Introduction

This was a study on collaboration between five master students and a lecturer, myself. First, the collaboration was a partnership, with the common aim of conducting research. One of the two research areas of the project was mathematical modelling in a critical mathematics education perspective. Second, the collaboration itself was studied from an educational point of view. The collaboration was voluntary and not part of their master education. The second research area is thus on the significance of collaborative academic writing as an approach for teaching and learning.

As an introduction to critical mathematics in a master course, a graph (Figure 1) produced by IPCC (Intergovernmental Panel on Climate Change) on predicted temperature anomalies was discussed. The discussion was followed by a lecture on selected literature from critical mathematics education, where links between the initial discussion and the literature were collectively drawn. The sessions where the students participated were recorded, and the students were invited to write an academic paper on an analysis of their discussion together with me. The analytical tool to be applied was part of the curriculum. Five of the nine students present in class accepted the offer.

The described approach matches education based research as it promotes insights in curriculum related topics and literature through research. The approach also matches research based education as the students achieved experiences and insights in research related to the master course. The approach also matches what Alrø and Skovsmose (2002) call the inquiry cooperation model: Research is an inquiry based activity in itself, demanding a critical attitude, and the intention was to collaborate with the students on more or less equal terms by giving them responsibility. The collaboration was an experiment on a new approach to teaching, where the aim was to give more responsibility to the students, to give them a task that also counts outside the classroom and which is meaningful to them in new ways. This can be described in accordance with the model on developing teaching by Skovsmose and Borba (2004): I arranged a teaching situation based on pedagogical phantasy with an ideal situation in mind. One purpose of the project and the collaboration was thus to study this teaching situation and its approach: What are the learning potentials for both students and lecturer? How do the students reflect on their own competence? The other purpose was to develop a paper on an issue that has received limited attention in the academic literature on mathematics education.


Figure 1. Temperature change. The graph shows modelled developments of the global annual mean temperature change (IPCC, 2013, p. 1054). The change is calculated relative to the period 1986–2005. Four future scenarios are presented with mean (the coloured trajectories) and 90% confidence interval (the coloured shadings). The colours represent different possible emission levels of greenhouse gases (RCP - representative concentration pathways). The red shading represents status quo.

Society is facing great challenges concerning national economies, social issues, natural resources, food production and energy supply. Such issues are often associated with complexity, uncertainty and conflicting stakes. Mathematics supporting political decisions are therefore advanced, but also associated with uncertainty. Skovsmose argues that mathematics has a formatting power in society in that "new insights may change interpretations, but also in the sense that mathematics colonizes part of reality and rearranges it" (Skovsmose, 1992, p. 6). He further argues that the ability to recognize this formatting power, and reflect on it, is an essential democratic competence in order to balance the experts' influence on politics and society.

There are mathematics and science educators who argue that the issue of climate change should be brought into classrooms. Suggestions on how and why include (i) student discussions on the role of mathematics in climate science and policy making to facilitate critical reflections (Barwell & Suurtamm, 2010), (ii) student discussions on graphs and the predictive power of mathematical models (Hansen, 2012) and (iii) student tasks on controversies surrounding features of climate science to understand aspects of science in society (Erstad & Klevenberg, 2011). Barwell (2013) argues that mathematics education has a responsibility for developing ideas on how to discuss climate change in classrooms. This is a research area which has received limited attention. The issue of climate change is perhaps too complex for students to develop alternative mathematical approaches that can pinpoint socio-political consequences of mathematized information. A relevant area of research would therefore be on what kind of critical reflection non-experts can provide on advanced mathematized information that is useful for critical citizenship. The purpose of the papers written together with the students was to analyse the classroom discussion on the IPCC graph (Figure 1) with regard to the kinds of critical mathematical reflections being made.


Method

The project consists of two parts which make use of different sets of data and methodological approaches. These are presented under separate headings.

The joint collaboration with the students

As an introduction to critical mathematics education in a master course, the nine students and the lecturer (myself) discussed a graph produced by IPCC on predicted temperature changes (see Figure 1). The discussion was followed by a lecture on selected literature from critical mathematics education, where links between the initial discussion and the literature were collectively drawn. The lecturer chose this figure because of its potential for generating reflections about mathematics and its use: (1) its political relevance and dispute, (2) it expresses uncertainty through statistical spread, different pathways of future emissions and implicitly through disagreeing model results, (3) the social construct of "global average surface temperature" and (4) the problem of measuring temperature in space and time. My intended role was to keep the discussion as student driven as possible, but at the same time ensuring that student reflections took place.

Before the discussion, I invited the students to write an academic paper on an analysis of their discussion together with me. The analytical tool to be applied was part of the curriculum. Five of the students accepted the offer and became co-authors of two papers (Vethe et al., submitted; Hauge et al., accepted).

The classroom discussion was analysed using Skovsmose's (1992) six reflection steps, of which five are later picked up by Gellert, Jablonka and Keitel (2001) and referred to as five reflection levels. The steps can be summarised as reflections related to the following questions: (i) Are the calculations right? (ii) Was the right algorithm used? (iii) Is the mathematical approach reliable? (iv) Could the problem be solved without formal mathematics? (v) How does the mathematical approach affect the specific context of the problem? (vi) Could we have reflected on this in another way? The six reflection steps refer to what Skovsmose calls steps towards reflective knowing and which is essential for critical citizenship. They represent steps of students' reflections while engaged in problem solving. The first pair of steps refers to reflective thinking within pure mathematics, the next within applied mathematics and the latter within a broader, societal, understanding of mathematics (Skovsmose, 1992). As these reflection steps are developed for situations where students reflect on their own problem solving, in contrast to our case, we had adjusted the reflection steps for our purpose. We chose (i) and (ii) to include our reflections on the mathematics expressed in the graph, which are developed by others. Since the underlying mathematics is hidden for us, we have concentrated on our understanding of the graph contents, for example: Have we understood the graph correctly? Have we understood underlying algorithms? We considered such reflections as a required basis for reflective knowing and critical citizenship and are therefore relevant to include in our analysis.

The two classroom discussions on the graph and on applying the reflection steps on the graph were audiotaped. Additional data include notes and the lecturer's PowerPoint presentation. We applied the authors' names in the excerpts. The other names are anonymised. The audiotaped discussions were transcribed and coded by the students in accordance with the six reflection steps. The coding turned out to be challenging as the steps were not straight forward to apply. This was discussed during meetings.

The study on the collaboration

Additional data and methods were used to study the collaboration itself and its potential for learning, including mutual learning. Also, more information on the collaboration was gathered. When the students were invited to join the research project, the minimum requirement to become a co-author was to transcribe five minutes of the audiotaped recording and to critically read through the final draft of the text. I also explained that further effort was rewarded by climbing up the list of co-authors. The project was presented as a cutting edge project due to the lack of academic papers on the area, and the paper was to be submitted to Tangenten and be peer-reviewed. In the beginning of the audiotaped class, all students seemed to be interested in participating in the research project. I therefore allocated five minutes of the recordings to each of the nine students, leaving half of the transcribing to me. After some time had passed, four of the students had informed me that they wished to withdraw from the project, while three of the remaining students asked for more data to transcribe. Finally, the students had transcribed all the data, leaving the quality check of the excerpts we used for the papers to me. I then provided a one and a half page overview of the first paper, consisting only of paper headlines accompanied with content cues and short explanations for these. The students could choose how they wanted to contribute to the writing process by choosing headlines and content cues. All of the five students transcribed, analysed data and wrote pieces of text.

It turned out that the students wished to select excerpts for the paper, carry out the analyses and to provide drafts of text for basically the whole paper. My role in the development of the manuscript was to provide the initial text structure and research question, to discuss and advise at meetings, to edit the final draft and to submit the manuscript to Tangenten. The six meetings were about i) organising work, ii) discuss excerpts, analyses and text and iii) to discuss the order of authorship. The students did most of the work and took many of the decisions, and I saw my role basically as ensuring that the paper would meet my publication standards.

The intention with the collaboration was to obtain as much student influence as possible, but at the same time offer them flexibility in how much effort they would like to make on the paper. I wanted to make the collaboration as authentic as possible in the sense that the research area and research question were interesting for the academic community and that the work was accompanied by scrutiny, critique, equality and mutual respect. Complete equality is not possible since I have more experience within research than the students and since the paper idea was mine and settled. I felt the overall responsibility for the project, and I wished the experience to be pleasant for the students.

The methodological approach to the collaboration is two-fold. The collaboration was an experiment on a new approach to teaching. Thus, I applied the model on developing teaching by Skovsmose and Borba (2004). The model consists of three teaching situations: the present, the ideal and the arranged situation. The arranged situation is what I studied, while the ideal situation was what I imagined. The ideal situation could be described as a cooperation where all the students in class would participate, take responsibility, enjoy the activity and find it meaningful in the sense that they would learn curricular elements more in depth and learn about conducting research and that the research task and their contribution would give them a feeling of authenticity which had meaning outside the classroom.

Within this approach the activity also fits within what Alrø and Skovsmose (2002) call the inquiry co-operation model (the IC-Model). This model denotes a way of communication and its elements: "the IC-Model consists of communicative acts among teacher and students that support learning in a particular way" (Alrø & Skovsmose, 2002 p. 62). The elements are i) getting into contact, ii) locating, iii) identifying, iv) advocating, v) thinking aloud, vi) reformulating, vii) challenging and viii) evaluating. Getting into contact includes taking responsibility as a listener and to have mutual attention in the classroom dialogues. Locating is about examining student perspectives and different approaches. Identifying is about mathematising an approach. This is different in our case since the research activity is not directly related to mathematics but to research on a mathematics class. Identifying must therefore be seen in this context. Yet, it is a process associated with inquiry. Advocating ideas or points of view and thinking aloud are also part of an inquiry and make it possible for others to investigate perspectives. Formulations of ideas and views can be reformulated for clarification purposes. Challenging and evaluating perspectives are also central parts of an inquiry.

The analysis of the collaboration is not yet completed, but so far, the analytical tools are developed from the model on developing teaching (Skovsmose & Borba, 2004) and the inquiry cooperation model (Alrø & Skovsmose, 2002). The data for this part of the project consists of different versions of texts and analyses from the collaborative paper project, notes, and the last meeting was audiotaped.


Discussion/conclusion

The collaboration with the students has resulted in a manuscript submitted to Tangenten for a peer-review (Vethe et al., submitted) and a conference paper, which has been peer-reviewed and presented (Hauge et al., accepted). The funding from MatRIC has made it possible for the students to present the first paper and their collaboration experience at the conference Kultur og kreativitet (etter- og videreutdanningskonferansen) on Hurtigruten last year and at the first MatRIC conference in Trondheim. A main finding of these papers includes that discussing a graph like Figure 1 can promote mathematics related critical reflections on a range of levels, in spite of the incomprehensible mathematics supporting the graph. The sorts of reflection the students made are important to critical citizenship because they show they are able to pose key questions about mathematics in a given context and on the connection between uncertainty aspects and its role in society and policy making. If critical citizenship is considered a value in and for society, mathematics education should take a role in facilitating the development of such capabilities.

A paper on the second part of the project is still under development. Research questions for this part are on the potential of such a collaboration for mutual learning. How do the students reflect on their competence during the process? The collaboration has shown to be an arena for co-learning on course curriculum topics, research competences and teaching.

The very first task the students were to do with the transcribed data was to mark utterances from the classroom discussion in accordance with Skovsmose's six reflection steps, our analytical tool, and to highlight utterances that they found particularly interesting in this regard. Some of the utterances they highlighted were outside the scope of the research question, and there was no complete agreement on all the reflection step labelling. The students had for example highlighted who was right in a certain part of the discussion or a particular role I had taken as a lecturer in a certain situation. These points might have been interesting in a certain context, but they did not fit our research question or analytical tool. The students showed progress during our collaboration in being able to regard the data though the lenses of the research questions. This is certainly a key competence in conducting research.

The disagreements on which reflection step to apply in the labelling exercise, generated insights in regard to both the curriculum and to conducting research. Applying the steps in the analysis made the students reflect more thoroughly on the steps themselves, which were part of the course curriculum. This gave them a deeper understanding of the reflection steps. Since the students had transcribed all the data and had chosen excerpts for the analysis, they became very knowledgeable on the details of the data and how they could be used for our research. It happened several times that a student disagreed with my advice and provided argumentation based on more in depth knowledge about the data. The students were thus a valuable resource to the research project, which also meant that we managed to obtain equality in that we were experts in different parts and aspects of the paper and paper production. We discussed the reflection steps a lot because they were not easy to apply. The students experienced that analytical tools can be, and sometimes must be, adjusted to fit a purpose. While I suggested some amendments, first in the classroom lecture, and later during the first meetings, the students became eager in defining an additional step. They expressed an ownership of this seventh step. The discussions on the reflection steps promoted not only insight in their curriculum, but also on conducting research: how to apply an analytical tool and to reflect on its suitability and its purpose.

The students' academic writing improved greatly during the collaboration, the text on the analysis in particular. The five students were all involved in writing analysis sections. Structuring the analysis showed to be challenging in the beginning, including evaluating what kind of information is necessary to include and what can be neglected. Their improvements turned to be a learning situation for me as well as I understood more in depth what was challenging for the students and why. This is very relevant for my role as a supervisor for master students in general.

The students expressed that writing a paper was a new kind of experience in several ways. They found paper writing different from writing student essays: in the purpose of literature references and the role of theoretical perspectives within the research and text. The students had all chosen their area of research for their master theses, but they still had to finish full time master classes. This meant that they had not collected data for their theses or spent much time on it. They reflected on the differences between a paper and a master thesis, and we discussed it somewhat during meetings, for example the level of details in the various parts of a paper and a thesis.

The students reflected on their own learning process during the collaboration. They experienced an increased ownership of the curriculum literature. What they perhaps expressed most strongly was that they felt more prepared for their master thesis because of the paper project. One student said that the collaboration made him more prepared for his work on the thesis than any course at the university college. They stated that how to apply an analytical framework, in practice and through text, was their most useful learning outcome. They also highlighted the usefulness of having transcribed recordings and that they had become more open minded in the process. More surprisingly, they claimed they had never cooperated better in a group before during their four years of education. They explained that they had taken more responsibility for each other's contributions and that for the first time they were in a group where critical reflections to their own and others work were normal, accepted and not offending. Of course I am very satisfied that the students had this experience. On the other hand, I was disappointed on behalf of the students about their education not having given them such an opportunity earlier. It made me start thinking about what the conditions are for successful group assignments. This is surely a very complex question. They expressed a pity for the students who declined the offer and said with regret that the other students should have taken the opportunity. When I asked the students whether I should make the collaboration task compulsory in the future course, they all answered "no". The best part of the collaboration was that it was voluntary, they expressed. My interpretation of their answer is that it made the task more authentic and meaningful.

My own learning process and outcome was multi-faceted. Both the students and I found the collaboration a success, which means that I have plans for something similar next year. The students were offered a responsibility, and they took far more responsibility than I ever expected. Although I had a supervisor role, it was different than supervising a master student. We managed to obtain more equality than I had pictured before the collaboration because the students got a better grasp of the data than what I had. During our collaboration, the analytical tool had to be explored and adjusted for our purpose, and the students contributed greatly to the inquiry of the tool and reflected critically on each other's and my contributions. The collaboration was real in the sense that the students and I were mutually dependent, the product was something real: a paper to be published, the paper was of great importance to me and I demanded a certain standard. I learned that all these aspects were important conditions for the students' learning processes.

I also learned something about my own teaching. Simply listening to my voice on a tape gave me some ideas on how I can develop as a lecturer. One example is that it made me see how often implicit statements were expressed in the classroom where I could have taken the responsibility to facilitate further specification, explanation or exploration. Another major insight was a necessity to better explain the context and purpose of curriculum ideas and content. This became first clear to me at the initial stage of the collaboration when we were to apply recently presented curriculum ideas. Also at the final stage of the paper writing, I was somewhat surprised that the students had some difficulties in expressing the point of the paper, which I found so closely related to their curriculum. This made me aware of the necessity of meta-reflections on course content on a much more extensive level than what I had carried out in class.

As a final remark, I would like to comment on a couple of the success factors of the collaboration because they made me think more broadly about our education system. First, is the extensive use of project assignments in teacher education a failure? How can we facilitate group work environments with the qualities my collaborative students experienced? I take for granted that mutual respect, students taking responsibility for all parts of an assignment and a group environment where students can critically reflect on each other's contributions are qualities in group work that any education system wishes the students to learn and to pursue. Second, what does it tell us about the students' education when the students find the collaboration a success because it is voluntary? I have to admit that it disappointed me, but at least it triggered my own thinking process.


Acknowledgements

I would like to thank the master students for the cooperation: Terje André Bringeland, Andreas Austlid Hagen, Marius Sætren Sumstad, Maria Astad Sørngård and Tor Inge Vethe. It has been immensely rewarding for me. Thanks to MatRIC, the students were given the opportunity to show their work to a broader community. I am also grateful to IPCC for granting a permission to reprint their figure.


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