So, I think the appropriate term is certainly "pick somebody up on something" per Longman definition (if I am not missing something) :

      Longman entry:       Pick

       Subentry : Pick up on  --- on the 3rd acception of the word:

       Definition_____to criticize someone slightly for something they have said. Ex; " The Senator picked him up on his use of the word "deception".    "

      Other way of seeing that is by saying "He was trying to rally support on his views that were (always) contrary to the one of the teacher, just because he had taken dislike of the guy."  

      His dislike of the teacher made him rectify/resist/emend/oppose to/attack every idea/statemtent/opinion coming from the him (the teacher).

      His disapproval (wether justified or not) of the teacher´s ways/style/character, made him dismiss/minimize every statement coming from him, afterwards.  Once the class was finished.

      He took a gratuitous offense, problaby because of some question he made, what generated some misgiving between them, so he criticizes him after his classes.

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       The way you asked would make we think and answer that way, in that direction and sense, no way of being different!   That´s why context is everything!

       I hope we have got it right now.