• Conjectures and Refutations in Grade 5 Mathematics

    David A. Cramer
    This article makes a contribution toward clarifying what mathematical reasoning is and what it looks like in school contexts. It describes one pattern of reasoning observed in the mathematical activity of students in a Grade 5 class and discusses ways in which this pattern is or is not mathematical in order to clarify the features of a pattern of reasoning that are important for making such a judgment. The pattern    involves conjecturing a general rule, testing that rule, and then either using it for further exploration, rejecting it, or modifying it. Each element of reasoning in this pattern is described in terms of the ways of reasoning used and the degree of formulation of the reasoning. A distinction is made between mathematical reasoning and scientific    reasoning in mathematics, on the basis of the criteria used to accept or reject reasoning in each domain.