After you have explored the Unit Prices problems, please answer the following questions:

• Which strategies did you use to work on these problems? Which were familiar to you? Which were new, or represented a fresh approach to the given problem?
• What insights into connections between multiplication and division did you gain while solving these problems? Did you use multiplication to check your answers?
• How can different problem-solving strategies help students understand operations with decimal numbers?
• What connections did you make to other mathematics and other problems as you worked on this set of problems?

Problem solving and the exploration of specific strategies, such as using diagrams or manipulative materials, can help make the connections between mathematical ideas more apparent. Rich problems and various solution approaches provide learners with a platform for discussing, investigating, and learning complex mathematical ideas.

As you analyze your own problem solving, you are experiencing the importance of reflection in the problem-solving process. The more experience you have in solving problems in a variety of ways and reflecting on your own approaches to the problems, the more effective you will become in understanding a variety of ways that students approach problems and in helping students as they develop new concepts through problem solving.

As teachers, it is important to frequently reflect on our own reactions to students' problem-solving experiences. Our own knowledge of more sophisticated solution methods can serve as a backdrop for reflecting on and supporting students' initial efforts, but we shouldn't necessarily share this information with students immediately. For example, once we know a concept or calculation procedure, using manipulatives or diagrams to represent a related problem may on first thought seem more complicated or confusing than the seemingly straightforward approach of teaching the procedure that we were taught. But on further reflection, the benefits to students of being able to make sense of new ideas through the use of tools, rather than only using step-by-step prescribed procedures, become more apparent.

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