Most public school districts around the country have publicly elected board members. They are very often parents and educators, although anyone in a community can seek election. In theory, publicly elected board members are the choice of their communities who have received mandates to carry out their ideas. If board members don't do their jobs properly, they lose their seats and the community chooses someone else it trusts. Elections mean direct accountability to the public.
While people trust public school board candidates have altruistic motives for serving and place the best interest of children first, there can be more at play. Elections, by nature, have politics involved and there is the concern that taking stances, networking and worrying about reelection can take board members' focus away from their duty to do the right thing. Additionally, there's always the possibility board members can receive the support of special interests and work not with students' best interests at heart, but to push through land sales or shift public assets in ways that may benefit businesses and private interests.
Some school districts such as New York, Chicago and Boston have school board members appointed by their mayors. The city of Baltimore's school board is appointed by both its governor and mayor in order to prevent board members from being too linked to any one official. The main idea behind appointing school board members is that the mayor or governor is elected and carries the worries and concerns of reelection--not the board members, who can instead focus on making tough decisions in a way consistent with their beliefs, guts and their perceptions of children's best interests. In theory, local politics don't come into play. Additionally, appointments ensure that board members are qualified to serve--usually having backgrounds in education, public finance or being involved parents.
Critics of the appointed school board system feel that appointments aren't any less political. Mayors and governors make decisions based on politics and people they appoint may be beholden to them for one reason or another. Appointments may also remove the voice of the people and take away their ability to directly impact the education of their children.