Divide praise spreading about old student (8)
I believe the answer is:
polarise
'divide' is the definition.
(I know this)
'praise spreading about old student' is the wordplay.
'spreading' indicates anagramming the letters.
'about' is an insertion indicator.
'old' becomes 'o' (abbreviation).
'student' becomes 'l' (as in L-plates for learner drivers).
'praise' with letters rearranged gives 'parise'.
'o'+'l'='ol'
'parise' placed around 'ol' is 'POLARISE'.
(Other definitions for polarise that I've seen before include "divide" , "Set or become set at opposite extremes of opinion" , "Cause to adopt extreme positions - lose pair (anag)" .)