Abstract
Professional organizations of mathematics educators and mathematicians are calling for major reforms in the teaching and learning of mathematics. Among those reforms are a shift in emphasis in curriculum from mastery of lists of unrelated mathematical concepts and skills to exploration of rich mathematical topics and problem situations, and a shift in learning from memorizing and replicating algorithmic procedures to investigating and solving complex problems. To help students achieve proficiency in solving problems, the curriculum must focus on development of the major concepts of mathematics, the enhancement and enlargement of students' repertoires of problem-solving strategies and reasoning methods, and the refinement of communication and collaboration skills. Because they present intriguing problems whose solutions or solution methods are not immediately obvious, and require the application of concepts from different areas of mathematics, and, in some instances, knowledge from other content areas, investigations are powerful vehicles for helping students achieve expertise in solving problems. The nature of investigations and their educational benefits are described. Three types of investigations, whimsical, real, and mathematical, are defined and illustrated. For each investigation, the mathematical content and problem-solving strategies are identified, and a solution method is presented. The responsibilities of the teacher, before, during and after an investigation are described.
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