The way to determine if he was correct is to convert the two mixed numbers into decimal first. 3 1/7 or 22/7 is 3.142857... and 3 10/71 is 3.140845... Subtract pi from the former and we have 0.00126..., a positive number and subtract pi from 3 10/71=-0.00075.
If a-b>0 and c-b<0, then b<a and b>c or c<b<a.
Putting b=pi, a=3 1/7 and c=3 10/71, 3 10/71<pi<3 1/7, so pi lies between the two mixed numbers.