'to assume something' implies that you make a judgement about something without examining all the evidence. For example, 'You shouldn't assume students have understood what you said, you should check first.'
See also 'to make an assumption about something'
'to assert something' = to state a fact confidently
'to claim something' = to say that something is true
'assert' and 'claim' can sometimes be used as near synonyms, though they are used slightly differently in different contexts.
Student or Learner