(I'm a software engineer at Khan Academy.)
Today, SRI International released a research report on a two-year study of the use of Khan Academy in schools. We're encouraged by the results.
It doesn't cover the use of Khan Academy instead of traditional teaching as your question suggests but instead its use as a supplemental tool. The full report is available on the SRI website; I'll copy here the main bullet points from the research brief:
Student Use and Perceptions
- Student perceptions of their time spent on Khan Academy was highly positive.
- Students’ engagement level was generally high during Khan Academy sessions.
- Students perceived that use of Khan Academy encouraged greater independence in learning.
- The amount of time students spent working on Khan Academy varied considerably across and within sites, and also by school year.
Teacher Use and Perceptions
- A majority of teachers were happy with their Khan Academy experience and planned to use the Academy with their students in the upcoming school year.
- Teacher perception of Khan Academy’s impact on students varied across different learning areas.
- Teachers reported that integrating Khan Academy into their instruction has increased their capacity to support their students in a number of areas.
- Teachers who viewed the Khan Academy reports regularly found them useful.
- A positive association was found between more Khan Academy use and more problem sets completed and two outcomes: (1) improvements in student test scores, and (2) improvements in three of the four self-reported nonachievement outcomes – math anxiety, math self-concept, and academic efficacy (i.e., belief in one's ability to succeed in academic endeavors).
The report goes into much more depth on these topics as well as information about what the educators found helpful and challenging about using Khan Academy, and I'd recommend reading it if you're interested in the results.