is this taking longer than expected, or not working the way expected? If the honest answer is “I can’t tell yet,” that’s real information too, not a failure to try harder.
- If it’s judgment or belonging, the fear of what a colleague, a committee, a referee will conclude, effort doesn’t touch this directly. A conversation usually does more than another draft: ask someone you trust what the actual standard is, rather than guessing at it alone, and if that trust or access isn’t there yet, building toward it is its own worthwhile project, separate from the work itself.
- If the honest answer is “there’s no more information to get, for anyone, right now,” that’s not a verdict on you. Pushing harder won’t manufacture information the field doesn’t have yet. Sometimes the right move is to set it down and come back, or move to a problem where the ground is more solid.
None of these four map cleanly onto career stage. A postdoc can be in the first case; a full professor can land in the last. What matters is catching which one you’re actually in before defaulting to the same response for all of them.
So what was the work actually doing, those years I told myself it was closing a skill gap that wasn’t really there? Partly, I think, it was standing in for something more honest and less controllable, I couldn’t govern how a result would be received or what a colleague would conclude about me, but I could govern my own effort, and there’s a real, if imperfect, logic to that: more attempts increase the odds that some subset of them lands, regardless of what you believe about your own malleable talent. That’s not growth mindset, exactly. It’s closer to playing the odds honestly under uncertainty about which piece of work will end up mattering. It only works, I’d add now, if the effort is aimed, responsive to what the last attempt taught you, rather than simply more hours spent the same way.
If you’ve done something similar, worked harder specifically because the thing you were actually afraid of wasn’t something work could touch, it’s worth asking, honestly, what the work was doing for you. Not whether it was wrong. Just what it was actually for.
What I can’t fully claim, and won’t pretend to, is that the hard work caused the good outcomes. I only ever got to observe the one career I actually had, never the one where I’d handled the uncertainty differently, asked for more feedback instead of producing more pages, say. That comparison isn’t available to me, and I think it’s worth saying so plainly rather than letting the story round itself off into a tidier lesson than the evidence supports.
If I could hand my younger self one thing, it wouldn’t be more confidence. It would be the question underneath the confidence: which of these three problems is this, actually, and does the thing I’m about to do all over again even touch it? I didn’t have a good way to ask that question at the time. I’m not sure I have a complete one now. But knowing there were three separate problems, disguised as one, would have saved me from applying the same tool to two locks it was never going to open, and it might have left me, on the two occasions the work wasn’t the answer, looking somewhere else instead of just working harder at not finding one. It’s the same question I’d hand you now, if you’re in the middle of one of these three fears rather than looking back at them: not “how hard should I work,” but “which lock is this, actually.”