Strategy games and their mathematical component

The Mathematical Optimization group, led by Prof. Dr. Kathrin Klamroth, uses games to teach student teachers real-life subject knowledge.

""I believe that everyone can learn mathematics,"" says Prof. Dr. Kathrin Klamroth, head of the Mathematical Optimization Group in the Faculty of Mathematics and Natural Sciences at Bergische Universität, and together with her team she is finding new ways of communicating mathematics that can inspire student teachers. In a KoLBi project (Coherence in Teacher Education) with the goal of improving students' understanding of mathematics, she and her colleagues Dr. Michael Stiglmayr and Konstantin Kraus discovered that board games in particular often have a mathematical component that is not always obvious but can be helpful in teaching.

Games can arouse enthusiasm

Klamroth knows that games in the leisure sector are fun, a varied activity, you can have a good time with friends and they also make you think, and Kraus adds that children in particular also learn many social skills unconsciously through games. ""We wanted to generate enthusiasm,"" says the scientist, ""we have many student teachers who will certainly get a job with mathematics, but not all students are equally enthusiastic about mathematics. We started seminars in which we wanted to use practical problems, i.e. problems from everyday life, to build a bridge to the mathematical topics covered in school. For example, one group went into the woods to determine a route so that many geocaches could be collected in a short time, while others worked out a route for a sports competition or evaluated the Nordbahntrasse with regard to safe routes to school. The board games then lined up very well with these tasks, and those responsible also noticed that ""many math students actually play"" explains Klamroth. ""The math department has a strong games club, so a lot of them are interested. And many games have a mathematical component, are strategic. Chess is perhaps the best-known example, and there is a lot of literature on it. But then we mainly looked at other board games."" The main focus was on games for which there was no extensive literature, explains Kraus, because the students were supposed to develop something new and valuable, which was also a new approach that required a lot of independence and commitment on the part of the students.

Mathematical components in board games

Board games are a great tool for teaching students mathematical concepts. Kraus explains it with the example of the board game 'The Settlers of Catan`. ""A group of students once used 'The Settlers of Catan`, designed a series of lessons and conducted them in 7th grade at a school. The game served as an introduction to stochastics, so to speak, because you roll two dice in 'Catan`"" he explains. ""For students, this was new knowledge that they acquired without realizing it."" Many of these games are so-called strategy games that work with graphs, maps that show any connections between places. ""In optimization, a graph is a classic construct on which you then look at optimization problems, for example, to take the shortest path from A to B,"" Kraus says. The topic of combinatorics, when you think of a deck of cards and have to consider probabilities, also comes up again and again in board games, he adds.
The variety of mathematical optimization possibilities in the gaming sector is large. In 'Agricola - die Bauern und das liebe Vieh`, for example, there is a guaranteed best strategy in the single-player game, in 'Settlers of Catan` only hints indicate a probability of successful settlement, and in the locomotive game 'Zug um Zug` one can find out the shortest routes. ""Lately, with 'Thurn und Taxis`, we've really been working on a scientific level as well,"" Klamroth explains. ""This is a mail game, which also has a real-world background. That's where you want to have as many possibilities as possible to further expand a route you've started, and that then also goes into mathematical research, building a bridge again.""

How do strategies work?

The research area Mathematical Modeling and Optimization can be well explained using board games, because many games are about strategies. Dr. Michael Stiglmayr, who also supervised the KoLBi project, explains it using the example of the game 'Kahuna`. ""'Kahuna` is a two-person game that has relatively simple rules. It's about connecting islands located in the South Pacific to build your own empire and defend it against your opponent. The game board can be mathematically described by a simple network structure. In this network, the islands represent nodes, and the connections between the islands that can be built are mathematically called edges. The players lay these connections and thus conquer the islands. It is always necessary to balance between conquering new islands and securing the ones you have already won. After all, these islands can easily be lost again through the actions of the opponent."" There is no luck component, because strategies are characterized by the fact that you can make long-term plans and still have to react adaptively to your opponent.

Teaching through games?

Games could be a proven means of explaining applications in school. ""I already believe that you can inspire children with them, because games are from their lives,"" Klamroth says. ""Real applications are often far too complicated, too. If you really want to model real applications, it's not easy. With games, you don't do much wrong either, you can get into it relatively easily, like with the stochastic example."" In Stiglmayr's opinion, modeling is also made even easier in particular by the fact that games are inherently set in a simple world. Of course, not all mathematical things could be realized via games, but ""games in themselves make simplified assumptions of reality. There is only a small set of rules and even complicated games are then more complex to model, but it is still only a small set of rules,"" he explains.

Games in teaching?

Approximately 34 million German citizens of all ages play board games at least once a month, says game researcher Jens Junge. Games thus have a whole range of functions. They are interactive, bring people together, get people talking. So the question inevitably arises as to whether this playful knowledge can also be used in teaching. The researcher has already done so in seminars and is also thinking about ways to integrate it into exercise classes. ""It is suitable in seminars where students also work in groups among themselves. That is in demand, but it is also very time-consuming for the students. A lot of things have to be worked out independently and they also have to find the appropriate literature themselves."" Stiglmayr sees another advantage: ""The students also do things they haven't done before.  In our project, many of them programmed independently and let the computer find the solution. That worked out wonderfully. That wouldn't have worked with an artificial problem."" Although the supervision involved additional effort, he says, because the learners first had to be introduced to the new topic, the result was very satisfying for everyone involved. ""For us it was exciting because at the beginning we didn't know where the journey was going"" Kraus explains the approach, ""because we didn't give the students a game and say, now you have to look at the following problem, but they chose a game themselves and looked for problems in it. It was very free."" One synergy effect that should not be underestimated is a publication Kraus is preparing that deals with the route in the game 'Thurn und Taxis`. ""So this also leads to new findings on the scientific side,"" says Klamroth. There are also mathematical problems in many board games that have yet to be found in the literature.

Boardgamegeek

The team doesn't want to recommend one particular board game. ""There are so many great games on the market and it's always very personal what kind of games you like, how the layout should be, whether they are games with rather complex rules or whether you play them with children,"" Klamroth says. There is, however, an Internet site that provides wonderful information about the diversity of the gaming world. Boardgamegeek is the magic link that provides information on a variety of game genres, shows user reviews of games and has a top 100 of the most popular games available.
""In our project, we mainly looked at this combinatorial and optimization aspect,"" Klamroth concludes, but, ""there are still a lot of fields there, and you can do a lot more.""

Uwe Blass (interview 11/22/2021)

Prof. Dr. Kathrin Klamroth heads the Mathematical Optimization group in the Faculty of Mathematics and Natural Sciences at Bergische Universität.
Dr. Michael Stiglmayr is a private lecturer in the Mathematical Optimization group.
Konstantin Kraus is a research associate in the Mathematical Optimization Group."