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Did you ever discover something on your own, lean back, and smile to yourself with a feeling of satisfaction from and appreciation for your discovery? What gave you that sense of satisfaction? Most likely, it was the sheer joy of uncovering a fact on your own and thinking, "Aha! I've got it!" This article presents ways to create and use guided-discovery lessons. As the name suggests, a guided-discovery lesson includes the important aspect of discovery. We call this discovery aspect the Aha! component. Such a discovery is an inherent part of the way that mathematics is developed and should be a part of the way that mathematics is taught in school. At times, because of curriculum, assessment, state mandates, and other influences, mathematics courses may be devoid of this Aha! component, which positions the study of mathematics as a livin, breathing science.
The purpose of this article is to have readers experience guided-discovery lessons from two vantage points: first, as someone engaged in a guideddiscovery activity and then as an author of a guideddiscovery lesson. This approach maximizes readers' ability to use and create guided-discovery lessons in the classroom.
WHAT IS A GUIDED DISCOVERY LESSON?
Guided-discovery lessons offer students an opportunity to become archaeologists on a mathematical dig. In a guided-discovery lesson, students sequentially uncover layers of mathematical information one step at a time and learn new mathematics. Before we delve any further into our definition of guided discovery, perhaps the best way to simulate the learner's role in a guided-discovery experience is for readers to complete such an activity. This opening activity puts readers in the role of the learner. We have tried to select a mathematical situation that might be new, so that readers can have a guided-discovery experience. At this time, we invite readers to complete "Fibonacci and the Preservation of Area," the guided-discovery activity that begins on page 11.
REFLECTING ON THE GUIDEDDISCOVERY EXPERIENCE
After completing the activity, readers should reflect on the following questions.
Was the problem new to you?
Did you know the answer after reading the problem?
What question best represented the Aha! component?
Did you understand the answer after completing the activity?
Could students complete this activity independently even on a day when the teacher is absent?