In contrast to these findings, Le et al. This finding was consistent across student ability range, and thus could not be attributed to differences in general mathematics ability. As an alternative hypothesis, Le et al. To test this hypothesis they repeated the study in a second semester calculus course. They found that those students who primarily seek to memorize material used the pause feature more often and performed worse in the course. This finding points to the need to help students develop more effective strategies for using and learning from online content.

A potential shortcoming in the Le et al. To in part address this shortcoming, Inglis et al.

## Volume 1: Content Knowledge

Consistent with Le et al. Iannone and Simpson a , b , in press investigated the ways in which undergraduate mathematics students perceive summative assessment. Indeed these students prefer to be assessed by assessment methods they perceive to be good discriminators of ability and see the closed book examination as one of the best of these discriminators Iannone and Simpson a. They also see oral examinations as a very good way to assess for ability in mathematics. In a follow up study Iannone and Simpson b show how English students appreciate the oral examination method for its immediacy of feedback and perceive this assessment method as requiring conceptual understanding.

Validity was investigated by correlating peer assessment outcomes with assessments by experts and novices as well as with marks from other module tests. High validity results suggest that the students performed well as peer assessors.

Difficulties associated with the transition from school to tertiary mathematics teaching and learning has been a focus of research in mathematics education since early s mostly by trying to understand the cognitive structures and processes that determine the difficult changes and restructurings that this transition involves e. Tall As Artigue et al. In this section, we briefly review some recent research results that focus specifically on this transition. Then, we consider the transition to abstraction and formal mathematical thinking.

We close the section discussing initiatives aiming to ease the way into tertiary studies through the first year. Recent research addresses difficulties experienced by students when starting studies at a tertiary level. These difficulties are related to differences and possibly conflicts between school and tertiary contexts. This includes teaching styles, instructional approaches, studying and learning strategies as well as views about mathematics, specific mathematical concepts, mathematical knowledge, and goals of learning.

Some studies suggest that beginning undergraduate students do not see university mathematics topics as continuations, extensions, or generalizations of topics previously studied at schoolâ€”they tend to regard these as completely different subjects. For example, Cofer reports that even prospective secondary mathematics teachers were unable to link school algebra and university algebra, despite their having already finished an abstract algebra course.

Suominen analyzed nine undergraduate abstract algebra textbooks, searching for explicit connections between abstract algebra and secondary school mathematics concepts. Results were organized according to an analytic framework based on categories adapted from previously established work by Businskas and Singletary : alternate representations; comparison through common features; generalization; hierarchical inclusion; and real world applications.

The author found that comparison of common features, hierarchical inclusion, and alternate representations appear more frequently than the other categories, including generalization. These results differ from earlier literature, which highlights connections related to abstract algebra as generalizations from school concepts e. That is, connections are established in these textbooks in a different manner.

The study is motivated by a reported decline in the proportion of Australian students opting to study higher-level mathematics. Data were collected through audio-recorded in-depth interviews of the mathematics leaders. These strategies were often described by the leaders in terms of encouraging students to aim high and challenge themselves growth mindset , or discouraging students from attempting a subject in which they were not deemed capable of succeeding fixed mindset.

Nonetheless, a majority of leaders indicated that students could override the recommendations and choose for themselves. A tension between the need to consider performance and the desire to promote progress was noticeable. There was also the sense that the leaders experienced tension related to differences between them and the career staff, between them and their mathematics teaching staff, or between them and parents.

Barnett et al. The authors found no consistent association, and distinguish a productive effort, that carries the expected benefits, from ineffective efforts, that is associated with negative consequences. The authors found that more reading of the course textbook was associated with worse college calculus performance for students with all kinds of high school preparation.

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Data comes from a longitudinal survey study with first semester mathematics students enrolled in a regular mathematics program or a financial mathematics program at the University of Munich, Germany. In particular, clearly extrinsic motives professional perspectives went along with less constructive learning activities. A direct connection between a motive to apply mathematics and reduced study success was found. These results underpin the view that only motives in line with the program under study can support in coping with the transition from school to university mathematics.

Kempen and Biehler investigate argumentation skills of the students at the beginning of the course, and identify common gaps or pitfalls in their argumentations. The participants were undergraduate pre-service teachers in Germany attending a bridging course. They were asked to verify a statement of elementary number theory, and were given a questionnaire with items concerning argumentation and proving, attitudes towards proving, the nature of mathematics, and the nature of mathematics teaching.

Results reveal that participants are not equipped with the argumentation skills required for proving. This suggests that mathematics at school does not provide the future students with adequate heuristics for problem-solving and basic proving skills.

These findings underline the importance of introductory courses, such as bridging courses, that introduce basic skills for arguing and proving. Here, it is important to emphasize the meaning of informal arguments in order to stress the quality of a given argumentation. If we highlight the possibility to formalize an informal argument we also underline the function and value of using algebra and variables in mathematics. Using Anthropological Theory of Didactics Chevallard , Job and Schneider consider the development of calculus as an epistemological transition between two types of praxeologies: pragmatic and deductive.

In this view, the authors discuss the dichotomy between formal and intuitive aspects of limits, as a mathematical activity that is expected to become rigorous on some formal definition. The authors highlight the notion of limit as a pragmatic model of magnitudes relying on mental objects. They argue that the prevalence of empirical positivism constitutes an obstacle to the learning of calculus, which is reinforced by the institutions, as a consequence of their inability to give credit to a pragmatic level of rationality.

Reported obstacles and pitfalls experienced by beginner undergraduate students in mathematics courses, as well as the acknowledgement of the first year of the university as a crucial step for long-term success in degree programs, led some institutions to implement initiatives aiming to ease the way into tertiary studies through the first year, to reduce retention and to increase graduation rates.

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Some of these initiatives were objects of research projects as we discussed in previous sections e. Bausch et al.

Furthermore, Engelbrecht and Harding report a multi-dimensional approach conceptualized and implemented at the University of Pretoria, South Africa, aiming to improve teaching and learning in first year mathematics courses. This effort was motivated by the national policy of increasing graduates in the sciences in the country.

These aspects regard mathematical preparedness, social transition, learning style, support, and conceptual understanding.

## Wasserman, Nicholas H. (nhw) | Teachers College, Columbia University

The author suggests that problems associated with the transition to the university are perhaps exacerbated because of the procedural approach followed at school and the practice of examination coaching. Therefore, students have to undergo a change in thinking approach, from procedural to more conceptual, and a culture of independent learning needs to be fostered.

Especially, in the context of mathematics teacher education programs several studies suggest innovative approaches at the entry phase aiming towards the improvement of professional knowledge by altering the conditions in university teaching see for example, TEDS program, Buchholtz and Kaiser In this section we will focus on some examples of theoretical and methodological advances we identified in our review, although by no means does this section intend to be a comprehensive summary of all the approaches we met. When the community is the main focus, the lens of communities of practice has been used in studies that investigate teaching and learning practices at the tertiary level see Biza et al.

In this review we reported the study of Biza and Vande Hey that endorses this theoretical perspective. Especially from the community of inquiry perspective we visited Jaworski and Matthews who address teaching mathematics to engineering students. Furthermore, the documentational perspective suggested by Gueudet and colleagues bring new theoretical constructs that can deal not only with the mathematics teacher actual instructional activity but with their activity outside the classroom in their preparation, evaluation and revision of their teaching resources as well as with their professional development Gueudet ; Gueudet et al.

Finally, Rasmussen et al. We return to the coordination of both individual and collective theoretical lenses later in this section. In our review we identified, also, a range of studies, which aim to liaise different theoretical perspectives in order to address their research questions. We consider this as a step forward in the investigation of teaching and learning issues at university mathematics education in a more holistic way. Inevitably the combination of different theoretical perspectives is not always straightforward, as it demands compatibility between the epistemological and ontological underpinnings of distinct perspectives Kidron To this aim the study deploys a combination of theoretical lenses by drawing on the Documentational approach Gueudet et al.

Another example is from the work of Tabach et al. To this aim they draw on the Abstraction in Context Hershkowitz et al. In this section we draw on two quite different innovative methodological advances we met in our review: storytelling and eye - movement methodologies. Results are presented in the form of a dialogue between two fictional, yet entirely data-grounded, characters: a mathematician M and a researcher in mathematics education RME.

The re-storying approach is presented in details and exemplified through an application of it in a small number of interviews, which were re-storied into an exchange of utterances between the M and RME characters on potentialities and pitfalls of visualization in university mathematics teaching. There are an increasing number of studies in teaching and learning of mathematics adopting eye - tracking methodologies.

Beitlich et al. They found that all participants paid attention to the pictures and tried to integrate information from text and picture by alternating between these representations. Chumachemko et al. Similarly, Obersteiner et al. Finally, the Hodds et al. Problems in teaching mathematics to non-mathematics students and suggestions for enhancement; application of mathematics and modelling to non-mathematics disciplines; and how mathematical concepts are addressed in non-mathematics programs.

Opportunities afforded by textbooks; how students use and read these textbooks; how they use online resources; and how they perceive or act in the assessment. Theoretical and methodological perspectives and liaison of theories that address issues in relation to the teaching and learning mathematics at the tertiary level. The spectrum of foci of the studies we reviewed indicates the increasing interest in the field as well as the prominent ongoing need for more robust research of issues pertaining mathematics teaching and learning at tertiary level.

We will conclude this review with a discussion of potential ways forward for future research in this field of enquiry. One example of an under-investigated area is the transition of mathematics graduates to postgraduate studies. In a different context, Nardi addresses the transition of mathematics graduates to postgraduate programs in mathematics education.

The study discusses an intervention into the practices of post-graduate teaching and supervision in the field of mathematics education that facilitates students in their shift on how to read, converse, write, and conduct research in the largely unfamiliar to them territory of mathematics education. It seems that more research is needed into the characteristics of the transition from undergraduate to post-graduate studies, especially when the epistemology changes, for example, from undergraduate studies in mathematics to postgraduate studies in statistics, engineering, mathematics education, or mathematics teaching for those who want to become teachers.

What do we mean by knowledge? How does this knowledge develop? How does this knowledge reflect on practice? Potentially we may want to investigate practice and knowledge together, e. Teaching practice development is another potential area. As Jaworski et al.