Knowledge in different research paradigms is discussed: mathematics, mathematics education and a paradigm of practical knowledge. I argue that the three paradigms as highly distinct, different and important. They try to cope with entirely different types of knowledge, all highly relevant for mathematics teachers.

While mathematics is deductive and mathematical education is evidence based, practical knowledge is a type of knowledge that professionals in any profession develop by experience and by exchange with other professionals. It is based on experience more than on written text. It is well known that it to a large extent is difficult to articulate. Such knowledge is also essential in important types of mathematical knowledge. We discuss the role played by vagueness in mathematics. We also discuss linguistic mathematics knowledge which typically is present but mostly unformulated, as the mother tongue.

I argue that mathematics, pedagogy and mathematics education suffers from drawbacks by being strongly rooted in the positivist tradition, in which knowledge can always be expressed in words – otherwise it is not knowledge. Central aspects of teacher’s day-to-day profession are too complicated to be captured in words. However, work has been done to allow such practical knowledge to be formulated among professionals. I would like to sketch a more fluent cooperation between the paradigms, in which the advantages of all the different knowledge types may interact and become increasingly useful to each other.

For such an idea to reach reality, an efficient meeting form is needed. The Dialogue Seminar is developed precisely to study and communicate difficult-to-articulate practical knowledge among experienced professionals from different areas, using analogue and metaphor as catalysts. This offers mathematicians, mathematics education researchers, mathematics teachers and teacher students, and others, an excellent opportunity to listen in depth to each other, and to have a dialogue.