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Last year I was offered full funding for my PhD program to TA two courses a semester for the duration of my program (education psychology). Key words, Teaching Assistant. I expected this to mean I would be working with a professor each semester and teaching a recitation section twice a week (one for each course). This was all fine and great in the fall.

Then this semester, I was told I would be instructor of record for two courses (aka, I would be teaching these two courses entirely on my own and responsible for everything).

I feel extremely overwhelmed by this workload. My research has been suffering as I've had no time to work on anything outside of TAing and coursework. Next fall I was also told I would be instructor of record for two courses again (these two courses I have never taught).

I e-mailed the person in charge and asked if it would be possible to actually TA a course as the workload was a lot to manage and was hit with

As TAs progress we expect them to take on full classes

From speaking with other TAs in the department, this is very common at the school. I was not aware off this before taking the TAship, as at the time I thought TA meant teaching assistant and not being expected to take on a whole class. Very few are only TAing (most of my colleagues are responsible for their own classes), so from my perspective it feels like an exploitation because they don't have enough professors.

Is this a common practice? Nowhere in my offer letter does it say instructor of record. It distinctly says TA. Any advice from anyone in a similar position?

You should consult with your department, asking explicitly:

How do you want to proceed, I feel like these conditions are excessive and were not part of the deal as far as I know.

Come to the meeting with your signed contract / TA paperwork.

I assume, your department cares about quality of the education they provide. Don't let them pin this on you, you are a student first of all. If they employ you, they are responsible for your performance as a teacher.

Other answers didn't pause here:

I feel extremely overwhelmed by this workload and my depression has made this worse I've cancelled class at least 5 times this semester because I couldn't get out of bed

I suggest you immediately seek professional help. Your university must have services available for students (and be paid-for by university's health insurance), especially given recent attention to mental health crisis in academia. That is your responsibility to take care of yourself. You are also a real person, not just a robot TA, don't let mental health issues fester.

Not to defend your high teaching load, but yes, heavy teaching loads like these are fairly common for TA's in the US in my discipline (mathematics) and in other disciplines that teach lots of students in general education courses (freshmen composition, world civilization, introductory courses in science, etc.)

Keep in mind that Ph.D.'s working as full-time instructors in the US often teach 4 or even 5 sections of such courses per semester. Thus teaching two sections of such a class at a time is roughly half of the workload of a full-time instructor. This is work that you should be able to accomplish in 20 hours per week, but you may well have to compromise your standards and carefully manage your time and effort to get the work done in that time.

The fact that your offer letter says "teaching assistant" without defining it probably does not contractually exclude being assigned as instructor of record (although you could check with a lawyer to confirm this). It's likely that the university does have some written policy somewhere that does define "teaching assistant" to include "instructor of record." Even if they don't, using grad students as instructors is a common practice at US schools and is evidently a common practice within your program, so it would be hard to argue that you were misled by the offer letter. Is it exploitative? That's a different question.

Moral 1: Before accepting a funding offer, always ask for a detailed description of your duties, regardless of what the funding is called. Many "fellowships" come with teaching duties.

Moral 2: The best way to know exactly what you're signing up for is to go to a school where the grad students are unionized.

The load seems heavy to me also. But there may be reasons why you are "trusted" to handle it and not all are necessarily bad. I was in grad school half a century ago and one was trusted to teach a course as instructor of record only in the final years of study when you had proven yourself. But at the time, the expected time to finish a doctorate was much shorter. We treated such situations as an honor, not a burden. I think your situation is actually one of the reasons why the time to completion has extended in the interim.

Perhaps you are seen by the university/department as being especially able or already advanced. Perhaps the courses are seen to be especially easy. Or, on the other hand, perhaps you are just being exploited because the department has insufficient regular faculty to handle its obligations.

You may be stuck and just need to learn to cope if the department won't help you. Cancelling classes seems to be a no-win situation and might lead to your expulsion or withdrawing your funding. You need a way to avoid that.

Let me make a couple of suggestions about how to handle the teaching load.

Look for available materials that the department has used in these courses in the past. Another grad student or a prof might be able to provide you with working notes to ease your preparation.

Put more of the responsibility for student learning on the students themselves than on your own shoulders. At the end of the day the students learn themselves. You don't "learn them". Group work helps in some fields. Fewer, larger, projects that you monitor can help. A flipped classroom can be a big help.

Find a way to schedule your time better. Yes, easier said than done, of course. Your university may, however, have a counseling center that can provide advice. Most of the advice is time scheduling for students, but they may be able to offer some suggestions. In particular, find a way to take advantage of "lost moments" when you could be doing simple, but productive, things. Five minutes waiting for a bus can be productive if you carry your notes with you at all times, or always have an important book that you can browse.

One aspect that you probably aren't considering. If you set aside time for your different activities and stick to that schedule, you will find that sometimes (maybe often) you don't complete what you thought you would at the end of the scheduled time. Don't extend the session, however, but go on to the next scheduled activity. Quitting before completion will be at least partially compensated by the way the mind works. It is likely that giving topic A a rest while you go on to B won't result in a loss for A, as your subconscious is still at work and is capable of filling in a lot of things without your conscious effort.

I expected this to mean I would be working with a professor each
semester and teaching a recitation section twice a week (one for each
course).

Generally, yes, unless specified otherwise. Usually it also included quiz/exam grading, holding office hours, organizing field trips etc.

My research has been suffering as I've had no time to work on anything
outside of TAing and coursework.

This is a red flag. You won't be able to use your TA workload as an excuse for your down performance in your research project. Therefore, I suggest you talk to your project supervisor as well as your TA supervisor about this situation. I was once in a similar situation, but because I talked to both of my supervisors, we were able to come to a solution that was mutually beneficial.

"as TAs progress we expect them to take on full classes". Is this a
common practice?

In my experience, no. I have been a TA at two different US universities, and in both I assisted the professor with aforementioned duties. Sometimes, I had to deliver lectures when professor had a meeting or so. But taking on a full course seems like too much for a PhD student. Especially, if it just say TA on the record.

However, that being said, if you want to stay in academia, this is in reality a blessing in disguise. Time management is a great skill to be successful in academia. But I would still talk to the person in-charge of the courses about the workload and explain how it is impacting your research.