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# Supporting the Transition Between Mathematics and Physics in the First Year of University

Pauline Hellio, Ghislaine Gueudet, Aude Caussarieu

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Pauline Hellio, Ghislaine Gueudet, Aude Caussarieu. Supporting the Transition Between Mathematics and Physics in the First Year of University. Canadian Journal of Science, Mathematics and Technology Education, 2025, 10.1007/s42330-025-00346-4. hal-05002683

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# Supporting the Transition Between Mathematics and Physics in the First Year of University

Pauline Hellio<fnref n="1" /> Ⓓ • Ghislaine Gueudet<fnref n="1" /> Ⓓ • Aude Caussarieu<fnref n="2" /> Ⓓ

Accepted: 28 November 2024

© The Author(s) 2025

**Abstract** Undergraduate science students face difficulties using mathematics in their physics courses. Choosing an institutional perspective, we consider that these students experience a permanent transition between mathematics in their mathematics courses and mathematics in their physics courses. We refer to the anthropological theory of the didactic and the notion of didactic contract to understand this transition. In France, the Maths4sciences digital resources have been designed to help students learn the mathematics used in physics. We investigate students' difficulties in the math-physics transition and the affordances and limitations of Maths4sciences resources to help them. We designed a physics exercise where students must recognize and solve a first-order linear differential equation. We interviewed three students who worked on this exercise and had access to a Maths4sciences tutorial sheet concerning such differential equations in a physics context. Through the analysis of these interviews, we observed that students faced different types of difficulties: recognizing mathematical types of tasks intervening in the technique for solving the physics exercise, performing types of tasks blending mathematics and physics, and making sense of physical notations, in particular. The Maths4sciences tutorial sheet only helped them with some of these difficulties. Beyond the cases studied, our work evidences the difficulties raised for students by different kinds of “recognition” types of tasks in physics and suggests directions for curriculum design and teaching mathematics for physics.

**Supplementary Information** The online version contains supplementary material available at https://doi.org/10.1007/s42330-025-00346-4.

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Pauline Hellio
pauline.hellio@universite-paris-saclay.fr

Ghislaine Gueudet
ghislaine.gueudet@universite-paris-saclay.fr

Aude Caussarieu
aude@caussarieu.fr

<footnote n="1">¹ UR Études sur les sciences et les techniques, Université Paris-Saclay, Gif-Sur-Yvette, France</footnote>

<footnote n="2">² UMR ICAR, École Normale Supérieure de Lyon, Lyon, France</footnote>
**Résumé** Les étudiants en sciences au premier cycle éprouvent des difficultés en ce qui a trait à l'utilisation des mathématiques dans leurs cours de physique. En choisissant une perspective institutionnelle, nous considérons que ces étudiants font l'expérience d'une transition permanente entre les mathématiques dans leurs cours de cette matière et les mathématiques dans leurs cours de physique. Nous nous référons à la « théorie anthropologique du didactique » et au concept de contrat didactique pour comprendre la nature de cette transition. En France, les ressources numériques « Maths4sciences » (Mathématiques pour les sciences) sont conçues pour aider les étudiants à mieux comprendre les mathématiques utilisées en physique. Nous nous penchons sur les difficultés des étudiants dans la transition entre les mathématiques et la physique et les capacités de suggestion ainsi que les limites des ressources « Maths4sciences » pour les aider. Nous avons conçu un exercice de physique dans lequel les étudiants doivent reconnaître et résoudre une équation différentielle linéaire du premier ordre. Nous avons interrogé trois étudiants qui ont tenté de résoudre cet exercice et qui ont eu accès à une fiche-méthode « Maths4sciences » concernant de telles équations différentielles dans un contexte de physique. En analysant les réponses données par les participants, nous constatons que ceux-ci ont rencontré différentes sortes de difficultés: la reconnaissance des types de tâches mathématiques impliquées dans la technique de résolution applicable à l'exercice de physique, l'accomplissement des diverses tâches qui combinent les mathématiques et la physique, et la compréhension du sens général des notations physiques, en particulier. La fiche-méthode « Maths4sciences» n'a aidé les participants que pour certaines de ces difficultés. Au-delà des cas examinés, notre travail met en évidence les problèmes que soulèvent différents types de tâches de « reconnaissance » en physique pour les étudiants et présente des orientations en ce qui a trait à la conception des programmes et l'enseignement des mathématiques pour la physique.

**Keywords** Curriculum resources · Mathematics for physics · Anthropological theory of the didactic · Didactic contract · University mathematics education

**Introduction**

This paper deals with the difficulties “non-specialist” science students (meaning non-mathematics majors) face during the first year of university when they attempt to mobilize mathematics to solve physics problems. The international research community has shown a growing interest in non-specialist science students (González-Martín et al., 2021), especially since mathematics proficiency is linked to success rates at university (for example, in engineering studies, see Heublein et al., 2017, pp. 96–97). Our study addresses the “Curriculum and Instructional Design” theme of the Special Issue; it takes place in France, where a specific mathematics online curriculum, “Maths4sciences”,¹has been designed in the context of a national project. We focus on the case of calculus and mechanics and on the experience of first-year science students who need to use calculus to solve exercises in their mechanics courses. The general aim of our study is to investigate the difficulties faced by these students and the affordances and limitations of purposefully designed resources to help them face these difficulties.

In what follows, we first refer to previous research on the learning of mathematics and physics in the first year at university and present the Maths4sciences curriculum. We then introduce the theoretical elements guiding our study and our research questions. We expose our methods for data collection and analysis, present our results, and, finally, discuss the answers to the questions studied and, more generally, the contributions of this paper.

<footnote n="1">¹ https://maths4sciences.ens-lyon.fr/</footnote>
# Learning Mathematics and Physics in the First Year of University

Recent research showed that “non-specialist” science students, for example, in physics or engineering study programs, considered that the mathematics taught to them in mathematics courses was too theoretical and disconnected from the teaching of other disciplines that use mathematics (Faulkner et al., 2020; Jablonka & Bergsten, 2022). Taylor and Loverude (2023) studied how students (physics and engineering majors) solved a problem in which they were given an “*x* as a function of *t***” graph and were asked to give the function *v* of the variable *t*. The students they interviewed struggled to relate this problem to a mathematical problem, as illustrated by this student’s quote: “I was originally trying to make this look mathematical with a whole function and everything, but if I had *f(x)* I guess I could do that… No that doesn’t make sense…” (Taylor & Loverude, 2023, p. 171). This illustrates how difficult it is for students to make links between physical and mathematical situations, which are needed to transfer what they learned in calculus courses to physics. This stems from differences between mathematics and physics but also, in a probably less visible way, from differences between mathematics in the mathematics course and mathematics in the physics course.

Karam et al. (2019), drawing on the history of mathematics and the history of physics, foregrounded deep epistemological differences between the two disciplines. They also acknowledged that the community of mathematicians and the community of physicists had different conventions. Redish and Kuo (2015) argued that physicists and mathematicians used different languages. Analysing university textbooks, Hitier and González-Martín (2022) identified inconsistencies between practices linked with the derivative in calculus and mechanics courses. Caussarieu (2022) studied the differences between mathematical practices in mathematics and physics exercises and grouped the differences she observed into four categories: different notations, differences in the objects manipulated, different techniques for similar tasks, and different types of tasks using the same notion.

All these differences evidenced by previous research have led us to consider the experience of first-year science students enrolled in mathematics and physics courses as a transition. We follow here Yerushalmy (2005), who defines transitions as “a necessity for entering into a different type of discourse [...] or more broadly as changing ‘lenses’ used to view the concept at hand.” (Yerushalmy, 2005, p. 37). Students experience a permanent transition (Gueudet et al., 2016), which is not linear in time (unlike the secondary-tertiary transition), as they move back and forth between mathematics and physics courses. To the best of our knowledge, the possible contribution of this transition perspective for understanding non-specialist students’ experience and for designing resources likely to support them has not yet been explored by research. From now on, we will use the expression “math-physics transition” to refer to this “transition between mathematics in the mathematics course and mathematics in the physics course.”

# Maths4sciences: Digital Resources to Help Science Students Master the Mathematics Needed to Succeed in Higher Education

The Maths4sciences project gathered 20 mathematics or physics teachers from secondary school and university and one didactics researcher between 2017 and 2022 to produce digital resources to help science undergraduates succeed at university. Their first goal was to write quizzes to test mathematics skills in a physics context for formative assessment.

However, discussions within the design group made them realize that mathematics and physics teachers solve the same exercise differently. This led them to write about 50 tutorial sheets on mathematics skills, such as derivation or solving a second-degree algebraic equation, used in physics. The mathematics involved ranges from secondary school to first-year university mathematics. Each tutorial sheet is made up of two to
three solved physics exercises and a description of the mathematical method used by the physics teachers to solve them. This design choice aims to make explicit how mathematics is used in the physics course. For example, the tutorial sheets present steps such as “identifying which letter in the physics expression plays the role of the variable x in mathematics” and discuss differences in notations between mathematics and physics. They were initially written to complement the online quizzes and to help students master the skills presented in the sheet by presenting methods and advice. The team of authors is engaged in a reflection about other possible uses and possible improvements. Our work takes place in this context.

# Theoretical Frameworks, Previous Work, and Research Questions

## The Anthropological Theory of the Didactic (ATD)

Chevallard (1999) introduces the concept of an institution in a wide sense, encompassing any legitimate social group. Adopting a sociocultural perspective, he considers that knowledge is shaped by the institutions where it lives. Following this perspective, we generally consider the mathematics course and the physics course in the first year of science studies at university as two different institutions.

The concept of praxeology describes how knowledge is shaped in a given institution and provides tools to analyse it. Four elements compose a praxeology: a type of tasks T (e.g., “solve a differential equation”) which gathers tasks with a similar aim; a technique τ to perform this type of tasks (e.g., “determine the solution of the associated homogeneous equation, determine a particular solution of the complete equation, add these two elements”); a technology θ, which is a discourse that justifies the technique, and a theory Θ, which is a more general discourse justifying the technology itself. Chaachoua (2020) complemented this construct by evidencing that the techniques τ are themselves composed of types of tasks (e.g., “determine the solution of the associated homogeneous equation,” “determine a particular solution of the complete equation,” and “add these two elements” can be considered as three types of tasks). These types of tasks are called ingredients of techniques.

Mathematical notions (e.g., differential equations) live in both institutions, mathematics courses and physics courses, but the corresponding praxeologies in each institution are different. The math-physics transition can, therefore, be seen as a transition between different praxeologies.

## Didactic Contract

Within his Theory of Didactical Situations (TDS), Brousseau (1997) defines the didactic contract as a set of rules, mostly implicit, describing the expected behaviours of the students and teacher in a given didactical situation. The expected students' behaviours can lead to wrong answers, but they can also correspond to a correct understanding. The didactic contract in the mathematics course and the didactic contract in the physics course are different. For example, the conventional notations used in the mathematics course and the physics course for the same object can differ (e.g., in France, the derivative is noted $f'$ in mathematics and $\frac{df}{dx}$ physics). The math-physics transition is, therefore, also a transition between these two different didactic contracts.

We will combine the concepts of praxeology and didactic contract to deepen our understanding of students' experience of the math-physics transition.

## Praxeologies in the Math-Physics Transition and Potential Difficulties

In the first phase of our work (Hellio et al., 2024), we identified potential difficulties through a text-book analysis. One of the aims of the present study is to observe whether the students experience these difficulties.
Using the ATD and particularly the concept of praxeology, we designed a method to identify similarities and differences between mathematics in mathematics courses and physics courses. Applying this method to a physics textbook and a mathematics textbook (aimed at the same first-year science students), we noticed several elements likely to raise difficulties for students. This led us to distinguish between several types of potential difficulties in the math-physics transition (Hellio et al., 2024).

We first noted that most of the mathematical types of tasks in the physics textbook only appeared as ingredients of techniques in physics. They were included in the technique for a physical type of tasks. For example, “solve a differential equation” was an ingredient of the technique for the type of tasks “find the velocity.” It is possible that the students do not recognize the presence of the mathematical type of tasks (Difficulty 1).

We also found some physical types of tasks that are solved with a technique that blends physics and mathematics. For example, “find the velocity” can require identifying that a given equation is a differential equation because it involves the velocity and the acceleration. This aligns with results obtained by White Brahmia (2023) about modelling in physics, which involves hybrid knowledge at the intersection of mathematics and physics. This hybrid knowledge is never explicitly taught. This can be the source of a second type of difficulty since students have to combine their mathematical and physical knowledge (Difficulty 2).

Last, the different notations in mathematics and physics are another difference between the didactic contract rules in mathematics and physics. For example, in physics, the velocity at time $t$ can be noted as $v$, whilst it would be noted as $v(t)$ in mathematics. The students can have difficulties using what they have learned in mathematics because of such different notations (Difficulty 3).

We do not claim that these are the only difficulties that students can face. They can, for example, have problems with algebraic calculation in their physics course, indicating the need for more practice on basic algebraic techniques in the mathematics course. We are especially interested in the difficulties cited above because they are directly linked with differences between mathematics in the mathematics course and mathematics in the physics course.

## Research Questions

The research questions we address in our study are:

* RQ1: What difficulties do first-year science students encounter in solving a physics exercise conveying different types of potential difficulties of the maths-physics transition?

* RQ2: How do resources from the Maths4sciences platform support or fail to support students facing these difficulties?

## Context of the study

Our experiment took place in a research-intense university in France, where students are enrolled in a “Sciences & Technologies” bachelor program. The first-year students in this program were “non-specialists” who attended, during the first semester, a mathematics and a mechanics course alongside other science courses such as engineering science and biology. The Maths4sciences resources were made available to them during the academic year, and they encountered mathematics exercises within a physics context throughout their first-semester 2023–2024 mathematics course. In their mathematics exams, the students were evaluated on mathematics problems; some of these problems were situated within a physics context.
# Methods

To answer our research question, we designed an ad hoc exercise which included the potential difficulties discussed above. This exercise includes a first-order differential equation, for which there exists a Maths4sciences tutorial sheet. We selected three students and organized two interviews with each of them during the second semester.

## Design and Analysis of the Exercise

We designed a physics exercise, referred to as “the Space Rocket” exercise throughout, which incorporates mathematics with the specific intention for the exercise to convey the three types of difficulties discussed above. The design of the exercise and its a priori analysis, focusing on the praxeologies and the potential difficulties, were conducted simultaneously. We collected students’ course materials in mathematics and physics and used them to determine whether the notations and, more generally, the text of the exercise were more typical of mathematics or physics for the students, as well as which techniques students had previously been taught.

## Analysis of the Tutorial Sheet

The tutorial sheet associated with the exercise we designed was named “Find the solution to a first-order differential equation (by identification).” We analyse the types of tasks present in the different parts of this sheet. Referring to the potential types of difficulties discussed above, we discuss the potential support that this tutorial sheet could provide to students trying to solve this exercise.

## Student Interviews

To select students, we first collected all first-semester mathematics exams and scanned for questions mobilizing differential equations, derivatives, and integrals. We asked for volunteers for the interviews. We crossed our list of 21 volunteers with their results to the exam questions and selected four students (one for a preliminary test of the interview protocol, three for the experiment) who showed only partial success in these subjects, with apparent difficulties in applying them within a physics context. The three students involved in the experiment are referred to below as St1 (male), St2 (female), and St3 (male). The Maths4sciences tutorial sheets had been handed out to the whole class at the beginning of the first semester. St2 reported having used some of the tutorial sheets in the first few weeks of the year before stopping altogether, whilst the other two students never used them. All students interviewed declared they had never used the particular tutorial sheet we tested.

We interviewed the selected three students individually for 40 to 45 min, twice, Interviews 1 and 2 being roughly a week apart. The interviews took place in March 2024 and were video recorded. The first author, who conducted the interviews, had taught a mathematics class that the students attended during the previous semester (we call her “R” in what follows). This constituted a bias that we were aware of: students viewed her as a mathematics teacher and likely expected her to want them to demonstrate mathematical knowledge.

Interview 1 was a task-based interview where the Space Rocket exercise was proposed to the students; they were asked to think aloud whilst solving it, first without the tutorial sheet and then with it. More precisely, after presenting the context of the interviews to the student, R gave them the exercise. Then, the student explored potential solution paths until they declared they were blocked, which took 5 to 10 minutes. During this first phase, R provided no help. When the student reached their blocking point, R gave them a list of five tutorial sheet names and asked the student to choose which one they believed
could be helpful and to explain why. After around 5 min of reading the list and discussing each item on it, the student made a choice. R gave the students the tutorial sheet on differential equations, whether or not they had selected it. The next phase of the interview consisted of the student solving the exercise with the help of the tutorial sheet and some prompts from R. During the entirety of the interview, R encouraged the student to think aloud and to express their thoughts and ideas, following the methods of the explicitation interview (Vermersch, 1994; see, for example, Maurel, 2009). All of the student's written productions were collected.

Between Interviews 1 and 2, we made a first analysis of the students' Interview 1, looking for aspects requiring clarification or which we deemed particularly relevant to our study. We extracted video excerpts from these moments.

Interview 2 was separated into two phases (see Interview 2 guidelines, Appendix 1). First, the student was asked a series of questions about the exercise and its possible origin, the tutorial sheet and its use, and their difficulties in mathematics and physics in general. Then, the video excerpts that we selected were played back to the student, and we encouraged them to explain further their experience solving this exercise. This data gives us access to students' activity, when they attempt to solve the exercise and when they use the sheet and thus allows us to observe whether they experience the potential difficulties (RQ1) and whether the tutorial sheet helps them to overcome these difficulties (RQ2).

## Interview Data Analysis

We started by analysing Interview 1 for each student, using the video and their written production. For each praxeology identified in the analysis of the exercise and the tutorial sheet, we noted whether the student performed the associated type of tasks, and if they did, with which technique and completed our analyses with the students' declarations during Interview 2. We separated the students' activity into two phases: their work on the exercise before they received the tutorial sheet and afterwards. We noted the difficulties encountered by the students relating to each type of tasks identified during the a priori analyses, their type (either one of the three types envisaged or something else) and their possible causes. We then investigated whether the tutorial sheet supported them and which aspects of the tutorial sheet were used.

# A Priori Analyses

## A Priori Analysis of the Exercise

We designed the Space Rocket exercise to convey the three difficulties on which we focus (Fig. 1). In particular, it was not designed to mimic an exercise that students could find in their mechanics course or their mathematics course. We now show how each type of difficulty appears through the analysis of the different types of tasks that present themselves when solving the exercise (see Appendix 2 for an example of a possible solution for the exercise).

The type of tasks that appears in the text of the exercise is $T_{detVelocity}$: "Determine the velocity." We identify this as a physics type of tasks which appears in the students' mechanics course but not in their mathematics course. This conveys Difficulty 1: in this exercise, mathematics appears as an ingredient of the technique to solve the type of tasks "Determine the velocity."

To solve this type of tasks, students must first recognize that the velocity is a function of time ($T_{recoVFT}$). Then, they must recognize that they have an equation to solve ($T_{recoEq}$) and that this equation is a differential equation ($T_{recoDE}$). These ingredients of techniques involving recognition are bypassed in the students' mathematics course, as the question in mathematics exercises is consistently "solve the differential
## Space Rocket Exercise

A space rocket is moving through the atmosphere at constant speed $v_i$. We set the time origin as the instant the rocket leaves the atmosphere to enter the stratosphere, at which point it accelerates. Its velocity then verifies:

$$v = -\tau \frac{dv}{dt} + v_i$$

Q. Determine the rocket's velocity with respect to time.

**Fig. 1** Space rocket exercise presented to the students (All translations to English are the authors'. The notation in the exercise is not related to the $\tau$ that refers to the technique in the praxeological quadruplet [T, $\tau$, $\theta$, $\Theta$].)

equation” followed by the differential equation. Here, in the case of $T_{recoDE}$, the term “differential equation” is voluntarily absent from the text of the exercise. Additionally, the left-to-right ordering of the terms in the equation and the fact that there is a coefficient ($-\tau$) multiplying the derivative are both uncommon in the mathematics course the students followed. However, it is a form that can be found in mechanics exercises, as it derives from the application of classical mechanics laws. This form may have an impact on $T_{recoDE}$. Moreover, during the solving of the differential equation, students must recognize that there is an initial condition $v_i$ ($T_{recoIC}$) and take it into account. There is no “at $t = 0$” in this exercise's text, nor is there an expression of the form “$v(0) = v_i$.” As is the case in most physics exercises from students' course materials, the students must, therefore, infer from the text of the exercise what the initial condition is. These types of tasks ($T_{recoDE}$, $T_{recoIC}$, etc.) all convey Difficulty 2, which we wish to study.

Finally, we used notations more common in the students' physics course for the following quantities: the velocity $v$, its derivative $\frac{dv}{dt}$, the initial velocity $v_i$, the limit velocity $v_l$, and the characteristic time $\tau$. Moreover, by design, there is nothing mentioned about the physical quantities apart from $v$ and $v_i$. These choices allow us to test Difficulty 3 relating to notations.

## A Priori Analysis of the Tutorial Sheet

The tutorial sheet is made up of two solved examples followed by a method, by which we mean a series of steps. The first example is called “Differential equation of the form “$y' = ay$” and the second is named “Differential equation of the form $y' = ay + b$.” An excerpt of the tutorial sheet we analysed is provided in Fig. 2 (for the full tutorial sheet, see Appendix 3). Since the exercise we designed proposes to solve a differential equation of the second form, we selected in this excerpt the second example and the method. We analyse it below, focusing on its affordances and limitations to support students working on the Space Rocket exercise.

Students must recognize a differential equation ($T_{recoDE}$) that has to be solved to perform the type of tasks from physics, which is to determine the velocity (Difficulty 1). At this stage, the tutorial sheet has little potential to support, first because the students cannot choose the relevant sheet if they do not recognize that there is a differential equation to solve to determine the velocity; second, even if the solution of the example starts with “we recognize a differential equation...,” how to perform this recognition is not explained. We also note that the term “differential equation” is present in the text of the example.

For Difficulty 2, addressing types of tasks mixing physics and mathematics, the tutorial sheet offers some support. To solve the type of tasks “write the differential equation under the chosen form” ($T_{writeForm}$, see Table 1 for a complete list of the types of tasks present in the tutorial sheet), the phrasing of the method emphasizes the need for algebraic manipulations, to reach the form “$y' = ay + b$.” This work is paramount in the following task of identifying the quantities $y$, $x$, $a$, and $b$ ($T_{recoTerms}$). This typically mixes mathematics and physics: the students have to identify, using knowledge from physics, the function $y$ and the variable $x$. The tutorial sheet does not explain how the students can perform this. The use of an initial (or “particular”) condition is also emphasized in
**Differential equation of the form $y' = ay + b$**

**Example**

The temperature $T(t)$ of an incompressible system satisfies the differential equation:

$$ \frac{dT(t)}{dt} = -\frac{hS}{mc}T(t) + \frac{hS}{mc}T_e. \text{ At time } t = 0, \text{ the system's temperature is } T_i. $$

Determine $T(t)$'s expression

**Solution**

* We recognize a differential equation of the form $y' = ay + b$, which solutions are of the form $y = ke^{ax} - \frac{b}{a}$ with $k$ an integration constant.

* By way of identification, we have $x = t$, $y = T(t)$, $a = -\frac{hS}{mc}$ and $b = \frac{hS}{mc}T_e$ therefore, $-\frac{b}{a} = -\frac{\frac{hS}{mc}T_e}{\frac{hS}{mc}} = T_e$ and $T(t) = ke^{-\frac{hS}{mc}t} + T_e$

* $k$ is determined using the initial condition on $T(t)$. At $t = 0$, we have $T(t = 0) = ke^{-\frac{hS}{mc} \times 0} + T_e = k + T_e = T_i$, form which $k = T_i - T_e$

* Finally, $T(t) = (T_i - T_e)e^{-\frac{hS}{mc}t} + T_e$

**Solving a first order differential equation**

**Method**

* Write the differential equation in the form $y' = ay$ or $y' = ay + b$ (the derivative must be "on its own" and on the left-hand-side of the equal sign)

* Identify, in the physical system studied, the quantities corresponding to $y$, $x$, $a$ and $b$.

* Infer from this the form of the solutions: $y = ke^{ax}$ or $y = ke^{ax} - \frac{b}{a}$ with $k$ an integration constant.

* Using a particular condition on the quantity studied, determine the integration constant.

<caption>**Fig. 2** Excerpt from the tutorial sheet</caption>

the example and the method, but the only explicit advice about how to recognize this condition is to use the description of the situation in physics.

Finally, both the notations $y'$ and $\frac{dT}{dt}$ are used to refer to the derivative. This corresponds to the choice of the authors, relating to Difficulty 3, to use diverse notations to support students' flexibility with them.

We note the lack of technological discourse in the tutorial sheet. We, therefore, expect that the tutorial sheet may support students on the type of tasks and technique aspect but not on technological or theoretical understanding.

In the next section, we investigate to what extent the tutorial sheet's potential to help students working on the Space Rocket exercise was realized.

**Results: Analysis of the Interview Data**

All of the students encountered difficulties with the exercise proposed; none of them was able to solve it without the tutorial sheet. The three of them eventually found the expected solution with the help of the tutorial sheet and some prompts from R.

<table><caption>**Table 1** Types of tasks present in the Solution of the worked example and the Method section of the tutorial sheet</caption><thead><tr><td>Code</td><td>Type of tasks</td><td>Solution</td><td>Method</td></tr></thead><tbody><tr><td>T<sub>recoForm</sub></td><td>Recognize the form of the differential equation</td><td>Yes</td><td>Yes</td></tr><tr><td>T<sub>writeForm</sub></td><td>Write the differential equation in the form chosen</td><td>No</td><td>Yes</td></tr><tr><td>T<sub>recoTerms</sub></td><td>Recognize which terms correspond to x, y, a, and b</td><td>Yes</td><td>Yes</td></tr><tr><td>T<sub>detGS</sub></td><td>Determine the general solutions</td><td>Yes</td><td>Yes</td></tr><tr><td>T<sub>detIC</sub></td><td>Determine the integration constant</td><td>Yes</td><td>Yes</td></tr><tr><td>T<sub>detFS</sub></td><td>Determine the final solution</td><td>Yes</td><td>No</td></tr></tbody></table>
As explained above, to analyse the interview data, we first looked at the students' activity without the tutorial sheet and then with it. For the presentation of the analysis, we keep this distinction. Moreover, we also split the activity of the students with the tutorial sheet into two parts: before they start solving the differential equation and when they solve it. Hence, this section is divided into three subsections. In each, we start with a table presenting the types of tasks from the list established in the previous section and tackled by at least one of the three students. We note, for each student, their technique, and then we discuss their difficulties and whether or not the tutorial sheet supported them.

## Students' Activity Without the Tutorial Sheet

For the type of tasks $T_{\text{undExercise}}$: “Understanding the text of the exercise,” St1 reads the text and observes things that he considers important: $v_i$, then an acceleration, and the “formula” characterizing the acceleration (see Table 2 for a complete description of the types of tasks and techniques students used). He comments on this formula using knowledge from the physics course: the acceleration is the derivative of the velocity (this means that he recognizes the notation $\frac{dv}{dt}$ as the derivative of the velocity; he also recognizes that $t$ denotes the time and that the velocity $v$ depends on $t$). But he declares that what he calls “the formula” already determines the velocity with respect to time. R has to intervene and help him, saying, “There is more work to do.” We note that it would be possible, at that stage, to transform it into $\frac{dv}{v-v_i} = -\frac{dt}{\tau}$ and to integrate both sides to solve the equation. However, he had not learned this method in either of his courses and stayed stuck. When R proposes to choose a tutorial sheet likely to help him, he dismisses the sheet about differential equations. In Interview 2, he confirms that he did not recognize a differential equation and says that in his courses, it is simpler since, within a given chapter, all the exercises use similar physical notions and similar mathematical tools. Thus, students can use didactic contract rules like: “We are working within the kinematics chapter. Therefore, the exercises contain differential equations to be solved.”

Concerning $T_{\text{undExercise}}$, St2 reads the text and observes the important information: “The constant velocity is $v_i$, the origin is when the rocket leaves the atmosphere, and the formula” (from Interview 1). Then, she reads the question and understands that the exercise asks to look for the velocity with respect to time. She seems to understand the physics context, but she does not use physics knowledge at this stage. She never says that the velocity depends on the time and seldom uses the term function; instead, she refers to the notation $v(t)$ that she describes as “$v, t$ within brackets” (Interview 1). St2 does not use the term “equation;” instead, she says she must “reshape the formula.” When R proposes to choose a tutorial sheet, she immediately says that the sheet about differential equations is useless. During the whole interview, she describes “$\frac{dv}{dt}$” as “the derivative of $v$ over the derivative of $t$.” She does not seem to consider $v$ as a function of $t$, nor does she identify $\frac{dv}{dt}$ as the derivative of $v$ with respect to $t$.

The case of St3 is very different. For $T_{\text{undExercise}}$, after reading the text, he draws a first graph representing velocity versus time (Fig. 3a).

St3 immediately recognizes $v$ as a function of $t$. He adds that the equation is a differential equation since “there is $v$, and there is the derivative of $v$” (Interview 1). He also immediately reads the notation $v_l$ as the “limit velocity,” which we interpret as a didactic contract rule stemming from his mechanics course. He relies on similarities he notices between the exercise and transient states learned in the electricity course to draw the second graph (Fig. 3b) and comments that $v$ will stay under $v_l$.

Then, St3 uses algebraic manipulations to transform the differential equation of the exercise into “a usual form.” Answering a question by R, he explains that the usual form of a differential equation is $\alpha f + f' + “something” = 0$ (Interview 1): we also interpret this as a didactic contract rule. St3 transforms the equation into “$0 = -\tau\frac{dv}{dt} + v_l - v$” and declares that this is a homogeneous differential equation since there is a “$= 0.” Once again, this is a didactic contract rule (“$= 0$” means the differential equation is homogeneous”) that creates difficulty for St3.
<table><caption>**Table 2** Types of tasks and techniques used by the students before their use of the tutorial sheet</caption><thead><tr><th>Type of tasks</th><th colspan="3">Technique used by the student</th></tr><tr><th></th><th>St1</th><th>St2</th><th>St3</th></tr></thead><tbody><tr><td>T<sub>undExercise:</sub><br>Understanding the text of the exercise</td><td>Reads & points out information</td><td>Reads & points out information</td><td>Reads, points out information & draws a graph</td></tr><tr><td>T<sub>recoVFT:</sub><br>Recognize that v is a function of t</td><td>Uses that the velocity depends on the time (physics)</td><td>No</td><td>Uses that the velocity depends on the time (physics)</td></tr><tr><td>T<sub>recoEq:</sub><br>Recognize that there is an equation to solve</td><td>No, then with the help of R: on the right side of the equal sign, d<sub>v</sub> depends on v</td><td>Unclear, speaks about "reshaping the formula."</td><td>See below: St3 recognizes a differential equation</td></tr><tr><td>T<sub>recoDE:</sub><br>Recognize that the equation is a DE</td><td>No</td><td>No</td><td><sup>dv</sup>&frasl;<sub>dt</sub> is the derivative of v, the relation given is a differential equation</td></tr><tr><td>T<sub>recoForm:</sub><br>Recognize the form of DE</td><td>No</td><td>No</td><td>Transforms the equation of the exercise to obtain the "usual form" α<sub>f</sub> + β<sub>f</sub> + "something" = 0</td></tr></tbody></table>

<figure><img src="image_4.png" /></figure>
Pre-solving Phase of Students’ Activity

This phase of the students’ activity starts when R gives the tutorial sheet to the students and ends when
they start solving the equation of the exercise. Table 3 presents the techniques used by students.

We find that the type of tasks $T_{\text{recoDE}}$: “recognize that an equation is a differential equation” was diffi-
cult for two out of three students and, as expected in the a priori analysis, that overcoming this particular
difficulty was not supported by the tutorial sheet. Indeed, St3 had recognized it before being given the
tutorial sheet, St1 recognized it when R gave him the “Differential equations” tutorial sheet, and, dur-
ing Interview 2, St2 declared she did not recognize it as such even after being given the tutorial sheet.

After having started reading the tutorial sheet, St1 and St2 recognize the expected form between
the two presented in the tutorial sheet ($T_{\text{recoForm}}$ in Table 3). These two students rely on the equations
presented in each of the examples to do so, which are $\frac{dN}{dt} = \dots$ and $\frac{dT(t)}{dt} = \dots + \dots$, and compare these
with $v = -\tau \frac{dv}{dt} + v_l$. St2’s reasoning relies on the existence of the + sign in the second equation as well
as in the equation given in the exercise. St1’s reasoning for choosing the second example relies on a
didactic contract rule: a term dependent on time (here, $v$) must be followed by a “$(t)$,” as is the case in
the expression “$\frac{dT(t)}{dt}$” in the tutorial sheet.

St3 starts by trying to identify the different terms before choosing one form between $y' = ay$ and $y' = ay + b$. He first chooses $y' = ay$. Then, he counts the “number of terms” that make up the equations in the exercise. He declares that $v = -\tau \frac{dv}{dt} + v_l$ is made up of four terms, as is $y' = ay + b$, but that $y' = ay$ is only made up of three terms. He concludes by choosing the second form.

All students try to recognize the terms making up the differential equation ($T_{\text{recoTerms}}$) before reor-
dering the terms of the equation to write it under the desired form ($T_{\text{writeForm}}$) but fail. Indeed, St1 and
St2's technique is to identify terms through their order in the equation and the corresponding form
$y' = ay + b$ from left to right when reading. St1 then notices an incoherence in his result: $y = \frac{dv}{dt}$ and
$y' = v$ "is not logical; it's the opposite," he says (Interview 1), which leads him to reorder the terms
to identify a and b, arriving at the equation $-\frac{1}{\tau}v + \frac{1}{\tau}v_l = \frac{dv}{dt}$.

Only St3 holds a discourse on the physical meaning of the terms. Indeed, to perform $T_{\text{recoTerms}}$, he remarks that he needs to know what depends on the time or not: this means implicitly identifying the status of the different letters (function, variable, parameter). He declares then “$v_l$ is the limit speed, it does not depend on the time” (Interview 1) and “for tau, I don’t know if… tau only has one

<figure><img src="image_5.png" /><figcaption>**Fig. 3** a, b Graphs drawn by St3</figcaption></figure>
<table><caption>**Table 3** Types of tasks and the techniques used by students with the sheet before solving the equation</caption><thead><tr><th>Type of task</th><th>Technique used by the student</th><th></th><th></th></tr></thead><tbody><tr><td>T<sub>recoDE</sub><br>Recognize that a given equation is a differential equation</td><td>St1<br>Recognizes it when told the tutorial sheet to use is the “Differential equations” one</td><td>St2<br>No</td><td>St3<br>Before the tutorial sheet is given (see above)</td></tr><tr><td>T<sub>recoForm</sub><br>Recognize the form of DE<br>With the sheet, only two possible choices</td><td>St1<br>Chooses between (dN/dt = -λN(t)) and (dT(t)/dt = -(h<sub>S</sub>T(t) + h<sub>C</sub>T<sub>e</sub>) / (m<sub>C</sub>T<sub>e</sub>) the formula with (t) on the left-hand side because the velocity is a function of time</td><td>St2<br>Chooses between (dN/dt = -λN(t)) and (dT(t)/dt = -(h<sub>S</sub>T(t) + h<sub>C</sub>T<sub>e</sub>) / (m<sub>C</sub>T<sub>e</sub>) the formula that also has a "+" sign</td><td>St3<br>Chooses between y' = ay + b and y' = ay the formula that has the same number of terms as the Eq. (4 terms)</td></tr><tr><td>T<sub>recoTerms</sub><br>Recognize the terms making up the equation</td><td>St1<br>Identifies y, y', a, and b through the left-to-right order of terms<br>Brings meaning into his reasoning (v must be y and d<sub>y</sub>/dt, not the opposite)</td><td>St2<br>Partially identifies through the order of terms at first<br>Identifies x, y, a, and b, also through the order of terms</td><td>St3<br>Identifies y, y', a, and b through the order of terms<br>Questions the physical significance of τ whilst identifying<br>Algebraic operations (with mistakes)</td></tr><tr><td>T<sub>writeForm</sub><br>Write the differential equation in the desired form</td><td>St1<br>Algebraic operations (with mistakes)</td><td>St2<br>Algebraic operations (with mistakes)</td><td>St3<br>Algebraic operations (with mistakes)</td></tr></tbody></table>
<table><caption>**Table 4** Types of tasks and techniques used by the students for solving the DE with the tutorial sheet</caption><thead><tr><th>Type of tasks</th><th>Techniques used by the students</th><th></th><th></th></tr></thead><tbody><tr><td>T<sub>detGS</sub><br>Determine the general form of the solution</td><td>St1<br>Applies the formula given in the example of the tutorial sheet</td><td>St2<br>Applies the formula given in the example of the tutorial sheet</td><td>St3<br>Applies the method presented in the tutorial sheet</td></tr><tr><td>T<sub>recolC</sub><br>Recognize that there is an initial condition in the exercise</td><td>Yes, seems to do so after glancing back at the tutorial sheet</td><td>No<br>Translates the notation T<sub>i</sub> of the sheet's example into v<sub>i</sub></td><td>No, then, after a request by R, reads the example of the tutorial sheet, notices that he missed something, and rereads to the text of the exercise</td></tr><tr><td>T<sub>detC</sub><br>Determine the solution satisfying the initial condition</td><td>Reads the example of the tutorial sheet and follows the steps<br>Follows the technique presented in the example of the tutorial sheet</td><td>Follows the technique presented in the example of the tutorial sheet</td><td>Reads the text of the exercise, goes back to his graph<br>Follows the technique presented in the example of the tutorial sheet</td></tr></tbody></table>

<figure><img src="image_6.png" /></figure>
value, so it is also a constant.” (Interview 1) These declarations are grounded on the notations used in his physics courses; hence, they come from didactic contract rules. After this, he successfully identifies each term.

In each case, the tutorial sheet did not motivate the students to write the differential equation in the desired form. Indeed, St1 and St3 are motivated by the failure of their first identification attempt, whilst R must ask St2 to perform that task. This leads us to infer that, although the method section of the tutorial sheet states, “Write the differential equation in the form $y' = ay$ or $y' = ay + b$ (the derivative must be “on its own” and on the left-hand-side of the equal sign),” it doesn’t support students in solving differential equations which are not already in the desired form.

Throughout these steps, many of St2’s praxeologies are founded on two didactic contract rules. First, she must do what she thinks R (who was her teacher) wants her to do. Second, if the tutorial sheet is given as help, then the exercise must be just like the examples of the tutorial sheet. She is also very limited by her belief that if there are no parentheses, then there is no function, which prevents her from considering $v$ as a function.

## Solving the Differential Equation

In this phase of their activity, all three students used the tutorial sheet; the types of tasks and techniques they used are described in Table 4. They first write a solution of the form $v = ke^{-\frac{x}{t}} + v_l$, using slightly different techniques. St1 and St2 use the formula given in the example of the tutorial sheet and replace $a$ and $b$ with their expressions. St1 first wrote “$y = ke^{-\frac{x}{t}} + v_l$,” using the notations $y$ and $x$ of the tutorial sheet. He changed these notations after a remark by R. St3 reads the method presented in the tutorial sheet and says, “Ok, this is, in fact, simple!” (Interview 1) Answering a question by R, he says that he forgot that when $b$ does not depend on the variable (“the variable for me is $t... b$ does not depend on anything in fact...” (Interview 1)), a simple identification is needed to find the solution, instead of the use of a particular solution. St3 computes the general form of the solution and checks that it satisfies the differential equation.

After finding the general expression of $v$, all three students pause for a brief moment. St1 and St2, who closely read the tutorial sheet, notice that there is a next step: using the initial condition to determine $k$. St1 says: “We have to do the initial velocity.” (Interview 1) St2 still follows the sheet very closely; she translates the notation $T_i$ of the sheet to $v_i$ but never speaks about an initial condition. St3 thinks that the question is answered; then R asks him to look at the tutorial sheet. He reads the method and observes the presence of the next step. He commented that so far, “we have all the possible solutions, but in the context of the exercise, we have $v_i$.” He goes back to the text of the exercise and to the graph he drew at the beginning and identifies $v_i$ as an initial condition in this situation.

After this, the three students tackle the type of tasks $T_{detFS}$. St1 and St2 once again closely follow the example of the tutorial sheet, using the notation “$v(t = 0)$.” St1 first thinks that he needs to determine $k$ and $v_l$; R must tell him that $v_l$ is not unknown (and indeed, nothing in the text of the exercise indicates the nature of $v_l$). St3 reads the example of the tutorial sheet and declares that it is difficult for him to use an example instead of a method. All three students finally obtain the expected formula: $v = (v_i - v_l)e^{-\frac{x}{t}} + v_l$.

## Conclusion and Discussion

In this section, we first draw on the results presented above to answer our research questions. Then, we discuss the limitations of our study and its contributions to research and curriculum design.
# Elements of Answers to the Research Questions

Our research questions were:

RQ1: What difficulties do first-year science students encounter in solving a physics exercise conveying different types of potential difficulties of the maths-physics transition?

RQ2: How do resources from the Maths4sciences platform support or fail to support students facing these difficulties?

Concerning the difficulties, we recall that the exercise designed conveyed three types of difficulties: (1) recognizing mathematical types of tasks hidden as ingredients of technique, (2) addressing “recognition” types of tasks using techniques which mix mathematics and physics, and (3) managing the presence of notations of physics in a mathematical type of tasks. St1 and St2 indeed encountered the three difficulties. First, they did not recognize that there was a differential equation to solve, which we relate to both Difficulty 1 because the type of tasks $T_{recoDE}$ was an ingredient of the technique for $T_{detVelocity}$: “Determine the velocity,” and Difficulty 2 because to solve this task students had to blend their mathematics and physics knowledge. Moreover, these students had difficulty performing $T_{recoDE}$ because they had not mastered the significance of the notation $\frac{dv}{dt}$, which we relate to Difficulty 3. Moreover, St1 and St2 had difficulty recognizing the correct terms making up the equation ($T_{recoTerms}$, Difficulty 2).

The case of St3 is different. He did not show having any of the three types of difficulty. However, he was searching for a “usual form” he had learned in the mathematics course and that he had forgotten. He reordered the terms, but the unusual form of the equation provided in the exercise led him to think that the differential equation he obtained after reordering was homogeneous. This shows how another difficulty can be at hand in students’ struggle to transition between mathematics and physics: in our case, the mathematical objects present in physics tended to be present in more diversified forms. In this case, differential equations in the students’ mathematics course were always in a specific form, which may have led him to incorporate a didactic contract rule like “if there is a ‘= 0’, then the equation is homogeneous.” This does not apply when the equation is in a different form, as can be in the physics course. Further investigation is needed to elaborate on this hypothesis.

The tutorial sheet helped all three students solve the exercise. Not only did they declare so in Interview 2, but they also all used the technique presented by the tutorial sheet. They were helped by the tutorial sheet for identifying the form of the differential equation ($T_{recoForm}$) and determining the general form of the solutions ($T_{detGS}$). In the case of St3 specifically, he would have stopped solving the exercise after having found the general solution, and the tutorial sheet led him to recognize he needed to find an initial condition and calculate the final solution ($T_{recoIC}$). However, we do note that although St2 produced the correct answer, she declared she did so by imitating the expressions from the tutorial sheet without understanding the meaning of the physical situation. For example, she did not recognize that $v$ was a function of time, nor did she identify the initial condition from the text of the exercise.

# Limitations of the Study, Contributions to Research, and Curriculum Design

We are aware of some of the limitations of the study presented here. We only interviewed three students working on a single exercise. The students used a single Maths4sciences tutorial sheet during a brief period and were sometimes helped by prompts from R rather than by the tutorial sheet or were only helped by the tutorial sheet after R had pointed them to it. Other students might have
other difficulties, and long-term engagement with Maths4sciences resources could be much more beneficial. Nevertheless, our study has contributed to research and curriculum design.

Undergraduate science students experience a permanent transition between mathematics in their mathematics courses and mathematics in their physics courses. Our analyses evidenced different types of difficulties that these students face whilst moving through this transition. We also highlighted the ubiquitous presence of “recognition” types of tasks in the maths-physics transition. Indeed, six recognition types of tasks appear in our analyses. Students must recognize that an equation has to be solved ($T_{recoEq}$) and then that it is a differential equation ($T_{recoDE}$). These mathematical types of tasks are hidden as elements of techniques in physics exercises. Moreover, the different notations can prevent some students from recognizing the differential equation. White Brahmia (2023, p. 70) notes that students have difficulties “activating appropriate mathematical tools without prompting” in physics class. Our results confirm this observation and suggest investigating the link between “recognition of a type of tasks” and “activation of a mathematical tool.” Students also needed to recognize that the velocity is a function of time ($T_{recoVFT}$) and that they know a particular value of this velocity (initial condition, $T_{recoIC}$). The techniques to perform these types of tasks are grounded in physics knowledge since the notations $v(t)$ or $v(0)$ are not used in the text of the exercise. Finally, the students had to recognize the form of the differential equation ($T_{recoForm}$) and its different terms ($T_{recoTerms}$). The techniques here include mixed interpretations of the physical situation to determine what depends on time and what a parameter is, as well as algebraic transformations of the equation. In their work in physics education, Kuo et al. (2013) have evidenced that in expert solving of physics problems, mathematical manipulations and physical reasoning are blended, and our results are consistent with their study.

The Maths4sciences tutorial sheets have several features that are likely to support the students in performing recognition tasks. First, they comprise multiple examples of physics exercises that use the mathematical method presented. Second, they deliberately use different letters and notations to encourage students’ flexibility through these recognition tasks. Finally, they explicitly discuss some of the differences between mathematics and physics notations and methods. Nevertheless, we observed limitations to the support of the tutorial sheet offered to students. Modifications of the tutorial sheets can overcome some of these limitations. For example, in our case, sentences can be added about the notion of the initial condition and how to find it in a physical situation. Moreover, the tutorial sheet supports students in performing the tasks, but, as said in the a priori analysis, it does not have information about *why* the technique works or, in other words, on the technology justifying the technique. To support students in a more all-encompassing understanding of both the technical and technological aspects of differential equations, the tutorial sheet would need to incorporate more elements of justification. The Maths4sciences team of authors is working on an improved version of the resources; in particular, tutorial sheets will be complemented with quizzes asking to recognize whether the method can be applied to solve a given exercise.

Nevertheless, modifications to the tutorial sheets cannot solve all the difficulties we observed; our study also suggests the need to consider them in the mathematics and physics courses. For example, $T_{recoEq}$ and $T_{recoDE}$ could be addressed in the mathematics course with problem-solving activities outside of particular chapters, where students have to recognize equations and differential equations without drawing on didactic contract rules (e.g., “In the differential equations chapter, equations are differential equations”). For exercises in a physics context (in the physics or the math course), identifying the status of the different elements intervening in the text could be presented as a starting point: $t$ is a variable, $v$ is a function of $t$, $v_i$ is a constant whose value is known, etc.

Identifying other recognition types of tasks, finding a classification of those and investigating their link with mathematical modelling in physics is a research direction that can lead to theoretical and empirical results and inform instruction. This line of research will be central to our future work.
**Acknowledgements** We thank members of both the administration and pedagogical teams at Institute Villebon – Georges Charpak for the time and resources they allocated. We also thank the four students who volunteered for the experiment.

**Funding** Open access funding provided by Université Paris-Saclay.

**Data Availability** The data of our project is not publicly available.

**Declarations**

**Competing Interest** The authors declare no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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