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Trail of the month: Lindau Island MathTrail

The Trail of the Month for April was created at a picturesque location on Lake Constance, on the island of the city of Lindau in Bavaria. Together with his P-Seminar, a special type of course in the Bavarian gymnasiale Oberstufe, teacher Jan Neuendorf created the “Lindau Island Mathtrail” (Lindauer Insel Mathtrail), which is available in […]

The Trail of the Month for April was created at a picturesque location on Lake Constance, on the island of the city of Lindau in Bavaria. Together with his P-Seminar, a special type of course in the Bavarian gymnasiale Oberstufe, teacher Jan Neuendorf created the “Lindau Island Mathtrail” (Lindauer Insel Mathtrail), which is available in the MCM app under the code 376526 and in the MathCityMap web portal here.

Along the harbor and through lindau’s old town, the trail winds its way across the entire island and integrates various sights of the city, making it very interesting not only mathematically, but also architecturally and historically. The trail contains a total of ten tasks that focus in particular on the content of the eighth and ninth grades.

An interview about the background of the trail is given by Jan Neuendorf in the following interview:

How did you come across the MathCityMap project?

I first heard about the project from colleagues who had spoken about it in various training courses. Afterwards, I found out more about the MathCityMap project on the Internet. This gave me the idea to offer a P-seminar in mathematics, which had the goal to develop a Math-Trail on the island of Lindau and to make it accessible to interested people via the MathCityMap-App. The P-Seminar is a special feature of the gymnasiale Oberstufe in Bavaria. It supports students in their study and career orientation and focuses on the planning and implementation of a subject-related project.

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

The trail is located on the island of Lindau in Lake Constance. With its historic old town, narrow streets, medieval buildings and picturesque harbor with lighthouse, lion and mountain view, the island provides a unique backdrop for the elaborate math trail. Therefore, it was also an exciting challenge to discover and develop suitable mathematical tasks on objects on the island. Thus, the trail combines sightseeing with math activities, which is an exciting combination.

How do you use MCM and why?

So far, MCM has served as a guiding idea for the P-Seminar in mathematics. The goal of the participating students was to plan and implement a math trail on the island of Lindau. In the future, the trail will be used in grades 9 and 10 as a subject outside the classroom or as part of our project week. It is certainly also desirable that other schools in the Lindau area will use the trail for classroom excursions and class action days.

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

My favorite task of the trail is the task to the Mangturm (Mangtower) at the Lindau harbor. On the one hand, the task is to be solved directly at the harbor in the heart of Lindau, which gives the task an exposed place within the trail. On the other hand, it is a suitable task from the field of geometry, in which mathematics is applied in practice and in which geometry as the science of measurement can be understood in its most original form.
The task is solved with the ray theorem. The fascinating thing is that this theorem can be used to determine lengths that are difficult or impossible to measure.
If you form the 2m long meter stick into an isosceles, right-angled triangle and place it on the harbor railing in such a way that you can aim at the top of the tower via the tip of the leg of the meter stick that is far from your eye, you are not far from the solution. After you have measured the horizontal distance of your location to the Mangturm with the help of the railing elements, you add the height of the railing to this quantity and thus obtain the height of the tower.

MaSCE Meeting in Lyon

After two years of online meetings, the team of the current MathCityMap EU project MaSCE3 (“Math Trails in School, Curriculum and Educational Environments of Europe”) could finally meet again in person and plan the activities for the final year of the project. The meeting was hosted by the project partners of the University Claude Bernard […]

After two years of online meetings, the team of the current MathCityMap EU project MaSCE³ (“Math Trails in School, Curriculum and Educational Environments of Europe”) could finally meet again in person and plan the activities for the final year of the project. The meeting was hosted by the project partners of the University Claude Bernard in Lyon, Christian Mercat and Patrick Berger, who ensured a successful exchange and a structured organization.

Within the MaSCE³ project, fantastic innovations for MathCityMap have already been realized. For example, new answer formats such as the vector, the cloze or the information station have been introduced. Furthermore, the possibility of adding so-called subtasks to existing tasks has been implemented. Furthermore, the project developed the already widely used and effective “Digital Classroom” with its manifold application possibilities. Of course, you can read about the many other project results on the project’s website.

We are very excited to see what other developments MaSCE³ will bring to MathCityMap this year, which we will of course inform you about on our website!

A mathematical culture trail

In this article we present a very interesting development in Ober-Ramstadt, a city in the south of the German state of Hesse. There, Daniel Reckhard, a student teacher at the Georg-Christoph-Lichtenberg-Schule, has developed a special kind of math trail with MathCityMap. The aim of the so-called mathematical culture trail is to combine the culture of […]

In this article we present a very interesting development in Ober-Ramstadt, a city in the south of the German state of Hesse. There, Daniel Reckhard, a student teacher at the Georg-Christoph-Lichtenberg-Schule, has developed a special kind of math trail with MathCityMap. The aim of the so-called mathematical culture trail is to combine the culture of the city with interesting and creative mathematical discovery opportunities and thus to gain a new perspective on mathematics. Further information about the trail can be found on the website of the city of Ober-Ramstadt and an interview about the background of the mathematical culture trail with the creator can be read below.

How did the idea for combining the topics of mathematics and culture come to life?

Mathematics is one thing above all: an art. Very eloquently and with the necessary leisure Paul Lockhart describes this in “A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form”. An abridged version is freely available as a PDF on the Internet and is absolutely worth reading.

Math as art is meant to be very inclusive, that math is so amazingly suited to describing our universe is one aspect of it. My favorite subjects are the STEM sciences and within mathematics my favorite is statistics. In statistics, too, you are not at home until you have filled its concepts with experience and thus brought them to life, whereupon you can create. So also in statistics aesthetics in the literal sense (aisthesis as “sensual cognition”) is central.

The typical school mathematics could hardly be more opposite. I am forced to cram through a crowded curriculum, necessarily superficial and hasty. I have to drill students on exam math and fill them with mindless arithmetic, dry formulas, and meaningless concepts. So my pubescent students came back to class from distance learning and already the “mathematics competition” of the state of Hesse loomed on the horizon, a de facto comparative test in which an immense range of topics is asked. And this (thoroughly international) rigidity of the curriculum is commonly thought to be the true face of mathematics. No wonder, there is no time for real understanding.

The central reason for the connection, then, is as a resolute antithesis of instruction characterized by control. A second reason is that mathematics is in any case inseparable from our culture as a part of it. For example, I had attended the lecture “Mathematics in Context” by Prof. Burkhard Kümmerer and had enjoyed studying the genesis of mathematics.

Why did you choose MathCityMap to implement the idea?

The idea came from the supervisor of my educational thesis, Steffen Burk. I had the idea of laying a geocache, and he thought there was something better because it was tailored to schools. My second subject is computer science, I like to work my way into new techniques. And here I also found it exciting to see how I can implement my ideas in a very closed learning environment. For example, how do I represent imagination, how do I enable mathematics to unfold?

Because the core principle behind MathCityMap is the same as with the Skinner Box: learners interact with a machine, but outside, i.e. on real objects. And of course, Prof. Ludwig does not intend MathCityMap to replace teaching, but recommends it for certain scenarios, especially as a deepening of the application of already learned concepts. So a real encounter of the learners with something fundamentally new was a challenge. I hope that I succeeded well. In any case, my students had a lot of fun, as did my colleagues who had previously tested the trail.

What I and the students really liked was going out into the world. It naturally led to cooperation with the city administration and to opening up the trail to the general public. That’s another aspect of cultural school: connecting the school community with the communities that surround it. That also enlivens the math. What I also really liked is that the user-friendliness allows students to create tasks. This allows them to be immediately creative.

For example, I could also imagine implementing open-ended task formats: for example, “How should (mathematical object) be designed?” The students use the funnel principle to make assumptions or choose suitable real objects, and their closed MathCityMap task thus represents a solution to the open task.

What is there to discover and learn on the trail?

The goal of this trail is to show the cultural and aesthetic side of mathematics, its diversity, hidden patterns, symmetries, but also how mathematical cultural techniques shape our world. Compared to other trails, the tasks are deliberately kept quite easy. After testing, I have adapted the tasks and aids so that many students can solve the tasks well. As with the Computer Science Beaver, the trail is intended to enable students to gain a positive experience with mathematics without a great deal of prior knowledge, which creates a desire for more.

At my favorite station, students discover what numbers (for example, their favorite number) really look like because, for example, “4” or “four” are just names for a number, they are not the number itself. I implemented this via a geogebra activity. At another station, they learn why scaffolds always contain triangles, as they recreate them and thus “grasp” math in the literal sense. The material for this is on file at the school. Those outside the school can borrow the material from the town hall with a deposit of 10€.

At yet another station, they stretch a twelve-knot cord and have to understand how many stone slabs are spanned, thus discovering the decomposition principle and thus the decisive basic idea for the area of any polygon. One station (appropriately enough, the library) activates the imagination and asks what would be created if one were to think of the mirrored object in addition to the given object (a decorative window).

And quite incidentally, something about cultural history is told, for example, why a stream leads to a mill wheel. I found it impressive to calculate with which enormous force this wheel is turned. Finally, the students are taken to hidden cultural sites, for example, our small town has its own museum, which many people are not aware of.

Trail of the Month: Counting, measuring, calculating and estimating between Sechseläutenplatz and the city border

The term mathematical walk could hardly be more apt than for the trail of the month March. This one comes from the largest city in Switzerland, Zurich. Here, Roland Wiss, a member of the school management and executive board of LIPSCHULE Zurich, has created the trail “Counting, measuring, calculating and estimating between Sechseläutenplatz and the […]

The term mathematical walk could hardly be more apt than for the trail of the month March. This one comes from the largest city in Switzerland, Zurich. Here, Roland Wiss, a member of the school management and executive board of LIPSCHULE Zurich, has created the trail “Counting, measuring, calculating and estimating between Sechseläutenplatz and the city border” (Zählen, Messen, Berechnen und Schätzen zwischen Sechseläutenplatz und der Stadtgrenze), which can be accessed in the MCM app under the code 257781 and is available on the MathCityMap web portal here.

The trail stretches over a total length of 2.8 km and, as the title suggests, leads from the centrally located Sechsläuteplatz along a walking path alongside Lake Zurich to the city border. A total of twelve interesting mathematical tasks with a wide variety of content can be found along the route, which not only offers a chance to marvel at the beautiful nature surrounding the city of Zurich. What most of them have in common, however, is that there seems to be too little data to solve them.

Roland Wiss explains among other things the concept of the trail in more detail in the following interview:

How did you come across the MathCityMap project?

I am always interested in different ways to show my students the beauty and excitement of mathematics. This includes mathematical problems from everyday life and especially outside the classroom. For this reason, I regularly search the internet for exciting math projects. In doing so, I came across the MathCityMap project, which immediately appealed to me.

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

I walk every morning in Zurich from Stadelhofen station to Lipschule and since I am a big fan of Fermi questions, I had the idea to design this trail along my way to work. I call a Fermi question an estimation about a problem, where the students seem to have no or insufficiently accurate data at a first superficial glance. However, when their explorer and detective eyes are awakened, students notice that they can decompose the question into several sub-problems for which they can find exact or approximate solutions. Combining and completing the partial results, they arrive at an overall result that is very close to the actual value. My students like to deal with Fermi questions and they learn a lot. So it was obvious for me to create a trail on the topic “Counting, Measuring, Calculating and Estimating”. Since the Lipschule is a comprehensive school with different age groups, I wanted to create a trail that contains tasks for many age groups. In addition, there is the wonderful location at the lake and the possibility to extend the trail to a day trip with (at least in summer) a swim in Lake Zurich.

How do you use MCM and why?

We regularly have a mathematics project week. One workshop of this week is called “Mathematics outside” and has the following content: “Mathematics is everywhere. We explore the surroundings from Sechseläutenplatz to Lipschule with a mathematical eye”. The MathCityMap app is perfect for this. I especially like the fact that the students are outside thanks to the app and solve many tasks in teamwork. They are also actively involved and have to solve the tasks and problems using appropriate tools. The students learn not only mathematics, but also teamwork and the use of clever solution and organization strategies.

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

My favorite task is called “Area of a hexagon” because it can be solved in different ways and by different ages. On the one hand, the older students can use the area formula for the hexagon 3*√3*s*s/2 and count the paving stones along the sides to determine the number. On the other side, there are hexagons of equal area in the square, which are filled with paving stones. Younger students, who do not yet know the area formula, can determine the number in a clever way by counting and estimating. Since the hexagons filled with pavers all have similar patterns, the students can also think about the method the paving contractor used to lay out the pavers. It is therefore a place and a task that can stimulate a variety of thinking processes.

The new trail world record goes to…

After almost two years, the time has finally come. A new Mathtrail takes the title of the most downloaded trail in the world! The previous record holder, the trail MCM@home (Ffm a. M.) by Matthias Ludwig with 477 app downloads, has now been replaced by a great trail from Jakarta in Indonesia. The trail Banteng […]

After almost two years, the time has finally come. A new Mathtrail takes the title of the most downloaded trail in the world! The previous record holder, the trail MCM@home (Ffm a. M.) by Matthias Ludwig with 477 app downloads, has now been replaced by a great trail from Jakarta in Indonesia.

The trail Banteng Berhitung was created by Yunas Chandra and has already been downloaded 569 times in the MCM app since its release on 09.12.2021.

Congratulations to the new record holder and we are curious when we will set a new record.

A short interview with the creator of the Mathtrail follows now in this article. Have fun reading!

How did you come across the MathCityMap project?

To be honest, I never guessed that my Mathtrail will be downloaded that many times. It’s a pleasure for me. This Mathtrail probably is the output from “Bimtek Penguatan Keterampilan Numerasi Guru Dikdas melalui Math City Map” which was held by Ministry of Education. It was a teacher training at which all the participants had to go outdoors to create a task and trail and “Banteng Berhitung” is my trail which consist of tasks of myself and other participants. So, I thought this reward must be declared for all the participants in that “Bimtek” especially who made the tasks that I used in my trail.

Please describe your Mathtrail.

“Lapangan Banteng“ is a historic square located in a historic area formerly known as Weltevreden. Formerly students could learn history and enjoy the beauty of “Lapangan Banteng” and now they can also learn about numeracy. Students can apply their math knowledge in real life so they can maintain and improve their numeracy skill.

How do you use MCM and why?

I really love to use this app, it can help us as teachers to make more interesting learning settings included in an outdoor activity. So far, students rarely use their math skills in their life problems. With MathCityMap they can learn how to apply them and they deepen their knowledge about math.

Describe your favorite task of the trail. How can it be solved and what can students learn from it?

“The City of Collaboration” is my favourite task. It needs numeracy skill to solve this, because if students don’t have it they will never get the answer. Its very simple to answer but people can easily type in the wrong answer too. They have to determine the area to be installed ceramics in and then divided it by the size of ceramics that is supposed to be use.

100 Trails in Slovakia

Great news from Slovakia! 100 math trails with MathCityMap have now been published there. The trails were created mainly by teachers of mathematics didactics, PhD students and student teachers at Constantine the Philosopher University in Nitra. In addition, MathCityMap has a strong community within primary teachers in Slovakia. The 100th public trail in Slovakia was […]

Great news from Slovakia! 100 math trails with MathCityMap have now been published there. The trails were created mainly by teachers of mathematics didactics, PhD students and student teachers at Constantine the Philosopher University in Nitra. In addition, MathCityMap has a strong community within primary teachers in Slovakia.

The 100th public trail in Slovakia was created in the small town of Želiezovce by Réka Veszprémiová. The trail entitled “Matematická prechádzka v Schubert parku v Želiezovciach” (Mathematical Trail in Schubert Park in Želiezovciach) consists of five tasks, is created for the 6th grade and covers a wide variety of topics, such as calculating the area of rectangles or finding the greatest common divisor.

Keep up the good work Slovakia!

Portugals second MCM partner school

Only a few weeks after the presentation of our first MCM partner school in Portugal, we can already welcome another school in this great program! The “Escola EB1 do Cálvario” has successfully passed the application process and is now also an official MCM partner school. The process was initiated at the Portuguese elementary school by […]

Only a few weeks after the presentation of our first MCM partner school in Portugal, we can already welcome another school in this great program! The “Escola EB1 do Cálvario” has successfully passed the application process and is now also an official MCM partner school.

The process was initiated at the Portuguese elementary school by three student teachers who came into contact with MathCityMap during their studies.

Again, the package with the measuring instruments and the official partner school badge is on its way to the school and we are looking forward to more applications from Portugal, but of course also from all other MCM countries.

You can read a brief report from one of the student teachers about the application process further down in this article, and all other information about the partner school program can be found in the article about the first MCM partner school.

We are a group of pre-service teachers doing our internship at Escola EB1 do Calvário, in Viana do Castelo, Portugal. We developed an interest for MCM during our master’s course and decided to implement three trails with our students in the schoolyard. For the 1st graders we created two thematic trails related to the contents they were addressing in their lessons at the time, the number space to 10 and geometry, as for the 3rd graders we created a more diversified trail involving different mathematical topics.

Testing out the trails was a successful activity that was done with both groups of students during math class. They were able to apply what they were learning, were motivated to get out of the classroom and collaborate using technology (tablets). The students had no difficulty using the app and recognizing the location of the objects, with the exception of the first graders who needed the help of the supervising teacher to read the tasks. For us as pre-service teachers, it was a very rewarding experience to perceive the positive impact of the MCM activity with the students and to have the opportunity to design tasks in a real-world context, making them more meaningful to the students than in the traditional classroom setting where they previously spent too much time. We will certainly continue to use MCM!

Trail of the month: Math Trail di Fort Rotterdam Macassar

The Mathtrail of the month February comes from Indonesia, more precisely from the city of Makassar on the island of Sulawesi. Here, teacher Jamaluddin Tahuddin created the trail “Math Trail di Fort Rotterdam Makassar” a special math trail that leads through the historic fort of the city of Makassar with a total of six tasks. […]

The Mathtrail of the month February comes from Indonesia, more precisely from the city of Makassar on the island of Sulawesi. Here, teacher Jamaluddin Tahuddin created the trail “Math Trail di Fort Rotterdam Makassar” a special math trail that leads through the historic fort of the city of Makassar with a total of six tasks. The trail can be accessed on the web portal and in the app under the code 157539.

You can find a short interview with the creator of the trail below. Have fun reading it!

How did you come across the MathCityMap project?

Every year, students go on a study tour in Fort Rotterdam. They work on a project assignment to make a report given by the Indonesian teacher. After attending training on how to strengthen numeracy skills through the MathCityMap application, I was interested in making a Math Trail in Fort Rotterdam. In addition to doing historical tours, students will also be able to do numeracy activities at Fort Rotterdam. Thus, this activity can involve many subjects, including Mathematics, Indonesian, English, History, and Science.

Please describe your Mathtrail.

Fort Rotterdam is one of the historical places in the city of Makassar. Everyone including students in Makassar know this place. So far, they have only seen Fort Rotterdam from a historical perspective. But now they will also be able to look at Fort Rotterdam from a numeracy point of view. Inside the fort, I’ve selected several objects that can serve as numeric contexts. So that people who visit Fort Rotterdam will not only do historical tours, but can also do numeracy tours.

How do you use MCM and why?

Students can use the MathCityMap application for activities to practice numeracy skills outside the classroom.
Students are organized into several groups and each group consists of 3-4 students. Each group only needs 1 smartphone so that all students can be involved even though not all have smartphones. The slow speed internet connection is also not a problem because every math trail that students will complete can be downloaded first so that it can be used offline. Teachers can also know how students solve each of the numeracy problems through a worksheet which can be downloaded through the MCM application.

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

My favorite trail task is “Gerbang Gereja“ on the Math Trail in Fort Rotterdam City of Makassar. In addition to its unique shape, at the Church Gate students can also learn from the context of numeracy. In this task, students will calculate the maximum height of a box car that will carry cultural heritage objects into the building, the car has a width of 167 cm. To solve this problem, students must know the relationship between the radius of the circle, the slope, and the distance of the circle from the center of the circle.
Where the width of the box car is the minimum segment length and the distance from the center of the circle is the maximum height of the box car. So, to solve it, students must measure the width of the gate which is the diameter of the semicircular gate first.

Introducing the next MCM patner school

We are more than happy to welcome the next MCM partner school! The “Jaime Cortesão Secondary School” has successfully passed the application process and is now the first MCM partner school in Portugal. The teachers of the school were inspired by the MCM MOOC last year and the trails were created by master students working […]

We are more than happy to welcome the next MCM partner school! The “Jaime Cortesão Secondary School” has successfully passed the application process and is now the first MCM partner school in Portugal.

The teachers of the school were inspired by the MCM MOOC last year and the trails were created by master students working at the school and tested with the students.

The package with the measurement tools and the official partner school badge is now on its way to the school and we are very much looking forward to receiving more applications from schools.

A short report from the school on the application process can be read further down in this article and all further information on the partner school program can be found in the article on the first MCM partner school.

Together with the supervising teacher, the interns developed the activity “MathTrails” for the students from Escola Secundária Jaime Cortesão. This activity was created using the platform MathCityMap, explored in one of the interns’ subjects from the Math Teaching Masters Degree Course from the University of Coimbra.

The created trails are called “Matematicando por Coimbra” (Code: 497843) and “ π–sando as ruas de Coimbra” (Code: 697842). Our students transferred the app into their phones which gave them access to the trails and to the digital classroom that was used by the interns to control them and check their scores. The students were given a participation diploma after completing all their tasks. The winners of the activity were also rewarded with medals created by the interns.

In general, the students liked the activity, stating that the tasks had the right amount of complexity and that said tasks allowed them to recall some of the concepts learned in previous years. They also agreed that the activity was useful in order to cement their new acquired math knowledge.

The interns’ goal was to show the students that math is everywhere through a different and dynamic activity where they could explore the world that surrounds them. According to the students that goal was successfully achieved.

MathCityMap: News from Indonesia

In 2014 MathCityMap was introduced to Indonesia with a pilot study conducted by Goethe University Frankfurt in collaboration with Universitas Negeri Semarang involving several schools in the city of Semarang. Just three years afterwards in 2017, the Mobile Math Trails Research Group was established to become a center for the study of math trails […]

In 2014 MathCityMap was introduced to Indonesia with a pilot study conducted by Goethe University Frankfurt in collaboration with Universitas Negeri Semarang involving several schools in the city of Semarang. Just three years afterwards in 2017, the Mobile Math Trails Research Group was established to become a center for the study of math trails with digital technology in Indonesia. Since then MOOCs are offered and teacher trainings are organized in collaboration with teacher organizations and the Education Offices in several cities. At some universities, MathCityMap is part of the courses taught to prospective mathematics teachers and is a topic investigated in several theses.

At the end of 2021, the Ministry of Education, Culture, Research and Technology invited the Mobile Math Trail Research Group to collaborate in organizing a teacher training on strengthening literacy and numeracy skills using MathCityMap. The speaker in the training was Professor Matthias Ludwig as the founder of MathCityMap who explained the MathCityMap concept, and the technical details of the webportal and the app in collaboration with Dr Adi Nur Cahyono, the coordinator of MathCityMap in Indonesia, and his team.

Participants in this training were 240 teachers from 34 provinces in Indonesia. Primary and secondary school teachers who were selected from 6500 applicants to the teacher training. The training was conducted in four stages and each stage was carried out in 4 days in Jakarta and Bandung. The training was realized with offline sessions which consisted of an indoor theory lesson and an outdoor practice in the famous city parks in the two cities, namely the Bogor Botanical Gardens, Jakarta Old Town, Banteng Field, Kencana Park.

As a follow-up, the trainees returned to their respective areas and held multiplier trainings for their colleagues. They also created a mathtrail in their area with the local cultural context and implement it with students and the public. Communication with the teachers is maintained to discuss and share implementation experiences through the MathCityMap Indonesia Community forum.

MathCityMap trails and users in Indonesia have increased significantly and continue to grow as a positive impact of dissemination through MOOCs, lectures at teacher education institutes, teacher trainings and communities. In 2022, the Directorate General of Higher Education of the Ministry of Education, Culture, Research and Technology of the Republic of Indonesia will provide training on numeracy and literacy skills using MathCityMap for 22,000 university students who will undertake internships in schools throughout Indonesia.