Learn Turkish: Öğretmen kitap gösteriyor mu? - Is the teacher showing a book?

The question particle follows the 4-way vowel harmony rule based on the very last vowel of the word right before it. The last vowel in gösteriyor is o. Because o is a back rounded vowel, the particle must be mu.

How do we get gösteriyor from the dictionary verb?

The dictionary verb is göstermek (to show). If you drop the -mek, you get the root göster-. To make it present continuous, we add the suffix -iyor. We use the i version because the last vowel in göster- is e, which takes i under 4-way vowel harmony.

Why is there no word for "a" before kitap?

In Turkish, you often leave out the word for "a" (bir) when talking about a general type of object. When a noun like kitap is placed directly in front of the verb without any endings, it acts as a general category. In English, we naturally translate this as "showing a book" or "showing books".

Why is mu written with a space before it?

The yes/no question particle (mı, mi, mu, or mü) always changes its vowel to match the word before it, behaving just like a suffix for pronunciation. However, it is a special particle that is always written as a separate word with a space.