Due to the possessive determiner in front you have to apply the mixed inflection pattern for the adjective:
„Unser nächster Lehrer…“ (mixed)
But: „Der nächste Lehrer…“ (weak)
There is an easier alternative. Think of possessive 1st and 2nd person personal pronouns as weak ein words. Then you can apply the Strong form case
Source : http://gregreflects.blogspot.in/2015/02/how-to-memorize-german-cases.html
unser can be the the determiner for a masculine or neuter possessed item, but I wasn't referring to the inflection of the possessive determiner.
Instead I wanted to point out that the inflection of the adjective changes depending on whether there is a definite article (use „weak inflection pattern“) or a possessive determiner (use „mixed inflection pattern“) in front of it.
Another way of saying this and something that is noteworthy is that unser declines as ein does (also similar words like kein mein dein etc do that) and not like the definite article (der in this case).
Note that unser is the most basic form of the word so it is: unser-- which can be: unser, unserem, unseren etc. and not unserer (that's possessive) nor does the -er- in unser show masculinity (it's just the most basic form and corresponds to ein Lehrer in that it does not show gender).
You would've thought that it would be just like "welcher nachste Lehrer" in which case only an e is enough like you say but here welch declines like a definite article.
There are different versions of the possessive determiners in German.
One of them is when you use the determiner in front of the noun:
„Unser nächster Lehrer ist ein Mann.“ – “Our next teacher is a man.”
One of them is when you replace the noun, without the usage of an article.
„Euer nächster Lehrer ist eine Frau. Unserer ist ein Mann.“ – “Your next teacher is a woman. Ours is a man.”
And the third variant is to replace the noun but with the usage of an article.
„Euer nächster Lehrer ist eine Frau. Der unsere ist ein Mann.“ – (probably translated as above)
You also use the masculine gender if you're talking generally or you don't know. E.g. Jeder wird beantworten = Everyone will answer. Jeder is used to convey everyone in the general sense or if you don't know - there might be a group of women in there as well.
I believe Unser nächster Lehrer functions the same way - up to that point, you're talking about the next teacher be it a man or a woman.. then comes ist ein Mann - that the teacher is a male.
Both are correct, it just depends if it's high or low German, Low German pronounces as ks, High German pronounces as the "common" ch-s.
The standard German we are learning here is heavily based on High German so the "correct" would be ch-s.
Low German is also used "only" on North Germany so if you want to make more people happy use ch-s. Another good reason to pronounce this way is that nobody speaks Standard German on everyday use, they speak their local variant. As you are an outsider it's better to start using the standard pronunciation.
The ch-s pronunciation seems to be the correct one, as one would speak it this way in situations where comprehensibility matters, such as dictating. But I guess in usual conversations you'd encounter the word mostly pronounced with a sound closer to ks, or x, similar to the sound in Porsche Boxster.
"Our next teacher is a man" is the suggested answer from Duolingo, although it will undoubtedly have "one man" in the database of acceptable answers as well (I think it adds that automatically, even though it sounds strange). Because it's in the database, there's a chance it will be suggested as a correct answer when you make a mistake in your answer. Perhaps you made an unconscious error or encountered a bug.
"Lehrer" is also used as the "generic masculine" to refer to a teacher of unspecified or irrelevant gender. "Unser nächster Lehrer" would normally be understood as "our next teacher," and not "our next male teacher." (The same goes for other masculine nouns referring to people: the masculine form can also be used generically.)