In this article the authors explain that when engaging in a problem-solving conversation with a child, their goal goes beyond helping the child reach a correct answer. They want to learn about the child's mathematical thinking, support that thinking, and extend it as far as possible. This exploration of children's thinking is central to their vision of both productive individual mathematical conversations and overall classroom mathematics instruction (Carpenter et al. 1999), but in practice, they find that simultaneously respecting children's mathematical thinking and accomplishing curricular goals is challenging. In this article, they use the metaphor of traveling down a road that has as its destination children engaging in rich and meaningful problem solving like that depicted in the Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010). This road requires opportunities for children to pursue their own ways of reasoning so that they can construct their own mathematical understandings rather than feeling as if they are mimicking their teachers' thinking. Knowing how to help children engage in these experiences is hard. For example, how can teachers effectively navigate situations in which a child has chosen a time-consuming strategy, seems puzzled, or is going down a path that appears unproductive? Drawing from a large video study of 129 teachers ranging from prospective teachers to practicing teachers with thirty-three years of experience, the authors found that even those who are committed to pointing students to the rich, problem-solving road often struggle when trying to support and extend the thinking of individual children. After watching teachers and children engage in one-on-one conversations about 1798 problems, they identified three common teaching moves that generally preceded a teacher's taking over a child's thinking: (1) Interrupting the child's strategy; (2) Manipulating the tools; and (3) Asking a series of closed questions. Because these three teaching moves were almost always followed by the taking over of a child's thinking, the authors came to view them as "warning signs," analogous to signs a motorist might see when a potentially dangerous obstacle lies in the road ahead. By identifying these warning signs, they hope that teachers will learn to recognize them so that they can carefully examines these challenging situations before deciding how to proceed.