Author: Philipp Larmann

November will be sporty with the trail of the month. The UR Sports Centre (UR Sportzentrum) mathtrail is located on the sports grounds of the University of Regensburg and was created there by several student teachers from the university.

The trail consists of a total of 10 tasks that combine the topics of sport and maths using various sports equipment and facilities. The trail is available on the web portal here and can be accessed in the app with the code 0116250.

Anna Hendlmeier, one of the students who created the trail, gives us an insight into working with MathCityMap at the University of Regensburg and the special features of this exciting mathtrail in an interview:

How did you come across the MathCityMap project?

I’m studying maths and sport to become a secondary school teacher and as part of my maths degree I chose a seminar that was about this project. As part of the seminar, we created 2 trails on the campus of the University of Regensburg, always in a group of 5 people. I found it very interesting and see it as a good opportunity to teach maths in a more practical way. The project was initiated by senior academic advisor Andreas Eberl, who is responsible for the didactics of maths at the university.

Please describe your mathtrail. What is special about the trail?

Our trail is located at the University of Regensburg’s sports centre and all the stations actually have something to do with sports facilities or equipment, which is special. Many students are more motivated because we have also created tasks that ask questions that you can ask yourself in everyday life. The trail is also located on the university campus, which can be very interesting for students as it relates to their possible future after graduating from high school.

How do you use MCM and why?

I used MCM in the course of the seminar and am very convinced of its usability and that it can be an enrichment for schools and teaching. I am therefore going to write my admission thesis on MCM and create a trail for a secondary school. With a view to my future career, I now know another way to get students excited about maths that can hardly be covered in normal lessons.

Describe your favourite task on the trail. How can it be solved?

My personal favourite task is “Game, set and match!” (Spiel, Satz und Sieg). In this task, 5 friends get together to play tennis and want to organise a tournament in which everyone plays against everyone else. They have booked the tennis courts for 90 minutes, the question now is how long a game can last so that all games are the same length. I really like this task as I think that application-based tasks are particularly important when it comes to combinatorics. This question has perhaps often been asked by someone in a different context. If you stand in front of the tennis courts, you can see that there are two tennis courts, so two games can take place at the same time. In the “everyone against everyone” mode, player A competes against players B, C, D, E: Makes four matches. Player B then has to compete against players C, D, E: Another three games. Player C against players D, E: Two more games. Player D only has to play against player E. Then everyone has played each other once and we get a total of 10 games. (4 + 3 + 2 + 1 = 10) So there are five rounds. If these game rounds are to be evenly distributed over the 90 minutes, a game round, i.e. a single game, must last 18 minutes.

Once again, there is a new MathCityMap partner school in Germany! The Dürer-Gymnasium in Nuremberg is the fourth German school to join our network and we were very pleased with the creative tasks and trails that were created for the successful application.

“At the beginning of our project seminar at Dürer-Gymnasium Nuremberg, we set ourselves the goal of becoming a MathCityMap partner school. For this purpose, a total of four trails were created in the immediate vicinity of our school. Prior to the publication of these trails, three different classes in grades 5-9 have already completed them with the help of our supervision. These students mostly enjoyed working on the tasks in class and we hope to have inspired both teachers and students at our school for MathCityMap,” wrote teachers Anne Wagner and Quang Bach Duon, who were largely in charge of their school’s application.

The different trails can be found in the web portal and in the app as follows:

“Fürther Straße” with the code 0413379

“Renovierung des Wiesengrunds” with the code 2313070

“Reutersbrunnenstraße” with the code 1714621

“Der Fürtherstr. Rundkurs” with the code 7813414

The package with the official partner school badge and the MCM measuring instruments has already been delivered and we are looking forward to receiving more applications from all over the world.

All further information on the partner school programme and the requirements for application can be found both in the article on the first MCM partner school and on the homepage of our MaSCE³ project.

The fact that creating mathtrails is sustainable is proven by the mathtrail “Berner Altstadt” (Bern Oldtown), which with 99 downloads is one of the most downloaded trails of the past weeks and thus fits perfectly into our category “Popular Mathtrails”. Already created two years ago, it has now been revived by teacher Damaris Burri, who told us in a short interview how it came about and what new activities she is planning with MathCityMap.

How did it happen that the trail was downloaded so often?

The mathtrail “Berner Altstadt” was created as part of my master thesis. In the meantime, I have been working at the school in Grindelwald for a year and since this August I have been the class teacher of the 7th class B. Last month we visited the BAM ( Career and Education Fair) in Bern, together with the parallel class and the 8th grade. As a school class, we were assigned a time slot. For us it was the afternoon. It takes 2 hours by train from Grindelwald to Bern, so we wanted to spend the whole day in Bern. This was ideal to try out my mathtrail with a class. The other two class teachers thought this was a great idea. So we visited the Mathtrail “Berner Altstadt” with 58 students.

In addition, it may well be that other schools used the BAM, which lasted a whole week, as an occasion for a trail visit.

What will you work on next with MathCityMap?

In the last two years I have not done much with MathCityMap. A little more than a year ago I revised my 1st private trail “Von Roll Areal” so that it can be published. After that there was a break until now. After the fall vacations I will start a trail for Grindelwald with the students in the gifted program. This should be for 5 – 6th grade. The children are already excited about the project.

The trail of the month August is located in a city that is always worth a visit and now can also be rediscovered through mathematical eyes. We are talking about Venice and the mathtrail “Altstadt Venedig” (Oldtown Venice), which, as the name suggests, leads through the narrow streets and over the bridges in the old town.

The trail, which can be found on the web portal here and in the app under the code 2616643, was created by Günter Kämpfert, a math teacher from the city of Hanover in Germany, and the students in his upper secondary school class.

In a short interview, Mr. Kämpfert gives us an insight into his work with MathCityMap so far and the creation of this wonderful mathtrail.

How did you come across the MathCityMap project?

A few years ago, I came across the MathCityMap project through a didactic article in the magazine “mathematik lehren”.
After initial experiments with existing trails in Hanover, the students in my upper-level course were asked to create their own tasks.
After getting to know an existing trail in Hanover, we first tested the procedure by creating small tasks in the school and on the school grounds.

Please describe your mathtrail. What is special about the trail?

For the study trip to Venice in October 2022, the students in my upper-level course were to create their own MathCityMap tasks in the old city of Venice. To do this, they worked in small groups on tasks that they then had to present and explain in a short written paper. The existing mathtrail was created from the existing tasks of the whole course. The old town of Venice offers many historical buildings, archways and bridges, which can be discovered and viewed with mathematical eyes in a completely new – and different – way. Creating mathematical tasks should encourage people to perceive their own environment as a place where mathematics can occur.

How do you use MCM and why?

The existing app can be easily used by all students on their mobile phones. Photos that are needed for an assignment can be created directly with the cell phone and inserted into the app. Working with the app is largely intuitive. For working on assignments, we have set up a group in the app so that all students can access the assignments shared in it. In addition, this also enables everyone to support each other. For working on the tasks that are created in school, we use the helpful trail code, which also allows the tasks that have not yet been published to be used.

Describe your favorite task on the trail. How can it be solved?

The “Bridge Arch” task deals with what is probably one of the most impressive experiences of any Venice tourist walking through the old city: countless bridges and canals. The Ponte del Teatro is located in the heart of Venice, crosses the Rio San Luca and is a modern bridge in the style of ancient Venetian architecture. The context of the task (a necessary modification of the bridge arch) is fictitious, but conceivable. To solve the task, the bridge arch can be modeled in a simplified way using a quadratic function. For this purpose, suitable measured values must first be collected and the function equation of the corresponding parabola must be determined. A stretching factor may only shift the high point of the parabola so far that the slope at the edges of the arc does not exceed 45 degrees. This condition results in the stretch factor we are looking for, which can now be used to calculate the position of the new high point of the parabola. With these coordinates, the permissible elevation of the bridge arch high point can then be calculated.

The first half of 2023 was also an exciting time for all students and teachers who wanted to get to know Mathtrails in Portugal. In May and June, a total of 300 students and teachers participated in events as part of our MoMaTrE project. These events were accompanied by Professor Amélia Caldeira from the University of Porto and a special guest was certainly the Councilor for Education, Fernando Paulo, who could convince himself of the enthusiasm that mathematics can trigger outside. In the context of the events a great video was created, which we of course do not want to withhold.

Further events and trainings for MathCityMap and our partner project ASYMPTOTE are already planned for fall 2023 and spring 2024 in Portugal. These will be especially promoted by SPM (Sociedade Portuguesa de Matemática) to all schools in northern Portugal.

Not one but two new partner schools in Slovakia are introduced this week! Gymnázium Štefana Moysesa in Moldava nad Bodvou and Základná škola in Bánovce nad Bebravou both successfully applied and are now partner schools five and six in Slovakia.

Gabriela Kövesiová, a math teacher from Gymnázium Štefana Moysesa told us that: “All math teachers of this school decided to organize a mathtrail for the students of this small town in the east of Slovakia. Together they prepared two trails near the school, one for the 6th grade students and one for the 7th grade students. They also tested the trails with the students of two classes, and then these students created their own math problems in small groups. In this way, the third walk was created, with the tasks coming from the learners. Mathtrails are a great opportunity to see math in a different light. Here, many students learned how to use measuring devices and understood that math is all around us. They enjoyed working in teams and solving different problems together and understood that together they can achieve anything. We thank you for the opportunity to be part of this great project, and we wish that the network of practical mathtrails will grow with more and more high-quality routes, providing unforgettable experiences to its users.”Also the Základná škola in Bánovce nad Bebravou gave us a little insight into their application process through the teacher Marián Igaz:

Also the Základná škola in Bánovce nad Bebravou gave us a little insight into their application process through the teacher Marián Igaz:

“One of the reasons we were interested in creating Mathtrails was to take a different approach to teaching math. Tasks written in books focus to a small extent on the application of acquired mathematical knowledge in practice. We welcome the use of tablets or folding rulers in the classroom and find this very interesting and motivating for children. A few initial trails have convinced us that the children enjoy doing these tasks. We have developed our own tasks to bring more quality to math lessons and we think we have fulfilled our intention.

MathCityMap impressed us because we saw great potential in combining theoretical knowledge with everyday situations. The children have the opportunity to work in groups and compete with each other, which they find very amusing. At the same time, they have the opportunity to gain practical experience by measuring lengths in geometry. In addition to measuring, they also learn about the approximate distance between different objects. Mathtrails are very attractive to students and add variety to the teaching process.”

Of course, the package with the official partner school plaque and the MCM measuring instruments is already on its way to both schools and we are very much looking forward to receiving further applications from all over the world.

All further information about the partner school program and the requirements for the application can be found in the article about the first MCM partner school as well as on the homepage of our project MaSCE³.

A mathtrail that fits perfectly into our category “Popular Mathtrails” is the trail “Prechádzka na ZŠ Duklianska pre 5 ročníkn” which was created by the experienced MathCityMap author Silvia Haringová in the Slovakian town of Bánovce nad Bebravou. Consisting of a total of six tasks, the trail gained 119 app downloads in June.

In a short interview, Silvia tells us how this came about:

What is the trail about? What is your favourite task of the trail?

The trail is designed to repeat the curriculum of the 5^th grade. In the trail pupils solve, for example how many times longer does it take to decompose a plastic bag than paper or they find out how many euros it cost to replace the posts on the fence that borders the playground.

My favourite task in this trail is “Volleyball court”.

This is an application task in which pupils help the janitor find out the sum of the lengths of all the yellow stripes of the volleyball court, because the janitor wants to repaint them.

Why do you think the trail was it downloaded that often?

The trail was downloaded often, because at the end of the school year, the teachers let all the 5^th grade students solve this trail, but it was also solved by the 9^th graders, who gradually solved the trails for the 5^th to 8^th grade. The teachers let them repeat the curriculum of the entire second grade of elementary school in this way.

Already the seventh partner school in Portugal! We are very pleased that the Garcia de Orta school association has successfully applied to become the first partner school in the northern Portuguese city of Porto.

The application was made with two of the numerous trails that have since been created and published at and around the school:

“Um passeio na Foz do Douro” which can be found in the app under the code 013089.

Also, “Escola Garcia de Orta“, a classic school trail that has the code 592281.

In their application, the school writes: “We are committed to creating and using this challenging application, in the creation of mathematical trails, to use with our students, with the aim of motivating and enhancing the study of Mathematics.” In addition, it makes an appeal to other schools, which we of course support: “Get to work, Mathematics is everywhere!”

The package with the official partner school plaque and the MCM measuring instruments is already on its way to Portugal and we are looking forward to receiving more applications from all over the world.

All further information about the partner school program and the requirements for the application can be found in the article about the first MCM partner school as well as on the homepage of our project MaSCE³.

As our Trail of the Month for June, we would like to introduce the Mathtrail Freiburg, which, as the name suggests, is located in the southern German city of Freiburg. The trail can be found in the app under the code 1412574 and in the web portal under this link.

The trail consists of a total of eight tasks on diverse mathematical topics and was created as part of the international ICSE Academy project. What ICSE is and what makes this trail special is described in a short interview with one of the authors, Silvia Haringová.

Where is your trail located? What is special about your trail?

The trail is situated in the historic centre of the city of Freiburg im Breisgau. This trail was designed as an illustration for partners of the project ICSE Academy by Silvia Haringová, Janka Medová and Martin Cápay from Constantine the Philosopher University in Nitra, Slovakia.

What is the project ISCE and why was working with MCM benefitial for the project?

The International Centre for STEM Education at the University of Education Freiburg, Baden-Würtenberg, Germany (ICSE) is a research and networking centre focusing on a future-oriented development of STEM education in Europe. It is working closely together with the ICSE Consortium and other Partners in European projects to further develop and disseminate innovative teaching approaches in STEM education. Currently, they are working on various projects concerning environmental socio-scientific issues (MOST, ENSITE), the empowerment of girls in STEM (GEM) and the development of transversal skills in class rooms (STEMKey), as well as the professional development of teachers (3C4Life).

The ICSE Academy is a project where universities, schools and school authorities from 13 European countries work together aiming at development of STEM teacher education, both initial teacher education and  professional development of in-service teachers. As shown by our work in Slovakia, MCM trails have big potential in stimulating teacher collaboration and co-design. Furthermore, MCM trails also fit with European priorities as defined in ICSE Academy project, they prepare pupils for life in digital era, teach them to collaborate and communicate their ideas, provide reasoning and also offer the opportunities to bring environmental issues in mathematics education.

Describe your favorite task on the trail. How can it be solved?

We create trails in different cities and states and the triangular shape is usually hard to find, so we were very happy to come across it in the city centre. Because of this fact, my favourite task is called Kornhaus passage.

The task is related to a triangle made of stone blocks. The user has to determine the circumference and the area of this triangle. For this, corresponding measurement data must be collected. Afterwards, one only needs the formulas for the perimeter and the area of a triangle.