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This article is concerned with tensions that teachers who are studying for a graduate degree in mathematics education experience when conducting a research project. Specifically, we focused on the tensions that emerge in situations where teacher-researchers encounter the mathematical misconceptions and difficulties of their interviewees. The analysis of participants' research reports and reflections revealed that these situations are loaded with tensions between their desire to help their interviewees with mathematics and to collect data for their projects. Based on the modified analytic induction, we explained participants' decision making with situated learning theory. Possible sources of participants' tensions are discussed in terms of their methodological knowledge.

This article is concerned with exploring the mathematical knowledge and reasoning that can be engaged in providing explanations for the convention "a"[superscript 0] = 1. Posts from online open forums are used to illustrate the variety of mathematical ideas that can be utilized in the explanations. The explanations are examined in terms of quality criteria of definitions and proofs.