Ways of thinking associated with mathematics teachers' problem posing in the context of division of fractions

Twenty-four mathematics teachers were asked to think aloud when posing a word problem whose solution could be found by computing 4/5 divided by 2/3. The data consisted of verbal protocols along with the written notes made by the subjects. The qualitative analysis of the data was focused on identifying the structures of the problems produced and the associated ways of thinking involved in constructing the problems. The results suggest that success in doing the interview task was associated with perception the given fractions as operands for the division operation and, at the same time, the divisor 2/3 as an operator acting over 4/5. The lack of success was associated with perception of division of fractions as division of divisions of whole numbers and using the result of division of fractions as the only reference point. The study sheds new light on the teachers' difficulties with conceptualization of fractions.