Mathematical Rigor From Within

There is a certain feel that is unique to the rarefied context of rigorous mathematics. These poems constitute an exploration of my experience of mathematical rigor when I am in the midst of exercising my skills as a research mathematician.

A CASE STUDY ON HOW PRIMARY-SCHOOL IN-SERVICE TEACHERS CONJECTURE AND PROVE: AN APPROACH FROM THE MATHEMATICAL COMMUNITY

This paper studies how four primary-school in-service teachers develop the mathematical practices of conjecturing and proving. From the consideration of professional development as the legitimate peripheral participation in communities of practice, these teachers’ mathematical practices have been characterised by using a theoretical framework (consisting of categories of activities) that describes and explains how a research mathematician develops these two mathematical practices. This research has adopted a qualitative methodology and, in particular, a case study methodological approach. Data was collected in a working session on professional development while the four participants discussed two questions that invoked the development of the mathematical practices of conjecturing and proving. The results of this study show the significant presence of informal activities when the four participants conjecture, while few informal activities have been observed when they strive to prove a result. In addition, the use of examples (an informal activity) differs in the two practices, since examples support the conjecturing process but constitute obstacles for the proving process. Finally, the findings are contrasted with other related studies and several suggestions are presented that may be derived from this work to enhance professional development.

Is Learning Data in the Right Shape?

In this short thought-piece, I attempt to capture the type of freewheeling discussions I had with our late colleague, Mika Seppälä, a research mathematician from Helsinki. Mika, not being a psychometrician or learning scientist, was blissfully free from the design constraints that experts sometimes ingest, unwittingly. I also draw on delightful conversations with the German research mathematician, Heinz-Otto Peitgen, a polyglot whose work includes advances in medical imaging and explorations in fractal geometry for K–12 students. Together, they taught me to reconsider foundational assumptions about learning, how to describe it, and how to grow it. Accordingly, I use this set of papers as a prompt for examining assumptions that numerical precision ensures scientific insight, that linear models best capture growth in learning, and that relaxing a fixation with time (exemplified by the reification of pre- and post-testing) might open up new topologies for describing, predicting, and promoting learning in its myriad manifestations.