Mr. Paul will be teaching the students mathematics next year. In this sentence, the correct answer is "will be teaching." Let's break it down to understand why.

1. Future tense: The sentence is talking about an action that will happen in the future, specifically next year. This indicates that we need to use the future tense.

2. Continuous tense: The action of teaching is ongoing or continuous. Mr. Paul will be engaged in the act of teaching for a period of time. To express this, we need to use the continuous tense.

3. Subject-verb agreement: The subject of the sentence is "Mr. Paul," which is a singular noun. When using the future continuous tense with a singular subject, we need to use the helping verb "will be" followed by the present participle form of the main verb.

4. Present participle: The present participle form of the verb "teach" is "teaching." When using the future continuous tense, we use the present participle form of the main verb.

Putting it all together: So, to express the idea that Mr. Paul will be engaged in the ongoing action of teaching the students mathematics next year, we use the future continuous tense and say "Mr. Paul will be teaching the students mathematics next year."

Summary: In summary, the correct answer "will be teaching" is used to express the future continuous tense with a singular subject. It indicates that Mr. Paul will be engaged in the ongoing action of teaching the students mathematics next year.