Triangulation is an important idea – I’ve been reading about it in Visualising Research by Carole Gray and Julian Malins.
Mine is sort of a back door approach; I’m interested in figuring out exactly what mathematics is and what an outsider can know of it by looking at the way that it’s done, how it’s passed from person to person, rather than theorising about the ideal of it (I’ve started to find that Wittgenstein’s pretty relevant, but grappling with the beast that is the philosophy of mathematics is not my aim).
My main source of data then is looking at maths presentations at conferences and analysing them – seeing if there’s anything I can find that’ll offer a clue, or something interesting to follow. I’ve just finished a playful paper looking at why maths presentations sometimes sound like cookery programmes, but I can’t publish that just yet. I’m using methods from linguistic pragmatics and looking at representations and diagramming, carrying out qualitative textual/content analysis and interpreting what particular features that stand out can tell us about the practice of mathematics (any insight, for the outsider, is potentially interesting..).
My other main method – in a sense, a form of data analysis – is my creative practice, which I’m using to experiment with diagramming and test out any ideas I come up with in my analysis of these talks. As well as theorising about how another person made their decisions in choosing different modes and forms of representation, I’m trying it out through embodied thinking and first-hand experience. That’ll require some rigorous recording of reflections and sensations to try to answer questions about what works and what doesn’t. So that’s my second viewpoint, reasoning through making.
Thirdly, I’m making as many links as I can with maths practitioners so as to be able to keep up exchanges and run what I’m doing past the people who work with mathematics all the time. That way I can make sure that the research isn’t just flying completely out of the window, and is remaining relevant/interesting to mathematicians (plus I’ve found in the past that some maths people really enjoy approaching what they’re doing from a completely surprising angle). My third viewpoint then is through the eyes of others who know mathematics from the inside.