Mathematics and other disciplines, and the role of modelling: Advances and challenges

Alejandro S. Gonzalez-Martin, Ghislalne Gueudet, Berta Barquero and AveniIde Romo-Vazquez

Introduction

Ten years ago, Artigue, Batanero, and Kent (2007) stated that a major part of university mathematics education (UME) research had been established “having in mind more or less explicitly the mathematical education of ‘pure mathematicians’ or mathematics teachers and their particular needs” (p. 1030). Although in recent years more papers have examined a wider variety of student profiles, there is still much to do: in the opening session of the first INDRUM conference, Artigue (2016) stated that the practice of the mathematician is still an implicit reference in most research studies on UME. This indicates that many facets of UME continue to be under-researched. However, research examining how non-specialists learn and are taught university mathematics could yield many benefits, considering the number of students enrolled worldwide in mathematics service courses as compared with specialist mathematics courses. The high number of students in mathematics service courses is an important issue, due to the high number of students who drop out of their programmes after failing calculus (Rasmussen & Ellis, 2013). In particular, many engineering students abandon their programme after failing a mathematics course (Faulkner, Earl, & Herman, 2019); tellingly, some engineering programmes blame mathematics courses for as many as one third of their dropouts (Faulkner et al., 2019).

In recent years, UME researchers have shown a growing interest in mathematics for non-specialists (as reported in various research surveys on UME, e.g., Biza, Giraldo, Hochmuth, Khakbaz, & Rasmussen, 2016; Winslow, Gueudet, Hochmuth, & Nardi, 2018). Many studies report that non-specialist students encounter difficulties with mathematics, identifying a common phenomenon that may be aggravating these difficulties: there is often no explicit link between the content of mathematics courses and the content of professionally-oriented courses specific to other fields of study (such as engineering). In fact, mathematics course content can differ significantly from the content of specialist courses (e.g., Harris, Black, Hernandez-Martinez, Pepin, Williams, & TransMaths Team, 2015) and often seems disconnected from students’ future professional practice (Wake, 2014). Certain pedagogical interventions are therefore needed: teaching mathematics in a way that provides links with other fields (e.g., biology, economics or physics) and that draws on problems encountered in these disciplines or in the workplace. This raises the issue of modelling and the need to investigate how modelling activities can play a central role in such interventions.

The above also applies to engineering, which has attracted more research than other fields in recent years, giving rise to a whole area of study: mathematics education for engineers. Research on the professional practices of engineers (e.g., Artigue et al., 2007; Gainsburg, 2006; Kent & Noss, 2003) shows that, in many cases, engineers do not recognise the mathematics they use. The following statement illustrates this phenomenon:

It is notable in talking to experienced engineers that the mathematics that is useful and relevant has in many cases long since come to be thought of as part of engineering, whereas mathematics that is not used is regarded as “mathematics”

(Artigue et al., 2007, p. 1034)

This “invisibility” of mathematics can lead students (and decision-makers) to question the prominence of mathematics in engineering education. However, the aforementioned authors point out that this invisibility is due to the fact that most professional tasks requiring the explicit use of mathematics are carried out by specialists or are performed using sophisticated computer programmes that function as a black box. The use of tables and metalanguages that facilitate the implicit use of mathematics, often entwined with knowledge of the engineering sciences, is also evidenced. However, it is important to highlight that the professional needs of engineers are continuing to evolve, with the use of mathematics growing in importance. Examples include interpreting the multiplicity of computer-generated results, or using mathematical knowledge and language to communicate modelling tasks and results to other engineers (Quéré, 2019).

The above issues were raised in many of the works discussed in Thematic Working Group 2 (TWG2, “Mathematics for engineers - Modelling - Mathematics and other disciplines”) at the INDRUM 2016 and INDRUM 2018 conferences. TWG2 had seven papers and two posters in 2016 and 10 papers and four posters in 2018; it offered an opportunity to present and discuss ongoing doctoral studies (e.g., Feudel, 2016; Tetaj, 2018) as well as projects at their beginning stages (e.g., Fredriksen & Hadjerrouit, 2016; Tetaj, 2018; Viimian, 2016) that later became GERME or journal publications. In what follows, we summarise some of what we see as the main contributions of INDRUM to the above themes. We focus mainly on contributions that directly address issues specific to the training of non-specialists (including the role of modelling in this context). Keeping in mind the content of TWG2 contributions, we have structured this chapter as follows: those contributions that mainly address teaching and learning issues (including curriculum issues and the learning of specific topics), and those that mainly address interventions (such as the use of modelling or other types of intervention). We conclude the chapter with some reflections on potential ways forward for research on these themes.

Teaching mathematics to non-specialists: curriculum, resources, and learning

As mentioned above, research on engineering studies is attracting a growing number of mathematics education researchers. For this reason, this section is divided into two subsections: the first is devoted to engineers while the second looks at other cases involving non-specialists. In the context of INDRUM, these “other cases” focus on two fields of study: biology and economics.

Mathematics in engineering training

Around the world, engineering training programmes generally reflect the model of the École Polytechnique, founded in France in 1794 (Romo-Vazquez, 2009). The French mathematician Monge was a key player in its foundation, and he conceived of a curriculum organised like a tree, mathematical and physical knowledge forming two main branches (Belhoste, 1994), with geometry (where analysis was studied through geometrical applications) forming the core of the mathematical branch. In this way, “the curriculum takes on an encyclopaedic structure, where pieces of knowledge are linked one to the other, from the principles of science to practical applications1” (p. 13). In 1795, geometry was declining while analysis was on the rise, with Monge, Lagrange, and Prony teaching courses in analysis, which took on greater importance. Most of the characteristics of the analysis course were developed by Lagrange, who saw analysis as “a general method that can be applied to geometry and mechanics” (p. 22). This “reduces all mechanics and geometry to analysis, itself reduced to a purely algebraic calculation, and its implementation, both pedagogical and scientific, calls for a deep reflection on its principles.” (p. 22) Thanks to Laplace, the course eventually encompassed series, derivatives, and integrals. It is a model that continues to influence the structure of many engineering training programmes today, with basic and specialised training usually offered in different course blocks (see, for instance, Artigue et al., 2007; Faulkner et al., 2019). Mathematics dominates most foundational courses in engineering programmes, which usually require the use of algebra, linear algebra, calculus, discrete mathematics, differential equations, etc. This educational model has created a gap between the abstract content of mathematics and its application in professionally-oriented courses (Christensen, 2008) and in professional practice, as illustrated decades ago in the ICMI 3 study (Howson, Kahane, Lauginie, & Tuckheim, 1988) and in more recent works (e.g., Wake, 2014).

A first step in addressing the gap mentioned above involves analysing the content of courses in engineering programmes to identify the ruptures and connections between courses and professional engineering practices. The studies considered in this section take this approach, investigating how mathematics is taught to future engineers or used in advanced engineering courses. This raises a number of questions, such as: in mathematics courses, is the content linked to other disciplines and/or to problems encountered in the workplace? In courses of other disciplines, what mathematics is introduced and used, and how does this relate to the content taught in “pure” mathematics courses? In the context of INDRUM, these questions have been addressed using an institutional approach: the mathematics courses, the professionally oriented courses, and the workplace are seen as different institutions, shaping the mathematics taught and used through different practices and needs. We discuss two papers: Gueudet & Quere (2018) and Gonzalez-Martin & Hemandes-Gomes (2018). These papers analyse mathematical content in curriculum resources: online courses in the first paper, and textbooks in the second.

Gueudet and Quere (2018) analysed trigonometry-related content in mathematics courses for engineers and the use of trigonometry in electrical engineering courses. They looked at several electrical engineering courses available online and analysed their use of trigonometry. They then analysed how content concerning trigonometry was presented in two online mathematics courses for future engineers: a MOOC designed to teach mathematical foundations, and an online course entitled Mathematical Tools for Physics. The analysis was conducted in a global way through an institutional lens (Chevallard, 2006) and in a more local way by considering the notion of connectivity. This construct of connectivity has been defined by Gueudet, Pepin, Sabra, Restrepo, and Trouche (2018) in the context of e-textbooks analysis. They distinguish two types of connectivity: 1) connectivity at a micro-level, which concerns a particular mathematical topic and includes connections between different representations, links with other topics or other disciplines, different strategies for solving problems, etc.; and 2) connectivity at a macro-level, which concerns connections outside the textbook, such as connections between users, between users and the textbook authors, between the textbook and the users’ resource system, etc. In the context of teaching mathematics to engineers, the issue of connections - or lack of connections — between mathematics and other disciplines, or between mathematics and workplace scenarios, is central. Hence, the use of the connectivity construct is likely relevant for the analysis of mathematics taught to future engineers.

Gueudet and Quere (2018) provide examples of their results at the micro-level, for instance concerning the alternating sinusoidal cunent. The authors show connections between the functional representation of the sine function s(f) = /l'2sin(wf + u(t) = U2sin(ft>t) and the cun ent f(l) = h/2sin(tof — may be considered as functions, and represented by graphs (Figure 9.1, left). At the same time, they may be represented as vectors (Figure 9.1, right).

In examining connectivity at the micro-level in the case of trigonometry, Gueudet and Quere (2018) observe that electrical engineering courses naturally connect mathematics with electrical engineering; these courses also draw links between different mathematical concepts (namely functions and vectors) and various representations.

Signals in the alternating sinusoidal current and phase difference. On the left

FIGURE 9.1 Signals in the alternating sinusoidal current and phase difference. On the left: a functional frame; on the right, Fresnel vectors in the vectorial frame (Gueudet & Quere, 2018, p. 139)

In contrast, mathematics courses for future engineers offer fewer connections; the authors point out that: “within mathematics, some exercises are limited to a single register: the study of trigonometrical functions, for example, does not always require thinking in terms of angles. Using trigonometry in engineering coutses, on the opposite, always requires such connections.” (p. 142)

The way calculus is used in engineering courses has been discussed by Gonzalez-Martin and Hernandes-Gomes (2018). Specifically, they examined the notion of first moment of an area, which appears in Strength of Materials courses. Like Gueudet and Quere (2018), these authors also consider elements of the anthropological theory of the didactic (ATD), specifically the notion of praxeology and the circulation of praxeologies between institutions (Castela & Romo Vazquez, 2011; Chevallard, 1999). Through the lens of ATD, mathematics and engineering courses are seen as different institutions (with textbooks seen as a way to have access to what is done in them): mathematical praxeologies are taught in mathematics courses, while in engineering courses these praxeologies are used under different rationales, adapted to the nonns and needs of the new institution. This construct serves as a powerful tool to analyse practices in different courses, in particular their different components, both explicit and tacit. It is also worth noting that the authors collaborated with an engineer, who helped pinpoint topics specific to engineering in which calculus is used and the courses where these topics are presented (we return to the issue of collaboration in the next subsection). The notion of first moment of an area can be defined as follows:

In civil engineering, for example, to solve bending problems one must take into account some specific geometrical characteristics of cross-sections of a bar, which is the general term for structures that include beams (Feodosyev, 1973). In this situation, the notion of the first moment of an area is used to calculate the centroid of an area and the shearing stresses in transverse bending. The centroid of area A is its geometrical barycenter and is the point C of coordinates x and y such that the following relationships hold true: JxdA = Ax and f.tydA = Ay. (Gonzalez-Martin & Hernandes-Gomes, 2018, p. 116)

The authors analyse a classic, international engineering textbook - Mechanics of Materials (Beer, Johnston, DeWolf, & Mazurek, 2012) — used in many Strength of Materials courses. They highlight that first moments and centroids are defined using integrals and, in calculus courses, the content related to integrals is usually divided into two closely related local praxeological organisations (i.e. two major themes): one focuses on the definition of the indefinite integral and the techniques that allow students to obtain it, and the other focuses on Riemann sums which constitute a more solid theoretical basis for defining the definite integral and interpreting it as an area, as well as for presenting the Fundamental Theorem of Calculus and the techniques to calculate integrals using Barrow’s rule. Regarding the engineering textbook, Gonzalez-Martin and Hemandes-Gomes (2018) show that although first moments of an area are defined as an integral, their use does not require the calculation of integrals; students are instead prompted to use basic area calculations and ready-to-use formulae to substitute values. The analyses reveal that tasks and techniques are completely different in the Calculus course and the Strength of Materials course (although the notion of integral appears in both), leading to two different praxeologies and two ways of working for the student. Moreover, in solving tasks in the engineering book, students may not realise that the notions they are using are based on the notion of integral. These results are consistent with previous work concerning the use of integrals to define bending moments in beams (Gonzalez-Martin & Hernandes-Gomes, 2017).

Two significant issues are addressed by Gueudet and Quere (2018) and by Gonzalez-Martin and Hernandes-Gomes (2018): 1) there are relationships between mathematics and engineering, at different levels, and they can be identified through the study of mathematics and engineering courses (notions such as connectivity and praxeology are useful tools for conducting such analyses); 2) The way that mathematical notions (such as integrals, or trigonometry concepts) are presented in upper-level engineering courses (such as Strength of Materials, or Electricity) is such that the rules of the course are imposed on these mathematical notions in order to adapt them to their needs. That is to say, importing a mathematical concept or praxeology into non-mathematics, specialised courses generates, in effect, a transpositive process. Both papers also show how the use of sociocultural and anthropological tools helps address specific issues related to the use of mathematics in service and professionally-oriented courses, bypassing some of the obvious limitations of cognitive approaches to these issues (Artigue et al., 2007).

We end this section by highlighting that recent works (e.g., Faulkner, 2018; Quere, 2019) have also analysed engineering courses, with similar findings: although mathematical content is present in these courses, their use and the tasks that call for this content can be very distant from what is usually presented in mathematics courses. This calls for more research regarding the professional needs of engineers, to further the debate about what should be taught in mathematics courses, but also to help propose modelling activities that are closer to the students’ future professional practice.

Mathematics for other disciplines: the cases of biology and economics

In this section we highlight the use of mathematics in two other fields of study -economics and biology - which, although fairly unexamined in the international literature, have attracted attention in recent years (e.g., Mkhatshwa & Doerr, 2015; Viirman & Nardi, 2017, 2019).

In the case of economics, Feudel (2016, 2018) found that students seemed to consider the economic interpretation of the derivative as less relevant than pure mathematical procedures (such as differentiation rules). Derivatives can have a very specific interpretation in economics, in particular because of their link with the concept of marginal cost. Feudel (2016) notes that the derivative C'(x) of a cost function C(x) “is often interpreted as the additional cost while increasing the production from x to x + 1 units” (p. 182). Economics textbooks usually introduce derivatives from a purely mathematical point of view, then derive the approximation formula /(x + /і) —fix) * f(x)-h for It close to 0, and finally use the argument that h = 1 is small in economics to justify the identification of C'(x) and C(x + 1) -C(x). Based on this, Feudel (2016) investigated which aspects of expected knowledge concerning the derivative are considered important by economics students. He asked students to summarise the important aspects of derivatives, and analysed the responses of 146 economics students, observing in particular:

  • • Aspects related to differentiation rules and derivatives of elementary functions (algorithms) appeared in 86.3% and 79.5%, respectively, of the summaries.
  • • Aspects related to definition, slope of the tangent line and rate of change (representations) appeared in 74.7%, 58.2%, and 16.4%, respectively, of the summaries.
  • • The economic interpretation is the least present category (8.9%), whereas aspects related to the term “marginal,” meaning derivative, and the approximation aspect appeared in 48.6% and 15.1%, respectively, of the summaries.

These results led the author to conjecture difficulties in justifying why C’(x) can be identified with C(x + 1) - C(x), which could explain the poor performance of economics students involving the use of derivatives for approximation, as reported by Bingolbali and Monaghan (2008). As a follow-up, Feudel (2018) interviewed eight economics students who had successfully completed their calculus course about the connections they could make between the derivative and its economics interpretation. Using the construct of concept image (Tall & Vinner, 1981), the results show that the students had trouble making adequate connections, even though derivatives had been covered in their calculus course. These results reflect those obtained by Mkhatshwa and Doerr (2015), who also showed that many economics majors mistake the amount of change and the rate of change interpretations, when solving economic problems.

In the case of biology, mathematics is becoming increasingly important, leading to international efforts to improve biology students’ mathematical skills (Viirman &

Nardi, 2017). Discussions at INDRUM 2016 and INDRUM 2018 looked at ways to integrate mathematics into the biology curriculum through, for instance, the use of mathematical modelling. Mathematical modelling was chosen in part because research indicates it can foster more positive attitudes towards, and self-perceived competence in, both biology and mathematics (Viirman & Nardi, 2017, p. 2275). Using a commognitive approach, Viirman (2016) presented preliminary results involving 12 first-year biology students who had sessions on modelling tasks in parallel with their one compulsory mathematics course. Some of these tasks included, for instance, estimating the population density of rabbits close to a highway, or estimating the size of an extinct species of bird by comparing data on dimensions of fossilised bones with similar data from contemporary species of birds. The preliminary results reported commognitive conflict concerning the use of assumptions and the fact that the students’ engagement with mathematical modelling discourse differed from that of the teacher. It is also important to emphasise the fact that this process led to the development of new routines (for instance, concerning assumption building and graphing) and that the participants were able to shift from ritualised to exploratory routine use (Viirman & Nardi, 2017, 2019). The role of the teacher in facilitating this shift also appeared crucial. The study of biology students’ ways of thinking and reasoning while engaging in biology problems that use mathematical models as their representation was the focus of Tetaj’s work (2018), who also found that although biology students enrolled in an Evolutionary Biology course engaged with the biological content of the tasks in an exploratory manner, they employed the necessary mathematical routines in a ritualised way (Tetaj & Viirman, 2019). For instance, in a problem involving the heritability equation, students had to either remember the correct form of the equation or construct a mathematical model for the problem. In the second case, the students did not remember the biological meaning of any of the variables present in the equation and worked in a purely process-oriented manner. This might be because the course focused more on using mathematical models rather than on constructing them. Therefore, students seem to be able to use mathematical models, but not to argue the validity of a particular model or come up with reasons why a particular model makes sense or not. These results are consistent with those of Hester, Buxner, Elfring, and Nagy (2014), who report that biology students have trouble spontaneously transferring their mathematical knowledge into biology problems. This suggests it would be useful to propose courses in which students can be actively engaged in the construction of the mathematical models.

In the next section we discuss the potential of modelling activities to train nonspecialist university students. Regarding the two above examples from biology, it is worth noting that they are the fruit of a collaboration between research institutions in mathematics education (MatRIC) and biology education (bioCEED). Some of the projects discussed (Gonzalez-Martin & Hernandes-Gomes, 2018; Tetaj, 2018; Viirman, 2016) involve collaboration between specialists from different fields. This exemplifies the fact that research on the use of mathematics by non-specialists could benefit from such collaboration. In the next section we present research that looks at interventions, many of which also involve collaboration.

Designing and assessing interventions in the context of mathematics for non-specialists

In this section we focus on studies that examine the implementation and evaluation of interventions for non-specialists. As mentioned in the introduction, these interventions are most often designed to improve links between the mathematics taught and the speciality of the students by drawing on problems native to a given discipline. This raises several issues, such as the challenge of designing tasks with an appropriate level of difficulty or of developing students’ modelling skills. An increasing number of such interventions are based on the use of ATD (Chevallard 1999, 2006), which proposes a theoretical and methodological framework for designing such interventions through the construct of Study and Research Paths (SRPs). For this reason, this section starts with a subsection devoted to SRPs. The next two subsections are devoted to other types of interventions: some of these interventions also draw on modelling, while others (keeping a more traditional approach) focus more on an innovative organisation, drawing on technological means. We note that most of the papers we mention examine engineering education, but other degree programmes are also analysed, such as business administration.

Mathematical modelling within ATD: study and research paths

There is international consensus shared by researchers and practitioners on the need for alternative teaching practices that go beyond the mere transmission of mathematical knowledge, and that use mathematics as an essential modelling tool to study real problems. This contrasts with the usual way of organising university mathematics teaching (as evidenced, for instance, by many authors in Holton, 2001). As discussed in the previous section, this traditional way of teaching may create disconnections and gaps between mathematical content and its uses and functionalities in other scientific disciplines. In order to overcome the constraints imposed by this isolation and lack of functionality of mathematics, university institutions usually propose mathematical modelling to help (re)establish links between mathematics and other disciplines, and more importantly, to reposition mathematics as an essential modelling tool for studying extra-mathematical problems.

The integration of modelling into current university systems has been the subject of numerous investigations but remains a major challenge (Burkhardt, 2008). Mathematical models can be found in the mathematics course syllabi of many university-level natural sciences programmes (Barquero, Bosch, & Gascón, 2013); however, the teaching of mathematical models often comes at the end of the process, if at all. Moreover, at university level, the dominant ideology (which the authors characterise as applicationism) is such that in most cases, modelling represents a mere “application” of pre-established knowledge, leaving little room for the process of questioning, building, and validating mathematical models. Some important innovations could address this issue.

Considering ATD, the notion of mathematical modelling has been linked to the notion of mathematical activity since the framework was first developed, assuming that doing mathematics essentially consists of producing, transforming, interpreting and arranging mathematical models (Barquero et al., 2013; García, Gascón, Ruiz-Higueras, & Bosch, 2006). Chevallard (2006, 2015) has proposed the construct of SRPs as an epistemological and didactic model to move towards a new pedagogical paradigm, where questions become central to teaching and learning and where mathematics appears as a modelling tool to provide answers to these questions. According to Barquero and Bosch (2015), the starting point of an SRP is a question of real interest for the community that studies it (students and teachers). The study of this generating question (Qo) evolves and opens many other derived questions Qb Q2, ..., Q„. The continuous searching for answers to Qo (and to its derived questions) is the main purpose of the study and an end in itself. As a result, the study of Qo and its derived questions Q, leads to successive intermediate answers A, that can be helpful in elaborating a final answer /1 to Qo- This allows the creation of a question-answer map (Jessen, 2014; Winslow, Matheron, & Mercier, 2013).

In what follows, we discuss different papers concerning SRPs, but with different foci. The first type discusses internal elements that facilitate students’ engagement and researchers’ work with teachers; the second type discusses the effects of SRPs on content and the way it is usually organised; and the third discusses the possibilities of transferring SRPs to different university settings.

The first type is represented in the research presented by Barquero, Serrano and Ruiz-Munzón (2016) and Monreal, Ruiz-Munzón and Barquero (2018). These authors present the a priori analysis, implementation, and a posteriori analysis of an SRP proposing a comparison between forecast and actual number of active monthly Facebook users in a complete process of modelling, after two successive implementations with first-year business administration students. The implementation took place in a “Math modelling workshop” divided into six sessions, including personal work outside the classroom. The authors sought to identify what instructional strategies can prompt university students to engage in a coherent and self-sustained study and research activity (Barquero et al., 2016; Barquero, Monreal, Ruiz-Monzón, & Serrano, 2018). Two main dialectics play a key role in analysing the internal dynamics of the SRP (Chevallard, 2008; Winslow et al., 2013):

  • • the questions-answers dialectic helps describe the arborescence of questions and answers proposed by students and teachers, which allow researchers to sketch the modelling trajectories followed by the participants;
  • • the media-milieux dialectic focuses on what different media (such as web resources, content of the course, GeoGebra or Excel simulation tools, etc.) are available to students and how students and teachers can create the appropriate means to integrate them into their activity.

Barquero et al. (2016) and Monreal et al. (2018) show that an SRP needs to combine open moments of inquiry with more transmissive moments, where new mathematical or extra-mathematical content is presented depending on the questions opened by the SRP. Moreover, Monreal et al. (2018) show that the two dialectics cited above can enrich teaching and learning practices; they also help open discussions with mathematics teachers about the intervention and integrate the teachers in the research.

The second type is represented by Florensa, Bosch, Gascon and Mata (2016), who focus on the a priori design of an SRP to organise a course on general elasticity in a mechanical engineering programme. Their research evidences the “traditional” organisation of this course, mainly described in terms of the elements of content taught (relatively independent from each other) through a predetermined number of lectures and problem sessions. Considering previous research on the role of modelling in engineering professional practices (Gainsburg, 2006; Kent & Noss, 2003), two particular challenges have been identified in engineering training: 1) practitioners have to understand the phenomenon to be modelled (which usually remains inaccessible to engineers); 2) practitioners have to be explicitly aware of the assumptions and hypotheses of the model used. In this complex context, Florensa et al. (2016) propose an SRP that focuses on generating questions about how to choose one material (with unknown mechanical properties) from a set of three, in designing a specific part for a bike (brake lever, crank, gear and bike lock key). This SRP was implemented in eight two-hour sessions over four weeks with two groups of 25 students each (working in teams of three or four). In this paper, the use of question-answer maps plays a prominent role in analysing the expected mobilised knowledge, from the mathematical and engineering points of views, related to the domain of general elasticity'. Two elements of the SRP are highlighted in this paper. First, the teachers were explicit about the elements to include in the final reports (such as dimensional plans, estimated loads, choice of material/s, safety factors, computer simulations of the models used, etc.), which raises the issues that students need to consider in order to give appropriate answers, while also providing a simulation that more closely mimics a real-world scenario. Second, the lecturers asked the working groups to deliver weekly reports in which students were responsible for planning a number of elements that the traditional didactic contract does not contemplate: time planning, questions they planned to work with during the week, descriptions of the tasks carried out even when the results obtained were incorrect, and proposed follow-up questions. Note that some of these elements (such as time planning and budgeting) are aspects that usually remain outside scholarly knowledge related to general elasticity, even though they are very much present in the professional practice of engineers.

Finally, the third type concerns the design and successive adaptations of an SRP in different university settings. The main aim of the research presented in Barquero

(2018) is to analyse what the authors define as the “ecological relativity” of modelling practices; that is, the variety of constraints that appear when modelling is implemented in different university settings. The students concerned are enrolled in a Mathematics programme; but this issue is clearly also relevant for non-specia-lists. This paper presents an SRP based on an urban bike-sharing system launched in Barcelona in 2007, implemented in two different courses in three different university settings. The initial question of this SRP concerns the difficulty of getting a homogeneous distribution of bicycles throughout a city with many sloping streets, requiring models (with matrices) to describe the daily flow of bikes between stations and challenging students to find ways to predict the bikes’ redeployment needs. Barquero (2018) discusses how different institutions established different relationships with the teaching of modelling, identifying conditions that may help students. For instance, in the first case study some initial modelling activities were planned with the aim of building, together with students, a common discourse to analyse modelling practices. This was not considered as important in the second case, where students worked on their modelling projects in small groups throughout most of the course, without participating in very many common activities. Consequently, students were less well equipped with appropriate tools to talk about, justify or validate the modelling activity they had to report.

It is important to emphasise that, while it is explicitly stated only in this last paper, the study of conditions (both favourable and unfavourable) for the inclusion of mathematical modelling in teaching and learning practices is an overarching question linking most of the research developed using SRPs. It generally appears as a central axis to which research from different perspectives can contribute, from analysing the ways in which curricula and educational policies can favour the implementation of modelling and applications in school, to analysing teachers’ beliefs in relation to the teaching of mathematical modelling.

Modelling-based interventions for non-specialists

Modelling is also central in many other interventions. The idea — present throughout this chapter — of connecting the mathematics taught in mathematics courses with the mathematics taught in other courses or used in the workplace, is still central to these interventions. Earlier we mentioned the work by Viirman (2016) regarding biologists; we now consider other examples.

The first example is strongly driven by APOS theory (Amon et al., 2014), which has been used over the past ten years in studies proposing modelling activities regarding differential equations and linear algebra. The use of modelling activities intermingled with activities based on a genetic decomposition has proved to be effective in helping students to construct new concepts and in identifying relationships between different concepts. It is also consistent with the APOS theoretical framework since it provides students with opportunities to bring forward their previously constructed schemas - including mathematical knowledge and previously constructed knowledge in other disciplines or life experiences - and use it in constructing new schemas. An example of such a use is provided by Vázquez, Trigueros and Komo-Vázquez (2016), who designed and tested an activity to introduce linear algebra concepts to second-year students (future engineers, economists, and applied mathematicians): matrix of a linear transformation, linear system, and inverse of a matrix. The starting point of the teaching was a signal processing problem proposed to the students, the Blind Source Separation (BSS) problem, which made a connection to a “professional” topic (signals). In this scenario, a meeting taking place in a room has been recorded with several devices. The students had access to the recordings and to a seating map showing the location of the participants and the recording devices in the room. Using this information, the students were asked to match voices with participants. The 24 participating students worked in teams of six. They had access to a specific online tool that produced pure tones at various frequencies, and to GeoGebra software, which was used to display the results from the application of a mixing matrix. In a first phase, the students identified a linear model: an input is a vector of n sources, considered as pure tones, and a mixing matrix transforms it into an output. In a second phase, the students worked to determine mixing matrices in several cases. Finally, in the third phase, they worked on determining the inverse matrix. Student learning was analysed using APOS theory and the proposed conceptualization of a modelling activity. All students found the adequate model and, by working on the activities, they also found (unaided) an algorithm providing the inverse of a given matrix. The authors concluded that these students constructed an action conception of a matrix transformation and its inverse. Some students, in the case of a 2x2 matrix, found the input vector by solving a linear system, deducing the inverse matrix. They demonstrated a process conception of a 2x2 linear system. Confronted with other cases, some of these students were able to generalise the use of a linear system to study the existence of the inverse matrix and to compute it when possible. The authors concluded that the students constructed a process or an object conception of linear systems. All students were motivated and interested by the context of the problem.

This kind of motivation was also observed by Gjesteland, Vos and Wold (2018), who designed a project-based teaching activity (the tracker project) concerning the motion of an object. The students (working in teams of two or three) chose a moving object, filmed it with their smartphones, transformed the videos into graphs using tracker software, and proposed a mathematical model for the movement, presenting it on a poster. The authors evidenced that this process increased the students’ motivation and that such teaching approaches can be applied to large class sizes (346 students in this study). The latter is significant, considering the classroom conditions of engineering programmes in many countries, which makes the ability to upscale innovations especially critical. In their work, Schmidt and Winslow (2018) have also discussed an approach used with many students at the Danish Technical University (approximately 1,100 students each year). During the last four weeks of their basic mathematics course, first-year students work in teams on projects, producing a written text (of 20 to 50 pages) and giving an oral presentation. A specific feature of these projects is that they are “authentic problems from Engineering which can be solved using the mathematics to be taught in their course” (ibid. p. 166). The authors studied the conditions and constraints of the transposition process (Chevallard, 1991) that would allow such projects to be integrated into the engineering courses’ institution. They defined ten didactic variables concerning, for example, the mathematical content covered (breadth of the content; old/new content), the use of Maple software, and the branches of engineering related to the project. These variables allow for the analysis of existing projects, sorting isolated applications of mathematics from practical uses of mathematics in solving real engineering problems. They are also likely to constitute a useful tool for designing more projects in other institutions, and are linked with the general issue of articulating mathematics and engineering in the curriculum evoked above in the previous section.

Using technology to design innovative teaching approaches for non-specialists

Several of the modelling-based interventions discussed above make significant use of various technologies: GeoGebra, Maple, other specific software for signal processing or motion analysis, smartphones, etc. Yet the use of technology and its effects on teaching and learning processes are not central issues in these studies. Its use seems more or less self-evident: technology is seen as a natural tool in this context. In this section we discuss papers where the use of technology plays a central role: we start with one paper where technology is considered to be a mediator in students’ understanding of mathematical content, and continue later with other contributions where the use of technology does not affect the mathematical content.

In the study by Lagrange and Kiet (2016) on the teaching of probabilities for non-specialists at a university in Vietnam, a spreadsheet (Excel) and the statistics software R were used by students to design simulations of a probabilistic model. This teaching approach is innovative, compared with traditional approaches to teaching probabilities in Vietnam, and certainly in other countries. The simulations offered by the software help develop students’ probabilistic reasoning skills. This reasoning is indeed specific. For example, Batanero, Chemoff, Engel, Lee, and Sanchez (2016) state that “probabilistic reasoning is a mode of reasoning that refers to judgments and decision-making under uncertainty and is relevant to real life” (p. 9). Probability is an interesting example of a mathematical topic taught to many non-specialist students. Making connections with other fields (such as medicine or economics) fosters students’ engagement in modelling activity — with the specific features of the probabilistic model. In the teaching approach designed and tested by Lagrange and Kiet, the software plays a central role in the modelling process because it allows for the simulation, and hence the evaluation, of different models.

Other intervention studies, by contrast, consider the use of technology as a means for introducing changes likely to favour the learning of mathematics by non-specialists, but focus on changes concerning the pedagogical organisation. For example, Fredriksen and Hadjerrouit (2016) tested and analysed a Flipped Classroom (FC) organisation in the context of an engineering course at a Norwegian university. Out-of-class sessions with videos and quizzes were proposed to the students before each in-class session, which is quite different from a “traditional” teaching approach. This change leads to different kinds of tensions in the system. Some students had difficulties understanding the mathematical content of the videos (or simply did not watch the videos) and were therefore unable to engage in group work. Although the FC model has the potential to engage students in more active learning, the new proposed work paradigm was not adopted by all the students (Fredriksen, Hadjerrouit, Monaghan, & Rensaa, 2017). Online quizzes on a Moodle platform were also used in the study by Gaspar Martins (2018): the students were assigned weekly online quizzes as homework. The students appreciated this formative assessment, which supported regular personal work. While these innovative didactical organisations seem promising, they are not a priori specifically from the engineering context; studying which specific features could make them efficient for engineering training is an interesting perspective for further studies.

Conclusions and perspectives

In this chapter we present an overview of the INDRUM contributions to the international literature concerning the teaching and learning of university-level mathematics for non-specialists. These contributions include:

  • • Analysis of resources and curricular material presenting mathematical content in specialist courses and of students’ difficulties dealing with mathematical content as presented in different courses. This helps explain some of the gaps students may experience, but can also lead to recommendations for mathematics courses (for instance, which connections should be reinforced) and hint at possible modelling activities (by identifying professional problems where mathematical content is used).
  • • Designing modelling-based interventions. These contributions provide pathways towards a more meaningful learning of mathematical content and more connections with professional activity. Some of these interventions integrate the use of technology.

We note that these issues are coherent with problems already seen as critical by recent surveys on UME regarding mathematics for other disciplines (e.g., Biza et al., 2016; Winslow et al., 2018): the need to consider different student profiles, the professional environment where mathematics will be used, and the integration of modelling activities in the teaching of mathematics (which is particularly crucial for non-specialists).

We also note that the papers discussed in this chapter tackle these issues using a variety of approaches: APOS theory, concept image, commognition - with a significant number of papers relying on ATD, using an institutional perspective. The increase in the use of sociocultural approaches in recent years is evident, and is related to the fact that different kinds of courses (in different professional learning pathways) are seen as different groups or institutions, as already pointed out by Artigue et al. (2007). Moreover, sociocultural approaches have provided interesting insights into issues proper to UME (Nardi, Biza, Gonzalez-Martin, Gueudet, & Winslow, 2014). We highlight the potential of several constructs to address issues related to non-specialists: praxeologies, connectivity and discursive shift.

Many perspectives for future research on university mathematics education for non-specialists arise from the synthesis presented in this chapter.

Firstly, concerning the theoretical approaches mentioned above, further studies are necessary to deepen our understanding of the contributions of these constructs. Another interesting avenue for future research involves theoretical networking, as illustrated by Gueudet and Quéré (2018), associating praxeologies and connectivity, for instance.

Secondly, the use of mathematics in professionally-oriented courses and in the workplace also needs further study. Only a small number of courses have been analysed in the literature, and studies examining professional needs in the workplace are scarce. We note some recent studies in this direction (e.g., Faulkner, 2018; Quéré, 2019) that call for more systematic research on these issues. Moreover, as emphasised by Winslow et al. (2018), studies questioning current curricula, the place of mathematics in them, and the content taught are also necessary. The use of technology in the mathematical training of non-specialists also deserves further research: technology is integrated into the daily practice of many professions and its use in teaching can help introduce more realistic activities.

Thirdly, regarding modelling, the second part addresses some issues that merit further research: how to facilitate students’ engagement in modelling activities, how to collaborate with teachers to integrate modelling into their courses, how to reorganise “traditional” content with the integration of more realistic tasks (e.g., budget planning), and how to transfer modelling activities to different contexts. The discussed papers provide some insights that can inform future research. In connection with the previous paragraph, we believe that studies analysing the use of mathematics in other fields, as well as epistemological analyses on the origin of this use, are crucial in order to adequately address questions concerning interventions.

Finally, we also mention that the papers synthesised in this chapter show the importance of collective work, bringing together mathematics education researchers, mathematics teachers (who can be mathematicians but also engineers or specialists in other areas), teachers in other disciplines, and professionals in other fields. As discussed, research on the use of mathematics by non-specialists poses challenges to mathematics education researchers, and, in many cases, requires collaboration. We see this as one example of what Artigue et al. (2007) forecast ten years ago:

We end this chapter with the feeling that research on mathematical learning and thinking at post-secondary level is entering now a new and fascinating phase, with difficult and new challenges to face, challenges that will require to be solved to extend interactions and collaborations beyond the traditional community of research in mathematics education at advanced level, (pp. 1044-1045)

Such collaborations offer an important perspective for future research. It is also crucial to consider the fact that, in the context of mathematics courses for nonspecialists, it is common that teachers with different backgrounds teach the same courses (as discussed by Hernandes-Gomes & Gonzalez-Martin, 2016), and that the consequences of this have not been explored in great depth (Artigue et al., 2007; Gonzalez-Martin & Hernandes-Gomes, in press).

The synthesis of recent research presented in this chapter suggests many directions for future research, and we have mentioned only some of these. We anticipate that the way mathematics is used for and in other disciplines, and mathematical modelling at university, will be studied in the years to come by an increasing number of researchers at the international level, leading to fruitful implications for practice.

Note

1 In this chapter, all quotations from sources in French are our own translations.

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