The board voted unanimously for the two at its meeting, and their one-year terms begin immediately.
Brandman, who also served as president in 2008, said she plans to work with the board and Superintendent Joshua Starr to ensure a quality education for all Montgomery students.
"I am honored to serve as president of the board during this time of transition in the history of Montgomery County Public Schools," Brandman said in a statement. "... We must seize opportunities to strengthen our collaboration with the community and other elected officials as we face the ongoing challenges of limited resources and growing needs."
Brandman has been on the board since 2006 and has served three terms as vice president. She was recently elected secretary of the Maryland Association of Boards of Education.
Barclay, the outgoing board president, served one term as vice president in 2009 and has been on the board since 2006.
Brandman is an at-large board member, and Barclay represents District 4 and the Downcounty Consortium.