Upon receipt by the educational service district superintendent of a resolution adopted by the board of directors or a written petition from a first-class or second-class school district signed by at least twenty percent of the registered voters of a school district previously divided into directors' districts, which resolution or petition shall request dissolution of the existing directors' districts and reapportionment of the district into no fewer than three directors' districts and with no more than two directors at large, the superintendent, after formation of the question to be submitted to the voters, shall give notice thereof to the county auditor who shall call and hold a special election of the voters of the entire school district to approve or reject such proposal, such election to be called, conducted and the returns canvassed as in regular school district elections.
If approval of a majority of those registered voters voting in said election is acquired, at the expiration of terms of the incumbent directors of such school district their successors shall be elected in the manner approved.