We have reflected on problem-solving examples involving division and multiplication. These examples demonstrate that solving problems in a context can help students develop and deepen their understanding of mathematical concepts as they find connections between various applications of the same concept. Let's now take a look at the teacher's role in fostering learning through problem solving.

A powerful method that teachers may use to make a mathematical process visible is to draw and use diagrams. Students won't necessarily discover diagrams on their own. For example, the young students in the "What's the Price?" video first followed the teacher's modeling of a diagram before easily moving to drawing boxes and solving problems on their own. Diagrams can provide an entry point for students to solve many different types of problems.

Another method teachers can employ is using manipulatives. Students clarify their understanding of problems when using a variety of manipulative materials to stand for things or quantities while solving problems. The use of manipulatives is most powerful when the teacher facilitates the sharing, discussing, and critiquing of various solution methods, and when the teacher also occasionally models a solution method.

Asking questions is an important teaching strategy. Students benefit from questions that help them recognize connections between various solution approaches. For example, even though it might seem obvious to an adult, not all students see connections between actual objects and a diagram that uses generic symbols to stand for the objects. Similarly, a teacher can guide students as they explore a question, such as, "How do the parts in our numerical work match the parts of our diagram?" and "How can we use numbers and operation symbols to show the actions we just made with our manipulatives as we solved the problem?"

  The teacher's role in monitoring and reflecting