text:
Are books peer-reviewed?
I am unsure for books but I know for certain that selected journal articles are peer-reviewed.  (This just shows I am not in academia!)
Do books go through a peer-review process? If so, how does this happen?
If one is approached by a small publisher, does it matter if this publisher does not have a peer-review service (if there is one for books).

Some are.  Some aren't.  What kind of book are you talking about?

related answer: http://academia.stackexchange.com/a/9298/386

"Books published by university presses almost always go through a process of peer review." Source: https://ncu.libanswers.com/faq/168497

Typically, after an author presents a book proposal to a publisher, the publisher will circulate the proposal to some selected reviewers to vet the content. This is not like peer review in the usual sense: the reviewers only get to see the outline and maybe a chapter or two.
Once the publisher decides to go ahead with the book and it goes through the editing process, it might undergo further review, but nothing like a journal review.

Journals (generally) have too many papers, and need to cut down. Book publishers (in my experience) have too many people who never finish books.

@Jeremy If you are implying that peer review is about the quantity of publications: it's not only about quantity, but also about quality.

@root - not sure I understand. Peer review of books or journals? The process is pretty different.

I guess this is field-dependent because my experience (in math) is different from what is presented in the other answers. At the very least, some book series by some publishers have a peer-review process similar to that of journals for books that are about new research (not textbooks or similar material). The editors would give the whole book to several referees and ask each of them a report about the mathematical correctness, the context, the presentation, etc. There can even be a similar editorial process as the one for articles, with a back-and-forth of corrections and new reports. Sometimes referees would only each be asked about some part of the book, but each part would be covered by at least one referee. I would guess that this does not concern textbooks or "survey" type books, but my experience there is limited.

The major textbook publishers pay for "subject matter expert" reviews of completed books prior to publication, and even of new editions of previously-reviewed books.  Some reviewers apparently just submit the publisher's questionnaire.  Others, like myself, submit extensive comments.  I've proposed corrections that were accepted for books by well-known authors.


text:
Updating a preprint when it is under review
I know that preprints deposited into a preprint server should be in a polished state. However, what should happen once it starts the review process? Should those copy-edits be updated on the preprint? What about updates when replying to reviewer's comments? What if the manuscript is rejected and I am rewriting the manuscript to be resubmitted to another journal in another format?

In principle, anything that happens to the content of a paper before printing (ie, formal publication) should be reflected in the pre-print.  So yes, if you correct a bug while the paper is in review, or revise the paper for a new submission, then you should update the preprint.
However, some journals object to authors' posting post-copy-edited (or even post-refereed) revisions.  And posting a revision while a paper is under review could interfere with the reviewing process.  When in doubt, ask your editor.
Frequent updates may earn you a reputation for being sloppy, especially on a system like the ArXiv that publishes the preprint's revision history.  (Why didn't you fix those bugs before you uploaded the first time?)  But that's still better than leaving a buggy preprint out in the wild, thereby earning you a reputation for not even knowing (or caring) that you're sloppy.  The right answer, of course, is to debug your papers before you post them!

I agree. We should do it right the first time. However, if the author keeps improving his paper(not necessarily fixing bugs), should he revise the paper everytime or wait until he believes he finds the best way to present the paper?

I've clarified the question a bit more in response to your section on sloppiness. I think @Scaaahu makes an excellent point since after all, a paper isn't truly polished and best presented until it finally reaches print.

I tweaked my answer.  Changes to the paper's content should be reflected in the preprint.  For stylistic issues, I think it's better to wait until you've found the best way to present the paper.  (But to repeat myself: Shouldn't you do that before you upload the first time?)

In general, there are more than one way to prove a theorem in a math paper. If I find a better proof, what should I do? It boils down to another question, how many revisions would be considered too many by researchers? This probably should be a separate question. I just want to avoid  duplicates. Thanks to the OP for asking the question and to JeffE for answering it.

If you "find a better proof" more than once or twice, you probably should have thought harder instead of posting the first one.

It should be noted that if the journal chose "reject with permission to resubmit" rather than "major revisions" they have no legal right to block you from updating the pre-print. They rejected the old manuscript, it doesn't exist as far as they are concerned, and you can use the peer-reviewer feedback and post the edits. If they chose revise, then you need to follow whatever rules you agreed to with the journal.

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text:
How should I properly mention URL links in a thesis?
For example, I have an appendix X, in which I wrote:
Looking on different websites, a list of models of X was compiled for Y purpose. These websites are:
\begin{description}[noitemsep]
  \item \url{https:.....}
  \item \url{https:.....}
  \item \url{https:.....}
  \item \url{https:.....}
\end{description}

To clarify, models of X often have a short lifespan.
To determine if they are obsolete or not, the following websites were used: \url{https:.....} and \url{https:....}.

Is it okay to put the links like that, or should I make a reference in the bibliography for each link? How is the right format? For instance:
Looking on different websites, a list of models of X was compiled for Y purpose. These websites are [1], [2], [3], and [4].

To clarify, models of X often have a short lifespan.
To determine if they are obsolete or not, the following websites were used: [x] and [y].

In the bibliography:
[1] Website 1 URL
[2] Website 2 URL
[3] Website 3 URL
[4] Website 4 URL
[x] Website X URL
[y] Website Y URL
I've been considering the idea of naming the title of the website and adding a citation to it, something along these lines:
Appendix:
"...the following websites were utilized: "Title or Name of the Website" [x]. and "Title or Name of the second Website" [y].
Bibliography:
[x] [Title] Available at: www.example.com
I am thinking about removing the list of websites and leave only two links.

How long are these URLs? Are they super ugly?

what style guide does your school use?

@AzorAhai-him- the longest one is: https://gr-satellites.readthedocs.io/en/latest/supported_satellites.html

@RichardErickson They only said: "According to the examination regulations, there are no specifications regarding the formatting/design of the thesis. In case of doubt, please refer to DIN 5008"

@O.R.Mapper I think so

In lieu of any formatting prescriptions, remember that citations serve two purposes: to provide proper intellectual credit and for the use and convenience of your reader.
A format that makes your thesis hard to read is bad.
In my opinion, a list of websites used for data are distinct from other intellectual citations, so I think they would be most useful compiled and listed separately from your other references/bibliography. If they are in an appendix and you clearly refer to the appendix when it's appropriate to do so, that seems like it would be easily digestible to a reader. They only need to seek your appendix if the list of websites is important to them, and if the list of websites is important to them then they are organized in one clear place. You should also make clear to your reader how these particular websites were chosen, as well as all the information necessary to contextualize your use of those websites, like the date accessed.

While it is a bit hard to understand what you want, I think the essential thing is that each URL should be listed with the date at which you accessed it last ("...often have a short lifespan...").
This should always be done with any volatile resource or one associated with a given date/time.
Alternatively use a link archived at the Wayback Machine.

Yes, if they are transient sources, or might be, part of the biblio entry should be "downloaded on ", or similar.

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text:
Should a marker TA handle students' marking issues by email?
I am a university marker TA. The instructor requires TAs to meet students if they have marking issues. But I found most students just come to meet me for better grades, not to ask questions, and some are tough to handle; they just insist they should get better grades.
I feel that is annoying. Is it a better idea to handle the cases by email, and let them talk to the instructor if they have further questions?

Tell the professor in charge of the course that some students just come to meet you for more marks, not to ask questions, and some are tough to handle, insisting they should get more marks.  Hopefully the professor will tell you that if a student will not let it go, after 10 minutes of you attempting to explain the correct solution, and explain how yours falls short, then you should direct the student to the professor.  That is the ideal way to address this, and hopefully the professor will think so too.

See also what to do about grade grubbing

@Ben Crowell I grade exam papers.

While the situation is indeed annoying, think of it as a training opportunity: by dealing with these insistent students you are getting really good practice in standing up for yourself in a professional environment (moreover, in a low stakes situation where nothing much other than your ego is at stake, and where you are the person in authority, which makes things a lot easier). The skills you will develop through this practice will be really valuable for you later on in life when the stakes can be much higher -- say, when you are asking for a promotion or a raise, or arguing an important technical point with a colleague or superior. So, the next time you hold office hours think of it as "I am going to my `assertiveness class'" instead of "I am going to be attacked by hostile students". This subtle change in perspective could end up making the experience a lot more tolerable, even fun.

It may be a way to practice, but I found many TAs only do one term of TA job in their graduate study. So this practice might not pay back.

@Tony what do you mean by "pay back"? What form of payback are you expecting?

@Tony I think what Dan was implying was that this a chance to practice for future situations in the professional or academic environments. The merits of such experience are much more than just "helping you be a better TA in future semesters"

@Tony the payback is the rest of your life, not just your academic career. Maybe you have never met a car salesperson face to face yet!

@alephzero What if the students' attitude is awful and arrogant? I feel uncomfortable dealing with such students. Can I just tell them to get out?

I strongly recommend publishing and adopting the following policies:

Publish detailed grading rubrics describing how you assign partial credit.  Follow them slavishly when handling regrade requests.
Regrade requests must be submitted in writing at most X days after the graded work has been returned.  Each request must include a brief written explanation of why the grade is incorrect.  (Note: not "...why they deserve more points.")
While you are happy to discuss answers and even grades in person, you will not change any grades in the students' presence.  Or maybe even within 24 hours of meeting with the student requesting the regrade.  Maybe add an exemption for arithmetic / recording errors.
Your responses to regrade requests are final.  Further appeals will be automatically forwarded to the instructor.  (If the instructor bounces them back to you, that means they're happy with your earlier decision.)

Finally, ask the instructor to announce these policies on the course web page.  Better yet, ask the instructor to forbid you to change grades in the students' presence, or to regrade the same submission more than once.
As for demands from students for higher grades:  Look them straight in the eye and tell them that giving them anything less than the fair and honest evaluation that they (or their parents) have paid for would be grossly unfair, not only to other students or future employers, but to the students themselves.  Refer all complaints about your grading rubric to the instructor.  Unless they want to discuss the substance of their work, gently but firmly kick them out of your office.

I always went with "sorry, but I cannot give you marks you did not earn".
If they gave me a hard time (and I had a few... "well if you would just TRY to find somewhere to give me extra marks! I can't believe you won't try to help me pass!")
I just say sorry, but it isn't my responsibility to do the work. It's yours, and I can only give you marks where you've earned them.

These headaches are a reason us educators have a job.
Depending on the person and the work I like to look at their exams again and explain why they lost marks and where they could do better, in the rare case that this doesn't do them justice I make it clear that on this exam, today, right now they do not deserve the marks they're asking for. I'll typically throw a few tips at them and tell them to prepare better so they can make up the marks next time.
I like to be open-minded when it comes to a kid's marks because I know a lot of incredibly bright students who make silly and sad mistakes I know they could've avoided. Totally shutting them out and saying things like "marks are not up for negotiation" can really destroy the self-esteem of these students and make them lose hope throughout your course.

+1 This is very useful advice. Providing a constructive perspective is really the best way to go.

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text:
How can I have productive meeting with a very prominent researcher?
Summary: I am a shy/introverted PhD student and I have a collaborator at another university. They are setting up a meeting for me with an important researcher in the field. What should I even say in a meeting like that? How can I make it productive?

I am a PhD student and I have a collaborator from another university which invited me to visit. I came here and this collaborator suggested me to meet with one quite important researcher there in the field we work, with whom he works as well. By the way we had a very brief interaction once and he is very nice.
Now, I would certainly like very much to meet such an important researcher, but the thing is that I have no idea what to say in a meeting like that. I'm pretty much a beginner, and right now I don't have any bright ideas to put forward nor any useful comment.
In that case I feel pretty much confused about what to do. I feel like not going to such a meeting might be losing one opportunity and, moreover, might feel very discourteous. On the other hand I'm afraid of going, having not much to say and just being in an awkward situation.
Given that I'm a visiting researcher in another university and I'm invited to meet some other important researcher in the field, what should I even say in a meeting like that? How can I deal with this in the most appropriate way to not lose this quite nice opportunity and also not being in an awkward situation?
One small detail about myself: I'm very shy and introvert, so I'm not much the talkative type really.

Browse thru' the questions in this forum, select a few, and use them to initiate conversations.

@VitaminE Better yet, pick at random and greet them with gems like "Can you do research during a strike?" or "Sometimes, I want to punch people in the face, what do I do about it?" ;) Are there any legitimately good conversation starters on academia.SE though?

++++1. I'm a postdoc and still feel this way. My number 1 tip is don't be afraid to ask questions about the topic they're telling you about, even if you think the questions are really stupid/low level/naive. This way you can keep the conversation going, and try to learn something. They're not expecting you to come up with the next groundbreaking idea in the field, so don't place this expectation on yourself either :)

Your collaborator proposed this meeting; what did he say the purpose of the meeting would be? Advice for you will be quite different depending on whether the purpose of the meeting is for the researcher to learn about your work, or for you to learn about their work, or for the three of you to talk about working together—or perhaps it's just a social meeting so that you'll know each other in the future.

Rather than put forward your own ideas, which might still be a bit naive (as you suggest), ask for advice. Let the superstar take the lead. If you've read any of their papers, you can mention that.
But, for a beginner, it is enough to express some interest in your own areas, even if that is different from that of the other.
You can always ask questions about what interesting research threads you might want to look at over the next year or so.
But, don't present yourself as something that you are not.
And if the meetings are informal, the discussions might well turn to non-academic topics: in the US, the drama of the impending baseball season, for example.

Hi @Buffy, thanks for the advice ! Yes, I have read his last paper (which turned out be in one particular subject in the field I'm very interested in), though I have not yet fully understood it because it employs a formalism which it's new to me.

@Aegon Then the things you haven't understood yet will form natural questions!

DO:
Try to keep up with the ideas and look for openings where you could potentially contribute something interesting.
Take this intimidation caused by a perceived power and skill imbalance out of the equation and focus on the content of the research instead.
Realize we all start somewhere and this one encounter would not make or break your career. People tend to overestimate the importance of such singular events and then the impostor syndrome kicks in.
Believe in yourself. Others (your collaborator) do. They think extending the collaboration might be fruitful - so, in a sense, you are "deemed worthy". Instead of questioning it, try to accept it and move on to, well, the actual research.
DON'T:
Pretend you understand something you do not. It gets you nowhere and more often than not is blatantly obvious anyway.
Over-prepare for the meeting. This can be outright creepy; some may find joy in this flattery but it is way more often than not bound to be a faux pas.
Retreat into the "coziness" of being timid and borderline subservient. Prefacing nigh every sentence with the likes of "Of course, I could not possibly understand this on the level you do, but maybe..." is disastrous and totally unacceptable for a topic you are actually working on. This is only potentially useful to establish some knowledge boundaries in interdisciplinary research and even then is more annoying than anything.
Hang onto this conversation long after it is over. See above: you might describe this event as life- or career-changing in some 30 years or you might forget about it entirely in a couple of weeks. You might even end up not having much in common at the end of the day, and clinging to this possibility for the sake of possibility is highly unproductive.

Preparing:
If you are anxious about a meeting like this, make sure you are prepared. Some things you can do in preparation

Be prepared to explain what you are working for your PhD. What is the overall goal? What have you found so far? For an initial explanation try to stick to the big picture. Initially try to be concise (i.e have an elevator pitch ready). Also be prepared to adjust the level of technical detail of your explanation. Just because the person your are talking to is a big shot in your field, doesn't mean that they are up to speed on the nitty gritty technical details of the work you are doing. Therefore start at a relatively low technical level, and adjust the level based on the question they ask and the amount of interest they show.

If you have produced nice figures that illustrate some of your results or particular aspects of the approach you are taking, it is a good idea to have them somewhere you can easily access them (possibly on your phone, or on a tablet/laptop if you happen to be bringing one to the meeting).

Read up on what the person is working on. Look through their recent publications.

Be aware of the major achievements of the person (e.g. look at their most highly cited papers).

During the meeting:

It is OK to let the person you are meeting and your collaborator lead where the discussion goes.
Don't feel pressured to ask a highly technical question about their work.
To keep the conversation going, some fairly standard "small talkish" questions to ask, include asking them about what they are currently working on. In general, this is the favorite topic of most scientists. (Although, in some fields people may be more tight lipped about ongoing research.)
Remember that despite their reputation the person you are talking to is also a human being.

("your are talking to" → "you are talking to" (flag this for deletion after use))

You should tell from your current research, so that the other person get an impression on what you work and how far in the process you are. He might think of you in the upcoming months and you can profit from this. It could be a recently published paper by a third party, he might mention you in a meeting with somebody else, you might be able to write him a question later on.
Often they have questions, as they know the field and they want to clarify what route you are heading for.
Beside that, just let it flow. He might be telling something, your contact might drive the discussion, or you are ending up just getting known each other without too much talk about his or your research. He might be a star in your field, but never forget he is also human. He might be shy, too.

I want to add something that I think is implicit in the other answers (which are good):
I would not expect a particularly "productive" first meeting. I would definitely not put pressure on yourself to make the meeting "productive" in any concrete sense.
It sounds like the researcher is happy to meet you and get to know a bit about you. That's probably all they are expecting, and it will be easy to meet those expectations -- just be yourself. On the other hand, if you set ambitious goals like starting up a research collaboration or saying something impressively smart, that's more likely to backfire and create unnecessary pressure on yourself.
Sure, it's a good idea to be prepared for some possible topics of conversation. You should know how you're going to describe your history and what you're currently working on. You can be prepared with some questions, even simple ones like "what are you currently interested in?" But I wouldn't expect any concrete result other than you two getting to know each other a little, which is "productive" or useful to both of you in a long term sense.
The other possibility, I guess, is that the researcher has a particular idea or project to pitch to you, but if so, you'll find out -- no particular prep is likely to help.

Imagine their situation. Would you like to get praised the whole time? Perhaps not. Treat them as a human being and not as an important researcher. Have a conversation on eye level. Ask how they are doing, how their travel went etc. If you're unfamiliar with small talk, copy what your colleagues say in such situation.


text:
Can I say no when my principal investigator asks me to help cover lectures next semester?
I'm a postdoc at a US university, and my principal investigator (PI) asked me to cover some of his lectures next semester. I do not want to do any lecturing as I want to focus solely on my research. I am involved in two funded projects, and I therefore have a lot to do research wise.
Can I say no to covering lectures or would that be considered a mark against me?

Are you planning for a career as academic, involving teaching?

Next up: "Can I say no when my ungrateful post-doc asks for a letter of recommendation?" - working with (and for) other people requires a balancing of obligations. A focus solely on your own research interests to the exclusion of all other considerations is not a good strategy going forward, regardless of your desired career path.

If you plan to stay in academia, the teaching experience will look very good on your CV.

The title asks a different question than the post: The title sounds like "can I decline to cover a particular lecture on a particular day", while the post is "can I decline to cover lectures always, for no reason". My suspicion is that these two questions have opposite answers. Can someone edit the title to make it less confusing?

Is your PI asking because they have to miss some lectures, or because they specifically want you to give some lectures?

What the heck does "PI" stand for?

@Notso "Principal Investigator". In other words, advisor, supervisor, boss, or whatever one calls them :)

Apropos of @Jon's comment, I actually heard the line "this candidate would rather quit a job than teach" from a reference once. Mind you, the person I was talking to is habitually blunt to the point of rudeness, but it was still a surprise.

I may be in a similar position to you soon, and also have time worries (and nerves). In my case because the course is established and there are notes and slides already written it would be a good soft-start to lecturing, thus it seems like an opportunity more than a burden.  Do you know if you'd have that same advantage, or if you'd end up writing a section of the course, an exam question or two...?

@JonCuster: "Ungrateful"? The PI is not doing him a favor, he's asking him for a favor.

@einpoklum PI's provide extra training, advice, support for postdocs. PI's often share their methodology, their knowledge, their network with postdocs and in many other ways go far beyond their contract obligations.

@DmitrySavostyanov: Your team leader or other kind of professional mentor in a commercial company also advice and support, and sometimes training, as well as share their (and their company's) knowledge, network of contacts, methodology etc. That's part of the process. The post-doc, or the new employee, reciprocates by doing his/her job well, and extending the same courtesies to others when s/he is the more experienced on. That doesn't mean that you would be ungrateful if you didn't, say, fill in for him in some stockholder's meeting.

@einpoklum In you are really comparing academia to a commercial company, sure thing, academia is going to lose in almost every possible regard. Lower salaries, higher workloads, non-existing work-life balance, writing guilt and impostor's syndrome as a year bonus. It's a bad environment, but not exclusively to postdocs.

@DmitrySavostyanov: All fair points. And I agree it's bad not only for post-docs. But I think there's enough of a parallel to explain how I don't believe the post-doc would be ungrateful.

@penelope is this an Americanism? I've not heard the phrase once and I am currently in my 2nd year of postgraduate studies... Edit: I see that it has already been added

@Notso I don't know about the US, it's used in the UK. Technically, here it has nothing to do with a PhD student - advisor relationship and pertains to the person who is responsible for the grant (the PI on a funded project).

I think it'd be really helpful if you'd clarify just how much teaching you're being asked to do.  If it's three lectures, I'd say "just do it".  If it's fifteen, I'd say "you're being abused".  Without this information, I think every answer is making their own assumptions, and this accounts for the wide disparity in advice you're getting.

I don't think a flat "no" would go over very well.  Covering classes is something that colleagues are expected to do for one another, within reason.  Yes, everyone has their own projects that this takes time away from.  But it's really the only way that faculty are able to travel during the school term, which most find essential to keeping an active research program.  Saying "I don't want to do any lecturing" will come across as unreasonable and possibly selfish.  Keep in mind that "focusing solely on research" is a luxury that basically nobody ever has.
However, you can certainly negotiate, and maybe reduce the amount, while still showing that you're willing to help.  Keep in mind that the PI presumably has an interest in your research success, so if the teaching would impact this in specific negative ways, beyond a generic "this is an hour of my time that I could be spending on research", you can point this out and suggest a compromise.

"I'm happy to help out, but seven classes over the semester is kind of a lot.  I am going to need to [achieve specific research goals] next semester and I think this might start to get in the way.  Maybe you can find another person who can cover some of them, and we'll split them up?"
"I can do January 18, but on March 7 there is a conference that I was really hoping to attend.  Perhaps someone else could cover that date?"
"February 10 would be fine.  April 3 is the week before [important deadline] and it might be kind of a crunch, so maybe someone else could teach on that day."

You can also ask for materials or other help that will lessen the amount of work for you.

"Sure, I can teach on January 30.  Can you point out the specific sections in the book that I should cover?  Will you be preparing notes for me to follow, or do you have notes from a previous semester?  Can you suggest any particular examples that I should present?  [etc...]"

With such assistance, covering a lecture should not end up taking an excessive amount of time away from your research.  You really should only need to prepare and deliver the lecture - you likely won't have to deal with creating assignments, grading exams, supervising TAs, holding office hours, answering emails from students, or the other things that end up taking the majority of time when teaching.

This is an excellent answer!

"Covering classes is something that colleagues are expected to do for one another, within reason" Absolutely.  And often times, people try to balance out, with the general idea being if someone covers for you, you'll be top of their list for who they ask to cover their class.

First, I agree with everything Nate Eldridge wrote in his excellent answer.
To add to that: you may think your job is only to do research, but the funding agency that provides the funds that pay your salary may not see things that way. Your PI pays you out of (I assume) a grant he got from the National Science Foundation or other similar government agency, as part of the grant’s “training” component, that is, the PI promised the agency to help train the future generation of researchers and educators (and the NSF cares about such things, including the teaching part, see here). One could make a very plausible argument that giving you some teaching experience is an essential part of that training. So, even aside from the fact Nate and others already explained that not being willing to do any lecturing will indeed come across as a black mark against you on an informal basis, you may also have a hard time arguing even at a purely formal level that lecturing is “not in your job description”.
Another thing to consider is the mindset of your PI, who as a regular faculty member likely teaches around 100 hours a year and also has many administrative duties, and yet manages to do productive research. From his point of view you have an incredible luxury to be able to focus essentially all your time on research. Depending on his personality and the level of stress he experiences from having such a workload, he may even be a bit resentful of you for having such unbridled freedom in comparison to him. From such a vantage point, a refusal on your part to take on just a few hours’ teaching load - a trivial amount, relatively speaking - is almost certain to appear selfish and immature.
Finally, to balance my answer a bit, I will go in the other direction and mention the possibility that a normally innocent request by a PI to his postdoc to teach a few lectures can in fact cross a line into potentially abusive behavior if the extent of the teaching grows above a certain point, especially if this is done without the knowledge and approval of the PI’s department. I have heard of cases of faculty who got their postdocs to do essentially all their teaching for them, and got in trouble for it. It would certainly be reasonable of you to be on your guard about a potential abuse of authority of this type, and to set a reasonable limit on the amount of teaching you are willing to do, and also ask some questions about whether this has been cleared with the department.

I would check with an NSF program officer or your Contracts & Grants office.  If teaching is not written into the grant, this sounds like an unacceptable way to have the NSF pay for the university's instruction.

@ChrisK checking with people who know the rules is generally a good idea, but I disagree with your second sentence. The training of a postdoc assumes training them for an academic (and/or industry) careers, so some amount of teaching may be implicitly assumed even if teaching is not explicitly mentioned. Of course, the amount of teaching would be expected to roughly correspond to the emphasis put on teaching in the grant, but saying that any amount is “unacceptable” if teaching isn’t mentioned in the grant sounds incorrect to me.

I'm sure no one would care about a lecture or two here or there, but I've seen a university that had to pay a postdoc a week of salary for their teaching while they were otherwise supported by an NSF grant.

Your personal relationship with your advisor is probably a very important consideration. This might depend on your assessment of how he/she would react. But in general, it is probably a good idea to get the experience of lecturing in a way that involves only a short term commitment. While I think you can refuse, and most legal systems would back you up, it may not be wise to refuse.
But you might want to try to structure it so that you benefit maximally from the experience. For example, asking now for the opportunity to discuss things with your advisor after the first such lecture would be an advantage. Also, it puts an obligation on the PI.
Your future may depend on your relationship, but also on the breadth of experiences that you have as a student. You could think of it as an opportunity, rather than an obligation. The PI seems to trust you. That is a valuable thing in itself.

The short answer is: it depends.
The objective of a postdoc is to obtain sufficient research experience so that you are ready to be a 'grown-up' academic and university professor. This is achieved mostly by doing amazing research, but your PI's assessment of your work is a significant part of this. So I would say that even if you feel that this is taking up a significant chunk of your time, it's better to keep a good relationship with your PI, even it means a few hours lost to teaching.
That said, almost any academic position would require some teaching, and perhaps this is the PI's way of assessing your teaching capabilities so that they can make an educated statement in a future reference letter.
If it is just a few lecture hours, I personally would not make a big deal out of it. If they are asking you to grade exams/consult students or something more time consuming, perhaps it would be worthwhile to diplomatically discuss it ('are you sure that this is the best use of my time given that I am involved in two projects?').
If you feel like this is part of a growing pattern of assigning you a bunch of tasks not related to your research, it may be good to raise this issue.

You can.
But how do you see your career progressing? Perhaps a combination of research and a small amount of teaching could be good for you and, for the future students.
Your choice, so good luck anyway.
I had the Professors coming to me and saying, will you do X lectures, we have the pay rates already agreed and signed... And the material, examples, handouts all prepared, printed and ready to go...

Yes you can. How you do it is the key point you should worry about.
I was in the exact situation once before. As a 1st timer postdoc my supervisor came to me informally to ask whether I could lecture in his place in case he'd go for "trips". Immediately my internal alarms went off. Why did he drag me into a separate room, and why was he smiling so much? Clearly he was pushing beyond limits.
Still, I was quite friendly and open. I asked how did that contribute for my record with the department and project? I wanted certificates per each class given, and beforehand some rough estimation of how many classes to expect and more or less when. I emphasised on how time-consuming the project and writing of papers were sure to be. And I asked whether officially my postdoctoral funding rewarded or at least demanded giving out classes (e.g. how many hours).
He immediately realised the resistance. He said "Well, in case you don't want to help with classes, it's OK", to which I said, "of course I am available, but I must understand the official details before making plans."
In the end, I lectured about 6-8h in total, out of 3 years as a postdoctoral fellow in that lab. Surely it could have been more. My impression is that he was wary of "leaving for trips" and relying on me to cover him up. I bet he wasn't really communicating absences, where talking about official procedures gets a bit unsettling.
Thus, my advice is: don't let others abuse you, but make concessions where you see some clean opportunity for experience or "points" with the department. You do not depend on your supervisor for jobs later -- this is usually a myth and they won't move a finger in any direction.

You may not need the support of your supervisor to get a job after the post-doc, but you may well need it to get a particular job you want.

Agreed, especially if you want to stay exactly where you are as a postdoc, such as sharing the office with that person officially. Still, my experience has shown that typically a minion may stay as such indefinitely while the supervisor helps random others, if anyone at all. Never rely on someone proposing some exchange of favours, especially a superior at work.

But exchanges of "favors" is just how human groups operate. There are all kinds of things that have to get done at a job that nobody really wants to do. The person that refuses to do any of it to 'focus solely on their research' is quickly going to become isolated and ignored.

Unless you have a sociopath as boss, running accounts about services rendered vs. services obtained is not going to go down well. As PhD and postdoc I was generous with my time and was equally treated with generosity. And, if you do have a sociopath as boss, you should change jobs, anyway.

"The person that refuses to do any of it to 'focus solely on their research'" -- I agree, as you can read from my answer. But being 'always ready to do the work of others' can make someone quite popular... in the wrong way.

@CaptainEmacs This ex-supervisor was no sociopath. In fact he's remained as an old friend until this day. He's just pushy at times, and extremely lazy.

@Scientist Fair enough. Perhaps he felt guilty and does not mind getting push-back. However, with many people, if someone starts making accounts, others open accounts, too.

You should respectfully tell him to "Insert Coin" (but be careful)
Most post-doctoral positions involve research work under the supervision of a more senior researcher. They do not include teaching or institutional-administrative duties. This is related to their fixed- and short-term nature - and to their significantly lower pay.
With due respect to your PI - making precarious researchers serve as a backup teaching workforce is a further degradation of their (or should I say our) position in academia. So while - as @NateEldridge suggests - accepting a backup request is the collegiate thing to do, it is also collegiate for your PI to make sure you are then recognized as having been employed as a teacher, even if for a short period of time, and paid for it (with consideration of the extra relative overhead per class hour when you are teaching just once or twice).
With that said,

It is very possible that refusing will hurt your relationship with the PI and even threaten your continued employment - this depends on the specifics of the situation. It is still possible, though less likely, that even requiring formal recognition of this duty and payment will hurt your relationship.
Careful management of your rhetoric and order of actions (see @Scientist's answer) is important, regardless of what exactly you choose to do.
Ignoring the questions of status and rights, I would be disposed to help a colleague in need, as @NateEldridge suggests. You mentioned your wish to focus only on research - remember that, collectively, as academics we have an obligation to teach students, so it is important that someone competent cover for your PI.

Makes excellent point about "undermining time / compensation for teaching or institutional-administrative duties".

It's a nice thought, but I don't think it will work well in practice.  In US academic culture, as I said, this is something that colleagues expect to do for one another, without extra pay.  As such there do not exist any funds to pay people for covering lectures, and even if there were, I'd expect the bureaucracy needed to make it happen would be prohibitive.  So asking for payment will be seen as asking for something unreasonable and out of the ordinary, a more annoying way of saying "no".  The PI will say "Forget it, I'll ask someone else", and will retain a memory of you as unhelpful.

@NateEldredge: When both colleagues get paid, then sure. When one gets paid significantly more than the other, then - perhaps in US Professorial culture post-docs are expected to do this. And post-docs may expect being pressured into doing this. Also, note I didn't suggest that you ask the PI for money, just that you ask him/her to make it official, so that the university (which pays him/her) will need to pay you. And I did acknowledge this could be poorly received.


text:
To what extent is the students' perspective on the lecturer credible?
I am giving a master course with another colleague, but since I am more experienced in the subject, I took the most difficult lectures.
My lectures contain a lot of mathematical equations and require deep knowledge in algebra. I tried to make them as easy as possible and even made extra videos to explain parts which I felt were difficult. I checked all available lectures to make sure I explain well. Therefore, I find my lectures among the most intuitive and informative ones.
It seems that my students (or most of them) are not that good in mathematics and found the lectures very difficult. They are rating my lectures with very bad scores although I made a lot of effort in these lectures (I have been teaching this subject for several years) this year as they are online. I see my colleagues just recording their old lectures.
As a lecturer, I try to transfer knowledge, so satisfying my students is my priority. However, I feel that my efforts were not acknowledged and I have lost motivation to prepare good lectures. I am planning to simply record lectures and read the slides as most of the lecturers at my university are doing.
Did anyone go through this? Also, how informative is this evaluation to me? In other words, how credible is it to validate if there is really an issue with my lectures or if it is just a biased opinion?
EDIT
It seems that the students are comparing my lectures and the lectures of my colleague with whom I am sharing the course. I assume they think the difficulty of the lectures is related to the lecturer and not the topic.

Good to see that you care about teaching; many academics just simply do not care.  I think your problem could be the prerequisite of your subject.  Another on-going challenge is that you are teaching mathematics.  Many (or most) students are not taught maths properly, so they absolutely hate it no matter how you sugar coat it.  If you cannot change the pre-requisite, then you need to aim lower.  Start from their level and built it up.  This may mean the students will never get to where you want them to be.

Have you asked for specific feedback?

Perhaps the students are not sufficiently aware of what they'll need to know for the course assignments, tests, final exam, etc. That is, they may not be fully aware that if you don't include the mathematical parts of your subject in your lectures, then they will have to learn it on their own for upcoming tests. Or maybe they feel that the mathematical parts should not be covered or tested on, as they feel it is outside the scope of the subject? And relevant to all this is whether other instructors of the same material at the same intended level teach with the same level of mathematics.

@Prof.SantaClaus: the problem is often not the prereq but that the students have been barely passing each prereq, without gaining any effective knowledge. Then you find yourself trying to teach calculus to people who don't know what a function is, don't know what the graph of a function is, cannot do any algebraic manipulation, think that the only solution of x^2=x is x=1 and that sqrt{x^2}=x; they have zero familiarity with trigonometric functions, and they don't know that 3/2 and 1.5 are the same number; and, of course, addition of fractions is out of the question.

@MartinArgerami: I'm ashamed how long it took me to find the other solution of x**2 = x.

Curious what the subject is, exactly. You say it requires, " deep knowledge in Algebra". Do you mean school (high school) algebra? Abstract algebra?

With all the positive feedback around here, I'd like to give a caveat: we don't know if your class is taught well. I feel there is to much bias to comforting you in the answers. I do know teachers that give their best and, well, it still isn't good. Now please note, I'm not saying this is the case here. But you should ask for honest feedback someone who can actually judge your class.

I'm not sure it's allowed but, perhaps you could give us a link to one of your lectures?

"I find my lectures..." - judging one's own work comes with a huge amount of bias. Some judge their work too harshly and some judge it too generously, and it's quite difficult to know how someone else might see it (especially for teaching, where you know the content and they don't). Many professions involve having work reviewed by someone else (or working together) for this (and some other) reasons. An explicit review step isn't so much a thing in lecturing as far as I know, but student ratings or reviews can serve the same purpose (if you can contextualise it properly).

I think vital info that you failed to exclude is how the students did in the course. Did they learn the material well?

The class is online - 'nuff said. Learning mathematics purely online without specific, realtime feedback is already an extremely difficult task - made more difficult by the new content of your lectures. Seems to be a symptom of the learning environment, not your content.

Intention is not enough, the impact has to be there. You could attempt to teach less as long as the material is adopted well - not to suggest you would do this. But offering assistance outside of lectures (even going so far as to offer additional classes outside of course hours) might encourage those who are struggling to take the initiative. In any case, breaking down the concepts and explaining the way of looking at problems helps - I find in maths those who know assume those who are listening know (we don't, we weren't taught to comprehend /why/ maths works in school just that -it does-).

Did you ask for specific feedback? Are the lectures difficult because the maths are hard to understand, or because you're hard to understand? I have had some profs who were terrible at public speaking but completely oblivious to that fact. And because they're experts in the material they teach, bad slides and explanations seem very clear to them.

What I don't see in your post is what is the nature of the subject and what the background of the students are. If you are a mathematics lecturer and you are giving a class to business students expecting them to react the way a class of math students would, then you will likely be disappointed.  I did my undergrad in Chem Engineering.  Most (not all) math profs hated teaching engineers and treated our lectures as punishment for past sins.  Most chemistry profs felt the same way - though one first-timer felt we were so different from Chem students that he thought it was a blast.

Student feedback can sometime be a difficult thing for a teacher, and it is sometimes  inappropriate that teaching quality is measured in terms of student feedback.
Let me explain this by means of anecdotes. Imagine a colleague who is a good performer, much as a stand-up comedy act. They give good jokes, tell stories, show interesting films. How would many students rate them? Quite a few students might give them a good rating on the basis that they had a good time. However ask them in five or ten years time what they got  out of that class, and they might remember one of the jokes or stories.
Now imagine a colleague who works their class. Makes them do test and examples. Makes them go places that makes them think, makes their brains ache from the thinking they had to do. It is possible that quite a few students might really give some negative feedback that this was a class they did  not enjoy and might have liked to be changed. However, ask some of these students in five or ten years what they got out of that class and it might surprise you. It might be that they remember what they were taught and use it in their profession.
Good teaching can be preparation for the future and sometimes students can only discover that when they reach their own futures. I started to find this out when I would get unexpected emails saying "remember me". "I didn't believe you when you taught me that thing, and now I see that lesson playing out in front of my eyes every day. Thank You."
We have to balance those two aspects of teaching. We have to give them something for their now and something for their future. In fact something for our future and our grand-children's future, because that is what we build by good teaching.
As has been said by others: if you care, you will find ways to improve. You are doing the right things. Keep going. Best of luck.

"Ask them [the students] in five or ten years" is, in my opinion, the right way to evaluate teachers, but of course it may be difficult to find the students and get their answers long after they've left the university. My department at least asks them as they're about to graduate.

It's also worth remembering that most students are adults and can also understand the difference between a jokester that doesn't teach them anything and someone who gives them useful knowledge and effectively explains why behind the knowledge.
Let's remember this isn't 5 year olds we're talking about. At my organization the student grading roughly matches the perception of effectiveness and the most easy going teachers do not hold best student grades.

@Mavrik In the past, there were societies where people were not considered to be "adult" until they were 50 years old. Changing the age threshold to 21 (or 18, or 16) hasn't changed human behaviour much.

@Mavrik even older adults often have a hard time evaluating teachers for something they don't know yet. In addition, emotional responses to character types or just as a mechanism to externalize guilt when some concepts don't get easily in your head are pretty universal independent of age. While the skill to recognize this may grow once you have been through a certain degree of learning, I think you will find it in all age groups. I've often seen very mixed evaluations for teachers that teach topics that are conceptually different from the other ones (and laying groundwork in that area).

This really doesn't change the fact that treating a whole generation of educated adults, who voluntarily enrolled into higher education, as children is horriblty condescending.

I'd add that this problem of not recognizing the value of what's in front of you is not unique to the education setting. It is almost invariably the case that any review given during or immediately after some kind of dense experience is going to be worthless. Just ask young Mark Twain what he thought of his father. Perspective takes time -- not necessarily years, but anything less than 24 hours of distance and reflection, and the opinion isn't worth the paper its written on.

@Mavrik Horribly condescending but unfortunately mostly accurate. Also, "voluntarily" doesn't feel entirely appropriate; a large majority of students is just following expectations while giving the bare minimum of effort or thought to what they are doing, or why they are doing it. Chalking up a poor grade to "bad teacher" is one way of doing this.

This assumes that there does not exist a golden middle, where you teach solidly, challenging and rigorous AND manage to captivate your audience while doing it. Feynman did both.

This answer comes across incredibly patronizingly. Student reviews are not just based on popularity like this answer implies

As a student I disagree with this standpoint. I generally don't conflict being funny with good teaching. When I think about the best teachers I've had I think about an organized course, good use of board/powerpoint/whatever, teaching at an appropiate level such that we can understand and made challenging but fair tests. The worst teachers rambled on during the lecture, didn't seem to care if we understand or had poor time management (went overtime). But I do agree that students may not always be the best judges.

@AccidentalTaylorExpansion I think there are two classes of student--the ones who only look at easy/hard, and the ones that look at actual learning.  Back when I was a student I considered the opinions of good students worthwhile, the opinions of poor students were inversely related to the value of the class.  I'd love to see a teacher rating system where it was organized by GPA.

Anecdotal, but the belief with student feedback at one university was "bad ratings mean a lot, good ratings mean little", as the belief was indeed that 'a hard class is maybe going to get "meh" ratings, but it still shouldn't get outright bad ratings as there are enough students who appreciate them', whilst 'an easy or funny class is often going to get very good ratings'. So only the most negative cases were actively looked into, whilst the feedback was completely ignored beyond that.

I'm aware of a variant of this: the same course is given by two different instructors in back to back terms; one of the instructor is "hard" and tenured,  the other is "soft" and a sessional.  Over the years the students have increasingly avoided the class of one to the benefit of the other.
Maybe this quote from Daniel Webster is slightly messianic, but nevertheless:

I should indeed like to please you; but I prefer to save you, what
ever be your attitude toward me.

In the end, you should do what you think is best.  You should still reflect on the difficulty of your lectures (is it really necessary?) but it's a pretty universal observation that students disproportionately prefer "softer" instructors simply because of the short term workload differential.  Regrettably many students (and their parents) are more interested in getting a degree than getting an education.

For completeness: the students favored the softer one?

It's perhaps also worth remember that a teacher that's being avoided by students cannot really teach anything useful.

@Mayou36 yes.  The scheduling favours the regular faculty member but despite this students naturally gravitate to the “softer” instructor.

@Mavrik That’s true literally - if there is 0 student in the class you cannot teach anything useful - but I would challenge this in general.  An instructor could teach nothing and give high grades, and be better liked than one who would shy away from doing this.  Such a situation happened in a developing country (because of low salaries): the instructor had another better paying job and just didn’t show up at the lectures yet gave every student A+.

I think the point raised is rather another (or I bring that up now): the amount and difficulty taught and what is received by the students is a function with a maximum somewhere. Teaching simple, slow won't make the student remember a lot. However, teaching too hard things too fast won't make them learn a lot either. The optimum is somewhere in between. So you can simply overshoot as well! Like with sports: use to light weights and you won't train, use to heavy weights and you can't even do a single exercise. The right amount of difficult is the best teacher.

@ZeroTheHero Honestly, my experience doesn't quite match your statements - the best graded teaches at my faculty aren't the most easygoing ones, but the ones that are kind but still demanding and effective teachers. The notion that teachers giving cheap grades are best rated ones simply doesn't hold water when looked at empirically.
Still, my point stands: if students actively avoid enrolling to your class, you won't each anything. There's a tradeoff there and you can be demanding, kind and effective teacher. Those are orthogonal properties.

@Mavrik I agree in some cases but teaching evaluation is dependent on lots of other factors as well.  I’m not suggesting that kind=“soft” or unkind=“hard”, simply observing that students (with exceptions) prefer good marks over pretty much anything else.

@Mayou36 agreed; this is why the reflection of the OP is quite important.

You say:

My lectures contain a lot of mathematical equations and require deep knowledge in Algebra. I tried to make them as easy as possible and even made extra videos to explain parts which I felt were difficult. I checked all available lectures to make sure I explain well. Therefore, I find my lectures among the most intuitive and informative ones.

You put a lot of effort in, but is it effective? The problem is that you are trying to determine whether the material is hard before, and after, watching your lectures. But you already know this material thoroughly, and have lots of subconscious background knowledge. It's impossible for you to truly judge your work as it'll be seen through the eyes of student for whom it is all new.
You also say:

Did anyone go through this? Also, how informative is this evaluation to me? In other words, how credible is it to validate if there is really an issue with my lectures or if it is just a biased opinion?

I'd say it's credible, but not informative. The students are having difficulty with your lectures, they're probably honest about that. They haven't offered very useful feedback for you to improve however.
I think you can kill two birds with one stone here. Involve your students in improving your lectures. Every lecture, contact some of the students to review the lecture. Assure them that they can speak freely and that you're intererested in their detailed feedback. Explain to them where you want to go, and ask them what they would need to get there. Then use that feedback for the next lecture. This should accomplish two things: (1) you get much better estimates of what is hard for them and where they need more help, and (2) the students feel involved in the quality of the lectures, which is very likely to boost their reviews.
This does increase your upfront workload, at least initially, because you can't pre-record all your lectures at the start of the semester anymore. However, it will also allow you to target your efforts more effectively.
Although the other answers aren't wrong, I think they're a bit too negative about students and feedback. I think there's a substantial share of students that do want to learn a lot, and that get quite enthusiastic and helpful if you involve them. Not all of them, but enough to make a difference.

Yes! This (in a much less comprehensive manner with worse examples) is what I wanted to write before I got to your answer. :)

I'd use an anonymous online form for feedback instead of contacting "some" of the students (which ones? why them?) or "assuring them that they can speak freely": this is vague and I am not sure what this is supposed to imply (that you won't grade them differently? IMO you can't promise you won't have biases).

I think having an actual face to face (well, over Skype or something) dialogue is more useful. It doesn't have to be about feedback on the course alone. In my experience as TA and student in the past year, 15-30m one on one talks with students have been extremely valuable. You can't really predict what topics will come up. Usually the students have at least some issue they need help with that wasn't originally on the agenda, but which might have never come to light otherwise.

Student perceptions are perfectly credible viewed from their own perspective. But they may not have universal validity.
Yes, you are suffering from the comparison with the other instructor based on the difference in difficulty of the material as perceived by the students. I suspect the exercises you give are also harder for them. Hopefully others will understand that such a thing is natural and that students can confuse such things as you note in your edit.
But teaching is about student learning and that takes hard work on their part, which they don't always recognize.
You didn't ask for a suggestion, but I'd recommend that you mix up the material a bit more. Your colleague would probably also benefit from the challenge of teaching harder material. You can assist them in this, of course.
But I agree with Prof. Santa Claus that you are doing well to care about teaching and the students. You might mention the issue to the department head to give them some foreknowledge of what is going on.

There is plentiful research to show that student evaluations are not a good way to measure the quality of teaching. See for example Clayson, 2008. That doesn't mean they are completely useless however. I am lucky that my department doesn't use teaching evals in our annual appraisal. I use evals in two ways: I look to explain a large difference in scores between years. Did something go wrong/right this year? Did I substantially change the material this year? If I'd made the material harder, I wouldn't neccessary worry about a drop in score, but if the score dropped when I hadn't then it might be worth investigating.
I also tend to look at the free-text feedback to find specific and actionable feedback. For example, when I started teach stats 101, I got a lot of feedback from student saying they didn't understand why it was relevant to their biology degrees, so I changed the program to spend longer talking about why it was important, and spent some time showing how each example was relevant to real biological problems.

how informative is this evaluation to me?

The information you got is that students think your lectures are "difficult" and "bad."  This is not informative.
I suspect your real question is: "How do I get useful student feedback?" which is a very broad question.
Some starting points are:

Identify learning objectives.
Form a hypothesis about how students achieve the objectives.
Ask students questions that test the hypothesis.
Change your teaching based on the results.

You should also take care to:

Assess students' abilities before you teach them.  This avoids the situation where it turns out at the end that students were "bad at math."
Do more than "knowledge transfer."  Have students practice learning objectives.  At the university level, some of those objectives should be more than just "knowing."

There are a variety of places you can get formal training that will help you.  If you do not have one locally, you might try http://cirtl.net/.

Its also important to take on board that some learning objectives may simply not be possible given where the majority of your students are starting.

I propose that the student feedback saying the lectures “difficult” and “bad” are informative. They say that OP needs to make a change to his teaching style. This answer offers some good suggestions to do that however

Students are not perfect judges. They are not even impartial. In fact, they are often very, very biased!

Differing motivations. Perhaps they are interested in your course. Or perhaps they are merely satisfying requirements.
Grading. The higher the grade you give them, the more positive their response will be.
Lack of reference point. Students may not have a reference point for what constitutes a "good teacher" for the course.
Perception. If students feel like they've been given a better deal in some factor (e.g. final grade, amount learned, effort spent) than they would have otherwise, you might get a more positive response.

When performing (or presenting) to an audience, the content is not necessarily the most important thing. A magician that performs a simple card trick well might be more interesting and memorable than another magician that escapes from a straitjacket while hanging upside down over a bed of nails. It depends greatly upon the presentation and how it made the audience feel.
Perhaps simplified content is not enough to earn the love of an audience. (Without a reference point, they may not even be aware that it is simplified!) You also need to make yourself appear to be a good teacher. There are various ways you might go about communicating this. Some positive ways include: more interactive lectures, questions, discussions, greeting the audience, smiles and warmth.
Perhaps some of these views are a bit cynical, but we are dealing with humans after all. And humans are quite illogical.

Coming from an masters student perspective: I would  agree with you and say that course evals are generally of little use (except when they are wayy below or above the median)
Indeed it's been proven that using methods that improve learning often makes students frustrated and confused while normal lectures  make the students learn less but are more enjoyable, more familiar and students have also been brainwashed to think good lecture equal good class. Famous experiment with a video teaching students physics, many students, thanks to confirmation bias and only paying half attention, felt like they learned but learned the wrong things.
Do the same thing where an actor in the video gives the wrong explanation and is corrected the students will learn more but leave the classroom more frustrated and confused.
The best thing to do from a student eval perspective is to inflate your stats by being funny and lecturing well and yes, even making things easy.
You gotta remember the median students want an A not subject mastery. Though this isnt as bad as it sounds, because most of education is signaling anyway, so its not like by being funny and not maximizing the learning and whatnot your short changing students, quite the opposite.
Personally, I think the best teachers play to both types of students simultainously, which is of course, tricky.

"Indeed it's been proven that using methods that improve learning often makes students frustrated and confused while normal lectures make the students learn less but are more enjoyable". Who proved that? Is that some known pedagogy fact?

it appears to not exist. woopsies.
I could swear that there was some meta-study that found that undergraduate students prefer passive learning to active learning, but I cannot find it, so it probably does not exist.

It's probably a combination of the entire course being difficult and you taking the difficult topics. The students don't know your lectures are inherently tougher, so they assume you're worse at explaining things.
I'd do 2 things. Look at the previous courses. They may be weaker than you'd assume, or simply terrible. I remember a Computer Programming-II course where it took me a few weeks to learn their CS-I was "principles" and a few small programs, so I toned things down. Then I had a section that I couldn't reach at all -- turns out they'd never written a program in their CS-I or even seen the language. I had as much trouble with a 3rd "cadre" and later found out it was them -- every other instructor also found them very unmotivated.
I'd also check whether it was an elective or required. I've found electives tend to get more serious, motivated students and if they aren't happy, it's probably my fault. Then also how it was advertised. Maybe it sounded like a simple class (after an experience I've always wondered if students see College Algebra and think "oh, I had that in my Jr. High").
Second. I'd compare notes with the co-instructor. I've only co-taught where each has their own section. Swapping lectures seems especially difficult. The other instructor may be slowing down when students aren't understanding, covering less. There may be a slight mismatch of terms, style of examples... which the students are picking up on, making it seem like 2 separate courses.
With my each-their-own-section co-teaching we had hour-long weekly meetings. With alternate lectures I'd almost want to meet for a bit after each lecture "how'd it go, where did you leave off?"

A few points in additiont to the existing answers:

since I am more experienced in the subject, I took the most difficult lectures.

IMHO in this particular situation, it may be good to tell the students beforehand that you are supposed to cover the "hard" topics. This may help them to differentiatiate topic from presentation of the topic.

IMHO the more specific feedback you ask, the easier to sort out general good/bad feelings from constructive feedback.
E.g. ask for examples of what was explained well and what wasn't and why (if possible), and do so timely (i.e. directly after each lecture).

I find my lectures among the most intuitive and informative ones.

I've also honed some explanations over time to be very concise and hopefully intuitive. However, I think that particular ways of explaining (mind models) may be intuitive for one group of learners but to others some other way may be more intuitive.  I think it possible that at this point your lecture would gain more from adding alternative ways of explanation rather than from further honing of the way that is intuitive to you.
Unfortunately, finding such alternatives is not easy at all. I think you've already done a good step in this direction by looking at how your colleagues approach the topic. However, I've once heard a lecture by someone from math didactics who made some (to me) very important points. One of them was that she told us (an audience coming entirely from STEM fields) that we are likely very similar in which (maths) explanation approaches do make intuitive sense to us: we're very likely self-selected with a liking for e.g. mathematical thinking. So are teachers (of any field). However, the student population may not be, meaning that there may be groups of students who'd do better with other approaches.
I can personally confirm that as undergrads in chemistry we had a physical chemistry professor who came from physics and we decidedly had the impression that his explanations just didn't "click" as well with our chemist mind-set as they'd have done with physics undergrads. So this may happen already with rather closely related fields.
And I've gotten feedback in a somewhat similar situation where a student heard both me (chemist) and a physicist colleague on the same topic and told me that it was completely worth while since we took quite different approaches to the same topic and thus they learned a lot by now having more and different "links".
I may add that I also enjoy hearing colleagues' explainations for the same reason: there cannot be too many different links in knowledge.
Now, lecture schedules usually do not allow time for multiple explanations, but

Over here (Germany), courses usually do not follow one textbook, but students are referred to a a bunch of (text)books. When I was a student, we were encouraged to work with whatever textbook on the topic we found suitable for us (also the library typically had different textbooks available).
You could consider presenting a different approach when a student asks about a topic.
The math didiactician told us that one of their most valuable tools is asking students to write down (e.g. as homework) how they's explain the topic, or have them explain it to each other and take notes of this.

Somewhat related, I took the teacher training with the carpentries where I learned that well-designed multiple choice questions can be used to get a quick overview over existing misconceptions/misunderstandings. In contrast to the explain-to-each-other appoach, multiple choice can of course test only for misunderstandings you are aware of.

First, congratulations for caring about teaching.

My lectures contain a lot of mathematical equations and require deep knowledge in algebra.

You did not mention the topic of your lecture, but (and really, I am not trying to argue about something I do not know) do they really need to know that?
If they need because otherwise it is not possible to have the course (say - not knowing what a derivative is and explaining what velocity is in a physics course for physicists) then please just skip this answer.
Otherwise:
My teaching experience
When doing my PhD in Physics I was giving lectures to students of biology. This was a course that was part of their curriculum (actually two: physics and biophysics).
The people who decided on the curriculum probably had not much contact with the real world outside of academia that will welcome these students once they leave the university - in practical terms, they will never ever need to understand how the Zeta function is used in some physical context. (They will never use the word Zeta while we are at it).
So I simplified it to the extreme. I went for very, very basic math in order for them to understand the core of what they need to understand. I lied (ok, simplified) many times so that they get a grasp of the topic. I suffered by telling things that are only rough approximations but at least they understood basic things.
Note that they would have never used the original content in life, at least now they can read their electricity bill.
I was very hard on exams, though - because if someone does not make the effort to understand the topics (not to learn them, to actually understand) at the basic level then there is no pity (they are either lazy or dumb. Or I may be a bad teacher).
They had usually good marks because they cared about understanding something they could understand.
My life after academia
I left academia after my PhD and went into industry. It is now 25 years and I am faced with math and physics again when helping my children.
I forgot everything in physics but is it is easy to get back on track because I had two teachers like you: they made me understand. The advanced stuff I had to pass an exam on looks like a miracle today, but the bases are there and I am a good teacher to my kids (because I understand what I am saying, and use the right level of abstraction).
To summarize

thanks for making the effort
make sure that what you are teaching is what they need to know
there is a real chance that they will see you as a great teacher if they are taught the things they actually need
some will tell stories about you 30 years from now

Should they be able to do that maths?
In my uni course (electronic and electrical engineering), we had compulsory maths in the first and second years.  In theory, this was supposed to teach us the maths skills we'd need for the rest of the course.  In fact, no-one had told the maths lecturer what we'd need.  The result was that basically the whole of second-year maths was 100% wasted effort with no application to our other modules; whilst two modules in the third year (modern control theory and electromagnetism) required matrix operations which we had never seen before, and which the lecturer did not have time to teach properly whilst also teaching the actual course content.
We all appreciated that the lecturers were in a situation not of their making.  However on course feedback we all gave negative feedback on those modules, because the engineering department as an organisation had failed to teach in a way which allowed their students to learn properly.
If your students are having problems with this, I suspect there is a very high likelihood that you are in the same situation.  Your first step should be talking to your students to find where they learnt how to do this maths.  If in fact they never did, or if the teaching was not rigorous enough to allow them to succeed in your class, then you urgently need your department to review the teaching structure.

To answer the question specifically I think the student feedback is valid and you should put it in context of your own evaluation of their learning. What you have learned is that they do not find your lectures helpful and find your teaching difficult to follow. If this is coupled with your students not doing well on assessments then it seems you did fail to teach them, plain and clear.
You mention the caveat that you think they disliked your lectures based on the algebra that you used to on them. I personally don’t think this demonstrates that your students arent prepared for the course but rather that you didnt use algebra in a clear manner. In my experience taking undergrad and graduate courses it is very easy to obfuscate the point of your teaching by wasting time doing algebra in front of them. This is simply fatiguing to follow and might not be the best way to teach.
Other answers suggest that the feedback wasnt helpful because they didnt give you suggestions to improve but I personally think that the student reviews do not serve that purpose , but instead give just an indication of how the students perceive the class. Then paired with the students objective performance in the class provide an indication of whether you failed or succeeded to effectively teach

I would say that your experience is one of the most basic reasons why professors used to have tenures. Tenure was an acknowledgement that a professor is expert in its field and able to teach. With tenures becoming rarer and rarer, more lecturers are falling in the trap you found in. With universities caving in to student opinion, this may go horribly wrong in the future.
Myself, I do care about student opinion, because university makes me to think about it basically every day. I am successful and apparently highly respected by students, but still unsure whether I should be harder on them and care less about what they think of me.
This is not much of help in your current situation, I understand. My best advice would be to avoid such situations where students can directly compare you to other lecturers, as these comparisons can be highly unfair. There is also another point you may think about: how does self-selection bias influence student opinion about you? I have recent experience where there was a questionnaire sent out to the students about the quality of online lectures in time of COVID. The feedback was pretty bad, but there was only one student who submitted any kind of feedback! I have the option to do such survey myself, anonymously for students, and when half of them responded, the feedback was actually pretty good - they were satistfied with the lectures and did not even to bother to provide any kind of feedback to the University, when asked before.
So, think of this too - who is complaining, and whether they are representative sample of your student body.
EDIT: I am not sure why this is getting negative grades, so I will clarify that: It is university's job to employ good lecturers, and student's opinion should not have too much weight precisely because the dilemma described by the OP. Placing too much weight on student opinion is University's way of acknowledging that their committees who select professors are not doing their job and really puts University's administration in a bad light.

Based on what you are judging my experience? Why you link being tenure or professor to give good lectures and to ask questions. I think even if I am close to retirement, it is always good to improve the quality of teaching and ask questions.
Btw, sometimes, my PhD students give better lectures than me and other tenures and professors.

Based on your question. If you don't like honest answers, don't ask questions. My answer was honest, borne out of experience and intended to help you. No need to get hostile on me. And to add explanation: tenure is a way for professors so they can lecture without a fear to getting "bad grades" from students that even should be in his/hers class. And to add the final answer: it is the university's responsibility to choose and employ good lecturers. So the "opinion" of the students is NOT crediblem precisely due to reasons YOU argued. I agree with you, but I am not sure you understood that.