To reason about
Solution 1:
The phrase "reason about" does in fact have a precedent for being used, and in the snippet you've posted, it is used correctly. Not surprisingly, the question of whether a person has heard this before is wholly dependent on the forms of writing and speech to which that person has been exposed. Your teacher—of what nationality is she? The phrase "reason about" is far more common in Great Britain than it is anywhere else in the world, and I'd not be wholly surprised to hear some American know-it-all claim it to be wrong.
However, being that your English teacher is an English teacher, she should be well ashamed. She should understand perfectly well that the fact that she's never heard a thing before certainly is not adequate proof for concluding that the thing in question is wrong, and anyway, one such precedent for use of this phrase is very well known amongst English readers and teachers. As soon as I saw this question, a snippet of Lewis Carroll's long poem "The Hunting of the Snark" came immediately to mind. You will find the following bit in the section "Fit the Fifth: The Beaver's Lesson":
"Taking Three as the subject to reason about—
A convenient number to state—
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight."The result we proceed to divide, as you see,
By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.
Solution 2:
The pairing is not only good plain English understandable solely by considering reason and about, but given explicitly in the OED:
- intr. To think in a connected or logical manner; to employ the faculty of reason in forming conclusions.
a. With from (a premise or datum); about, †of, upon (a subject).
Solution 3:
This does not seem to be correct to me in BrE.
I would, however, expect to see "...with how one should reason it out"
reason something out
to figure something out; to plan a reasonable course of action. Now let's be calm and try to reason this out. Let us reason out our difficulties.
http://idioms.thefreedictionary.com
Solution 4:
The thing is, when you use a preposition after a verb, people tend to parse it as a phrasal verb and look for the idiomatic meaning.
In this case, none exists. That's why it doesn't "sound right".
As blgt pointed out in their comment, the use of reason as a pure verb here is grammatically correct.
But still, to avoid the ambiguity, you should rephrase it. One way to do that is reason it out, like Marv said. EDIT: Reason it out may imply more people than you mean