Paul Hernandez-Martinez (a), Stephanie Treffert-Thomas (a),

Yuriy Rogovchenko (b) and Olov Viirman (b)

(a) Loughborough University, (b) University of Agder

We are reporting on the MatRIC funded research project "Discourses of Mathematical Modelling in Higher Education in Norway and England". The aim of the project was to investigate mathematical modelling (MM) practices based on mathematicians' views of modelling and of teaching modelling. We are interested in mathematicians' views and use of MM in their professional practice as well as in their teaching practice. We are also interested in their aims for using MM in university teaching or possible reasons for not using it. We report on the data collected from Norwegian lecturers which have been analysed fully to date. This has been submitted for publication in the journal *Nordic Studies in Mathematics Education*. Analysis of data from the English sample of lecturers is on-going and we have included in this report only the results that were available.

### Background and Literature

Mathematical Modelling (MM) is important in many subject areas (outside of mathematics) such as physics, engineering, the social and natural sciences, among others. Research into MM and the teaching and learning of MM is an active field in mathematics education that has spanned more than 25 years already. It is generally held, and supported by research, that the teaching of MM is important and necessary (Blum and Niss, 1991). Yet MM is quite often not mentioned and appears not to be a part of the curricula of mathematics degree programmes. Research has also shown that students find MM difficult (Soon et al, 2011) which may be a factor in lecturers' apparent reluctance to introduce MM in teaching. While this is known, it is less clear how modelling 'should' be taught: on its own, as a separate course; incorporated into an existing course; or possibly in some other way. If we agree that MM is an important part of students' academic preparation, these questions need to be explored.

MM is defined and used in a variety of ways in the mathematics education research literature (e.g. Garcia, Gascón, Higueras & Bosch, 2006; Blum, Galbraith, Henn & Niss, 2007; Jablonka & Gellert, 2007; Frejd, 2011). Kaiser and Sriraman (2006) developed a categorisation of international perspectives on modelling in mathematics education. They studied published research and produced a classification based on the central aim for teaching modelling (often with reference to school level teaching). As a result of this meta-analysis they offered five main perspectives as follows.

1) *Realistic (or applied) perspective*. In this perspective the focus of MM, and of teaching MM, is on solving authentic problems as they occur in real life and to prepare and equip students with the skills for doing so.

2) *Epistemological (or theoretical) perspective*. In this perspective MM is theory driven rather than based in realistic or real life situations. All mathematical activities, both in pure and applied fields, are regarded as modelling.

3) *Socio-critical (or emancipatory) perspective*. In this perspective MM is seen as developing citizenship and recognition of the role and influence of mathematics in society.

4) *Contextual perspective*. In this perspective MM is regarded as specific to the subject area and the context of the modelling activity. In particular, modelling is regarded as driving individual psychological concept development.

5) *Educational perspective*. In this perspective the aims of using MM are pedagogical. Two sub-categories are distinguished: MM and the modelling cycle, in particular, are tools for structuring the learning process (didactical modelling), or MM is a strategy for motivating students and developing mathematical understandings.

### Methodological perspectives

The five perspectives presented by Kaiser and Sriraman (2006) provided us with a useful categorisation for our research project. It encompassed a broad view on MM as it exists in the research literature to date. While we adopted their categorisation as a framework for this study we were aware of mathematicians' reflections on their mathematical activity and particularly the enjoyment and pleasure that they said to experience when working mathematically (see, for example, Hardy, 1940; Hadamard, 1945; Poincaré, 1952). Therefore, we complemented the existing categorisation with a new category, the "enjoyment (or affective) perspective" in which the aim of modelling is the intrinsic satisfaction derived from engaging in this activity.

To gather lecturers' views we developed an on-line questionnaire that was sent to lecturers and teachers of mathematics in university mathematics departments in Norway and England. The questionnaire had three distinct parts.

In the first part all questions were designed to obtain demographical data, such as work experience, gender, age, and research area. We also asked whether the respondent had any experience working with modelling in research or industry.

In the second part of the questionnaire we focused on views on modelling in relation to mathematicians' professional activities. There were different styles of questionnaire items. For example, there were Likert scale items where participants were asked, on a scale of 1 to 5, to agree or disagree with a statement. Most of these statements were developed from our reading of the MM research literature as well as MM textbooks such as Dym (2004), Giordano (2003), Shiflet and Shiflet (2014) or Velten (2009). At other places in the questionnaire participants were asked to rank statements about the aims of modelling that best corresponded to their own views, or to provide written comments in open comment boxes.

The third part of the questionnaire was concerned with lecturers' views on the teaching of modelling. Here we asked whether lecturers had experience of using MM in their teaching, how it was taught, and to describe what would be their preferred way of teaching modelling. As in the second part, we included Likert scale and ranking items.

Almost all of the questionnaire items were designed so that they could be easily analysed using the Kaiser and Sriraman (2006) perspectives augmented with the enjoyment (or affective) category as stated above.

The questionnaire was piloted twice at the conferences of the Norwegian Centre for Research, Innovation and Coordination of Mathematics Teaching MatRIC. On both occasions piloting led to changes in the questionnaire design and to the development of the project overall.

The survey was conducted using the online questionnaire software SurveyXact. An email invitation was sent to 498 mathematicians in Norway and 1304 mathematicians in England asking them to participate in the survey. Individuals were included in the email if they were an academic member of staff, working at a Norwegian or English university mathematics department. This survey was aimed at university mathematicians rather than teachers of university mathematics more generally. All universities in Norway were contacted with personal emails to potential participants. All universities in England were contacted via the Head of Department, which resulted in too few responses. Hence it was decided to contact potential participants directly in around half of the English universities. These were selected by geographical proximity. As a result, we received 119 responses to the questionnaire in Norway and 103 responses in England. This corresponds to a 24% response rate in Norway and 8% in England. In our email we clearly stated that we were equally interested in responses from lecturers who did use MM and those who did not, and never had used MM. We do not know why the response rate in Norway was higher than in England. However, it may be explained by MatRIC's involvement in initiatives and research collaborations across all educational levels in Norway, including in universities. The activities of MatRIC and its aims and reach are less known in England.

Both quantitative and qualitative methods of data analysis were used to analyse our data and identify response patterns among different groups of respondents. Qualitative methods such as interpreting, coding, and categorising (Cohen, Manion, & Morrison, 2008) were used to analyse the written responses that participants provided in comment boxes.

### Findings so far

Data from the Norwegian universities were collected faster and were analysed ahead of the English data. Hence findings reported relate mainly to the Norwegian cohort. We have included results from the English data analysis wherever possible.

The first part of the questionnaire was aimed at obtaining demographic data including, for example, respondents' PhD specialism. Here we found that respondents were fairly equally divided between pure mathematics, applied mathematics, and other subjects, such as physics, statistics and mathematics education for Norway with a larger percentage in applied mathematics in England. Most participants (around 90%) indicated that they were active in research, predominantly in mathematics and statistics. For both countries the gender balance of male to female was around 4:1.

In the second part of the questionnaire, we inquired into mathematicians' views on modelling as a part of their professional activity. We analysed responses and found that most Norwegian participants held a realistic perspective. In particular, 45% selected all three realist statements that were offered on the survey and a further 35% selected two of the three statements. The data from England came from a much smaller sample and showed an increased tendency towards a realistic perspective.

In the third part of the questionnaire, we inquired into mathematicians' views on the teaching of MM and on the aims of teaching using MM. Participants had to rank statements. In relation to questions about the aims of teaching MM, a varied response was found. Nearly half of respondents showed commitment to one of the perspectives with the realistic perspective being represented most, by 35% of respondents. However, all six perspectives were represented at least by one participant showing a greater diversity than we had found with views on MM in professional practice. 53% selected all of their three statements from different categories and hence showed no commitment to any particular perspective. This greater variety became apparent also when considering participants' first choice.

In Norway the realistic perspective is clearly dominant. However, all of the other perspectives are also represented among the sample and in a more pronounced manner compared to England where the variety is greater more generally. Around 1/3 of participants indicated a realistic perspective, and a further 1/3 a contextual perspective.

In relation to using MM in teaching, 74% of participants in Norway (and 86% in England) stated that they did use modelling in their teaching. As Table 3 shows, there were some differences in responses. In Norway 46% of respondents indicated that they used MM to support the teaching of mathematical theory with similar numbers of responses in England. However, a larger percentage of responses in England indicated that MM should be taught as a separate course.

So far we have reported mainly on quantitative data. We also conducted a qualitative analysis of the written comments by those who stated that they did not use MM in their teaching. We distinguished 4 categories of comments: on the nature of mathematics, on institutional issues, on the teaching of skills and comments relating to students. The answers from the first two categories were encountered more often than from the latter two. For example, respondents stated that MM was not relevant in their course, was not a part of the curriculum, or they had no time or no opportunity to teach MM.

While we have no results for England yet, we can say from the statistical analysis of the Norwegian data that those respondents who had experience of using MM in research were six times more likely to use it also in their teaching (P<0.005) compared to those who had no such experience. For the most part we found no significant differences between any subsets of the respondents. For example, there were no significant differences in responses between participants who held a strong realist view compared with those who did not. However, we did find some less surprising results, for instance, that respondents with a PhD in applied mathematics had used MM in their research to a significantly greater extent than the rest of the respondents.

In summary, most respondents, both in Norway and England, clearly indicated a realistic perspective according to our categorisation based on Kaiser and Sriraman (2006). However, when it comes to mathematicians' views on the teaching MM and/or on using MM in their teaching, the responses were more varied. We found that reasons for not using MM in teaching were mainly based on the nature of mathematics and on institutional factors.

### Future work

Analysis of the English data is on-going. So far a substantial amount has been analysed quantitatively and we will conduct the qualitative analysis shortly. This will provide insights into mathematicians' views on teaching MM in England. A comparative analysis of the Norwegian and English data will show how views and use of MM differ between the Norwegian and the English respondents. We have plans for research outcomes to be disseminated at future MatRIC conferences and to be published in a mathematics education research journal. We have had interest in this study from researchers in other countries. We have hence decided to continue our line of enquiry and will be applying for further funding from MatRIC and/or other funding bodies to do so.

### Acknowledgements

This project was funded by a research award from MatRIC, the Norwegian Centre for Research, Innovation and Coordination of Mathematics Teaching, project number 150401. We would like to express our thanks for this support.

### References

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