In class, our teacher talked about different ways to use consequently. We can use it at the beginning of a sentence, or use it after colon(;) to connect two sentences. However, our teacher say that we can put consequently between commas to connect two sentences.(e.g.: The weather is good, consequently, we can go picnic today.) The teacher said consequently can be used in the third way for academic paper. But I am just not sure it can be used this way. Can anyone help me with this question?
Your teacher is incorrect; "The weather is good, consequently, we can go (have a) picnic today." uses a comma splice, which is considered an error, particularly in the strict context of academic writing. Stick to the first two methods.
I'm sure some people write that way — it's not unusual to see a comma where a semicolon would be more appropriate — but that's not specific to consequently, it's not specific to academic writing, and it's not a good model to follow.
Your teacher may have been led astray by (standard) usages like "Everyone seemed to agree with the proposal; I felt, consequently, that I had no choice but to go along with it." In this example consequently is surrounded by commas because it's embedded inside the "I felt that […]" clause.
"This research shows that dogs are more likely to eat brand one over brand two. Consequently, owners are feeding their dogs brand one, even though it stunts dog growth."
Use it in the exact same way!