They speak deliberately slower to - me.
Well, if we're going to get all technical here, the word "slower" is an adjective, and an adverb is what you want. So one should write "deliberatrely more slowly"--slowly is the adverb.
Definition of SLOW
Usage Discussion of SLOW
Some commentators claim that careful writers avoid the adverb slow, in spite of the fact that it has had over four centuries of usage <have a continent forbearance till the speed of his rage goes slower — Shakespeare>. In actual practice, slow and slowly are not used in quite the same way. Slow is almost always used with verbs that denote movement or action, and it regularly follows the verb it modifies <beans…are best cooked long and slow — Louise Prothro>. Slowly is used before the verb <a sense of outrage, which slowly changed to shame — Paul Horgan> and with participial adjectives <a slowly dawning awareness…of the problem — American Labor>. Slowly is used after verbs where slow might also be used <burn slow or slowly> and after verbs where slow would be unidiomatic <the leadership turned slowly toward bombing as a means of striking back — David Halberstam>.
It seems like a poorly structured sentence to begin with so I'd probably ask the instructor where s/he was educated.