is that of a facilitator while

Question

Using a case-based approach engages students in discussion of specific scenarios that resemble or typically are real-world examples. This method is learner-centered with intense interaction between participants as they build their knowledge and work together as a group to examine the case. The instructor's role is that of a facilitator while the students collaboratively analyze and address problems and resolve questions that have no single right answer.

I cannot make heads or tails of the bold part, especially Italic one. What does that of a something mean?

Answer 1

That is a very overworked word, because it has at least three distinct uses:

1. As a subordinating conjunction^†

   John said that he would play the role of facilitator.

2. As a relative pronoun^†

   Facilitator is the role that John played.

3. As a demonstrative adjective or pronoun^†

   a. That role is the one which John played.
   b. That is what John did.

In your example, that is employed in use 3b: it acts as a demonstrative pronoun^†, and its referent (the entity it refers to) is the noun phrase The role:

  The instructor's role is    that    of a facilitator.
                               ↓
  The instructor's role is [the role] of a facilitator. 

While does not belong to this clause; it is a subordinating conjunction^† which introduces the following clause and defines its relationship to the preceding main clause. Its literal meaning is "at the same time as", but it carries a contrastive sense as well, emphasizing that in this "learner-centered" approach the instructor's role is reduced and the students themselves perform much of his traditional teaching role.

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^† These are the terms employed in the traditional grammar which you are most likely to have encountered. If you have enjoyed a Thoroughly Modern approach to English grammar you may know these as 1. complementizer, 2. relativizer, 3. determiner, and 3b. fused-head determiner; and you may have been taught that while acts as a preposition.