A Case for Flexibility in Classroom Instruction.

EVERY conscientious teacher of mathe matics attempts to create the basis for good teaching by attaining as deep and broad a knowledge of mathematics as he can. On the immediately practical level, he plans his lessons in as much detail as he feels necessary to guide him in the teaching of each class. A creative teacher anticipates student inquiry and provides opportunity for student conjecture and ex ploration. However, inevitably there are times when a student asks a question or poses a conjecture which has not been an ticipated and to which the answer is not immediately apparent. Over the years the writer has found that at these times it is beneficial in cases of genuine inquiry on the part of the students to depart from a planned lesson and follow up a students question or conjecture with class ?discus sion, thus capitalizing on the interest and motivation of the class. In the author's opinion, it is particularly in the working out of the answers to the questions to which neither students nor teacher know the answer, or the proof or disproof of the conjecture, that creative mathematics can be learned. These often prove to be the moments of maximum excitement in the classroom not only for the teacher, but for the students. No matter how small the "new piece of mathe matics^ is and no matter how well known it may be to some, to others it is new, and finding the answer, the proof, or disproof brings excitement and satisfaction. The example that follows illustrates this point of view. The following problem had been as signed to the class :