Constructivism is not reducible to a set of rules to follow or actions to perform. However, we suggest that it is possible to define beliefs that must be held by teachers if they are to create constructivist environments for learning. In the first part of this paper we put forward as critical four tenets of belief and follow this with descriptions and analyses of classrooms in which the teachers' intentions are to create environments based on these beliefs. We examine the mathematical understanding actions of pupils in these classrooms to determine the relevance and validity of our claims.