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For several years now, historians of mathematics have discussed the effects of historical writing on the representation of mathematics as a discipline. They have emphasized the fruitfulness of bridges established with history of science and history in general, which have contributed to strengthen local (as opposed to universal) aspects of mathematical knowledge and the dynamics of the discipline (Corry 2004; Ehrhardt 2010; Gispert 2015, foreword). Recent historiographical trends have contributed to shift the focus from the development of mathematical knowledge towards its broader cultural and social place. How is mathematical knowledge socialized, especially thanks to its teaching? what kind of values does it embody? how much prestige does it enjoy and for whom? what kind of mathematical practices do different social groups develop, and how are they influenced by historical events? These have become legitimate questions for historians of mathematics and science, which have resulted in showing the stratification of the mathematical milieu in different countries and in single national contexts (see for instance Belhoste 1998; Bottazini & Dahan Dalmedico 2001; Chabert and Gilain 2014).
In this paper, I aim at exploring further the French and English cases of the 1950s by focusing on specific sources which contribute to shape representations of mathematics - school textbooks. In doing so, I use textbooks as a strategic site for investigating mathematical conceptions in a given national context rather than for a study of classroom practices. Since the professional and personal profiles of textbooks authors are varied, these sources give indeed an access to the written productions of an extended mathematical milieu (Gispert 2015; D'Enfert 2012a). Moreover, textbooks are both reflections of on-going representations and active agents that contribute to (re)shape those representations. Far from being only an adaptation of conceptions held in the academic sphere, textbooks encapsulate and stabilize formulations that result from a process of book-making which confronts heterogeneous constraints and involves different professional communities (Radtka 2013, 303-364; Shapiro 2012, 2013). Once they come into being, they tend to materialize certain conceptions and to define what the subject they deal with should be for their readers. Mathematics textbooks thus contribute to building the discipline and to defining its identity - even though this textbook identity might be called into question by mathematicians working in...