Why research on proof-oriented mathematical behavior should attend to the role of particular mathematical content

When Students Prove a Theorem without Explicitly Using a Necessary Condition: Digging into a Subtle Problem from Practice

“…The construct of met-before provided us with a theoretical tool to make sense of both Jessica's content-independent and Rebecca's content-dependent proof structuring decisions. This approach contrasts to majority of proof studies that address either content-dependent or contentindependent aspects of proving (Dawkins & Karunakaran, 2016). By adopting a lens that allowed for flexible exploration, we were able to make two qualitatively different plausible explanations about the reasoning that produced the same common proof-structuring issue.…”

Posing New Researchable Questions as a Dynamic Process in Educational Research

“…The three studies in the area of students' justification schemes that we discussed in this part of the paper (Dawkins & Karunakaran, 2016;Stylianides, 2019;Weber et al, 2020) illustrate the idea that the conceptual terrain in this research area becomes more complex with advancements in research knowledge about the factors that can influence students' performance on proof construction tasks. Also, as the field's knowledge of new potentially influential factors grows, there is a consequential expansion of new researchable questions that can guide researchers' investigations towards a more refined, and presumably more accurate, body of research knowledge about students' justification schemes.…”

“…In this part, we will discuss briefly three recent studies that are relevant to the investigation of students' justification schemes (Dawkins & Karunakaran, 2016;Stylianides, 2019;Weber, Lew, & Mejia-Ramos, 2020), and we will consider how each of them can take RQ 2 to a new direction, thus giving rise to new researchable questions. We will label the new research questions as RQ 3a-c so as to indicate their conceptual (rather than actual) evolution from RQ 2 .…”

“…Moving on to the second study we will consider in this part, Stylianides (2019) drew attention to another factor that received limited attention by research on students' proof-related behavior, namely, the mode of representation (written versus oral) used by students as they communicate their proof constructions. In a way similar to Dawkins and Karunakaran (2016) but with a focus on the role of argument representation, Stylianides (2019) argued and illustrated the possible negative consequences of a representation-generic examination of students' proof constructions. Specifically, in the context of actual classroom practice, Stylianides compared the written arguments that secondary students produced and perceived to be proofs with the oral arguments that the same students presented in front of the class for the same claims, and he found that the oral arguments were more likely than the written arguments to meet the standard of proof.…”

“…Also, as the field's knowledge of new potentially influential factors grows, there is a consequential expansion of new researchable questions that can guide researchers' investigations towards a more refined, and presumably more accurate, body of research knowledge about students' justification schemes. For example, findings of investigations of new researchable questions RQ 3a and RQ 3b can contribute, respectively, to the development of content-specific (Dawkins & Karunakaran, 2016) and representationspecific (Stylianides, 2019) portraits of students' justification schemes, which in turn might help point out important trends that went undetected by prior research. At the same time, however, the field's research knowledge about students' justification schemes can become more fragmented: As research reports of content-and representation-generic findings give place to reports of content-and representationspecific findings, it is less justifiable for researchers to cluster together or even compare findings across studies that did not account for these factors.…”