This study begins with the assumption that the equal sign (=) in elementary school mathematics is not merely a symbol of mathematical logic. Rather, as the equal sign (=) appears in the school math curriculum, it orders children's thinking about equality by assigning identities to things of the world--as expressions of equivalences and non-equivalences. In this, it embodies principles of identity, equivalence, and difference that have cultural dimensions beyond pure mathematics. Understanding the relationship between "cultural" and "mathematical" systems of reasoning about equality is a central focus of the study. This is linked to a problematic implicit in contemporary reforms aimed at math for all--that is, in the impulse to include, reforms aimed at math for all also delimit and exclude the children who are not part of the all. An analysis of the equal sign (and the cultural reason it carries) is to examine how cultural norms and distinctions (re)present what constitutes the collective all. Looking historically across three major periods of elementary math reform, from the 1950s to the present, I discuss how the rules for representing identity and equivalence discursively intersect with psychological principles about who children are and should be as math learners. In this, the equal sign intersects with a cultural logic about how children are ascribed identity as certain kinds of people--who think and learn in particular ways. Interestingly, the inscriptions of identity and equivalence also serve as the norm by which to establish difference. By exploring how the equal sign is related to a theory of equality that is not merely mathematics, the dissertation aims to highlight the cultural norms of equality that organize--often in taken for granted ways--reforms in math education. More than that, this is to incite thinking about how the common sense of pedagogy carries cultural assumptions about how to learn math as well as political implications for who can learn it. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]