Old teacher’s concession welcome, by the way (7)
I believe the answer is:
sophist
'old teacher's concession welcome by the way' is the definition.
I can't tell whether this definition defines the answer.
'concession welcome by the way' is the wordplay.
'concession' becomes 'sop' (sop is a kind of concession).
'welcome' becomes 'hi' (both can be used as a greeting).
'by the way' becomes 'st' (I've seen this in other clues).
'sop'+'hi'+'st'='SOPHIST'
Can you help me to learn more?
(Other definitions for sophist that I've seen before include "One using clever but fallacious arguments" , "philosopher" , "One who uses clever but false argument" , "one might plausibly argue" , "Clever but fallacious reasoner" .)