This article is published *open access.*

# Survey on Research in University Mathematics Education at ICME 14

### Marianna Bosch

University of Barcelona, Spain### Reinhard Hochmuth

Leibniz University Hannover, Germany### Oh Nam Kwon

Seoul National University, Korea### Birgit Loch

University of New England, Armidale, Australia### Chris Rasmussen

San Diego State University, USA### Mike Thomas

Auckland University, New Zealand### María Trigueros

Instituto Tecnológico Autónomo de México, Mexico City, Mexico

At the 14th International Congress on Mathematics Education, which took place in a hybrid mode in Shanghai from July, 11 to 18, 2021, the survey team on Research in University Mathematics Education (RUME) presented an overview of their work.
As noted in the presentation, it is an exciting time for RUME.
There are now several major conferences every year across the globe, as well as the fairly new *International Journal of Research in Undergraduate Mathematics Education*, now in its seventh year.
The significant growth in the number of researchers focused on university mathematics education has led to the development of research groups and the consolidation of a diverse academic community; RUME is coming to age as a field of research that is beginning to coalesce and develop an identity.

To explore this identity, we surveyed 218 RUME scholars across the world, both well-established scholars and rising stars. We invited these scholars to respond to the following prompt:

What do you see as the most significant advances, changes, and/or gaps in the field of research in university mathematics education? These advances, changes, or gaps might relate to theory, methodology, classroom practices, curricular changes, digital environments, purposes and roles of universities, social policies, preparation of university teachers, etc. Please elaborate on just

one or two advances, changes, or gapsmost relevant to your experience and expertise.

We received 119 responses.
Our next step was to conduct a thematic analysis^{1}Special thanks to Antonio Martinez and Talia LaTona-Tequida, graduate students at San Diego State University, for their help in this analysis., which led to the identification of five areas in which there has been considerable progress (Theoretical Perspectives, Instructional Practices, Professional Development of University Teachers, Digital Technology, and Service-Courses in University Mathematics Education) and seven, non-disjoint areas in need of further research (Theories and Methods, Linking Research and Practice, Professional Development of University Teachers, Digital Technology, Curriculum, Higher Years, and Interdisciplinarity).
We then conducted a literature review, guided by the identified themes.
We hope that this brief report offers those less familiar with RUME an overview of the progress to date and spurs interest in areas in which the reader might want to contribute to the knowledge base.

One of the field’s major advances is that we now have a plethora of theoretical perspectives, and hence tensions among them can sharpen their constructs and methodologies and open the possibility of finding commonalities. This diversification has contributed to the development of new methods, research topics, and the development and research on theory-based teaching experiences. Recent years have seen the emergence of an interdisciplinary group of scholars interested in using a variety of approaches (logical, cognitive, historical, philosophical, etc.) to address questions which have always been of interest to RUME. Another theoretical advance that is of growing interest is the use of theories that enable insights into the interrelatedness of knowledge, identity, power, and social discourses [1 A. P. Adiredja and C. Andrews-Larson, Taking the sociopolitical turn in postsecondary mathematics education research. Int. J. Res. Undergrad. Math. Educ.3, 444–465 (2017) ]. While there is still much research that is needed here, we see this new direction as an important advance for the field of university mathematics education research.

The research of instructional practices at university level is another rapidly developing area of research. Much of the research on this topic relates to active or inquiry-based mathematics education [2 M. Artigue and M. Blomhøj, Conceptualizing inquiry-based education in mathematics. ZDM–Math. Educ.45, 797–810 (2013) , 6 S. L. Laursen and C. Rasmussen, I on the prize: Inquiry approaches in undergraduate mathematics. Int. J. Res. Undergrad. Math. Educ.5, 129–146 (2019) ]. Given the myriad calls for instructional reform in university mathematics classrooms, researchers and educators have challenged conventional lecture-based instruction by conducting studies that have provided evidence for the positive effects of innovative student-centered instructions on students’ cognitive and affective development. Active learning, broadly defined as classroom practices that engage students in activities such as reading, writing, discussing, or problem solving, that promote higher-order thinking, has repeatedly been shown to improve student success and to reduce the equity gap for women and underrepresented students [3 S. Freeman, S. L. Eddy, M. McDonough, M. K. Smith, N. Okoroafor, H. Jordt and M. P. Wenderoth, Active learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. USA111, 8410–8415 (2014) , 7 E. J. Theobald et al., Active learning narrows achievement gaps for underrepresented students in undergraduate science, technology, engineering, and math. Proc. Natl. Acad. Sci. USA117, 6476–6483 (2020) ]. For example, a meta-analysis of 225 studies that compared student success in traditional lecture versus active learning in postsecondary science, engineering, and mathematics courses and found that average examination scores improved by about 6 % in active learning sections, and that students in classes with traditional lecturing were 1.5 times more likely to fail than were students in classes with active learning; further, the effectiveness of active learning was found across all class sizes [3 S. Freeman, S. L. Eddy, M. McDonough, M. K. Smith, N. Okoroafor, H. Jordt and M. P. Wenderoth, Active learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. USA111, 8410–8415 (2014) ]. On the other hand, RUME has only just begun to deeply explore the culture, experiences, and gendered/racialized interactions in these classes, and how those social factors may be obstructing the students’ opportunities to learn [4 E. Johnson, C. Andrews-Larson, K. Keene, K. Melhuish, R. Keller and N. Fortune, Inquiry and gender inequity in the undergraduate mathematics classroom. J. Res. Math. Educ.51, 504–516 (2020) ].

Another area in need of research is on the learning and teaching of advanced mathematics. Historically, the work of Felix Klein is most relevant here. Core parts of his “Elementary Mathematics from a Higher Standpoint” [5 F. Klein, Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (Vol. 1). Courier Corporation (2004) ] actually refer to mathematics that many of today’s future teachers do not even get to know in the course of their academic studies. This applies, for example, to knowledge of Fourier analysis that goes beyond the basics, but especially also to knowledge of function theory, e.g. Riemann surfaces and value assignment theorems. Even when students hear about function theory, for example, they usually do not get as far as understanding what Felix Klein considered, more than a century ago, appropriate knowledge for prospective teachers. Klein considered this knowledge appropriate because it explains why, for example, certain elementary operations have to be restricted in certain ways for mathematical reasons (and not just for didactic reasons of reduction!), and related curricular decisions.

Also ripe for further investigation is the cooperation with mathematicians, engineers, economists, psychologists, etc. For many years, there have been many different kinds of cooperation, for example, agreements between faculties with regard to teaching. What does not seem to exist so far is, among other things, systematic research on these cooperations. What are the benefits of these? How do they take shape? How do they function? Possibilities, limits, etc.? Related to these cooperations is the relationship of mathematics to other sciences or the use of mathematics in other sciences. There are several places, such as philosophy or the history of science, in which such connections are examined and the question of what distinguishes mathematics itself and its respective role in other sciences is explored. Research on this is dependent on the respective ideological assumptions, and accordingly there are no unambiguous and generally accepted answers here. From the point of view of didactics, however, clarifications in this regard could certainly be regarded as desirable, since they would be of great help in answering the question of which goals, and how mathematicians and even more engineers, economists, psychologists, etc., are to be taught.

Last but not least are questions concerning mathematics itself. Mathematics, too, changes its inherent orientation, and to some extent its character, over time. New fields, such as Big Data and Data Science, are continually emerging. Correspondingly, there are new fields of application in other sciences, such as discrete mathematics in electrical engineering, numerical methods in psychology, etc. This leads directly to questions of what should be taught in service courses.

The video of the oral ICME presentation on the survey can be found here: https://drive.google.com/file/d/1LTBDl_KNZ371SL5ahvN2x_TA09PQQRlB/view?usp=sharing.

As we noted at the start of this brief overview, there is now much research-based wisdom, while at the same time there are exciting opportunities for new research. In particular, research mathematicians are welcome to join the systematic reflection and empirical investigation of university mathematics teaching.

Marianna Bosch is an associate professor of didactic of mathematics at the University of Barcelona.
She works in the development of the Anthropological Theory of the Didactic (ATD) created by the French researcher Yves Chevallard (Marseilles) and its contributions to the development of curricula and professional teachers.
In the case of university mathematics education, she investigates the conditions needed to link study-based and inquiry-based proposals through a specific instructional format called *study and research paths*.
She recently co-chaired the 3rd conference of the International Network for Didactics Research in University Mathematics (INDRUM).
marianna.bosch@ub.edu
Reinhard Hochmuth is professor of didactic of mathematics in the Department for Mathematics and Physics at the Leibniz University Hannover, Germany.
His research focuses on mathematics for teacher education, the use of mathematics in empirical sciences and the development of inquiry-oriented approaches to the teaching and learning of advanced mathematics.
The theoretical background of the research is the anthropological theory of didactics and a societally oriented subject-science.
He is director of the Competence Centre for Higher Education Didactics in Mathematics (khdm) and founding member of the INDRUM Network coordination group.
hochmuth@idmp.uni-hannover.de
Oh Nam Kwon is professor in the Department of Mathematics Education at Seoul National University, Korea.
Her research interests include curriculum development, teaching and learning of collegiate mathematics, and professional development of mathematics teachers.
She was a member of 2015 OECD/PISA Mathematics Expert Group.
She received a number of awards such as the Best Teaching Awards, Seoul National University in 2009, 2020 President’s Commendation, and the 2021 Svend Pedersen Lecture Award.
She serves currently as the co-chair of the 9th ICMI-East Asian Regional Conference on Mathematics Education.
onkwon@snu.ac.kr
Birgit Loch is Dean of Science, Agriculture, Business and Law at the University of New England, Australia.
Her research interests include open educational resources, blended and online learning, students as partners and innovative use of educational technologies.
She is a member of the International Steering Committee for Delta, the Southern Hemisphere Conference on Undergraduate Mathematics and Statistics Education, a national teaching award winner, Fellow of the Australian Mathematical Society and Principal Fellow of AdvanceHE (PFHEA).
Birgit.Loch@une.edu.au
Chris Rasmussen is professor in the Department of Mathematics and Statistics at San Diego State University.
His research investigates inquiry-oriented approaches to the learning and teaching of differential equations as well as the process of departmental and institutional change to improve introductory mathematics courses.
He is a founding editor for the *International Journal of Research in Undergraduate Mathematics Education* and was co-chair for the 13th International Congress on Mathematical Education Topic Study Group on Mathematics Education at the Tertiary Level.
crasmussen@sdsu.edu
Mike Thomas is an emeritus professor in the Mathematics Department, Auckland University, New Zealand.
His research encompasses advanced mathematical thinking, semiotic analysis of gestures and representations, versatile thinking and the role of digital technology in the learning and teaching of mathematics at school and university.
He was a founding editor of the *International Journal of Research in Undergraduate Mathematics Education* and led the 12th International Congress on Mathematical Education survey team on mathematical difficulties in the transition from school to university.
moj.thomas@auckland.ac.nz
María Trigueros is an associate professor in the Department of Mathematics at Instituto Tecnológico Autónomo de México.
She works on the development and application of APOS theory.
Through her research she has contributed to the study of the Schema notion in APOS Theory and to its use in investigating the learning and teaching of two-variable calculus and linear algebra.
She is also interested in linking APOS to other theories and in developing and testing modeling tasks in the teaching of linear algebra.
She now serves as chair of the 4th conference of the INDRUM-2022.
mtriguerosg@gmail.com

^{1}Special thanks to Antonio Martinez and Talia LaTona-Tequida, graduate students at San Diego State University, for their help in this analysis.

## References

- A. P. Adiredja and C. Andrews-Larson, Taking the sociopolitical turn in postsecondary mathematics education research. Int. J. Res. Undergrad. Math. Educ.3, 444–465 (2017)
- M. Artigue and M. Blomhøj, Conceptualizing inquiry-based education in mathematics. ZDM–Math. Educ.45, 797–810 (2013)
- S. Freeman, S. L. Eddy, M. McDonough, M. K. Smith, N. Okoroafor, H. Jordt and M. P. Wenderoth, Active learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. USA111, 8410–8415 (2014)
- E. Johnson, C. Andrews-Larson, K. Keene, K. Melhuish, R. Keller and N. Fortune, Inquiry and gender inequity in the undergraduate mathematics classroom. J. Res. Math. Educ.51, 504–516 (2020)
- F. Klein, Elementary mathematics from an advanced standpoint: Arithmetic, algebra, analysis (Vol. 1). Courier Corporation (2004)
- S. L. Laursen and C. Rasmussen, I on the prize: Inquiry approaches in undergraduate mathematics. Int. J. Res. Undergrad. Math. Educ.5, 129–146 (2019)
- E. J. Theobald et al., Active learning narrows achievement gaps for underrepresented students in undergraduate science, technology, engineering, and math. Proc. Natl. Acad. Sci. USA117, 6476–6483 (2020)

## Cite this article

Marianna Bosch, Reinhard Hochmuth, Oh Nam Kwon, Birgit Loch, Chris Rasmussen, Mike Thomas, María Trigueros, Survey on Research in University Mathematics Education at ICME 14. Eur. Math. Soc. Mag. 122 (2021), pp. 57–59

DOI 10.4171/MAG-42