Didactic transposition of loan repayments in economics and the place of the discrete-continuous interplay

Abstract

The study reported in this paper is motivated by an article published on a financial portal that presented loan repayments with a continuous approach, as opposed to the discrete approach of addressing loan repayments applied in mathematics courses for economists. Using the Anthropological Theory of the Didactic, we analyse the current didactic transposition of this content, based on a textbook for mathematics in economics and five interviews with teachers of mathematics and of economics, inquiring about their practices in teaching related to loan repayments in economics. The interviewed teachers indicated that this content is presented in a discrete way in mathematics courses for economics, although for reasons that do not seem directly connected to the content itself. Additionally, although the participants see the interest of considering the use of continuous variables and graphs, several constraints prevent them from their use in the context of loan repayments.

1 Introduction

Mathematics serves as an essential component in both economics education and the daily work of economists. Over time, numerous mathematical tools and principles have become integrated into the field, playing a key role in shaping various economic principles (Landgärds-Tarvoll, 2024). Yet, the question of what mathematics should be presented in university courses for economists and its underlying rationale remains open for discussion, and the literature in mathematics education has not yet addressed in detail the presence and need of university mathematics in specialized economics courses or in the workplace. In this sense, this area of research needs to catch up with recent advances in other fields, such as in the case of mathematics for engineers (e.g., González-Martin & Hernandes-Gomes, 2018, 2020; González-Martín, 2021; González-Martín et al., 2021). Furthermore, given that mathematics courses for non-specialists mostly serve as ‘service courses’ (Hochmuth, 2020), and as such may attract less attention and interest from students, it is reasonable to wonder how teaching can be tailored to accommodate these students without compromising the aims of academic training in a given programme. In this sense, it is essential in these contexts to better understand the didactic transposition processes that transform academic knowledge to shape the knowledge to be taught (Bosch & Gascón, 2006). By better understanding the mathematical content present in these programmes, its use and connections with actual professional practices, as well as the processes that lead to choices about inclusion (or exclusion) of mathematical content, research can better inform us in making decisions about what mathematical content to choose, in which form, for what purpose, etc. Therefore, by investigating and better understanding these processes, we can increase the impact of research-based knowledge in university mathematics education (Bosch et al., 2021). This can help to prevent mathematics from being perceived solely as abstract, rigorous and disconnected from the discipline students want to specialize in, and to avoid the pitfall in the education of non-specialists that ‘mathematical science takes on an artificial character’ (Poincaré, 1907, p. 21).

The research in this paper is inspired by an article published on a financial portal (Moj bankar, 2022), outlining two models of loan repayment for a broader audience. An excerpt of the article is presented in Section 5, Analysis of the task, and in Fig. 1. Some preliminary observations (Bašić & Milin Šipuš, 2023) suggest that this type of activity is tackled in a discrete way in economics. However, a continuous representation of loan repayments can also be found in professional resources.¹ These continuous representations mostly appear when using software for graphical representations, whether intentionally or unintentionally. We want to explore whether practices related to the teaching of loan repayments in economics enables students to handle the given situation and information related to loan repayments presented in a continuous way. Our interest is primarily in the didactic transposition of loan repayments, and its consequences for student learning and their ability to tackle activities related to loan repayments in a continuous setting. To do so, we: (1) explore how this content is presented in economics; (2) compare practices in economics related to this content with a practice based on the use of the continuous model; (3) interview teachers of mathematics in economics programmes and of economics to obtain their views about current practices and their raison-d’être and their thoughts on the possible use of continuous models. This work allows us to better understand current practices in the teaching of loan repayments, while identifying possible ways of (and constraints to) enriching this content by making connections with continuous variables (linear function graphs), despite the fact that loan repayment typically occurs in periodic, punctual instalments. We believe this analysis could facilitate teachers in making connections to integration, contributing thus to the considerations of González-Martín et al. (2021) that ‘mathematics course content [...] often seems disconnected from students’ future professional practice, [and] certain pedagogical interventions are therefore needed: teaching mathematics in a way that provides links with other fields [...] and that draws on problems encountered in these disciplines or in the workplace’ (p. 170).

2 Literature review

As discussed by Landgärds-Tarvoll (2024), economics became mathematized in the twentieth century. This may explain why undergraduate business and economic programmes usually include a significant proportion of mathematics content, mostly in the first year. Although this could be seen as an intention of programme developers to scaffold student learning in subsequent courses (Biza et al., 2022), the reality is that many economics students fail these courses, or dropout of their programme because of them (Landgärds-Tarvoll, 2024).

Additionally, there is a lack of research in mathematics education ‘outlining the mathematical needs—both in terms of content and level—of first year economics students’ (Landgärds-Tarvoll, 2024, p. 14). Additionally, most of the existing literature in mathematics education regarding economics focuses on derivatives or optimization. This could be related to the fact that business faculty members perceive concepts related to differentiation, optimization and rate of change as most necessary for students to succeed in subsequent business courses (Landgärds-Tarvoll, 2024). For instance, Mkhatshwa (2019) reported on student difficulties to set up the function to be optimized from a contextualized problem and their tendency to not appropriately consider the context of a task when interpreting quantities in their resolution of contextualized problems. Similarly, Mkhatshwa & Doerr (2018) reported student difficulties in dealing with instantaneous rates of change when solving problems related to marginal change in economics; in the same vein, Ariza et al. (2015) highlighted the need to grasp the relationship between a function and its derivative in the case of marginal analysis. Still related to derivatives, Feudel & Biehler (2021) discussed how derivatives are used to define marginal cost—the additional cost of the last unit. In economics, one unit is usually considered small. This consideration makes the well-known mathematical expression C(x + h) − C(x) ≈ C′(x).h, when h ≈ 0, becomes C(x + 1) − C(x) ≈ C′(x) when considering h = 1. We highlight the difference in mathematical meaning: while derivatives are a rate of change, the additional cost is an amount of change. Therefore, it becomes crucial that economics students are aware of this important difference and can use it adequately in economics contexts. However, Feudel & Biehler (2021) showed that this interpretation is not obvious to economics students, and the reasons for this interpretation even less so. We also note that a few studies have focused on optimization in several variables (e.g., Landgärds, 2023; Xhonneux & Henry, 2011).

Integrals are present in the scholarly knowledge in economics, for instance, to study ‘when an exhaustible resource is exhausted’ (Voßkamp, 2023, p. 62), as well as the use of continuous variables. However, we found no studies analysing didactic transposition processes related to the relations between discrete and continuous variables or to the use of integrals (in particular, their interpretation as accumulation and area under a curve) in economics, or studies identifying specific difficulties with these topics that can prevent economic students from grasping economics concepts. This contrasts with the literature about mathematics in other fields. For instance, in the case of engineering, González-Martín et al. (2021) and González-Martin & Hernandes-Gomes (2018) provided information in a Strength of Materials course about the shift from punctual loads on beams to distributed loads, which calls for the use of integrals. They observed that most techniques related to tasks involving bending moments and first moments of an area (which are defined using integrals) had no explicit connection to techniques learned in calculus in the chapter related to integrals. In the case of physics, Ivanjek et al. (2016) and Planinić et al. (2013) found that the concept of integrals poses an even greater challenge to first-year mathematics and physics students in Croatia than derivatives.

Finally, we note that Jones (2015) identified three different conceptualizations of the integral: as an ‘area under a curve’, as ‘antiderivative’ and as ‘adding up pieces’. His study showed that the two interpretations most frequent applied in calculus courses (‘area under a curve’ and ‘antiderivative’) are less helpful than the ‘adding up pieces’ interpretation to understand physics situations involving density, revolutions, pressure or force. In engineering, González-Martín et al. (2021) also showed that the ‘adding up pieces’ interpretation appears in the Strength of Materials and the Electricity and Magnetism courses, with the interpretation ‘area under a curve’ used in tasks related to calculating bending moments in beams. Therefore, in the analysis of professional practices related to the use of integrals, it is important to identify the interpretation that is mobilized in techniques and rationales.

All of the above raises interest in the didactic transposition of loan repayments in economics and possible connections with continuous models and integrals. This leads us to the following research questions:

RQ1. What is the didactic transposition of loan repayments in economics, and the practices associated with this transposition, and do they relate to the use of continuous variables and integration?

RQ2. What are the arguments given by mathematics and economics teachers about practices related to loan repayments in economics and about making connections to the use of continuous variables?

By studying these questions and current practices in economics, we wish to contribute to the growing literature in mathematics education about the mathematical needs of economics students. In addition, we elaborate on the relationship between loan repayments and the use of continuous models—connected to the use of integrals and the calculation of areas. These two topics are present in a mathematics course for economists examined in our study but are not connected in this course, and this connection has not attracted many studies focusing on economics students either.

3 Theoretical framework

Our study focuses on the didactic transposition of loan repayments, considering practices in economics. We therefore draw on the Anthropological Theory of the Didactic (ATD; Chevallard, 1999), which focuses on the institutional construction and transmission of knowledge. A main tenet of ATD is that all human activities are institutionally situated and, therefore, knowledge about a given activity (as well as why it is important and why it must be learned) is also institutionally situated (Castela, 2016). As a consequence, when studying practices, it is essential to situate them within their institutions. In this study, we consider three distinct institutions: the institution of calculus courses, the institution of economics courses and the institution of professional economics practices.

One key notion of ATD is that of praxeology, which is a model of human activity. Indeed, in ATD knowledge is linked to practices, and both need to be examined together. For this reason, praxeologies are formed by a quadruplet [T/τ/θ/Θ] formed of two blocks. The praxis block [T/τ] considers a type of task T, and a technique τ used to solve this type of task. The logos block [θ/Θ] includes a technology (or rationale) θ that explains and justifies the technique, and a theory Θ that provides a fundamental explanation of the technology. The institutional approach considers that any piece of knowledge produced in a given institution, when used in another institution, will most likely suffer some transformation, either in the praxis block, or in the logos block, or in both (Castela, 2016). Chevallard (1999) calls this phenomenon the ‘transpositional effect’ and it has an impact on the capacity of students and professionals to recognize knowledge connected to different tasks or techniques when used in another institution. In our case, we are interested in analysing the techniques and rationales associated to two ways of presenting loan repayments: 1) the algebraic, based on tables and discrete approach most often presented in mathematics courses in economics; and 2) the visual, graphical and continuous approach as seen in professional resources. Such an analysis can help determine whether students are prepared to interpret this information presented in a continuous way and to see what activities could foster this interpretation in mathematics courses for economists.

The consideration of these phenomena has evolved particularly in the case of studies focusing on the use of mathematics in engineering. Peters et al. (2016) proposed an extended ATD-model—that we simplify for this paper (see González-Martín, 2021)—which is related to Castela's (2016) proposal. In this model, the interaction of elements from mathematics and from engineering becomes more visible:

⁠. The model makes explicit that, in the resolution of a task presented in engineering, the technique may combine elements not only proper to higher mathematics (HM) but also proper to the specific field of engineering (ENG) in question. In the same way, the justification of this technique intertwines rationales that come both from HM and from ENG. A praxeological analysis then allows for identification of these implicit elements and their interactions.

Finally, we note that teaching or preparing a course can be seen as a very large task that can be subdivided into several sub-tasks, each with its own technique and rationale. In this way, a praxeological approach can be used to examine teacher activity when organizing the course and can also reveal the (usually implicit) rationales that underlie their choices. González-Martin & Hernandes-Gomes (2020) used this approach to study the practices of two teachers in supervising engineering capstone projects. This approach allowed the identification of certain shared tasks and techniques, though not always with a shared rationale, as each teacher relies on their professional background to justify their choices.

These theoretical considerations allow us to reformulate our research questions:

RQ1. What is the didactic transposition of loan repayments in economics, and the praxeologies associated with this transposition, and do they relate to the use of continuous variables and integration?

RQ2. What are the arguments given by mathematics and economics teachers about praxeologies related to loan repayments in economics and about making connections to the use of continuous variables?

We believe that, by studying the above research questions, our study contributes to the existing literature about the use of mathematics in economics programmes. As stated above, research in mathematics education about this topic has mostly focused on derivatives and related notions; we aim to analyse the current practice in a discrete context, where connections with continuous variables, integrals and areas under curves are possible. We hope that this study can contribute to the discussion about the didactic transposition of mathematical content in economics programmes, and possible missed opportunities. Additionally, most of the literature in this area of research has focused on student difficulties, instead of considering the views of professionals in mathematics and in economics. In this sense, our participants help to add layers to our analysis of resources, by discussing whether some connections could be possible with calculus courses for economics, and also discussing current practices in economics. Finally, although some recent studies about mathematics in economics have used an ATD approach (e.g., Xhonneux & Henry, 2011; Landgärds, 2023), other cognitive approaches are also commonly used; therefore, our use of ATD can help consolidate this approach for studies about economics, or help uncover phenomena not previously identified with other approaches.

4 Methods

4.1 Data collection

The article published on a financial portal that is the focus of our study was preliminarily considered in Bašić & Milin Šipuš (2023). We identified the course where this content is presented, as well as its textbook (Neralić & Šego, 2009) and exercise book. We then conducted the data collection in two phases, as follows.

In Phase 1, we analysed the presentation of loan repayments in the textbook, and the associated tasks and techniques, to grasp the usual praxeologies applied in economics. We then proceeded to the analysis of the article published on the professional portal, showing that economics practices—as shown in some online resources—may include continuous graphical representations, for which an enrichment of the logos block of the discrete case is possible.

In Phase 2, to gain insight into current practices related to loan repayments in economics, we conducted semi-structured interviews with teachers working at a faculty of economics in Croatia (see Table 1). This is a convenience sample of volunteers who agreed to participate in our study.

We interviewed five participants about their practices regarding loan repayments in teaching and in economics, and the role of graphs of different models of loan repayment, and whether the continuous model is presented or used in economics. The interviews were performed in two steps: MT1 and ET1 were interviewed in early 2023 to help us grasp the content and gain a first impression of the views of experts in mathematics and in economics, respectively (Bašić & Milin Šipuš, 2023). The remaining three participants (MT2, MT3, ET2) were interviewed at the end of 2023. The interviews followed a semi-structured protocol and lasted between 30 and 60 minutes. MT1 and ET2 were interviewed in person, and the others online. Two interviews took place in English (MT3, ET2), and the others in Croatian. The interviews were audio recorded and transcribed, with excerpts translated into English as necessary. The main themes in the protocol of interview are presented in Table 2. Note our interest in comparing praxeologies related to loan repayments in mathematics courses for economists to practices by economists. Also, guided by the results of Bašić & Milin Šipuš (2023) that indicated that these praxeologies consider discrete models, we decided to ask questions about the use of continuous variables and graphs to interpret (the differences between) these models.

4.2 Analysis

For Phase 1, we developed a praxeological analysis examining the introduction of content related to loan repayments in the textbook. We identified the practical block (tasks and techniques) and the knowledge block (rationales). We used open coding to identify text with the role of rationale (for instance, definitions), or explaining a technique, as well as tasks. For tasks, we identified the proposed technique and made connections with the rationale underlying this technique. The first and third authors conducted their analyses first and then conferred with the second author. Minimal disagreements were resolved, arriving to the analysis presented in this paper. We then proceeded to propose two models of praxeology for our task: a discrete model (typical in economics, and based on algebraic calculations, following our analysis of the textbook) and a continuous model (based on continuous graphs, connections to integrals and the geometric calculation of areas).

In Phase 2, we considered the answers from each transcript and grouped the actions and rationales shared by participants into categories (e.g., how the content is presented, the available techniques, etc.). We then used thematic analysis to group excerpts from the answers as tasks and techniques for teaching (or in the profession for economists), and the rationales behind these techniques. Further, we grouped the rationales provided by the participants into categories. Once again, the first and third authors performed their analyses independently, and then conferred with the second author; differences were resolved to arrive at a consensus.

5 Analysis of the task

Figure 1 shows the article for non-experts found on a financial portal (Moj bankar, 2022), graphically presenting (using continuous line graphs) two models of loan repayment: 1) by annuities,² and 2) by instalments.³ Since the article does not present the involved calculations, the reader is left to compare the two models using the graphical information provided.

5.1 Loan repayment modelled as an economics (discrete) praxeology

Loan repayment is a standard topic in financial mathematics, and it appears in the textbook of the course Mathematics (Neralić & Šego, 2009). The first chapter of the textbook covers matrix algebra (with sections on linear programming and input–output analysis as examples of applications), and the second chapter addresses applications of derivatives (with economics concepts related to average, marginal and total cost, as well as elasticity and the Cobb–Douglas production function), including optimization of functions of one or more variables. Integrals are presented in the third chapter, which starts with definitions, integration techniques and worked-out examples, and continues with Riemann integrals, defined by lower and upper Darboux (Riemann) sums, the Fundamental Theorem of Calculus and the Leibniz–Newton formula and ends with improper integrals and ordinary differential equations. As an example of application, integrals are used for the calculation of areas bounded by two curves. There are no examples in any context, including economics contexts, and the idea of ‘total accumulation’ is not mentioned. The fourth and final chapter concerns financial mathematics and focuses on discrete cases: interest and interest rates, anticipatory and decursive methods of interest calculation, simple and complex interest account, discrete decapitalization, loans and consumer credits. The tasks in this chapter require students to adequately choose a previously deduced formula and to simply calculate an unknown value given all other known values. Each section provides definitions and explanations of economics terms. All the topics of the fourth chapter are covered in a short time (two weeks). We see this fourth chapter as belonging to the institution of economics courses.

The two basic models of repayment (and their combinations) are discussed, and we refer to them as repayment in equal annuities (model 1, M1) and repayment in instalments (model 2, M2). After the explanation of the economics terms involved, based on recursive definitions, formulas for the repayment table (schedule) are derived (formulas for the interest amount at the end of a period, the amount of the repayment quota and the remaining principal amount), and a worked-out example with repayment schedule is calculated. A repayment schedule contains the data shown in Table 3.

Note that in M2, the concrete values in the repayment table can be calculated directly line by line. On the other hand, for M1, every a_i is expressed using the unknown parameter a and later it is necessary to set up an equation to calculate a, which then allows us to obtain concrete values to fill in the repayment table. Additionally, equal annuities (M1) are calculated from the following expression related to the sum of terms of a geometric sequence:

We note that this expression is given, taking for granted that students know the expression for the sum of the n first terms of a geometric sequence. Moreover, the expression appears next to an image with a vague explanation using a further economics concept of present value (Croatian: sadašnja vrijednost), assuming again that students are capable of independently interpreting this image⁴ (Fig. 2).

For the second model (M2), only the formula R_i = R is explained:

… if the model of nominally equal payment quotas is applied, it must be that C = R∙n, and therefore, with the assumptions given, the amount of nominally equal repayment quotas is |$R=\frac{C}{n}$| (Neralić & Šego, 2009, p. 418, our translation).

The calculation of the remaining values in the repayment table is presented in an example.

In the textbook, the total repaid amount is calculated simply as the sum of individual payments in the table and is not discussed further. The two models of repayment are treated separately and are not directly compared. Regarding the total repaid amount, the textbook states ‘We can observe that the sum of all interest plus the sum of all repayment quotas is equal to the sum of all annuities. Moreover, the sum of all repayment quotas is equal to the nominal amount of the loan’ (Neralić & Šego, 2009, p. 418, our translation). The total repaid amount is referred to as ‘the sum of all annuities’.

The techniques are illustrated with the following tasks:

1. (First repayment model: equal annuities) A loan of 100,000€ was granted to a company for 5 years with 10% annual decursive interest and payment in nominally equal annuities at the end of the year. Determine the amount of the nominally equal annual annuity. Produce a repayment table.

2. (Second repayment model: instalments) A loan in the amount of 100,000€ was granted to a company for 5 years with 10% annual decursive interest and annuity payments at the end of the year, with nominally equal repayment quotas. Produce a repayment table.

The tasks are posed in the economics context, with given information on the loan amount (the initial principal or nominal amount of the loan), interest rate⁵ (given as compound and decursive) and a period of amortization (repayment period). This way, the tables that are obtained in the case of equal annuities (M1) or of instalments (M2) are shown in Fig. 3.

We note that this type of mathematics is present in further economics courses (such as Microeconomics and Macroeconomics). This type of task includes a sub-task, which we identify as:

• T_Ec:

  Produce repayment tables for the loan and compare the total amount repaid in the two models.

• τ_Ec:

  1) By using ready-made formulas, produce the i-th term of each sequence (a_i, I_i, R_i, C_i). 2) Calculate the sum of n terms of an arithmetic–geometric sequence (M1) or solve a simple equation (M2). 3) Creation and validation of tables. 4)Interpretation of the results.

• Θ_Ec:

  1) The economics definitions used and the relations they define, which allow the determination of the necessary formulas and the interpretation of results. 2) Calculation of the i^th term of a sequence from its recursive form. Deduction of formulas for the sum of n terms by using known formulas for arithmetic and geometric sequences applied to recursively defined sequences for a_i, I_i, R_i, C_i, in particular for the calculation of the individual annuities (constant or non-constant, depending on the model).

• Θ_Ec:

  This task belongs to the economic branch of financial mathematics. Note how this area of knowledge is supported by mathematical principles related to sequences.

We see here a case of entanglement of elements from mathematics and economics. The techniques consist merely of simple mathematical manipulations, but they are supported by the economics definitions and principles. In other words: from the definitions of the basic economics notions (given in Table 3), it is possible to derive the formulas purely mathematically, but this is not the approach applied in the analysed economics textbook analysed. In this case, further economics principles are invoked, and the formulas are presented ad hoc as a consequence of these principles. However, note that the textbook does not justify why the given formulas work, i.e., they are taken for granted and provide good results for the given type of task.

5.2 Loan repayment modelled as a praxeology involving continuous graphs

We note that the graph provided in the article published in Moj bankar (2022) is not given as discrete (which would match the praxeology in the economics courses), but instead as a continuous line graph, as is the case for other websites concerning loan repayments. The graph of discrete data is a scatter plot and, by using software, it is possible to obtain a line graph (which the users of software may or may not know how to independently produce), but the connection between the discrete data and the line graph is not explicitly described on the professional websites or in the textbook we consulted. Nonetheless, we believe that such visual representations may support non-expert readers in grasping the difference between the two models (namely, that model M2 has higher initial annuities, but a lower total sum). To further develop this premise, we consider a new type of task (or variation) in the praxeology of loan repayment, which is to compare the two models using the given line graphs. Therefore, information regarding the total amount repaid can be compared (vaguely), read-off or estimated from the graph by comparing areas under the curves. The reader may immediately read off from the graph that the annuities are equal in the first model, while in the second model (instalments), the annuities are initially higher and decline steadily throughout the repayment period (see the graph in Fig. 1). Note, however, that it is essential to interpret the y-axis not as amounts of money, but rather as amounts of money per month. This technique is commonly used in physics, for instance, when working with kinematics graphs when calculating the distance travelled from a v-t graph; this has been the focus of several studies in physics education (e.g., Planinić et al., 2013; Ivanjek et al., 2016). From a calculus point of view, the task can be tackled by mobilizing two conceptualizations of the integral, as the ‘antiderivative’ (by calculating the equations of the lines) and as ‘adding up pieces’ (Riemann sums), which considers the area under the curves; they both are elegantly linked by the Fundamental Theorem of Calculus. The ‘adding up pieces’ interpretation can be seen as the ‘total amount of a changing quantity’ or accumulation, across various application contexts. It the case of loan repayment, it would appear as the total amount repaid.

To compare the total sums repaid, it is necessary to interpret total accumulation, drawing on the ‘adding up pieces’ interpretation of integrals, as the area under the graph, and then compare the two areas visually. In this case, this is easy as the areas differ in the two triangles between the lines, which are similar because they have equal angles. As one can observe from the graph that the intersection of the two lines is not in the middle, but shifted to the left, it is possible to conclude that the left triangle has a smaller area than the right triangle. This implies that the total sum is smaller in the second model (instalments). We describe this via the following praxeology:

• T_Graph:

  Based on the graph, explain which total amount paid is greater: payment in instalments or payment in annuities.

• τ _Graph:

  1) Compare the position of the x-coordinate intersection of the line graphs to the middle of the considered time interval. 2) Identify the two similar triangles between the lines and argue which has a greater area.

• θ _Graph:

  Interpretation of the area under a curve in terms of units and economics meaning as the total accumulation of values on the curve, via the definite integral. Properties of triangles in geometry.

• Θ _Graph:

  Financial mathematics, supported by single-variable calculus, in particular integral calculus.

By considering this praxeology, we see that the logos of the loan repayment praxeology can be enriched with the knowledge relating the total accumulation to the area, which is a piece of knowledge usually discussed in calculus courses in relation to the concept of the integral. Note that a variation of the technique would be to directly calculate the area of the trapezium (instalments) and the rectangle (equal annuities).

We go one step further and discuss that it is beneficial for economics students to produce such graphical representations without software and to analyse the information needed to produce them. For the first model (equal annuities), only the value of the annuities is necessary, but as discussed earlier, the calculation of that value requires a procedure of expressing all steps in terms of that value as an unknown parameter, and then setting up an equation from which it is calculated. In the second model (instalments), the first two annuities may be simply calculated, and from these two points the equation of the linear model can be obtained. It is therefore important to be able to read the formulas; for instance, the first instalment is a sum of an equal repaid principal of 100,000 euros in 120 months, and interest of 5/12% on 100,000 euros. Note that, in Fig. 1 of the article, these calculations are not necessary, since the graph is already provided. In the example from Fig. 1, the annuity in the first model is calculated as in the previous section and the line has the equation |$y=g(x)=1060$|⁠, while for the second model, the equation of the line is |$y=f(x)=-3.5x+1251.75$|⁠. Once these equations are obtained, the areas under the graphs may be calculated by using elementary geometry and the formulas for the area of a trapezium and a rectangle for the functions |$f(x)$| and |$g(x)$|⁠:

A_f = |$\frac{\left(f(0)+f(120)\right)\times 120}{2}=\frac{\left(1251.75+831.75\right)\times 120}{2}$| =125,010,

A_g = 1060 × 120 = 127,200,

or by using integrals:

The calculation will provide the same values as the calculation of the total sum in the repayment tables. For the calculation here, the values ​​read off from the graph are rounded.

We observe that these calculations are not necessary if the reader’s only interest is to compare the two models (which can be done visually and is the usual first aim of non-experts considering taking a loan). Additionally, if a person is interested in specific values of the total sum and the individual annuities, the described repayment tables should be calculated either way.

5.3 Interaction between the economics (discrete) and the continuous praxeologies

We emphasize that the praxeology involving continuous graphs is not present in the textbook, in either the chapter on integrals or the chapter on financial mathematics. However, we observe that professional resources for non-experts can contain visual representations of the two models of loan repayment as continuous graphs. The relation of the total accumulation and the area under the graph certainly enriches the logos block of the praxeologies on loan repayment, but this is not explained in any of the resources consulted. This motivates us to further analyse this part of the logos.

From a mathematical point of view, the shift from the discrete model to the continuous model can be done by graphically representing a sequence of rectangles, whose width corresponds to the length of the unit period (one year or one month) and height corresponds to the total amount of repayment in that unit period (see Fig. 4). The top sides of the rectangles form the graph of a step function representing the measured variable in discrete time.

The shift from the discrete model to the continuous model is possible by considering that the higher the number of data points in the discrete model, the better the approximation. These considerations may further lead to various mathematical ideas relating the discrete values (sequences) and a continuous variable, which are at the core of the definition of the Riemann integral of a (continuous) function. Such ideas are widespread in mathematical and statistical data analysis, and employ the ideas of interpolation, data smoothing and continuous approximation. This could also be present and exploited in advanced courses in economics education.

Hence, we see the potential of introducing both praxeologies in the context of the course, where integrals and financial mathematics are presented. Using the financial situation presented via a graph may support the development of modelling competences in economics students, to interpret the information provided and figure out an adequate mathematical tool to tackle the situation. Additionally, it may also serve as motivation and support in introducing the key idea behind the formal definition of the definite integral, which is the shift from Darboux (or Riemann) sums to the integral, that is, from the area of a finite number of rectangles to the area under a graph of an arbitrary continuous (positive) function on a segment. Moreover, it also emphasizes the passage in the reverse direction—using a continuous model as an approximation for calculating discrete sums.

Finally, we add that the use of graphs in resources for non-experts suggests that the continuous approach might be more informative for some people and, when information is displayed using a graph, the comparison between the total amount paid in the two models becomes quite simple. Our analyses show that the continuous technique to compare both methods of payment is quite simple, and in articles as the one presented in Fig. 1, the use of the discrete technique would be more complicated. However, our analyses indicate that the learning of T_Ec does not necessarily prepare students for T_Graph, since their technique and logos blocks are quite different; additionally, formulas in T_Ec are not justified mathematically, and the shift from the discrete to the continuous is not present. Moreover, since T_Ec is based on the use of formulas and the production of repayment tables, students may not be able to be flexible enough to tackle tasks as those in T_Graph, or even to associate them to content related to loan repayment.

6 Teacher views on both praxeologies

6.1 Views expressed by mathematics teachers

The three mathematics teachers confirmed that they tackle tasks related to the calculation of loan repayment schedules in their course, although this is done quite quickly:

MT1: [Yes], you can see the syllabus [on the web], so you can actually see how little we actually do, because it is actually 2 weeks or maybe even one and a half weeks for financial mathematics.

MT2: Yes, but in a span of four hours I have to explain the whole content of financial mathematics. […] Yes, we do discrete capitalization.

MT3: Yes, I teach those [two different models] for a week and a half in the first year, in the mathematics course.

The participants also confirmed that their approach matches the praxeology of economics courses: they discuss this type of task with a discrete model. They also clarify that they tend to make the rationales explicit and to deduce all the formulas:

MT1: …They have to know how to make these tables, and then everything is explained to them, what is happening with it. Formulas are deduced […] Of course, all the formulas are deduced there, and the illustration for the repayment [table], how it is…

MT2: I do the discrete case by an example, use all the formulas (which are deduced earlier) and produce a repayment table. I use very simple examples in terms of the number of rows; for example, the loan is repaid over four years at the end of each year.

MT3: We do that, we make the amortization schedules, the tables you use in this very same example. […] We deduce the formulas. I don’t [just] give them the formulas.

We see, therefore, that the praxeology of economics courses is recognized by the mathematics teachers and their practices include it. We also see in the participants a personal interest in not just giving the formulas, but in deducing them, and this is justified by a personal rationale that students need to be able to deduce the formulas. For instance, MT2 says that ‘it is very important for me that they know how to draw a timeline, and that they know how to indicate exactly where the annuities are through a discrete model, and then that they know how to derive the formula […] and make a [repayment] table’. In a similar way, MT3 states that ‘they have to be able to deduce the formulas and I do that because they need to understand’. We also note that the technique is applied by hand and that technology is not used to solve this type of task. The reasons for this choice are that the exams are solved by hand (MT1, MT2), or that students ‘have to know how to produce a table’ (MT1), and that ‘mathematical precision is important’ (MT1). Additionally, the three teachers confirm that they do not use graphs for this type of task, basing their practice once more on the fact that these tasks are tackled through a table that needs to be completed. The three teachers’ praxeology is essentially the same as listed in the textbook and reflects a usual practice in economics courses.

Moreover, they do not use tasks, such as T_Graph in the module related to integrals, although they agree there is great potential to make such connections:

MT1: I think that a concrete example would certainly fit right into this topic right now […] It would fit greatly […] with financial mathematics in the last chapter, that it makes sense to observe those areas under the graph. We don’t use it, but it would fit in well. […] The continuous case would fit in for the integral concept.

MT2: Many models are discrete, but I want to use the derivative, that is, I want to use continuity. I will then connect it, and I will see that the discrete [model] is an approximation, or I would say [that the discrete model is] a restriction of this [other model]. Well, it’s much more convenient for me to work with infinity, that is, continuity.

MT3: That’s a nice idea. Like that [using areas] would be a faster way to compute the total interest paid in the second [model], yeah. I think it has potential, really, and I actually can do it. […] [The] discrete [approach] is always done. But continuous functions have very important applications in economics.

The three teachers also see the value in using the continuous model: connections with areas and integrals (MT1), the convenience of continuous models (MT2) and their applications in economics (MT3), and that the related technique provides a faster way to produce acceptable answers when comparing the two types of repayment (MT3). However, the constraints also mentioned (little time to cover this content, a common exam solved by hand, the association of these tasks to the technique of constructing a table) may play a role in their teaching being restricted to what they see as the ‘regular’ praxeology from economics courses. This could explain that they had not thought about using this topic in the chapter on integrals. When thinking about it, they see important connections. MT1 sees an important link between the chapter on integrals and the chapter on financial mathematics, and MT2 and MT3 add comments about the order of chapters:

MT2: I think that it would be good to include, if nothing else, as two ways of solving or a parallel solving way, as enrichment. […] I would change [the order]. [I would] put the financial mathematics before the integral because […] we [currently] finish derivatives […] then we get to the integral of a function of one variable and then comes the financial [mathematics]. Then, maybe I would just change the order, so the financial mathematics would be explained before the integral, and since we are already on the economics [topics], now let's see through this example where I will need the integral exactly in economics. This might be very interesting, if that change of order is made, this example would be very good.

[…]

If we were to consider the intensity of the annuity, you have various models in finance, where various companies, for various reasons, they somehow try to invent new ways of repaying the loan, so let’s say these [new] lines, which are […] linear functions, but then would look like sinusoids […] or would look like, I don’t know, like a parabola for some reason. Then, the same could be said […] through integrals. That way it would be great!

MT3: I think actually it’s better to present it in the financial mathematics part and then, once they know the integrals, to show them this and show them, well, actually this is also an application of integrals. I think the order is really important. So, I think once like we are done with the integrals, so they have a little break from the integrals and then again, you reintroduce it. And I think this is going to reinforce like the learning… […] So, you again reintroduce this concept and you relate it to the financial mathematics with the integrals.

We see in the three a willingness to modify their practices in a way to make explicit connections between both praxeologies; however, the constraints mentioned above may prevent them from doing so. Moreover, we see in their rationale mathematical reasons (connecting integrals with financial mathematics), but also pedagogical ones as a way of helping students make connections between content. Additionally, the three share the rationale that it is important to show examples of applications of calculus content in economics, both from a mathematical point of view (see applications of content) and from a pedagogical point of view (help motivate students and improve their grasp on the content). Regarding this, MT2 adds:

MT2: If I had the freedom, that is, maybe I would write a new textbook in a few years. In your example… we had in a way actually how to motivate these new concepts from mathematics […] I think that the students remember more that way, they remember the examples. I would definitely do this in the textbook, I would change the order, so then actually [I would go] through financial mathematics.

[…]

One student said, ‘Congratulations to the mathematicians. Let them prove their theorems and we will believe them. Give us as many examples as you can’.

Finally, when asked about the interpretation of integrals (antiderivative, area, accumulation) that is more present in praxeologies in that chapter, the three agree:

MT1: Antiderivative. And we use substitution and partial integration.

MT2: Antiderivative. We determine the primitive function. We use substitution and partial integration.

MT3: Antiderivative. […] We use the simple integrals, so either you can use direct integration, substitution methods, or integration by parts. That’s it.

Finally, the three also agree that using graphs is important, although there are not many in the course. This could relate again to some of the constraints perceived, but also to their willingness to adhere to what they see as ‘regular’ practices from another field. To emphasize the use of given formulas and procedures, MT3 adds: ‘if you have the difference between mathematics and physics, well, I think the difference between mathematics and economics is ten times greater. The whole thing about economics is like trying to find the order within the chaos, then yes, that itself is a little bit chaotic’.

6.2 Views expressed by economics teachers

We started the interviews with the economics teachers by asking how representative the online article in question is a usual practice in economics. Although they agreed with the usefulness of graphs, they also agreed that using graphs is not a common approach for this type of task:

ET1: There are a lot of examples where we use graphs, with limit values, with profits, company balance sheets and the like, but in these matters, as I was taught, as far as I can see, it’s not that we work too much with graphs, it’s more of a pure mathematical language [referring to the ready-to-use formulas].

ET2: I think this [seeing information via a graph] is something that a working economist would definitely have to deal with on a daily basis, so anybody who works for a bank, for example, people who work in research departments of banks. They do things that are much more sophisticated.

[…]

[referring to the task] It is more like a technicality [referring to solve it just using formulas], and I think [students] more or less understand [these graphs]. The only problem [to solve for economists] is how to compute [the necessary values].

These initial answers confirm, in a way, that although graphs can be a part of professional praxeologies, T_Ec corresponds to practices in the field and that T_Graph is not associated with loan repayments. ET1 provided more insightful details about common practices:

ET1: I wouldn’t approach [this task] that way [with a graph]. As an economist, I would first ask myself what is better for me: an instalment or an annuity. Then, as an economist, I would think, starting with the Modigliani-Friedman’s consumer life cycle: what stage of life a consumer is in: is he at the beginning of his life and has a lack of income and larger debts, or is he somewhere in the middle and expects that, as time passes, he will have less income as he gets closer to retirement? So, I would think about whether an instalment or an annuity is better for me, that is, whether I can save more today, so that tomorrow will be more favourable for me… Maybe I would also think about what inflation will be like in the future. It influences on what the real amount will be… But what you say about the area, I wouldn’t have thought of that.

It seems that T_Ec is recognized as a professional practice, although other notions can be added to related tasks. Specifically, when asked about the continuous praxeology, ET1’s response was:

ET1: What I see [in this graph] is that, in later periods, my monthly amount will be less. But I don’t see a decision that is more favourable for me. I cannot conclude from this graph how much I will pay in instalments at the end.

We see clearly that in economics practices it is very important to understand the overall difference between both models (M1 and M2). Professional economists may resort to other techniques, but decisions are based on the main characteristics of the models (high initial annuities, lower total sum), and less on exact calculations, which contrasts with T_Ec. Neither economics teacher associated loan repayments with the calculation of areas under a graph, nor with connections with integrals and, although the graphical representation is evoked as an aid to grasp the models, they confirm that common professional praxeologies in economics do not consider the use of graphs or integrals for tasks related to loan repayments. Additionally, ET1 seems to indicate that the connection between the total accumulation and the area under the graph may be elusive for some economists, although both teachers value the use of graphs in their field. When asked about the usual practice in economics to solve tasks related to loan repayments, both immediately said ‘discrete’. Moreover, both agree that the specific technique is ‘to produce a repayment table’. ET2 adds that using Excel would be acceptable to quickly produce the table. Therefore, we see this usual professional practice as a rationale to teach the economics technique in the training of economists; in other words, the fact that T_Ec is recognized as a professional practice may have influenced its transposition to economics courses. ET1 added on the rationales and the perceived weaknesses of the graph:

ET1: When you repay your loan through annuities, and after 3, 4, 5 years, you see how much credit is left, and how much you have to pay. People don’t understand it at first. This graph is not informative concerning that. […] [It would be better to see it] in some chart with columns, or in tables. Tables are much more useful around these things. Repayment tables.

We could see, throughout the interviews, an important difference between ET1 and ET2 regarding the reading of information in the given graphs. This could be due to the difference in their base training: ET1 has had all their training in economics, whereas ET2 has had their base training in mathematics. We see a contradiction: both declare that graphs are present in economics practices, however, their use is not associated to loan repayments. This seems to indicate that this content has been connected with the use of continuous variables in some professional practices (as seen in several online portals), but its didactic transposition to economics courses has isolated it from the use of graphs, in particular with continuous variables, leaving it restricted to an algorithmic approach in the discrete case.

They both confirm again, later, that the repayment tables are both the usual technique for economists and the technique taught in their training. ET1 adds another rationale: ‘Always to produce the tables. It is important to be exact’. However, besides the fact that using a continuous approach and graphs is not a common practice related to loan repayments, they both see a rationale to include them in practices in mathematics courses:

ET1: These graphs make sense in introductory education, because many similar things come later, in higher years, for example in the analysis of different investment options, feasibility studies, and the like. Content in the first year of mathematics courses are crucial for a person to take them further […].

ET2: [It] has some potential in teaching mathematics for the first years […], for the students entering the calculus courses, so, not advanced students. Yes, although I’ve never seen it done.

ET2 added some rationales to their response: in higher level courses, phenomena are usually continuous. But, in the first year ‘it is this way [discrete]’, adding that they ‘think it would be maybe easier, easier for [students] to understand if it is discrete’. The comments of both interviewees agree with our impressions: continuity and graphs are important in economics professional practices, so there are reasons to work with students on the discrete–continuous shift. However, the didactic transposition of loan repayments seems to have focused on making things easier for students and presenting only the algebraic, discrete method, isolating this content from a graphical representation and the opportunities to work on a continuous interpretation. Additionally, both teachers see the continuous praxeology as a possible tool in the learning of integrals:

ET2: [Students] appreciate mathematical concepts being explained in a way that they know why they need to learn that, and in what context. I think it makes sense to link abstract mathematical concepts to some practical problem.

Both teachers agree that mathematics is necessary for economists, although they both provide different reasons to distinguish the mathematics studied in the first year from later years:

ET1: In economics, we say there are three primary languages: verbal, mathematical and graphical. When we cannot do it mathematically, because [students] are weak, we do it with the help of graphs.

[…]

Two things are important from the first year: these [types of task] and, secondly, something that only a few economics faculties here do, and that is mathematical modelling of economics problems. Especially in doctoral studies… If I have a problem that I know how to express verbally, or with a graph, and I want to write something for a strong journal, but I don’t know how to express it mathematically, there is no way to go further with it.

ET2: [Mathematics] is a powerful tool, yeah. But, at the same time, I mean it’s not easy for the economics students […]. They don’t like it. Nobody likes maths. More interesting problems like that, for example, if you know you have to give some story [to illustrate the content for which people know some] techniques.

[…]

So, in the context of research papers, all of that appears [derivatives, integrals]. I mean, I’ve seen these papers where they’ve integrated through three integrals [triple integration].

However, ET2 adds that ‘the learning of differentiation is something that [students] will really need when they come out of school and that will make them all money, they will become, you know, all eyes and ears’. Finally, although they agree that integrals are used in their field, they agree that ‘it is important to know it as an antiderivative, with all the techniques of calculating antiderivatives’ (ET2). Indeed, according to them, the interpretation ‘area under a curve’ seems less present in their field. As ET2 puts it, ‘I can imagine that you calculate in fact the area so, the area has some interpretation in it. Yes, but the interpretation in economics has a completely different meaning’.`

Our data indicates, therefore, that both teachers see graphs and the use of continuous variables as important for economist practices, even at the research level. However, these two elements are not seen as a part of praxeologies related to loan repayments, even if both teachers see their potential to better prepare first-year students, improve modelling competences, and motivate them to recognize the applications of integrals. The teachers evoke the weakness of students in mathematics, as well as student dislike of maths, and we believe these may be reasons that have influenced the didactic transposition of content related to loan repayment. The content seems to have been reduced to a discrete form, involving simple algebraic techniques, that allows for providing answers via exact calculations, but this form isolates this content from the use of graphs and continuous variables, which are important in later years and in professional practices. This may justify the fact that ET1 declares that graphs are important in economics, but has not developed practices to interpret the graph in the context of loan repayments.

7 Discussion

Regarding our first research question, our data shows that loan repayments are introduced in the training of economists mainly via T_Ec, which is also recognized as a professional practice. This is confirmed by our textbook analysis and both the mathematics and economics teachers, who all see T_Ec as the ‘regular’ practice in economics. This contrasts with the use of other approaches in other resources, such as those found online. T_Ec is set in a discrete case and provides formulas (without clear justifications) that allow for the calculation of exact values and responses. Therefore, by isolating T_Ec from connections to graphs and continuous variables, we conjecture that many students will not be able to tackle T_Graph. This seems to be the case of ET1, who highlights the importance of graphs and continuity in economics, but seems to not see how to use the graph provided in T_Graph to extract information. Indeed, the logos block of T_Graph requires the use of units and the interpretation of the area under a curve, which are not present in T_Ec and are often neglected in calculus courses to favour algorithmic approaches and the ‘antiderivative’ interpretation of the integral (Jones, 2015). Given the rates of failure in calculus courses for economics (Landgärds-Tarvoll, 2024), there is a need to make recommendations for such courses; in particular, the content about integrals needs further study.

Regarding our second research question, our participants see an important potential in establishing explicit connections between T_Ec and T_Graph. For instance, the mathematics teachers see the importance of introducing continuous variables, efficient techniques to solve new tasks, as well as the importance of introducing applications of mathematics in economics. Similarly, the economics teachers mention the importance of modelling and seeing applications in mathematics courses in economics, as well as the importance of graphs and continuous variables in economics practices, both at the professional and research levels. However, in spite of this potential and interesting possibility, all teachers mention important constraints to exploring this richness. The mathematics teachers agree on the limited time they have to introduce loan repayments, due to the dense content in their mathematics course. The fact that there is a common exam also prevents them from trying to make variations to the course content. Additionally, they seem willing to adhere to what they see as the ‘standard’ practice in a field—economics—that is not their field. This agrees with what the economics teachers declare: the standard practice related to loan repayments is restricted to T_Ec. The reasons evoked to transpose this practice as it is to economics courses are related to student weakness in mathematics and their general dislike of mathematics, in spite of the importance these teachers give to the use of graphs and continuous variables in their own field. Therefore, it seems likely that to attend to pedagogical reasons, content related to loan repayments has been reduced to an algorithmic approach, where formulas are mostly given, disconnected from the content that is important in economics practices. Our analysis allows us to see, regarding loan repayments, a substantial distance between practices in university mathematics (calculus in our case) and practices in economics courses; this gap may lead to questioning the adequacy of mathematics as a service course for economics. Our results seem to complement Landgärds-Tarvoll’s (2024) argument that concepts related to differentiation, optimization and rate of change are perceived as most necessary for economics students; these concepts are connected to the use of continuous variables, and our economics participants confirmed their importance in their field. We also note that continuity is associated to loan repayments on some professional portals, and it can provide techniques that provide fast responses when comparing the models M1 and M2; it is therefore important to train students to be able to tackle the information provided in these resources.

Our study also uncovers, in the case of economics, a general phenomenon identified at the tertiary level. Mesa & Griffiths (2012) showed how university mathematics teachers relied strongly on their reference books to prepare and teach their courses. Moreover, González-Martín et al. (2018) showed how the presentation of content in the resources shaped the teachers’ relationship with this content. We believe that this phenomenon seems to also apply in the case of mathematics teachers in economics and, additionally, their personal relationship with loan repayments seems strongly shaped by the content of their reference book. Moreover, González-Martín et al. (2021) showed that the strong influence of the textbook to prepare university courses also happens in engineering, and Hitier & González-Martín (2022) showed that it also happens in physics. As far as we know, our study is the first to reveal the same phenomenon in the case of economics. We conjecture that when teachers teach content from another discipline, the resources they use can influence their personal relationship and their teaching in a stronger way, so they can adhere to what they perceive as ‘standard’ practices in this other discipline. This phenomenon deserves further research, and it is also influenced by external constraints, such as curricular restrictions and the short time teachers have to cover financial mathematics. We highlight, though, the paradox that in the Mathematics for Economics course, the chapter on financial mathematics is the final one and is covered in barely two weeks. This seems to agree with an ‘applicationist’ approach to mathematics courses in other disciplines (Barquero et al., 2011): the focus is on introducing and developing mathematical content, with the applications of this content to the other disciplines at the end, if there is enough time. It becomes, therefore, important to reconsider the content and approaches of mathematics courses for economics by analysing practices in economics courses and professional practices, which can lead to a better balance between the mathematical tools needed and the actual practices in economics where mathematics is used.

Additionally, concerning the interpretation of the integral, the participants agree that the ‘antiderivative’ interpretation seems more present in economics. We believe this may explain particularly the case of ET1 (the only participant with complete training in economics) and their reactions towards the given graphs and not seeing how they could help in comparing the amounts to repay in M1 and M2. This links in with ET2’s comment, stating they ‘have never seen’ the task solved via graphs or integration, and that ‘it is important to know [integrals] as antiderivative’, whereas areas under graphs would have ‘a completely different meaning’ in economics. This would be an exception to findings in the literature: while the ‘adding up pieces’ interpretation has been identified as necessary for many physical phenomena (Jones, 2015), and also in engineering together with the ‘area under a curve’ interpretation (González-Martín, 2021), our participants suggest that the ‘antiderivative’ interpretation of the integral would be essential in economics.

From a theoretical point of view, the use of ATD allowed us to characterize the technique and rationale in the economics praxeology, T_Ec, which is present both in economics courses and in professional practices in economics, thus showing the entanglement of elements from mathematics and economics. This interaction, proposed by Castela (2016) and detailed in the case of integrals in engineering by González-Martín et al. (2021), appears clearly in our analysis. We believe this finding highlights the need for further research about the use of mathematics in economics, and this could lead to proposing extended ATD models in the case of economics.

8 Conclusion

Our study was motivated by our finding of an article on loan repayments published in a professional resource (website), which treated this content in a different way than in a course textbook. By investigating the teaching of this content in Croatia, our study contributes to the existing literature about the use of mathematics in economics programmes, providing information about the didactic transposition of content—loan repayments—and how it seems to have been isolated from other important content in economics, such as the use of graphs and continuous variables. Additionally, our study incorporates the voices of professionals in mathematics for economics, as well as economists, which is still rare in the literature about mathematics for economics. Therefore, our textbook analysis and the interviews add additional layers to our study about the didactic transposition of content for economics courses, providing possible reasons for why this content has been reduced to the use of algebraic techniques and isolated from other content (continuous variables and graphs) perceived as more ‘advanced’, but essential for economists. The teachers also provide reasons for this content being at an ‘equilibrium point’, where changes are not envisaged (even if connecting it with other content has advantages): the mathematics course is content heavy and loan repayments are only covered at the end, there is a common exam, textbooks present this content in that way, and this approach accommodates student difficulties in the first year.

Our results point to praxeological needs in economics related to the use of graphs, to the interpretation of areas under a graph, and to the use of integrals being of a different nature than for other fields, such as physics and engineering. If the interpretation of integrals that our interviewees see as more necessary is that of the ‘antiderivative’, then the teaching of this content could be revisited, and the use of technology in teaching practices be considered. This calls for further studies about economist practices in the workplace related to integration content.

Finally, as our economics interviewees put it, continuity and the use of graphs are important for economics, both at the professional level and at the research level involving writing research papers or in preparing the doctoral dissertation. However, didactic transposition can affect the introduction of some content in such a way that it appears disconnected from other content considered important by professionals. Finding ways to re-connect this content, to provide a richer logos and a larger repertoire of techniques (some of which are very efficient in a graphical context) seems like an important direction for research in the field of mathematics for other disciplines, in particular in economics. We plan to pursue our work in this context, and to collaborate with professionals in economics to better understand their needs and practices regarding mathematical content, with the aim of making recommendations for calculus courses.

Footnotes

References

Željka Milin Šipuš is a full professor of mathematics at the Department of Mathematics, Faculty of Science, University of Zagreb, Croatia. Her research interests include mathematics, specifically differential geometry, and mathematics education. In mathematics education, her primary focus is on university-level teaching, including the education of future mathematics teachers and mathematicians, teaching mathematics to non-mathematicians (e.g., physicists, economists), and inquiry-oriented teaching of mathematics.

Alejandro S. González-Martín works since 2006 at University of Montreal, Canada. He teaches courses in Mathematics Education for pre- and in- service teachers, and courses in research in education. His research interests include post-secondary mathematics education, with a special interest in calculus, textbook analysis, teachers’ practices, and the use of mathematics in other disciplines, in particular by engineers.

Matija Bašić is an assistant professor of mathematics at the Department of Mathematics, Faculty of Science, University of Zagreb, Croatia. His research interests include mathematics, specifically algebraic topology, and mathematics education. In mathematics education, his primary focus is on university-level teaching, including the education of future mathematics teachers and mathematicians, teaching mathematics to non-mathematicians (e.g., physicists, economists), and inquiry-oriented teaching of mathematics.