"mathematic","mathematics","Mathematic" & "Mathematics"

It should be mathematics.
FYI, names of school subjects are not capitalized except for language subjects, specific subjects and special courses.
E.g.:
-science, geography, visual arts, physical education, English language, civics, Malaysia history, etc.

COLOURED PENCILS or COLOUR PENCILS?

They are called as 'coloured pencils', not 'colour pencils'.
A photograph can be described as a 'colour photograph', not a 'coloured photograph' because of its natural colours.

Is it right to say "do correction" (especially for teachers)?

When a student makes a mistake or an error in his/her book, we should put a remark - "PLEASE MAKE A CORRECTION."
Informally, we can say 'Please do a correction'.
If there is more than one error, they need to make corrections.

So, it's wrong to say or write 'please do correction'.

#'Correction' is a countable noun
However, it can also be an uncountable noun if we refer to the general action of 'making something right'.
For example;
'Sentence correction is needed in his essay.'
'There are a few types of geometric correction.'