Collaboration between Mathematicians and Mathematics Educators is crucial in advancing knowledge on the teaching and learning of Mathematics, particularly in advanced Mathematics pedagogy. However, there is a need for the type of collaborations where Mathematicians and Mathematics Educators can find common ground, and the synergy of their expertise results in new, hybrid meanings and understandings that can benefit practice. This paper aims to help researchers from these communities come together by presenting a novel methodology for collaborative inquiry and qualitative data analysis--"dialogical inquiry"--based on Mikhail Bakhtin's theory of dialogism. In this framework, "truth" is conceived as never finalised but always coevolving through dialogue that involves participants with different "voices" and "languages". We describe the three principles that underpin "dialogical inquiry": motivation, power balance and a process for solving disagreements. These three principles interact together to create a space where critically productive dialogue allows for meanings to coevolve and new, hybrid understandings to emerge. We illustrate our operationalisation of these principles (i.e., the methods of "dialogical inquiry") in two areas: understanding solutions to linear ordinary differential equations and making meaning of the Bakhtinian concept of superaddressee. We reflect on our use of this methodology in Mathematics Education research and invite the readers to create their own dialogic spaces of collaboration.