Not a definitive answer, since I’m a postdoc with comparatively limited experience of such conflict situations, but a few suggestions. They aim to:

• maximise the change of getting better grading without ruffling too many feathers;
• give you solid documentation if you do end up escalating the complaint;
• minimise the embarrassment in case it turns out that you’re misunderstanding something and your lecturer’s grading is defensible after all.

If you confront the teacher again, do so by email rather than in person. This has several advantages. Firstly, it allows you to look over what you’ve written and make sure you’re phrasing everything as tactfully as possible. Secondly, it gives her time to (hopefully) get over any initial anger/embarrassment and give a considered response. Confronted on a sensitive topic in person, it’s easy to get flustered and defensive, and entrenched in a position it’s difficult to climb down from later. Finally, if she doesn’t respond constructively, you have the exchange in writing, so are on firmer ground for escalating the issue to the department head.

If possible, phrase the question/request so that your desired outcome is also palatable for her. If you argue that she’s fundamentally misunderstanding the course material, she’s pretty unlikely to accept that — admitting that one’s wrong about something is already difficult, admitting one’s incompetent is a whole lot harder again. But instead, you could say that (e.g.) you have learned some of the material previously, and so know it with a different viewpoint from hers (and maybe give a couple of examples here, ideally with sources in well-established literature) and you would like to check that this angle will also be acceptable for work on the course. I’m not saying you should say exactly that — but look for something that allows her to concede that your understanding is correct, without having to admit (to you or herself) that hers is wrong; and down the line this makes it easier for her to improve her understanding of the subject, rather than antagonistic towards your suggestions. On the other hand, if she does defend her current approach, this gives her a chance to lay out her case more clearly and carefully.

If you do this, and she still doesn’t engage constructively, you’re now on very solid ground to escalate the complaint. You have a written record of your good-faith effort to sort this out tactfully. You have specific examples where she has doubled down on her misunderstanding of the material. (I’d suggest double-checking these with your advisor or another faculty member to be absolutely sure you’re right about them.) Even if the head of department supports this teacher in general, it will be comparatively difficult for them to dismiss the complaint or paint you as a troublemaker.

Not a definitive answer, since I’m a postdoc with comparatively limited experience of such conflict situations, but I do have a few suggestions. They aim to:

• maximise the chance of getting better grading without ruffling too many feathers;
• give you solid documentation if you do end up escalating the complaint;
• minimise the embarrassment in case it turns out that you’re misunderstanding something and your lecturer’s grading is defensible after all.

If you confront the teacher again, do so by email rather than in person. This has several advantages. Firstly, it allows you to look over what you’ve written and make sure you’re phrasing everything as tactfully as possible. Secondly, it gives her time to (hopefully) get over any initial anger/embarrassment and give a considered response. Confronted on a sensitive topic in person, it’s easy to get flustered and defensive, and entrenched in a position it’s difficult to climb down from later. Finally, if she doesn’t respond constructively, you have the exchange in writing, so are on firmer ground for escalating the issue to the department head.

If possible, phrase the question/request so that your desired outcome is also palatable for her. If you argue that she’s fundamentally misunderstanding the course material, she’s pretty unlikely to accept that — admitting that one’s wrong about something is already difficult, admitting one’s incompetent is a whole lot harder again. Instead, you could say that (e.g.) you have learned some of the material previously, and so know it with a different viewpoint from hers (and maybe give a couple of examples here, ideally with sources in well-established literature) and you would like to check that this angle will also be acceptable for work on the course. I’m not saying you should say exactly that — but look for something that allows her to concede that your understanding is correct, without having to admit (to you or herself) that hers is wrong; and down the line this makes it easier for her to improve her understanding of the subject, rather than remaining antagonistic towards your suggestions. On the other hand, if she does defend her current approach, this gives her a chance to lay out her case more clearly and carefully.

If you do this and she still doesn’t engage constructively, you’re now on very solid ground to escalate the complaint. You have a written record of your good-faith effort to sort this out tactfully. You have specific examples where she has doubled down on her misunderstanding of the material. (I’d suggest double-checking these with your advisor or another faculty member to be absolutely sure you’re right about them.) Even if the head of department supports this teacher in general, it will be comparatively difficult for them to dismiss the complaint or paint you as a troublemaker.