In common with most other subjects, in mathematics it is rare to have a complete and full understanding of any topic. There is almost always more to learn, more connections to make, more insights to gain. Consequently it is a good sign when you become aware of a gap. The real question is what you do about it.

If learners only ever encounter fully knowledgeable adults who have the answer to everything, then they may develop an over-reliance on adult knowledge, and they may become disillusioned in their learning, feeling that there is no room for creativity.  Mathematics is perhaps the most creative of all the subjects taught in school, because there are always opportunities for self-expression and creativity. [see creativity in mathematics]. One of the aims of teaching mathematics is that learners become more and more independent rather than dependent [see dependent & independent learners].

When a learner says something that is unexpected, even surprising or startling, it may be that they have misunderstood or misconstrued something, but it may be that they have an insight which needs to be taken into account mathematically.

One strategy to use when someone says something unexpected is to label it as a conjecture, and to suggest that it be explored later at another time.  Some people use a sheet of paper to record conjectures which arise in class but which there is no time to pursue currently.  As long as the teacher indicates through his or her actions that such conjectures are valued (perhaps by returning to them occasionally and asking if anyone has thought more about them), such a list can serve as stimulus to further enquiry.  It also enables the teacher to continue with the lesson, but to have a record of things that might deserve further consideration.

Who can one turn to?  In a group of teachers who enquire and collaborate, colleagues can be a powerful resource, especially if the way of working together involves joint participation in mathematical activity rather than simply ‘those who know telling those who don’t’. [see collaborative practices].