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.

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.

For the type of tasks TundExercise: “Understanding the text of the exercise,” St1 reads the text and observes things that he considers important: \(_\), 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 \(\frac\) 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 \(\frac_}=-\frac\) 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 TundExercise, St2 reads the text and observes the important information: “The constant velocity is \(_\), 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 “\(\frac\)” 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 \(\frac\) as the derivative of \(v\) with respect to t.

The case of St3 is very different. For TundExercise, 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 \(_\) 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 \(_\).

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 \(\alpha f+f^+\mathrm=0\) (Interview 1): we also interpret this as a didactic contract rule. St3 transforms the equation into “\(0=-\tau \frac+_-v\)” 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.

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 TrecoDE: “recognize that an equation is a differential equation” was difficult 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, during 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 (TrecoForm in Table 3). These two students rely on the equations presented in each of the examples to do so, which are \(\frac=\dots\) and \(\frac=\dots +\dots\), and compare these with \(v=-\tau \frac+_\). 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 “\(\frac\)” 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 \(v=-\tau \frac+_\) 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 (TrecoTerms) before reordering the terms of the equation to write it under the desired form (TwriteForm) 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: \(y=\frac\) 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 \(-\fracv+\frac_=\frac\).

Only St3 holds a discourse on the physical meaning of the terms. Indeed, to perform TrecoTerms, 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 “\(_\) 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 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 \(^=ay\) or \(^=ay+b\) (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.

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 \(v=k^}+_\), 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 “\(y=k^}+_\),” 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 \(_\) of the sheet to \(_\) 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 \(_\) as an initial condition in this situation.

After this, the three students tackle the type of tasks TdetFS. 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 \(_\); R must tell him that \(_\) is not unknown (and indeed, nothing in the text of the exercise indicates the nature of \(_\)). 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: \(v=(_-_)^}+_\).