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This research commentary argues for more research that attends to the processes of organizational change in mathematics departments. It outlines both the ways that research on organizational change can benefit scholarship in mathematics departments, and how mathematics education researchers are needed to develop theories of change that are contextualized to the teaching and learning of undergraduate mathematics. This commentary closes with a research agenda for moving this emergent field of study forward. This agenda involves applying change theories to historical, ongoing, and new change projects in mathematics departments, while simultaneously attending to issues of equity and social justice.

In this paper we present the results from a study in which 21 mathematics trainee teachers engage with two practice-based tasks in which classroom management interferes with mathematical learning. We investigate the trainees' considerations when they make decisions in classroom situations and how these tasks can trigger their reflections on the teaching and learning of mathematics. In our analysis we used the constructs of "Social" and "Sociomathematical norms" (Cobb & Yackel, 1996) and "Teaching Triad" (Jaworski, 1994). Results indicate commendable norms trainees aspire to establish in their classroom, such as peer respect, value of discussion and investigative mathematical learning. However, they often miss the opportunity to engage students with metacognitive discussions and mathematical challenge as they focus on behavioural issues or endorse dichotomous and simplistic views of mathematical learning. We credit these tasks with allowing insight into trainees' considerations and we propose their further implementation in teacher education programmes.

Non-mathematics specialists' competence and confidence in mathematics in their disciplines have been highlighted as in need of improvement. We report from a collaborative, developmental research project which explores the conjecture that greater integration of mathematics and biology in biology study programs, for example through engaging students with Mathematical Modeling (MM) activities, is one way to achieve this improvement. We examine the evolution of 12 first-semester biology students' mathematical discourse as they engage with such activities in four sessions which ran concurrently with their mandatory mathematics course and were taught by a mathematician with extensive experience with MM. The sessions involved brief introductions to different aspects of MM, followed by small-group work on tasks set in biological contexts. Our analyses use the "theory of commognition" to investigate the tensions between "ritualized and exploratory participation" in the students' MM activity. We focus particularly on a quintessential "routine" in MM, "assumption building": we trace attempts which start from ritualized engagement in the shape of "guesswork" and evolve into more productively exploratory formulations. We also identify signs of persistent "commognitive conflict" in the students' activity, both "intra-mathematical" (concerning what is meant by a "math task") and "extra-mathematical" (concerning what constitutes a plausible solution to the tasks in a biological sense). Our analyses show evidence of the fluid interplay between ritualized and exploratory engagement in the students' discursive activity and contribute towards what we see as a much needed distancing from operationalization of the commognitive constructs of ritual and exploration as an unhelpfully dichotomous binary.

We examine teachers' use of resources as they prepare to teach the topic of numerical series of real numbers, in order to identify how their "personal relationship" with mathematical content--and its teaching--interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in terms of "documentation work" ("resources," "aims," "rules of action," "operational invariants"), a key construct from the documentational approach (DA, Gueudet & Trouche, in "Educational Studies in Mathematics," 71, 199-218, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource for teaching the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the "what" and "how" of their resource use as they organise their teaching of the topic) as well as the operational invariants (the "why" for this organisation of their teaching). We describe the teachers' documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series), and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers' documentation work) with the individual agency (teachers' personal relationships with the topic) provided by the former.

Storytelling is an engaging way through which lived experience can be shared and reflected upon, and a tool through which difference, diversity--and even conflict--can be acknowledged and elaborated upon. Narrative approaches to research bring the richness and vibrancy of storytelling into how data is collected and interpretations of it shared. In this paper, I demonstrate the potency of the narrative approach of "re-storying" for a certain type of university mathematics education research ("non-deficit," "non-prescriptive," "context-specific," "example-centred" and "mathematically focused") conducted at the interface of two communities: mathematics education and mathematics. I do so through reference to "Amongst Mathematicians" (Nardi, 2008), a study carried out in collaboration with 20 university mathematicians from six UK mathematics departments. The study deployed re-storying to present data and analyses in the form of a "dialogue" between two fictional, yet entirely data-grounded, characters--M, mathematician, and RME, researcher in mathematics education. In the dialogues, the typically conflicting epistemologies--and mutual perceptions of such epistemologies--of the two communities come to the fore as do the feasibility-of, benefits-from, obstacles-in and conditions-for collaboration between these communities. First, I outline the use of narrative approaches in mathematics education research. Then, I introduce the study and its use of re-storying, illustrating this with an example: the construction of a dialogue from interview data in which the participating mathematicians discuss the potentialities and pitfalls of visualisation in university mathematics teaching. I conclude by outlining re-storying as a vehicle for community rapprochement achieved through generating and sharing research findings--the "substance" of research--in "forms" that reflect the fundamental principles and aims that underpin this research. My conclusions resonate with sociocultural constructs that view mathematics teacher education as contemporary "praxis" and the aforementioned inter-community discussion as taking place within a "third" space.

Mathematics is the discipline that a significant majority of most incoming researchers in mathematics education have prior qualifications and experience in. Upon entry into the field of mathematics education research, these newcomers--often students on a postgraduate programme in mathematics education--need a broadened understanding on how to read, converse, write and conduct research in the largely unfamiliar territory of mathematics education. The intervention into the practices of post-graduate teaching and supervision in the field of mathematics education that I describe here aims at fostering this broadened understanding and thus facilitating newcomers' participation in the practices of the mathematics education research community. Here I outline the theoretical underpinnings of the intervention and exemplify one of its parts (an Activity Set designed to facilitate incoming students' "engagement with the mathematics education research literature"). I supplement the discussion of the intervention with comments sampled from student interview and student written evaluation data as well as observations of the activities' implementation. The main themes touched upon include: "learning how to identify appropriate mathematics education literature"; "reading increasingly more complex writings in mathematics education"; "coping with the complexity of literate mathematics education discourse"; "working towards a contextualised understanding of literate mathematics education discourse." I conclude with indicating the directions that the intervention, and its evaluation, is currently taking and a brief discussion of broader implications, theoretical as well as concerning the supervision and teaching of post-graduate students in mathematics education.

In this paper we outline the main tenets of the commognitive approach and we exemplify its application in studies that investigate the learning and teaching of mathematics at university level. Following an overview of such applications, we focus on three studies that explore fundamental discursive shifts often occurring in the early stages of studying Calculus. These shifts concern the lecturers' and students' communicative practices, routines of constructing mathematical objects and ways of resolving commognitive conflicts. We then propose that commognitive constructs such as subjectification can be deployed towards "scaling-up" the hitherto fine-grained focus of commognitive analyses. Finally, we conclude with observing how the commognitive approach relates to constructs from other sociocultural approaches to research in university mathematics education, such as "legitimate peripheral participation" from the theory of Communities of Practice and "didactic contract" from the Theory of Didactic Situations.

The study we report here examines parts of what Chevallard calls the "institutional" dimension of the students' learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests. (Contains 6 notes, 5 tables, and 11 figures.)

We describe a four-level spectrum of pedagogical awareness (SPA) that emerged from the analysis of six undergraduate mathematics tutors' (1) conceptualizations of their first-year students' difficulties; (2) descriptive accounts of their strategies for facilitating the overcoming of these difficulties; and (3) self-reflective accounts regarding their teaching practices. These were recorded in 45 semistructured interviews conducted during the 8-week Oxford University term and following minimally participant observation of their tutorials. We exemplify the four levels of SPA, across these three strands, in 12 characteristic examples where tutors discuss episodes from the observed tutorials. The design and analysis of the study reflect three underlying theoretical perspectives: constructivist, sociocultural, and enactivist. Finally, we draw on participants' comments to suggest that our methodology encouraged reflection, raised awareness, and demonstrated the pedagogical potential of collaboration between mathematicians and mathematics educators. We see such awareness and collaborative goodwill as prerequisites to conceptualizing and implementing reform of pedagogical practice. (Contains 5 tables and 5 footnotes.)

Examines how components of the concept of function (variable, domain, and range) and the process-object duality in its nature emerge as highly relevant to student learning in various mathematical contexts related to linear and abstract algebra. (Contains 22 references.) (ASK)

Identifies and explores the difficulties in the novice mathematician's encounter with mathematical abstraction. Observes 20 first-year mathematics undergraduates and extracts sets of episodes from the transcripts of the tutorials and interviews within five topics in pure mathematics. Discusses issues related to the learning of one mathematical topic, Group Theory. (Contains 26 references.) (Author/ASK)

Non-mathematics specialists' competence and confidence in mathematics in their disciplines have been highlighted as in need of improvement. We report from a collaborative, developmental research project which explores the conjecture that greater integration of mathematics and biology in biology study programs, for example through engaging students with Mathematical Modeling (MM) activities, is one way to achieve this improvement. We examine the evolution of 12 first-semester biology students' mathematical discourse as they engage with such activities in four sessions which ran concurrently with their mandatory mathematics course and were taught by a mathematician with extensive experience with MM. The sessions involved brief introductions to different aspects of MM, followed by small-group work on tasks set in biological contexts. Our analyses use the theory of commognition to investigate the tensions between ritualized and exploratory participation in the students' MM activity. We focus particularly on a quintessential routine in MM, assumption building: we trace attempts which start from ritualized engagement in the shape of "guesswork" and evolve into more productively exploratory formulations. We also identify signs of persistent commognitive conflict in the students' activity, both intra-mathematical (concerning what is meant by a "math task") and extra-mathematical (concerning what constitutes a plausible solution to the tasks in a biological sense). Our analyses show evidence of the fluid interplay between ritualized and exploratory engagement in the students' discursive activity and contribute towards what we see as a much needed distancing from operationalization of the commognitive constructs of ritual and exploration as an unhelpfully dichotomous binary. [ABSTRACT FROM AUTHOR]

We examine teachers’ use of resources as they prepare to teach the topic of numerical series of real numbers, in order to identify how their personal relationship with mathematical content—and its teaching—interacts with their use of a commonly used textbook. We describe this interplay between textbook and personal relationship, a term coined in the Anthropological Theory of the Didactic (ATD, Chevallard, 2003), in terms of documentation work (resources, aims, rules of action, operational invariants), a key construct from the documentational approach (DA, Gueudet & Trouche, in Educational Studies in Mathematics, 71, 199-218, 2009). We do so in the case of five post-secondary teachers who use the same textbook as a main resource for teaching the topic. Documentational analysis of interviews with the teachers led to the identification of their aims and rules of action (the what and how of their resource use as they organise their teaching of the topic) as well as the operational invariants (the why for this organisation of their teaching). We describe the teachers’ documentation work in two sets of aims/rules of action: scaffolding mathematical content (series as a stepping stone to learning about Taylor polynomials and Maclaurin series), and organising student learning about series through drill exercises, visualisation, examples, and applications. Our bridging (networking) of theoretical constructs originating in one theoretical framework (personal relationship, ATD) with the constructs of a different, yet compatible, framework (documentation work, DA) aims to enrich the latter (teachers’ documentation work) with the individual agency (teachers’ personal relationships with the topic) provided by the former. [ABSTRACT FROM AUTHOR]

Storytelling is an engaging way through which lived experience can be shared and reflected upon, and a tool through which difference, diversity-and even conflict-can be acknowledged and elaborated upon. Narrative approaches to research bring the richness and vibrancy of storytelling into how data is collected and interpretations of it shared. In this paper, I demonstrate the potency of the narrative approach of re-storying for a certain type of university mathematics education research ( non-deficit, non-prescriptive, context-specific, example-centred and mathematically focused) conducted at the interface of two communities: mathematics education and mathematics. I do so through reference to Amongst Mathematicians (Nardi, ), a study carried out in collaboration with 20 university mathematicians from six UK mathematics departments. The study deployed re-storying to present data and analyses in the form of a dialogue between two fictional, yet entirely data-grounded, characters-M, mathematician, and RME, researcher in mathematics education. In the dialogues, the typically conflicting epistemologies-and mutual perceptions of such epistemologies-of the two communities come to the fore as do the feasibility-of, benefits-from, obstacles-in and conditions-for collaboration between these communities. First, I outline the use of narrative approaches in mathematics education research. Then, I introduce the study and its use of re-storying, illustrating this with an example: the construction of a dialogue from interview data in which the participating mathematicians discuss the potentialities and pitfalls of visualisation in university mathematics teaching. I conclude by outlining re-storying as a vehicle for community rapprochement achieved through generating and sharing research findings-the substance of research-in forms that reflect the fundamental principles and aims that underpin this research. My conclusions resonate with sociocultural constructs that view mathematics teacher education as contemporary praxis and the aforementioned inter-community discussion as taking place within a third space. [ABSTRACT FROM AUTHOR]

The author introduces the research papers in the article, with topics that include the interaction between learners and teachers and their educational resources, communities of practice in mathematics education, and didactical practices used for teaching linear algebra.

In this paper we outline the main tenets of the commognitive approach and we exemplify its application in studies that investigate the learning and teaching of mathematics at university level. Following an overview of such applications, we focus on three studies that explore fundamental discursive shifts often occurring in the early stages of studying Calculus. These shifts concern the lecturers' and students' communicative practices, routines of constructing mathematical objects and ways of resolving commognitive conflicts. We then propose that commognitive constructs such as subjectification can be deployed towards ‘scaling-up’ the hitherto fine-grained focus of commognitive analyses. Finally, we conclude with observing how the commognitive approach relates to constructs from other sociocultural approaches to research in university mathematics education, such as “legitimate peripheral participation” from the theory of Communities of Practice and “didactic contract” from the Theory of Didactic Situations. [ABSTRACT FROM PUBLISHER]