Teaching Fellows Act of 2005 - Amends the Higher Education Act of 1965 to direct the Secretary of Education to make competitive matching grants to States for Teaching and Partnership Fellows programs of scholarships for prospective public school teachers.
Provides for scholarships: (1) under the Teaching program for high school seniors and second-year college students to encourage them to enter the teaching field and become career educators; and (2) under the Partnership program for individuals, such as teaching assistants and others currently in the education field, to complete a bachelor's degree with a State license or certification in education through partnerships among associate-and-bachelor-degree-granting institutions.
Requires scholarship recipients to repay specified amounts if they do not fulfill certain obligations to teach in their State's public elementary or secondary schools, under such Federal Teaching and Partnership Fellows program.