NCTM's Standards documents (1989, 2000) call for increased attention to the development of mathematical reasoning at all levels. In order to accomplish this, teachers need to be attentive to their students' reasoning and aware of the kinds of reasoning that they observe. For teachers at the early elementary level, this may pose a challenge. Whatever explicit discussion of mathematical reasoning they might have encountered in high school and university mathematics courses could have occurred some time ago and is unlikely to have included the reasoning of children. The main intent of this article is to give teachers examples of ways to reason mathematically so that they can recognize these kinds of reasoning in their own students. This knowledge can be beneficial both in evaluating students' reasoning and in evaluating learning activities for their usefulness in fostering reasoning.
