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2013c.00042 Schubring, Gert From the few to the many: historical perspectives on who should learn mathematics. Bjarnad\'ottir, Krist\'{\i}n (ed.) et al., ``Dig where you stand" 2. Proceedings of the second ``International conference on the history of mathematics education", New University of Lisbon, Portugal, October 2--5, 2011. Lisbon: UIED, Unidade de Investiga\c{c}\~ao Educa\c{c}\~ao e Desenvolvimento; Caparica: Universidade Nova de Lisboa, Faculdade de Ci\^{e}ncia e Tecnologia (ISBN 978-989-97487-2-9/pbk). 443-462 (2012). 2012 Lisbon: UIED, Unidade de Investiga\c{c}\~ao Educa\c{c}\~ao e Desenvolvimento; Caparica: Universidade Nova de Lisboa, Faculdade de Ci\^{e}ncia e Tecnologia EN A30 D20 history of mathematics education access to education Summary: Today we take for granted that everybody can and should be offered the opportunity to learn and study mathematics. Here, we discuss the fact that it was not until well into the twentieth century that ``mathematics for all" came to be regarded as an achievable goal. Indeed, the characteristic difference between the two essential social functions of mathematics, its utilitarian function and its capability to sharpen the mind and logical thinking, led in many cultures to opposed forms of teaching: in vocational institutions and in liberal education, where mathematics over long periods was regarded as less contributing to this function than classical languages. Mathematics proves to be, as its perennial characteristics, at the crossroads between liberal and professional training, and even to be the only school discipline with such a double orientation. Mathematics education was considered in earlier times to be important only for a few gifted individuals, but now it is accepted as a necessity for all within democratic societies that are more and more shaped by mathematical-technical applications. In particular, methodological issues will be discussed as necessary for meaningful investigations into this key question of the history of mathematics teaching and learning. The variance of meaning of the notion ``mathematics for all" will be analysed as well as sociological categories like the historical change from hierarchically organised societies to functionally differentiated societies. Changes in the functions of the state prove therefore to be essential to understand interrelations between qualification profiles and striving towards egalitarian structures in education.