The argument doesn’t involve the quality of teaching. Whether one teaches a subject well or not isn’t related to the premise or the conclusion.

The author tries to refute the math department’s view that it should teach the class by showing that one possible reason for that view — the fact that the class has math — is inadequate. This is flawed beacuse there could be other reasons supporting the math department’s view.

History is referenced to show that the fact a class has math doesn’t mean it needs to be taught by a math prof. We wouldn’t think a history prof. needs to teach a class just because it has history. What students know about math/history is unrelated to this line of reasoning.

The author doesn’t need to show that math profs can’t teach the class effectively. The author’s position is that they don’t need to have sole responsibility for teaching it. Maybe they can be effective, and others can be effective as well, and others should also teach the class.

The author doesn’t cite to any history “policy.” The reference to history is merely to illustrate the principle that the fact a class involves a particular subject doesn’t mean only profs from that subject’s department must teach the class. There’s no history “policy” referenced.