To add context to the exploration you've just completed, we'd like to offer two additional points.

The first is that problem solving, like all the Process Standards, does not occur in vacuum. A rich problem calls on communication, reasoning and proof, connections, and perhaps most critically on representations -- both our ability to interpret them and to create them. As good practices are established with respect to these other standards, improvements will occur in problem solving.

The second point is a self-reflective one. Now that you have worked on the Rods problem, can you recall what it was like not to know the answer and a solution method? When we have established a method for solving a problem, we may forget the open-ended feeling we had when we first approached it. A way of understanding this may be trying to recall what it's like not to be able to do some everyday task, such as reading or riding a bicycle. Once an effective method is learned, it may squeeze out or block the recollection of the challenge of the task.

For people who are not teachers, this may be no great hardship. Effective approaches are established and can be relied on. But for mathematics teachers, particularly during problem solving, this "blocking" effect is not helpful. To be effective in the classroom, you must be aware of your own practices in the context of the content you teach, but you must also be able to understand and guide students who have neither these practices nor this content, and whose approaches, correct and incorrect, will differ from yours.

Now we'd like you to write about your own classroom practice. As you make your journal entries, keep in mind the material you've just worked on as context.

• Examine the work you did to solve the problem, any notes you made. If you were now asked to hand this work in to your instructor, what might it reveal to that instructor about your problem-solving practices?
• How does your personal problem-solving approach influence your teaching and assessment practices?
• As a student, do you remember using problem situations to help you make sense of new concepts and procedures? What was the effect? Did this work help you make new connections?
• How can questioning be used to help students learn from errors or wrong turns in problem solving? What questions do you ask yourself when you are struggling while solving problems? Are these the same questions you would ask your students? Why or why not?

• Use pen and paper.
• Use a word processor.
• Use the form below.

Your work will be displayed in a printer-friendly format to enable you to print.

Thank you for writing in your journal. Please keep your entries in whatever format you choose -- you will find them useful for reference later.

  Learn how the Standards define problem solving