I plan to develop an open-source software application that speaks words in a foreign language while displaying pictures that represent the words. Can teaching methods (such as Rosetta Stone's language immersion software, which also displays a picture of a word while pronouncing the word) be patented? I'm concerned that someone might already have a patent over this method of instruction.
The US patent law does not exclude business methods from patentable subject matter. However, for software and mobile applications requires that the business method works in conjunction with a machine such as a computer or a mobile device. You can check check out the current case law In re Bilski http://en.wikipedia.org/wiki/In_re_Bilski regarding the same about software and business methods.
However, I'm not sure that you will be able to get a grant if you file a patent application before USPTO for an open source software.
To directly answer part of your question - yes methods of teaching can be patented. I did a quick search in the class that covers educational devices (434) and had over 5000 hits of patents with the word "method" in at least one claim. Also searched for any patent with both method and teaching in the clam wording. About 1000 hits.