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Research into didactics of calculus has maintained a long-standing interest in students' grasp of the relations between definite integrals and areas. This study comes to contribute to this line of research by unpacking how students use the concept of area to find definite integrals. Specifically, we focus on mathematical situations where the integral functions are represented graphically and change signs between the limits of integration. We conceptualize these situations through the lens of Fischbein's theories of mental models and contemporary modelling literature. This lens is applied to authentic data corpus collected in a large first-year mathematics course--video-recordings of solutions that students submitted as part of their coursework and students' responses in multiple-choice final exams. The analysis revealed two modelling paths that students' solutions followed: one based on areas that the function enclosed, and one based on the integrals of the assigned functions. While the key role of areas in the first path could be expected, area-based reasoning in the second path is noteworthy. In the exams, just above one-fifth of students' responses were consistent with adding the positive measures of the enclosed areas. We situate these findings in the calculus literature to draw educators' attention to silent challenges that are inherent in the integral--area relations.

Learning-support system is an umbrella term that we use for digital resources that assign students with mathematical questions and give automatic feedback on the inserted answers. Transitioning between questions and feedback is characteristic to students' work with such systems. We apply the commognitive framework to explore the role of within-system transitions in students' mathematics learning, with a special interest in what we term as "reroutinization"--a process of repeated development of conventional routines to be implemented in already familiar mathematical tasks. The study revolves around a digital module in integral calculus, which was designed to support undergraduates with finding areas enclosed by functions. The data comes from dyads and triads of first-year university students, who collaboratively interacted with the module. The analyses cast light on how transitioning within the module aided students to review familiar routines, amend them, confirm, and solidify the amendments. The transition process was not always linear and contained instances of students cycling back and forth between the assigned questions and feedback messages. We conclude with the discussion on the module's design that afforded reroutinization and suggest paths for further research.

This paper stems from calls for methodological advancement in two areas of research: primary mathematics education and dialogical education. We respond to these calls from the commognitive standpoint by introducing the "Grove of Realizations" as a tool for capturing and visualizing individual students' learning through collaborative work in small groups. We focus on a group of four students from a New Zealand Year 4 class (aged 8 and 9 years old), as they classified odd and even numbers and reasoned about their sums. This data is used to present the analytical affordances of the tool for (i) visualizing gaps in communication between group members, (ii) mapping their discursive development, and (iii) showing whether and how gaps are bridged within a group. The paper concludes with the discussion of the tool's affordances and directions for further development.

Drawing on the commognitive framework, we construe the secondary-tertiary transition (STT) as a distinctive element in the pedagogical discourses of various communities. Our interest rests with university tutors in light of the emergent recognition of their impact on undergraduates' mathematics learning in many tertiary contexts worldwide. We aim to understand the roles of STT communication in tutors' reflections on incidents that took place in their tutorials. Our participants were undergraduate students in the advanced stages of their mathematics degrees in a large New Zealand university and who were enrolled in a mathematics education course. Throughout the semester, the participants led tutorial sessions for first-year students and wrote reflections on classroom incidents that drew their attention. Our data corpus consisted of 58 reflections from 38 tutors collected over four semesters. The analysis revealed that STT communication featured in tutors' descriptions of classroom incidents, assisted them in making sense of unexpected events, positioned their instructional actions as replications of what was familiar to them from their own STT experience, and contributed toward generating new pedagogical narratives. We situate these findings in the literature concerning undergraduate tutoring and teachers' perspectives on STT.

Mathematics education research has been aware that calculus students can draw on single definite integrals as a model to compute areas (SImA), without minding whether the function changes its sign in the assigned interval. In this study, I take conceptual and empirical steps to understand this phenomenon in more depth. Building on Fischbein's theory, I conceptualize area as a figural concept and associate its calculation with intra-mathematical modelling routes that pass through the network of figures and integrals. To characterize these routes, I analyzed video clips that students in a large first-year service course in mathematics submitted as part of their coursework. This led to the construction of an analytical cycle explicating the models that students generated when implementing SImA. To appreciate how widespread SImA can be, I use the responses of nearly twelve thousand students in final multiple-choice exams in the same course over a decade. This analysis found that nearly thirty percent of the students chose answers that are consistent with this model of reasoning; in some exams the SImA-options were selected more frequently than the correct answers. Drawing on these findings, I make several teaching-oriented comments about the relations between areas and definite integrals.

In undergraduate mathematics education, students' collaborations have gained a reputation as a 'good' learning practice. A more complex image emerges once collaborations are construed as an arena where cognitive, social, and affective matters intertwine in ways that can fuel and impede learning. In this study, I take a close look at a twenty-minute-long interaction between two first-year students who engaged in a sequence of routine problems about dot products presented in a computer-based learning-support system. The commognitive analysis of the interaction revealed an asymmetrical structure: one student led the mathematizing, while his partner followed and invested considerable efforts to avoid making a mathematical contribution. The leader's attempts to switch positions and inquiry into the follower's recurrent refusals to mathematize entailed loaded exchanges that took a toll on both students in terms of identity and affect. I compare these findings to previous studies with school students to discuss possible sources of ineffective communication in situations where students are expected to learn through collaborations.

This study is concerned with the first experiences of in-service mathematics teachers in consuming scholarly mathematics education literature. Growing from the meta-didactical transposition model, we were interested in the praxeologies that may develop from teachers' engagement with research articles. The data were collected from a cohort of 13 teachers who read, interpreted, critiqued, and designed a follow-up to an empirical study as part of a master's level course in mathematics education. The teachers reported that such a consumption of research contributed to the expansion of their mathematical knowledge, instigated changes in their teaching of particular concepts, and provided them with ideas for enacting this change. Furthermore, we explore how engagement with an article shaped the teachers' understandings about research in mathematics education.

With this article, we aim to spur a conversation on research implementation in the context of collaborations of mathematics teachers and educational researchers, with a special interest in projects situated in university mathematics education. Implementation has been often discussed in relation to the education of school teachers, and we suggest that the conceptual, theoretical, and analytical apparatus that has been developed in this research area can be re-purposed to study implementation. Specifically, we propose that in collaborative projects, implementation can be construed as an interactive change-oriented process, in which teachers and researchers develop their praxeologies through interaction and joint work on boundary objects. We illustrate the usefulness of this proposal with a teacher-mathematician and a mathematics education researcher who joined forces to introduce scriptwriting activities into an undergraduate mathematics course. Drawing on the theoretical analysis and illustration, we make suggestions for future implementation studies in the context of teacher-researcher collaborations.

This article presents a research apparatus for investigating and making sense of stories that emerge from feedback that mathematicians provide on students' proofs. Using the commognitive framework, the notion of "didactical discourse on proof" is developed as a lens for conceptualizing mathematicians' practice of feedback provision. The notion is accompanied by a tentative organizational frame, within which didacticians and mathematicians can operate as partners. The methodological affordances of this apparatus are illustrated with a case study of a research topologist, who taught a small, graduate course in topology. The emerged characteristics of her feedback and discourse are situated in the literature and used to sketch future research avenues.

While the lion's share of mathematics education research in linear algebra has been concerned with the conceptual aspects of students' learning, this study focuses on procedural knowledge that undergraduates can develop in a typical first-year course. The study analyzes the methods that a large cohort of second-semester students applied in a final exam to solve three routine problems: finding a basis for a matrix column space, solving a linear system of equations, and devising a normal to a subspace. The results indicate an inverse relation between the efficiency of the methods that students used and the number of students who used them. This was especially evident in the case of theorems, the application of which was avoided by all but a few students in favor of compound algebraic procedures. Furthermore, the study shows that at the end of an instructional sequence, many students can struggle with choosing an appropriate problem-solving method and carrying it out without mistakes. The analysis of students' mistakes is leveraged to draw university teachers' attention to common difficulties in linear algebra.

Spurred by Kilpatrick's (1987) "Where do good problems come from?", this study explores problem-posing triggers of experienced problem posers for mathematics competitions. Triggers are conceived as instances of noticing, where an impulse draws a poser's attention and "triggers off" a mathematical re-action, one of the outcomes of which is a creation of a problem that gets accepted to some mathematics competition. The data were collected from 26 problem posers residing in nine countries, and who had experience in creating problems for national, regional, and international competitions. Three types of triggers emerged from the analysis: (1) Instances where the participants extracted mathematical phenomena from activities that are replete with modern elementary mathematics. These triggers were described in emotionally loaded terms that resonated with appreciation, surprise, challenge, and feeling of innovation. (2) Cases where the participants abstracted mathematical phenomena from common everyday-life tasks in which mathematical optimization was beneficial. These situations emphasized participants' desire to compensate for self-disappointment with an initially made decision and the value of finding a better alternative. (3) Situations where the participants were asked to pose a problem 'here and now'. The posers were unanimous in their dislike of such situations. They elaborated on how difficult it is to pose 'here and now' and highlighted the low quality of the resulting problems. The findings are situated in the body of knowledge on expert problem posing and educational literature on school students and teachers.

The goal of this paper is to present a framework of researcher knowledge development in conducting a study in mathematics education. The key components of the framework are: knowledge germane to conducting a particular study, processes of knowledge accumulation, and catalyzing filters that influence a researcher's decision making. The components of the framework originated from a confluence between constructs and theories in Mathematics Education, Higher Education and Sociology. Previously published interviews with professor Jeremy Kilpatrick and professor Michèle Artigue are used for illustrating how the framework can be utilized in data analysis. Criteria for framework evaluation are discussed.

This study embarks on the question in the title with the construct of epistemological status, which pertains to the solver's satisfaction with the way and the degree to which their solution had fulfilled their intellectual and psychological needs in a particular problem situation. The construct is used to hypothesize that a solver's decision to check the solution and the act of checking itself may be shaped by the epistemological status of a developed solution. Driven by the abduction methodology, this hypothesis is supported by two empirical illustrations. The first one comes from the final exam in a large first-year course for non-mathematics majors, where many students accompanied their solutions of a low epistemological status by written checks of final answers as a way to elevate it. No such checks were found in solutions with high epistemological status. The second illustration revisits some previously reported findings to propose that an application of conventional problem-solving methods may not be sufficient for students to endow their solutions with high epistemological status.

This study is concerned with the reasoning that undergraduates apply when deciding whether a prompt is an example or non-example of the subspace concept. A qualitative analysis of written responses of 438 students revealed five unconventional tacit models that govern their reasoning. The models account for whether a prompt is a subset of a vector space, whether the zero vector is included, the structure of vectors, their number in the formula for the general solution to the system of linear equations, and the corresponding coefficient matrix. Furthermore, a conception was identified in students' responses, according to which the algebraic structure of a vector space passes from a 'parent' space to its subset, turning automatically it into a subspace. For many students this conception of an inheriting structure was instrumental for identifying and reasoning around "sub"spaces. Polysemy of the prefix 'sub' and students' prior experiences in identifying concept examples are used for offering explanations for the emergence of the conception.