Mathematical modelling is a cyclic process that describes the translation between reality and mathematics in both directions. This process consists of certain sub-processes as shown, in an ideal-typical form, in Figure 1 (the modelling cycle from Blum & Leiß, 2007).

The sub-processes are to construct a situation model from a real-world problem, to generate real-world facts, data and relations, to simplify and structure the data, to mathematise (represent the data mathematically), to work within mathematics, and finally to interpret and to validate the mathematical results with respect to the real-world situation. A variety of illustrations of the modelling process exist in the literature (e.g. Stillman, Kaiser, & Lampen, 2020). The advantage of the illustration from Blum and Leiß (2007) is that individual steps (1–7) separate the phases of a typical mathematical modelling process. Students usually do not follow those steps in linear order when solving modelling tasks, but often ‘bounce’ between them. Each step can potentially be a cognitive barrier for students (Blum, 2011, 2015; Stillman, 2019; Stillman, Brown, & Galbraith, 2010). It seems important to develop mathematical modelling competency and sub-competencies related to the sub-processes of the modelling cycle.

Mathematical modelling problems are more challenging than traditional word problems, or application problems. Word problems (common in school textbooks) usually follow prior instruction for a specific theme in mathematics and relate only to a segment of the real world (COMAP-SIAM, 2016). Some authors refer to word problems as Level 1 modelling problems (Tan & Ang, 2012). When solving such problems, the modelling process is limited to mathematisation, mathematical procedures following on prior instruction, and direct interpretation. An application problem could be compared to a Level 2 modelling problem (Tan & Ang, 2012) as context and meaning is added to the word problem and required in the solution.

These problems are common in assessment tasks at the secondary level. However, students still do not have an opportunity to put their analysis back into a real-world situation (COMAP-SIAM, 2016). Also, the attempt to contextualise the mathematics may make the mathematical work seem completely irrelevant. Mathematical modelling problems, or Level 3 modelling problems (Tan & Ang, 2012), are open ended and generated from a real-world situation that requires the complete modelling process. No clear pathways are suggested in such problems and the modeller will seek for any mathematics to solve the problem and do research about the context and assumptions about the context. These problems might not be common in assessment tasks at the secondary level, but the benefit of incorporating such problems in the mathematics classroom has been established.

The following examples from the secondary environment, where students are learning how to write equations and draw graphs of linear functions given the slope and vertical intercept, can explain the difference between those three levels:

For various examples of modelling problems and an overview of the literature see Niss and Blum (2020).

The teaching and learning of mathematical modelling are difficult mainly due to the cognitive demand of modelling activities. Student teachers often have misconceptions about what mathematical modelling entails (Anhalt & Cortez, 2016), do not always understand the value of such activities, have mixed feelings about the topic and feel under-prepared to teach mathematical modelling (Blum & Borromeo Ferri, 2016; Ng, 2013).

An example of an exploratory study with a sample of mathematics educators on difficulties in teaching mathematical modelling in France and Spain is reported by Cabassut and Ferrando (2017). Results confirmed that most educators were positive about mathematical modelling, although some educators lack self-confidence. Most difficulties were experienced in relation to teaching the topic. These difficulties were specific to time (e.g. time to prepare for tasks, time on tasks), students’ involvement, and resources. Borromeo Ferri (2018) highlights four key competencies that should be developed in teacher education in order for teachers to teach mathematical modelling effectively and appropriately. The competencies are: (1) theoretical competency for practical work, (2) task competency for instructional flexibility, (3) instructional competency for effective and quality lessons, and (4) diagnostic competency for assessment and grading. Developing these professional competencies in student teachers might narrow the gap between research and practice in teaching mathematical modelling. One way of narrowing the gap could be through design-based research (DBR) which is linked particularly well with the teaching and learning of mathematical modelling.

DBR is a flexible methodology aiming at improving practices through iterative cycles of analysis, design, development, and implementation via interventions, and it focuses on the collaboration between researchers and practitioners with the intention to extend and identify new design possibilities (Abdallah, 2011). To identify key elements and to develop suitable design principles to guide the integration of mathematical modelling into the formal education of mathematics student teachers, Abdallah’s view of DBR seemed suitable. Seven key characteristics of DBR are widely agreed upon (Anderson & Shattuck, 2012; Reeves & McKenney, 2013; Wang & Hannafin, 2004) and provided a suitable validation to structure the study: first, DBR’s authentic nature and naturalistic context suggests a grounding in real-world contexts where participants can interact socially with one another, similar to everyday life. Second, DBR generates design principles that are contextually sensitive with the intention to improve practice. These principles are informed by theories, literature sources and the contextual variables, and are refined over multiple iterations. Third, DBR’s rigorous methodology faces the difficulty of incorporating a variety of factors. Thus, the outcome should be the culmination of the interaction between designed interventions, human psychology, personal histories or experiences, and local contexts. To improve the design, data collection and analysis are conducted simultaneously, while credible findings and meaningful solutions to the envisaged problem are produced by the rigour and reflection of the DBR process. Fourth, DBR’s fragile, complex and ‘messy’ nature requires mindful consideration from the researcher and a close collaboration between the researcher and participants to continuously refine the flexible design in the context. This close collaboration is possible through an appropriate well-structured or disciplined approach. Fifth, DBR functions via unique processes to transcend the local context, for example to refine the design continuously and iteratively (here the focus is on the processes). Sixth, DBR transcends the local context and attempts to provide an answer for why the problem occurs, what must be done and what we can learn from this to inform the practice of others (here the focus is on the elements related to the local context). Seventh, DBR generates credible evidence and useful knowledge that might also be used in another context. To ensure a purposeful and strategic intervention for student teachers to learn mathematical modelling, it is vital to consider these characteristics of DBR.

The product of a DBR study is to contribute to local (analogue to the situation and sample in this study) instructional theory through identifying key elements and fundamental principles.

Nieveen and Folmer (2013) emphasise that design principles, stemming from a DBR study, should provide insight into an educational intervention, and should communicate the purpose and context, key characteristics, design and procedural guidelines, as well as implementation conditions of the intervention. DBR has not often been used in studies to improve the professional development of teachers in mathematical modelling. However, one example of such a study to improve the didactical practices of primary school mathematics teachers through modelling was conducted by Biccard (2013). The study reported on in this article provided a unique opportunity to integrate mathematical modelling in the formal education of student teachers.

With the suitability of DBR having been theoretically established earlier, the research was conducted via three main phases (view Figure 2). The first phase (cycles 1–3) was in preparation for the experiment with a focus on needs analysis. This phase was informed by an in-depth literature review, the researcher’s personal experiences, ideas from other practitioners and specialists, and a pilot study. The second phase (cycles 4–6) focused on experimenting in the classroom keeping consistency and practicality in mind. This phase included all mathematical modelling activities structured over two iterations. The third phase (cycles 7 & 8) required a retrospective analysis of all qualitative and quantitative data with the intention to identify key elements and principles to prepare student teachers for mathematical modelling. Although the design of the three phases is supported by literature (see Reeves & McKenney, 2013), the phases reflected in Figure 2 included eight cycles, uniquely crafted for this study, that describe the design step by step. The intention with the eight cycles was to distil contextually sensitive design principles that might be suitable in further studies or in other contexts. Both qualitative and quantitative data were collected in cycles 2, 4 and 6. The qualitative data provided information on mainly the shortcomings and progression in student teachers’ mathematical modelling competencies and the quantitative data provided information on the change in student teachers’ attitudes towards mathematical modelling activities based on this exposure.

A pilot study was conducted (see cycle 2 in Figure 2) one year before the implementation of the main intervention. The sample and conditions in the pilot study were similar to the main study.

On all occasions the context was a lecture room, in a real-life setting. The intention with the pilot study was to test the validity of the data collection instruments, to confirm the identified needs of participants and to improve the researcher’s experience with a DBR strategy. Findings from the pilot study improved the preparation for the first iteration (cycle 4 in Figure 2), and some of its results have been published (see Durandt & Jacobs, 2014; Jacobs & Durandt, 2017). Similarly, findings from cycle 4 improved the preparation for the second iteration in cycle 6 and some results have been published (see Durandt & Lautenbach, 2020). Each data collection cycle (2, 4 & 6) was followed by a cycle where the results were conceptualised and draft design principles were identified and refined (cycles 3, 5 & 7). Through this cyclic process the findings from one cycle were incorporated in the design of the next cycle ensuring a continuous refinement of the key elements and principles relevant to the learning of mathematical modelling in this context.

The pilot study (cycle 2) consisted of one session of 90 minutes and participants acted as modellers. Iteration 1 (cycle 4) was similar to the pilot study. In iteration 2 (cycle 6) participants acted as both modellers and teachers of modelling. Four sessions of 90 minutes each were planned, two sessions allocated to each role. Rigour was maintained in all research phases by following all activities as planned and a methodological specialist attended the sessions. Multiple data collection instruments were used throughout the research phases to identify key elements and fundamental principles relevant when student teachers learn mathematical modelling. These instruments are related to learn the cyclic process of mathematical modelling (worksheet and poster documents), to identify shortcomings and capture experiences (open-ended questionnaire), and to determine student attitudes towards mathematical modelling (ATMMI). For the focus of this article, only data collected from cycles 4 and 6 will be reported on, which mainly informed the retrospective analysis.

In cycles 4 and 6 (see Figure 2) participants were exposed to modelling activities that ranged from solving easier tasks (like an application problem or Level 2 modelling example; see above for an explanation) to more challenging, open-ended and complex modelling tasks (like a Level 3 modelling example; see above for an explanation and below for an example). The intention with the range in complexity of tasks was to develop mathematical modelling competencies in student teachers, for them to learn sub-competencies and develop more positive attitudes towards such activities over time. The tasks asked participants to be modellers themselves, but also teachers of modelling. While acting as modellers themselves, they were expected to work their way through the modelling cycle in groups and record their work on a predesigned worksheet. These activities were included so that student teachers might develop both theoretical and task competencies as described by Borromeo Ferri (2018). One such example is discussed in a later section and for other examples of modelling activities and tasks used in the study, see Durandt (2018). In groups, they participated in discussions, designed their own modelling tasks, reflected on and evaluated their own and others’ work, prepared posters and presented proposed solutions. In these activities, participants acted in both roles, as modellers themselves and also as teachers, with the intention to develop instructional and diagnostic competencies as emphasised by Borromeo Ferri.

A population of 55 third-year student mathematics teachers from a large public university in South Africa participated in the main study that was completed in 2018. The participants did not share their mathematics class with other mathematics students (like engineers or computer science students) and this separation was automatically done by the university system. The participants were arranged in the same 10 comparable groups in all iteration phases by using purposive sampling procedures. Relatively small groups of four to six members were formed, with each group containing at least a high, a moderate and a low achiever. The achievement level was determined by their formal course marks in mathematics. The group selection was mandatory (determined by the researcher) and purposeful in planning for multiple dependent variables such as the complexity of the modelling activities, student teachers’ initial or limited exposure to mathematical modelling tasks, and the envisaged and expected valuable collaboration among participants.

Student teachers were exposed to mathematical modelling tasks to learn modelling themselves, to develop mathematical modelling competencies and sub-competencies and with the intention to develop positive attitudes towards the topic. One example is the ‘Location Problem’ (view Figure 3) where participants (in groups) were asked to use the data and make a recommendation to the Department of Town and Regional Planning on the best location for a day-care centre for toddlers.

Participants were expected to work through the processes of the modelling cycle, record their work on a predesigned worksheet with a four-step plan (view Table 1), similar to the idea of a plan used in other studies like the ‘solution plan’ used in the DISUM study (Schukajlow, Kolter, & Blum, 2015) and aligned with the main processes stemming from the modelling cycle (view Figure 1).

After solving the task, all groups participated in discussions, both in groups and as a whole class. Then, each group prepared and presented a poster to demonstrate visually how they used the modelling cycle to find a possible solution to the problem. See Figure 4 for an example from group 9 after solving the ‘Location Problem’ in iteration 2.

They also evaluated their own work and the work of others by using an evaluation sheet (selecting one of three criteria: high, medium, low). Later, at the end of each iteration, participants reported their experiences by answering an open-ended questionnaire individually.

The questionnaire included a section on biographical information of participants as well as a section on their perceptions of the mathematical modelling experience and support they might require. One example of a question from the questionnaire is: How can the university further support you (during your teacher training) in becoming an even more effective teacher of mathematical modelling (make concrete suggestions)? The qualitative data collection instruments (worksheets, poster and open-ended questionnaire) were designed at the end of cycles 1, 3 and 5.

Content analysis methods were used to analyse the data (Saldaña, 2016). Students’ worksheet and poster documents were marked according to a framework deduced from the sub-processes of the modelling cycle (see Figure 1) and compared through the cycles (see Table 2). Data collected through the open-ended questionnaire were analysed according to themes, and further separated in categories and sub-categories (see Figure 5). This was done via pen and paper and using Atlas.ti software. Strategies to maintain the trustworthiness of the qualitative component of this study included a thick description of the methodology (in section 3) and an external subject specialist confirming coding categories and qualitative findings (as suggested by Creswell, 2013).

The Attitudes Towards Mathematical Modelling Inventory (ATMMI) was used to gain information individually regarding student teachers’ attitudes towards mathematical modelling at the end of each iteration.

The ATMMI, adapted from Schackow (2005) that followed the original ATMI instrument from Tapia and Marsh (2004), is a locally tested instrument that consists of 40 Likert-scale items arranged from ‘strongly disagree’ to ‘strongly agree’ (5 possible responses). The items were grouped in four sub-scales: enjoyment (10 items to determine whether mathematical problem-solving and modelling challenges were considered enjoyable for participants), value (10 items, to determine whether mathematical modelling knowledge and skills were considered worthwhile and necessary for participants), self-confidence (15 items, to determine the expectations about doing well in respect of mathematical modelling, and how easily modelling was mastered by participants) and motivation (5 items, to determine the desire of participants to learn more about mathematical modelling and to teach the topic). Data were collected on paper at the end of iteration 1 (cycle 4) and at the end of iteration 2 (cycle 6). The Statistical Software Package for the Social Sciences (SPSS, version 24) was used to analyse the data. Internal consistency was confirmed by acceptable Cronbach’s alpha coefficients (in all sub-scales > 0.8). The results were similar to previous recorded alpha values by Schackow and Tapia and Marsh. Content validity was confirmed by mathematical modelling specialists from other South African universities, construct validity by previously recorded factor analysis (see Tapia & Marsh, 2004) and sight validity through the pilot study. A non-parametric test (Wilcoxon Signed Ranks test) was used to evaluate the change in attitudes of student teachers towards mathematical modelling after exposure to a series of mathematical modelling activities, and a parametric test (one-sample t-test) was used to determine the effectiveness of the methodological design. Due to the sample size (30+) and the robustness of statistical approaches (Pallant, 2010), the parametric test was a possibility.

Standard ethical measures were taken according to the literature (Creswell, 2013; Teddlie & Tashakkori, 2009) and the procedures at the university (ethics clearance number 2015-024). At the beginning of the modelling activities in cycles 2, 4 and 6, participants were briefly informed about the planned activities. Participation was voluntary and participants signed an informed consent form. The researcher had a dual role for the purpose of this DBR study, both designer and researcher. This is a limitation and was purposefully addressed by balancing the roles, and through the observations recorded by the methodological specialist during implementation.

Based on data collected from the worksheets, posters and open-ended questions (collected in cycles 4 & 6, view Figure 2) themes were identified reflecting student teachers’ shortcomings and progression in mathematical modelling activities.

All themes focused on elements and principles related to developing competencies (Theme 1), building perceptions (Theme 2) and providing resources and opportunities (Theme 3). The categories and sub-categories included in these themes intend to sketch a picture of the challenges participants experienced throughout the intervention. Figure 5 shows an overview of the themes, with categories and sub-categories.

The first theme, developing competencies, groups the needs (to a lesser or greater extent) of student teachers to develop particular competencies as they proceeded through the intervention in this study. There were four sub-categories: modelling competencies, mathematical competencies, utilising mathematical models and facilitator competencies (see Table 2). Modelling competencies refer to how participants proceeded through the different phases of the modelling cycle and interpreted their findings in terms of real-world relevance. Mathematical competencies refer to the mathematical representation of real-world problems, relevant mathematical content selection and the need for relatively accurate calculations. Utilising mathematical models refers to an investigation of model efficiency (by criteria from Meyer, 2012) and an exposure to real-life model applicability. Teacher competencies describe the preparation work (e.g. designing of a modelling task) and the facilitation work (e.g. facilitating class discussions reflecting on the modelling process) of participants as they took on the role as teachers of modelling. For example, in the ‘Location Problem’ (see sub-section 3.4) as participants took on the role of modellers themselves, they structured the information and represented the one-way streets and number of cars travelling along specific roads visually (like in the display in Figure 4), but they had trouble constructing a suitable mathematical model (a linear system of four equations with four variables) to represent the situation. Even if they managed to construct a suitable mathematical model, some groups struggled to solve the system of equations. Such difficulties caused blockages in the modelling cycle. One group made a calculation error (where the variables that represent the number of cars travelling along a specific road were equal to negative values) and they struggled to interpret their mathematical result within the context of the example. Table 2 represents the overall mean scores as percentages of participants in groups who mastered the specific competency.

The second theme, building perceptions, highlights the importance (based on former and current opinions) of developing a positive disposition in student teachers towards mathematical modelling activities (for example the ‘Location Problem’). This theme is divided into two sub-categories: ideas about mathematics and ideas about mathematical modelling (informed by data findings from the open-ended questionnaire). Ideas about mathematics refer to participants’ thinking about mathematics as a subject, the teaching thereof and valuing the subject. Ideas about mathematical modelling are related to participants’ disposition based on their exposure to modelling activities as modellers themselves and as facilitators of such activities throughout their formal education (for example, in this study their exposure to modelling activities in cycles 4 and 6; see Figure 2). Qualitative data collected from the open-ended questionnaire (in cycles 4 & 6) showed:

The third theme, providing resources and opportunities, indicates the required support highlighted by student teachers to develop in the teaching and learning of mathematical modelling throughout their professional education. This theme is divided into three sub-categories (informed by data findings from the open-ended questionnaire): pre-tertiary education, tertiary education and post-tertiary education. The categories pre-tertiary education and post-tertiary education are beyond the scope of this study although the researcher recognised from the findings that support during these phases could contribute towards the development of student teachers’ competencies in mathematical modelling and the further enhancement thereof. The category tertiary education refers first to the group work opportunities required by student teachers to gain modelling competencies (as modellers and facilitators) and to develop proficiency in collaborative actions, and second to particular curriculum considerations to ensure continuous contextualised exposure (to content and teaching methodologies) of mathematical modelling activities. For example, participants worked together in small groups to find a real-life solution for the ‘Location Problem’.

Qualitative data collected from the open-ended questionnaire (in cycles 4 & 6) showed:

Within each sub-scale, the researcher compared total scores and investigated cross-tabulation results for respective items. Due to the comparative nature of the analyses in DBR phase 3, only cases displaying the necessary information in both iterations were considered (enjoyment 44 cases, value 43 cases, self-confidence 44 cases, and motivation 43 cases). Hence, missing data were excluded pairwise. The Wilcoxon Signed Ranks test as statistical technique is regarded as suitable to answer the research question related to change in attitudes over time (Pallant, 2010). This test converts scores to negative or positive ranks and compares them at iteration 1 and 2.

Enjoyment: More positive ranks (26) than negative ranks (14) on the enjoyment scores of participants were detected, but no significant change in enjoyment mean scores following participation in the mathematical modelling intervention over two iterations. Table 3 shows z = –1.86 and p = 0.06, with a small to medium effect size (r = 0.20) according to the criteria by Cohen (1988).

Findings from cross-tabulation data in the enjoyment sub-scale revealed at the end of iteration 2 that most participants (over 80%) mildly to strongly agreed on the following aspects: (1) mathematical modelling is a very worthwhile topic and they wanted to further develop their mathematical modelling skills, (2) the process taught them to think and they recognised the importance of the topic in everyday life, (3) they were not sure how the topic can be utilised in learning mathematics although it seems important for mathematics students for any grades of teaching, (4) the usefulness of studying mathematical modelling at a higher education level, and (5) their belief that the topic will support them with problem-solving in other areas and a strong background in modelling could help any mathematics teacher.

Value: More negative ranks (24) than positive ranks (17) on the value scores of participants were detected and no significant change in value mean scores following participation in the mathematical modelling intervention. Table 4 shows z = –0.182 and p = 0.86, with a small to medium effect size (r = 0.20) according to the criteria by Cohen (1988).

Findings from cross-tabulation data in the value sub-scale revealed at the end of iteration 2 that participants mildly to strongly agreed that they: (1) got a great deal of satisfaction out of solving a mathematical modelling problem (70.2%), (2) enjoyed being involved in a mathematical modelling session (66.7%), (3) liked to solve real-world problems in mathematics (69.8%), (4) preferred a mathematical modelling task rather than writing an essay (73.2%), (5) thought they liked the topic (59.5%), (6) preferred real-world problems to other mathematical themes (29.3%), (7) found the topic very interesting (76.2%), (8) were comfortable expressing their own ideas on how to solve a modelling problem (69.8%), and (9) were comfortable suggesting possible solutions to a modelling problem (73.1%). Furthermore, participants (86%) mildly to strongly disagreed with the statement that mathematical modelling is dull.

Self-confidence: More positive ranks (26) than negative ranks (15) on the self-confidence scores of participants were detected and no significant change in self-confidence mean scores following participation in the mathematical modelling intervention. Table 5 shows z = –1.42 and p = 0.16, with a small effect size (r = 0.15) according to the criteria by Cohen (1988).

Findings from cross-tabulation data in the self-confidence sub-scale at the end of iteration 2 revealed that participants mildly to strongly disagreed with the following: (1) mathematical modelling is a feared topic (50%), (2) it created a feeling of dislike (69.8%), (3) their minds went blank when confronted with the topic (62.8%), (4) mathematical modelling made them nervous (51.2%), and uncomfortable (67.5%), (5) they experienced terrible strain in a mathematical modelling session (66.7%) and just thinking about the topic made them nervous (67.4%), (6) they were confused in a modelling session (55.8%), and (7) they felt insecure when attempting mathematical modelling (64.3%). However, participants mildly to strongly agreed with the statements: (1) mathematical modelling did not scare them (45.3%), (2) they had self-confidence with the topic (41.9%), (3) they would be able to solve such a problem without too much difficulty (26.6%), (4) their expectation would be to do fairly well in future modelling sessions (64.3%), (5) they will learn mathematical modelling easily (45.2%), and (6) they believed they were good at solving real-world problems (48.9%).

d) Motivation: More positive ranks (25) than negative ranks (14) on the motivation scores of participants were detected and no significant change in motivation mean scores following participation in the mathematical modelling intervention. Table 6 shows z = –1.71 and p = 0.09, with a small to medium effect size (r = 0.20) according to the criteria by Cohen (1988).

Findings from cross-tabulation data in the motivation sub-scale revealed at the end of iteration 2 that participants mildly to strongly agreed with: (1) they could pass a course on mathematical modelling for mathematics teachers (62.8%), (2) they were willing to learn more about mathematical modelling in future (65.1%), (3) the challenge of mathematical modelling is appealing (54.7%), and (4) they would be keen to enrol for a course on mathematical modelling for mathematics teachers (65.2%). Furthermore, participants mildly to strongly disagreed that they would like to avoid teaching mathematical modelling (67.5%).

Finally, the researcher investigated the percentage change in scores between iteration 1 and 2 for all sub-scales (by utilising the formula: Iteration 2 scores−Iteration 1 scoresIteration 1 scores×100).[Eqn 1]

The purpose of this test was to evaluate the effectiveness of the mathematical modelling intervention over two iterations on participants’ attitudinal scores and more broadly on the methodological design. Table 7 shows positive percentage changes in all sub-scales, and significant percentage changes in both the enjoyment (t[43] = 2.12, p < 0.05 two-tailed) and motivation sub-scales (t[42] = 2.16, p < 0.05 two-tailed).