Crookedly makes cuts, unedifyingly including education (10)

I believe the answer is:

lopsidedly

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'crookedly' is the definition.
(synonyms)

'cuts unedifyingly including education' is the wordplay.
'cuts' becomes 'lops' (lopping is a kind of cutting).
'unedifyingly' becomes 'idly' (I can't justify this - if you can you should believe this answer much more).
'including' is an insertion indicator.
'education' becomes 'ed' (abbreviation - e.g. sex ed).
'idly' placed around 'ed' is 'idedly'.
'lops'+'idedly'='LOPSIDEDLY'

'makes' acts as a link.

Can you help me to learn more?

I've seen this clue in The Everyman puzzle in The Observer.

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