During the last years, the student numbers grew rapidly while the supervision/mentoring capacities (lecturers and teaching assistants) remained rather constant. This led to bigger and bigger classes where individual mentoring and guiding is no longer possible.

One possible idea to tackle this challenge are interactive exercises with individual feedback. In Moodle, the platform that is most commonly used for lecture webpages/platforms, STACK provides such a feature.

STACK – which is also available in ILIAS and other similar platforms – provides a feedback rich, interactive and randomized exercise environment which has in its background a strong computer algebra system, MAXIMA, thus being a very powerful tool supporting on one hand the student’s learning process and on the other hand giving the lecturer insights on where the students struggled most.

STACK has already been successfully used at various Universities, including ETH Zurich, starting from the mathematics lectures in the first year (Basisjahr) and by now ranging also to other, non-mathematics courses.

**In our project, we extended these concepts to second year mathematics contents, in particular Partial Differential Equations (PDEs). Setting thus not only new standards in the mathematical education but also in the community of STACK users.**

The topic of Partial Differential Equations (PDEs) is fundamental for all students in natural and engineering sciences being one of the most common ways of stating a mathematical model. Mastering/Solving though such equations requires quite a bit of routine – often more extended that can be provided by classical exercise sheets. So, one first benefit from STACK exercises over classical exercises is that such calculation routines can be easily trained by randomized exercises, one of the core features of STACK exercises.

The second benefit from STACK exercises is that for longer calculations – or solutions that require several consecutive steps – **partial credit** together with **individual feedback** can be given. Moreover, correct results based on wrong intermediate results can be taken into account (Folgefehler).

In addition, if the students really do not have any idea how to solve a given problem, they can ask for the **correct solution** (by clicking on the corresponding button). And in the end, the students receive also as part of the overall feedback the entire step-by-step solution.

But STACK is not only an ideal training platform but can be used as well as an element which allows students to **individually learn new material**.

In our project we chose to provide a whole sequence of STACK exercises to guide the students through the modeling of the hydrogen atom from the underlying mathematical model to the discrete energy levels and concrete functions describing the corresponding states.

*The particular novelty of our project consists on one hand in pushing/expanding such STACK exercises to the realm on Partial Differential Equations and on the other hand to incorporate new feature as graphical components.*

*Figure 1:* Individual feedback and partial credit

*Figure 2:* Detailed step-by-step solution

*Figures 3 and 4:* Example of an exercise using novel graphical features

## Impact and future developments

STACK exercises do not only prove to be an excellent enrichment of our lectures but provide also a most valuable feature for exams. We strongly believe that such exercises together with classical multiple-choice questions and other formats provide an ideal infrastructure for (partially) **computer-based exams**. In addition, this exercise format can also be most valuably used in other – non-mathematical – courses and we are delighted to explore these grounds together with our colleagues.

If in the future STACK can be combined with AI tools, it will surely become even more powerful, in particular in the context of assessments.

## Course Description

- Name:
- Mathematics III: Partial Differential Equations
- Description:
- Classical tools to solve the most common linear partial differential equations (Examples of partial

differential equations. Linear partial differential equations. Separation of variables. Fourier series, Fourier

transform, Laplace transform. Applications to solving commonly encountered linear partial differential

equations (Laplace's Equation, Heat Equation, Wave Equation)).

- Department:
- D-MATH (for students mainly from D-CHAB)
- Level:
- second year Bachelor
- Format:
- 2V+1U
- Size:
- 200 students
- Assessment:
- session examination