In this chapter I focus on lesson fragments during which children come to articulate forms of knowing that are inconsistent with the mathematical ideas that the two teachers present in the classroom intend to foster. I do this not to denigrate what children know but rather to articulate this knowing as the very ground and material for all mathematical knowing that subsequently emerges. In their everyday of knowing, we actually find the very methods that allow any higher form of knowing to emerge in and through interactions with other members of society (their peers, teachers). In this, therefore, I radically depart from the approach expressed in Piaget’s work and in conceptual change research, both of which take deficit perspectives. For example, throughout Piaget’s work we can find characterizations of children’s where explanations such as “lack of exploration,” “a general deficiency,” “still passive,” “cannot analyze,” and “is incapable of abstraction” abound. In the constructivist conceptual change literature, we can find expressions such as “eradicating” children’s ideas and using pedagogical strategies so that they “abandon” what they know. Piaget and others regard children through the lens of adult scientific rationality that is the ultimate goal of a person’s development.