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I have been an (tenure-track equivalent) assistant professor for the past six months. A very large number of tasks need to be completed very frequently, divided between research, teaching, and service. I am getting more used to it as time passes, but I am feeling overwhelmed at least a few days every week. Effectively this has meant that the urgent tasks (class prep, student meetings, tutorials, committee meetings, etc.) are crowding out the important ones. I am unable to focus on research during the fragmented time that I can find. Those in academia I have spoken with either give up any personal time during weeknights/weekends or stop caring about research/teaching. If there's a more sustainable workaround that you have figured out, I would love to know about it.

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These problems are not unique to academia, in the spirit of "a more sustainable workaround," here goes:

There isn't any way to make more hours in the day, nor any productivity magic tricks. You have to triage your goals and work through your to-do list and adjust what you spend your time on if you do not think (or are told) that you are not meeting your goals. Two things to keep in mind while doing that triage:

  1. What matters to your pay-masters is probably more important than what matters to you. Make sure you really understand what your workplace wants, not what it says it want.
  2. You do not have to do well at most tasks, just well enough.

When you say "[t]hose in academia I have spoken with either give up any personal time during weeknights/weekends or stop caring about research/teaching," those people are showing you what their goals are. Since they are still in academia, their triage strategies appear to have worked to some extent.

I suspect, because you are "an (tenure-track equivalent) assistant professor for the past six months," that you are a high achiever who is accustomed to getting whatever passes for straight-As and somewhat new to the working world. If that is true, it might be useful for you to lower your standards for yourself and to make peace with those items on your to-do list that never make it close enough to the top or do not spend enough time there for your liking.

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  • These problems are not unique to academia - While the problems aren't unique, the job is rather so: typically less rigid schedules, deadlines, management/collaborative structure, ... Commented yesterday
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If your question is really about "feelings" then talk to a counselor trained in such things. Universities provide these for students in many cases, but maybe not for TT faculty. See what is available.

If the question is about efficiency and time management, then I have other suggestions, which I have frequently made here. Since you seem to be a mathematician (my field also) let me suggest that insight can come at any time. It isn't necessarily the result of long time segments devoted to a problem, but can be and often is a flash that comes when you have let a question/problem stew a bit in the back of your brain. It can even come at waking moments or just before sleep.

The trick is to capture these insights for later examination. My solution has been (and continues to be, even in retirement), the Hipster PDA. Simply put, carry a small deck of index cards (10 or so) and a pen/pencil. Some of these are blank and some are notes about things you are working on and/or interested in. Keep the writing sparse so that you can add ideas when the occur.

This can make otherwise unproductive moments more productive, say standing in a cafeteria line or sitting on a bus.

I prefer physical cards rather than electronic options as the index cards are easy to sort, discard, store, etc. Even just picking one out at random for a five minute (3 minute) review can lead to something.

Yes, I realize this isn't a complete solution, but it can help.

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We've had several questions that convey ideas about juggling day-to-day priorities, so to give some answer to the unique part of your question (namely, that you are in your first 6 months).

My experience (TT non-flagship R1 math past 3 years) is that you have two options:

  • Some semblance of work life balance while doing as much or a bit more than necessary to get promotion and tenure in terms of quantity & quality.
  • No work life balance but doing enough to look competitive for tenure track at R1 flagship+ or whatever the next step up the resource/prestige ladder is for you.

There are pros and cons to both choices, but both are livable. You may not have thought about things quite this way before if you're just starting a tenure-track job, as the rat race up to that point is simply "do maximally as good as you can" and the money is peanuts.

Secondarily, unless you are at a very wealthy place that can afford to give you very little and easy teaching assignments, I think most people find the first semester/6 months overwhelming. You have more tasks now than you generally will otherwise. New preps, less independent students, maybe even buying office furniture/computers, getting your new living quarters in order...I have met few people who do not slow down somehow in these circumstances.

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