Aran was born Zalman Aharonowitz in 1899 in Yuzovka in the Yekaterinoslav Governorate of the Russian Empire (now Donetsk, Ukraine), and received a religious education in a heder. He later studied agriculture in Kharkov. In his youth, he was active in Tze'irei Zion, and in 1917 became a member of the "Self-Defense Organization Committee" of the movement. He worked as a teacher and a statistician from 1918 to 1923. In 1920, after the party split, he joined the Zionist Socialists and was a member of its secret Central Committee from 1924 to 1925.
In 1926, he immigrated to Mandate Palestine, where he joined the Ahdut HaAvoda Party. He worked in building and road construction. In 1930, after Ahdut HaAvoda merged into Mapai, he was appointed the new party's General Secretary in Tel Aviv. From 1936 to 1947 he worked in the Histadrut Executive Committee as Treasurer and Director of the Information Department, and was one of the founders of the School for Histadrut Activists. He also became a member of the Zionist Executive Committee in 1946 and a member of its Presidium in 1948.
As Israel's Minister of Education, he introduced "Jewish Identity" and Jewish tradition into the curriculum and promoted the expansion of technical education. In 1955, the Knesset accepted his reform program for the Israeli education system and his demands for a secondary education diploma, as well as extension of Israel's compulsory education Law to the ages of 14 to 16. He also promoted the integration of children from different backgrounds into the same schools to accelerate Israel's melting-pot ideal and cut down socio-economic gaps in the Israeli society, including recreational activities for development town residents.
As a government minister in 1967, he initially supported the majority position which sought a diplomatic solution to Egypt's closure of the Straits of Tiran, rather than a pre-emptive strike, which he also felt posed a great risk to the home front and the Israeli Air Force. He also opposed the occupation of East Jerusalem.