Effective teachers develop and use sound knowledge as a basis for initiating learning and responding to mathematical needs of all their students.
What teachers know and believe about mathematics and what they understand about mathematics teaching and learning impacts directly on the way they organize their mathematics instruction in the classroom and upon the learning experiences for their students.
Teachers at all levels need to know their learners too, they need to be able to anticipate the difficulties that their students may encounter in their mathematics learning, to challenge and extend their students, and they should be able to describe learning trajectories and next learning steps. This demands a skillful response to teaching situations rather than simply an adherence to scripts or texts.
In some of the stories below, the writers highlight particularly well their understanding of the need to have sound pedagogical content knowledge in order to recognise and act on moment by moment teaching opportunities, to understand students’ thinking, to recognise misconceptions and to work with these, and to represent mathematical ideas appropriately in multiple ways.
I have done two postgraduate university papers in maths which provided me with in-depth pedagogical content knowledge. 2010 paper: Mathematical Literacy for Lower-Achieving Students.