Contingency in the mathematics classroom: Opportunities taken and opportunities missed

We describe and analyze three episodes from mathematics classrooms. In each case, the teacher was confronted by a "contingent" situation that they had not anticipated or planned for yet that offered interesting and fruitful learning possibilities if pursued. In two cases, we analyze the teacher's response; in the third, we speculate how they might have responded. In each case, we propose that the teacher's ability to capitalize on these contingent situations is underpinned by their knowledge and awareness of the mathematical potential of the unexpected opportunity and by an interest in, and commitment to, mathematical enquiry.