This article is great and potentially very useful to me. The author describes and laments the following problem:
In addition, the production of advanced mathematical texts, such as theses or journal articles, is often divorced from mathematicians’ experiences of creating mathematicswhich can lead them to a sense of personal alienation from their work.
He even brings in a bit of Marx.
Apart from introspective reports like Poincaré’s (1908) perhaps the most noteworthy example of an account of actual advanced mathematical activity is that of Tall (1980) who was in the almost unique position of being a mathematician as well as a mathematics educator. Tall’s approach could be described as analytical autoethnography in that he was ‘a full member in the research group or setting, visible as such a member in the researcher’s published texts, and committed to an analytic research agenda focused on improving theoretical understandings of broader social phenomena’ (Anderson, 2006, p.375). The lack of any similar papers to Tall’s since its publication, apart from Chick’s (1998) deconstruction of her own thought processes in advanced mathematics, indicates that such an approach is difficult for mathematicians to emulate as they generally lack the ability to qualitatively reflect on and interpret the significance of their actions.
Tall and Chick sound interesting.
Samuels outlines four data capturing techniques: Proof plan, activity transcript, concept map, and annotated draft and transcript, and argues that these could easily be integrated into ordinary mathematical work and could even be beneficial, although the analysis thereof should not be done by the mathematicians themselves. This is something I could make use of as a data gathering technique.
Here is an example of a proof plan:
An activity transcript is an account of an episode of mathematical activity. It combines notes from the mathematical activity with a journal style account. […] It is divided into four parts: a background to the mathematical activity (written eight days after it took place), notes from the activity itself, a written-up version of these notes (also written after eight days) and reflections on the activity (written after three weeks and written up after two and a half months). The time delay in writing up a mathematical activity appears to be important but an appropriate length of time may depend upon the writer’s context and personal preferences.
Next are concept maps:
Graphical tools for organizing and representing knowledge. They include concepts, usually enclosed in circles or boxes of some type, and relationships between concepts indicated by a connecting line linking two concepts. Words on the line, referred to as linking words or linking phrases, specify the relationship between the two concepts. (Novak and Cañas, 2008, p.1)
The final data capturing technique proposed here is an annotated draft and transcript. It is similar to the approach used by Valerie Eliot to represent the annotated draft of her late husband’s poem, The Waste Land (Eliot, 1971). I developed this technique as a means to capture my ‘thinking aloud’ as I re-read my internally published research reports.
However, as explained above, there was an additional non-mathematical motivation behind including the second and fourth examples. Rather, they exemplify how a mathematician might present content laden data, with each technique potentially providing insight into their thought processes when carrying out advanced mathematics research as well as conveying their mathematical meaning.