Do I need Prolog to teach logic concepts? I think that the use of some logic programing language like Prolog, for example, could be a good way to explain the symbolic/mathematical logic concepts but also apply this concepts in a real programming world, however, but I'm not getting it.

The problem is that Prolog are not largely used in real applications. I tried to use this IBM's NLP as an example, but this pattern matching exercise does not covers much concepts from mathematical logic.

The question is: How can I use programming to instruct students in topics related to symbolic/mathematical logic?
Logic is a tool for distinguishing true and false statements. If a statement is proved, then it is true. If its negation is proved, then it is false. Hence the core skill is checking whether a proof is correct and finding/inventing a proof of a statement.

Prolog is based on a subset of logic. A Prolog implementation finds proofs automatically. IMHO, anybody should learn how to find proofs by hand and have a bit of practice before using any automatic prover. Prolog is so weak that it can't find even simple theorems that are of interest, like theorems of Peano's Arithmetic. Moderately complex statements, like the axiom of induction, can't even be expressed in Prolog. No wonder that


  I tried to use this IBM's NLP as an example, but this pattern matching
  exercise does not covers much concepts from mathematical logic.


Prolog surely is an example application of logic, but it is too weak to show what mathematical logic is. A proof assistant is a better way. Prolog is not the only application of logic. Other options are reasoning about programs, hardware, and computer systems, proof assistants and provers, and Artificial Intelligence.