ETH Zurich is attracting more and more students, and all of them have mathematics lectures on their curriculum. But the number of lecturers and assistants, whose job it is to set exercises, correct them and provide feedback, is stagnating. Given that exercises and repetition are essential for mastering the fundamentals of mathematics, several lecturers are converting the exercises they do in courses into a computer-based format.
Meike Akveld and Andreas Steiger give lectures on the fundamentals at the Department of Mathematics and have amassed a vast collection of maths problems, including an integral trainer. These problems are based on STACK questions – a specific question type for mathematical expressions on the Moodle learning platform.
Ms Akveld, Mr Steiger, practising maths problems on a computer sounds like an obvious choice to me as a non-mathematician. What makes your approach innovative?
Akveld: It might not sound like much at first, but we’re not talking about a mere pocket calculator. STACK questions let us set our students problems that they can solve using mathematical expressions – for example, with polynomials. The program first checks the syntax– in other words, the “orthography” of the expressions – and then evaluates the answer at the touch of a button and provides feedback on any errors.
What are the challenges associated with such a system?
Steiger: The complexity lies in the didactics; that is, in the way we can structure the exercises. STACK allows us to mark individual steps in the working. It’s very good at recreating this constructive, step-by-step approach, which is so important in mathematics and would typically take place on the whiteboard. Building this kind of system takes a great deal of time and expertise. Unveiled back in 2005, it now seems to have achieved widespread acceptance.
How did the breakthrough come about?
Steiger: The push came from the pandemic, because lots of people were suddenly writing problems in STACK and started sharing them. Plus, the system has become very user-friendly.
What benefits are there for your students?
Steiger: Mastering the fundamentals of mathematics takes a great deal of practise. STACK exercises allow the students to practise as much as they need and whenever they want. And they get feedback right away. This is proving really popular, which is reflected both in student feedback about our courses and in the stats: students are practising frequently and for long periods.
Akveld: And that in turn helps us as lecturers: the more our students practise, the easier it is for us to see where they’re making mistakes. STACK brings all this information together in one place and, based on analyses, we can incorporate specific tips and hints into the exercises.
In addition to the collection of problems, you’ve also developed an integral trainer. What is that for?
Steiger: We’ve noticed a large disparity in skill levels when it comes to computing integrals. Depending on what students were taught at secondary school and what experience they have, some are proficient when they arrive, while others still need practise. When you’ve got 700 students in a lecture, having a tool that the students can use to practise exactly this aspect of mathematics is extremely welcome.
How does the trainer work?
Steiger: Using the integral trainer, students can practise the various methods they must master – first individually and as often as necessary, and with assistance in the form of tips or automated feedback. Later on, they can tackle less obvious problems that require them to identify and apply the correct method. We put a lot of thought into how we structure the exercises and the feedback before we programmed the trainer with STACK’s help. In this way, we showed that the potential of this tool extends beyond merely digitalising collections of problems to include actual training units with specific structures and processes.
Could the tool be suitable for use outside ETH Zurich?
Steiger: Yes, we’re getting enquiries from secondary schools in Switzerland and abroad. STACK questions have a very active community – the whole thing is open source (editor’s note: the source code is in the public domain) and there is a vibrant and immediate exchange of ideas. The advantage of STACK questions is that you can share them very easily. It’s all mutually beneficial.
How does ETH Zurich compare internationally in using STACK?
Steiger: Although we started using STACK only recently, our integral trainer has already attracted some interest. At the head of the pack is the University of Edinburgh, where Chris Sangwin, who developed STACK, teaches. The UK’s Open University also makes extensive use of the tool. In many countries, multiple universities collaborate to generate shared databases of maths problems for STACK. Bavaria wants to roll out STACK at its high schools. It’s a topic that’s now really gaining momentum.
Can STACK also be used for exams?
Akveld: Yes, just recently, first-year students took their first STACK exam as part of their analysis course. STACK may well prove essential for maths exams at ETH because we don’t have enough staff to keep pace with the increasing student intake. The conventional solution would have been to set more multiple-choice exams so we’d have the option of using automated marking. But the multiple-choice format limits the kinds of questions you can set. STACK gives us greater freedom in designing questions and we can still automate the marking. These are also easier to administer because, thanks to randomisation, each student’s questions have unique values.
What limits does the method have?
Steiger: STACK is great for practising and testing things that are based on mathematical expressions. And that’s the case for the fundamentals in particular. A mathematics degree, however, is really about modes of thinking and terminology and how these relate to each other. STACK is then less suitable than it is for the kinds of computational problems we set our engineering and science students.
How is the method changing the courses you teach?
Akveld: It’s putting students in a more active role. Every course features a few sequences that I use to demonstrate ways of solving particular problems. Presenting these sequences isn’t particularly interesting for the lecturer or for the students. I can substitute these sequences by sequences in STACK, and the students can then go through the different ways of solving them step by step with the inbuilt assistance features. They often grasp things faster doing it themselves than when watching me do it.
We can all remember that sinking feeling from maths lessons when we lost the thread around halfway through the explanation...
Akveld: Exactly – and even though you’re still taking notes you no longer understand a word. Now, if I offer a learning module with STACK questions for homework, you can just hit refresh and do it over and over again until you’ve got it. What’s more, if I provide students with STACK questions as prep, I can also integrate tips and assistance at those points where experience tells me they are likely to trip up. And then when they come to my lecture having done the prep, I can use the time better to delve deeper into the material.
You put a lot of time into this method. What drives you?
Akveld: I can use the tool to follow through on instructive ideas that I’ve been thinking about for ages. These are all about making mathematics more accessible to everyone. I believe people’s grasp of mathematics grows with each problem they solve. Everyone has areas they really have to grapple with, and STACK is a big help.
Steiger: It’s my job to take as many students as far as possible. And because they are all so different, we need a wide variety of means and methods. I want to give everyone the opportunity to understand the maths they’ll need in their chosen discipline. STACK is another tool we use to do just that.