This study is a part of an ongoing research that attempts to explain the relationship between the teachers’ instructional practise and students’ learning in the context of functions and function inverses. The question in this paper is how the use of technology as a pedagogical tool may contribute to the understanding of the inverse function concept. An engineering student group (n =17) was taught functions and inverse functions with the assistance of GeoGebra. In our theoretical framework we apply Variation theory together with the theory of Concept image and Concept definition. The data were gathered by doing a pre and post test concerning inverse functions. Our experiment revealed that students’ concept images in the post test were more developed compared with the results in the pre test.

Using Cultural-Historical Activity Theory, we analyze lecturers’ views on the aims and teaching practices of mathematical modelling (MM) education in Norway and England. We aim to expose the tensions that exist within the activity of teaching MM at university, such as those between multiple, sometimes competing, aims for teaching MM, or between the lecturers’ professional identities and the structure of university degrees. Our conceptualization of these tensions might help lecturers consider how to overcome obstacles in their own contexts.

This paper reports on critical aspects of three engineering students’ discourse in group work using a digital tool called Sim2Bil while solving mathematical tasks. Applying a commognitive perspective, where mathematical discourse is characterized by words used, visual mediators applied, narratives developed and routines established, we investigate how these characteristics are influenced by the technological environment. It is found that all of the aspects of the students’ discourse are influenced by Sim2Bil. For instance, a “trial and error” routine directly connected to the use of the tool is present in the students’ discourse.

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 paper reports on the views and use of mathematical modelling (MM) in university mathematics courses in Norway from the perspective of lecturers. Our analysis includes a characterisation of MM views based on the modelling perspectives developed by Kaiser and Sriraman (2006). Through an online survey we aimed to identify the main perspectives held in higher education by mathematics lecturers and the underlying rationale for integrating (or not) MM in university courses. The results indicated that most respondents displayed a realistic perspective on MM when it came to their professional practice. There was a more varied response when it came to their views on MM in teaching. Regarding conditions influencing the use or non-use of MM in teaching, these mainly concerned the mathematical content and the institutional practices.

In this paper, a developmental research project involving offering mathematical modelling (MM) activities to university biology students is presented, and a particular aspect is studied, namely the project as a collaboration between mathematicians and mathematics educators. The aim of the paper is to investigate what characterizes their participation in the project, and how the characteristics of the project and its development might influence this participation. Interview data as well as observation data from the MM sessions are analysed, and findings show that the mathematics educator served as a broker influencing the practice of the mathematician. It is hoped that the findings of the study can be of use when planning future collaboration between mathematicians and mathematics educators.

This study analyses what kind of conceptions teacher students and engineering students have about the function concept, and how these conceptions differ between the two groups. The study was conducted through questionnaires, and 34 students at a Swedish university participated. The function concepts of the students have been classified according to modified versions of models presented by Vinner & Dreyfus, Sfard and DeMarois & Tall. The study shows that the students primarily have operational concepts of function, with only a couple of students having a structural function concept. The study also shows distinct differences between prospective compulsory school teachers and engineering students, where the former have less developed functional concepts.

This paper reports on an ongoing study focusing on the teaching of functions in undergraduate courses in mathematics at three Swedish universities. In this paper excerpts from the lectures of three teachers at one university are analysed, using commognitive theory. Characteristic features of the teachers’ discourses about functions are presented. Definitions are found to be the central type of narrative, while theorems and proofs are largely absent, despite the fact that the teaching is of a traditional type, often connected to the “definition-theorem-proof” format. Three main categories of routines are found: substantiation, construction and motivation routines. It is also seen that the teachers are more concerned with questions of “why” to do things than “when” to do them. 

This paper reports on an ongoing study focusing on the teaching of functions in undergraduate courses in mathematics at three Swedish universities. In this paper two excerpts of lectures on linear transformations are analysed, using commognitive theory. The two teachers' discursive uses of the terms 'linearity' and 'linear transformation' are described, and potential consequences for student learning are presented. One lecture displays a strong mathematical content while lacking contextualization, while the other is well grounded in everyday experience, but with a lack of attention to mathematical detail potentially detrimental to student learning.

This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard’s commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories – explanation, motivation and question posing routines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines include known mathematical facts, summary and repetition, different representations, everyday language, and concretization and metaphor; the motivation routines include reference to utility, the nature of mathematics, humour and result focus; and the question posing routines include control questions, asking for facts, enquiries and rhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics.

The study reported in this paper investigates how notions of the nature of mathematical knowledge and mathematical objects are constituted through the teaching practices of three university mathematics teachers at a Swedish university. The data consists of video recorded lectures, and the analyses were informed by classifications presented by Lerman and Davis and Hersh. The results indicate that different epistemological and ontological positions are indeed constituted through the discourse. Although the discourse is generally highly objectified, the ways in which mathematical objects are introduced differ. Mostly the discourse was within an absolutist paradigm, but there were also examples of how the socio-historical nature of mathematical knowledge is emphasized.

This thesis concerns the teaching of mathematics at university level, with a particular focus on the teaching of the function concept. The main aim of the thesis is describing and analysing the teaching practices of university mathematics teachers regarding the function concept, and how this concept is constituted through these practices. To this end, video recordings of lectures by seven mathematics teachers at three Swedish universities were analysed using a discursive perspective, Sfard’s commognitive framework. The observed teaching was traditional in form, with teachers using “chalk talk” – simultaneously talking and writing on the board. The results show that the teaching practices of the teachers belong to two distinct but intertwined discourses – a mathematical discourse, and a discourse of mathematics teaching. Classifications of important aspects of these discourses are presented, and it is found that the teachers’ discursive practices, while sharing overall form, still display considerable differences. Other results include an analysis of the levels of objectification displayed by the teachers in their discursive constitution of the function concept. The study contributes to a small but growing body of empirical research on university mathematics teaching practice.

This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions, using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using “chalk talk”, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level, in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics.

This study analyses what kind of conceptions teacher students and engineering students have about the function concept, and how these conceptions differ between the two groups. The study was conducted through questionnaires, and 34 students at a Swedish university participated. The function concepts of the students have been classified according to modified versions of models presented by Vinner & Dreyfus, Sfard and DeMarois & Tall. The result of the study shows that the students primarily have operational concepts of function, with only a couple of students having anything resembling a structural function concept. The study also shows distinct differences between prospective compulsory school teachers and engineering students, where the former have less developed functional concepts.

This paper reports on a study of the teaching of functions in undergraduate courses in mathematics at three Swedish universities. In this paper excerpts from the lectures of the seven participating teachers are analysed, using commognitive theory. Characteristic features of the teachers’ discourses about functions are presented, focusing on routines – repetitive patterns characteristic of the discourse. The discursive practices are found to contain two intertwined discourses – mathematical discourse and discourse of mathematics teaching. Routines specific to the mathematical discourse are construction and substantiation routines, while the didactical routines include motivation, explanation, activation and recall routines.

The study reported in the present paper forms part of an ongoing project regarding the teaching of functions in undergraduate courses at three Swedish universities. The theoretical framework underlying the project is Sfard’s commognitive theory. In the present study, three lectures in calculus from two different universities are analyzed, focusing on the characterizations of the function concept presented by the discursive practices of the teachers. All three teachers are found to make extensive use of the principle of variation, but differences are found for instance in the emphasis put on different realizations of functions, and on the role of domain and range. Also, according to the type of content of the lectures, differences between more process-oriented and more object-oriented discourses of functions can be seen.

This study analyses what kind of concept images a group of engineering and teacher students have of the function concept, and how these concept images are related to the historical development of this concept. The study was conducted using questionnaires, and 34 students at a Swedish university participated. It is found that the students primarily rely on operational conceptions of the function concept, with only a minority of students possessing structural conceptions. The definitions given by the students mostly resemble an 18^th or 19^th century view of functions. The study also indicates that the character of the definitions given in the textbooks used by the students affect their concept images.

We report analyses from a collaborative, developmental research project between two Norwegian centres of excellence in higher education (MatRIC and bioCEED) in which biology-related mathematical modelling (MM) activities are introduced to biology students as a means to motivate their appreciation for, and competence in, mathematics. This phase of the project involved four sessions with 11 first-semester students. We report data and analyses from two activities: Yeast Growth and Digoxin. Our commognitive analyses trace the evolution of the students’ mathematical discourse in two episodes, revealing a scaffolding story about the gradual transition from ritualized to exploratory engagement with MM, and pointing to the crucial role played by the teacher in this process. We conclude with discussing some implications of our analysis for the design and use of MM activities for students of Biology, and other non-mathematics specialisms.

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.

The project we report from in this paper explores whether and how, biology students’ competence and confidence in – as well as appreciation for – mathematics in their discipline can be improved through greater integration of mathematics and biology in their study programme. Here, we examine biology students’ mathematical discourse as they engage with a biology-related Mathematical Modelling (MM) activity, the Digoxin task. We report commognitive analyses of data collected during sessions in which biology-related MM activities were introduced to undergraduate biology students (four sessions with 12 first-semester students).We focus on the interplay between students’ ritualised and exploratory engagement with the activity, particularly concerning graphing routines, and consider pedagogical implications.