1. The professor stood at the lectern and gave his lecture.

2. The teacher stood at the teacher's table and began to teach.

3. The teacher stood at the lectern and began to teach.

4. The teacher stood at the lecture table and began to teach.
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Are they al correct? What other words can we use? Which ones are commonly used?

All the sentences are correct. I've never heard reference to a teacher's table, although the phrase makes perfect sense.

You can also refer to the lectern as a podium.

Out of the four sentences provided, only sentence 3 is grammatically correct. The word "lectern" is commonly used to refer to the stand or podium where a speaker or lecturer stands and delivers a presentation or lecture.

Let's go through each sentence to see the issues:

1. The professor stood at the lectern and gave his lecture.
- This sentence is correct. The word "lectern" is appropriate to describe the stand at which the professor stood to deliver the lecture.

2. The teacher stood at the teacher's table and began to teach.
- This sentence is incorrect. While it is common to use "teacher's desk" or "teacher's table" to refer to the workspace of a teacher in a classroom, it is not typically where they stand to teach. A more suitable word for where the teacher stands and teaches would be the "front of the classroom" or "front of the room".

3. The teacher stood at the lectern and began to teach.
- This sentence is correct. The word "lectern" is commonly used to refer to the stand where a teacher or speaker stands to deliver a presentation or lecture.

4. The teacher stood at the lecture table and began to teach.
- This sentence is incorrect. "Lecture table" is not commonly used to describe the stand from where a teacher delivers a lecture. A better alternative would be "lectern" or "lecturer's stand".

In summary, the word "lectern" is commonly used to describe the stand where a teacher or presenter stands to deliver a lecture or presentation. Other suitable alternatives for describing the location where a teacher stands to teach include "front of the classroom" or "front of the room".