Introduction

It is often the case that beginning K–8 teachers emphasize basic skills and procedures in their classrooms, but rarely probe more deeply for understanding and connections. This is not surprising, considering that this is what they probably experienced in their pre-college courses. Future teachers who themselves were students in classrooms that emphasized skills and procedures often lack deeper understanding or have developed basic misconceptions that make it difficult for them to probe or to answer questions from their own students about concepts and processes. The MAA Committee on the Undergraduate Program in Mathematics' Curriculum Guide [54] and the National Council of Teachers of Mathematics' Principles and Standards [62] recommended goals and objectives include that students develop the skill of understanding, representation, connections, and reasoning. This means that the curriculum should be seen as related ideas and concepts that can be used to solve a variety of problems, not as a set of unrelated facts and algorithms to be memorized.

Faculty teaching math content courses for teachers have a short window of opportunity to provide future teachers with experience in a conceptually rich environment. Faculty in these courses should model not only higher levels of questioning, but encourage higher level thinking. Unfortunately, often students find it difficult to know what they do not know. That is, students are not even aware that they do not understand the concept to which they have just been exposed or what they think they understand is actually a misconception.