Number games for the young

9 September 2013

The purpose of the 'Hands-on Mathematics' ('mathematik begreifen') exhibition is to help school students better understand the world of numbers. Through play, youngsters are encouraged to discover the Pythagorean theorem, plan routes crossing all of Germany, and experience the wonder of the brachistochrone curve, the special effects of which are here demonstrated in the form of ball rolling tracks. Dr. Ekkehard Kroll of the Institute of Mathematics at Johannes Gutenberg University Mainz is in charge of the exhibits, which need a new home.

"There really is not all that much to see," says Dr. Ekkehard Kroll as he opens the wide red door. The mathematician wants to keep expectations low.

We are now in a basement room with a high ceiling where cables are being laid and cabinets installed. In the coming semester, school children will be invited to perform experiments here. This event is being offered by the Institute of Physics as part of the junior campus mainz (JCM) program designed to help bridge the gap between school and university.

However, this is not today's focus. Kroll goes directly to a large number of boxes and crates that stand like an island in the middle of the room. Stored in these are the 'mathematik begreifen' collection, i.e., around 70 exhibits that have already been used to help young people better understand mathematics.

Mathematics in a box

"Everything is packed up," explains Kroll, as he selects one of the exhibits. It is around two meters tall. The mathematician removes some brown tape and flips back a gray cover to reveal a map of Germany.

Pins mark about 20 cities all over Germany. Way up north, there is a string that is attached to Kiel. The challenge is to connect the marked cities using the shortest possible route. The cord is only long enough for the correct solution, which has to pass the cities of Hamburg, Bremen, Mainz, Chemnitz, and many other cities before returning to Kiel. If the cord is not placed along the optimal route around the pins, then it will be too short and won't reach as far as Kiel. Kroll tries it himself. He comes up a few centimeters short. "I must have made a mistake somewhere." He bends over the eastern part of Germany. "Somewhere around here."

The map makes math into a game, which is the entire point of the exhibition concept. "We need more young people to be interested in subjects such as mathematics, physics, and chemistry," says Kroll. "This applies to both boys and girls. We can only get them interested in mathematics once they see its practical uses. Mathematics has much that is playful, tangible. More theoretical aspects should come at a later stage. But schools often ignore the practical uses of math and tend to concentrate more on the theory."

2004 in the Natural History Museum

The game with the cord is known to mathematicians as the 'Traveling Salesman Problem'. At first glance, it may seem easy enough to plot the shortest route between a couple of cities and thus save the traveling salesman time and money. However, with just four cities, there are 24 potential routes; once you have ten cities, there are something like 3.6 million different options! "When we increase the number of cities, the amount of time we need to calculate the potential routes becomes enormous." But here, it's the game that counts.

The Hands-on Mathematics exhibition was originally conceived in 2004. "The interactive 'Mathematikum' exhibition in Gießen served as the inspiration." The Education Center of Rhineland-Palatinate – now the Rhineland-Palatinate State Education Institute – started to put the collection together so that it could be used for teaching purposes. Kroll and his colleague Professor Manfred Lehn of the Institute of Mathematics at Johannes Gutenberg University Mainz (JGU) acted as scientific advisors to the project.

The exhibition was on display in the Natural History Museum in Mainz for five months. Numerous school classes came to try out the various interactive exhibits explained by teaching degree students. The exhibits were arranged in six different topic areas designed to provide the visitors with access to various features of the world of mathematics. The map of Germany was in the topic area 'Precisely or Approximately?,' which dealt with aspects such as measuring, estimating, and optimizing. Other exhibits introduced visitors to the vast universe of probability calculations, the properties and patterns of numbers, and how statements can be verified by means of mathematical proofs. There was much onoffer, problems to be solved, or simply something that was surprising.

2011 in the City of Science

Kroll then points to an exhibit that is unpacked: a combination of ball rolling tracks. The center track is straight but inclined downwards; the tracks to the left and right dip in the center because they are curved. So the question is: on which track will the ball get to the bottom first? Around the year 1700, mathematicians were asking themselves exactly the same question: which is the fastest track? It turns out that the ball travels fastest on the curved track. The name 'brachistochrone,' which comes from the Greek and roughly means 'needing the shortest time,' was devised for this type of curve. "Another unique property of this curved track is that the ball always needs the same amount of time to get to the bottom, regardless of where the ball is started, whether at the very top or further down," explains Kroll.

When the Donors' Association for the Promotion of Sciences and Humanities in Germany named Mainz the 'City of Science' for 2011, the 'Hands-on Mathematics' exhibition went on a tour of schools. The exhibition was also put on display in the Mainz Neustadtzentrum, in the Institut français, and in the Youth Center. "We had a lot of space available there and were able to put all exhibits on display." There was also an accompanying 150-page catalog that not only described the experiments and games but also provided a lot of background information. The exhibition proved to be enormously popular.

Searching for a new home

The collection then had to go into boxes. "We lend some of the exhibits to schools once in a while and we also display a few items at the Mainz Science Festival," states Kroll. However, 'mathematik begreifen' will soon have to leave its current home in the basement. "Our dean is now searching for a new location," explains Kroll. "It would be nice to find a place where we could put on a rotating exhibition of just several pieces from the collection at any one time. We could use our Didactic Seminar to help with the design."

Kroll then goes on to show us a few more items: the wagon with incongruous square tires, which nevertheless can run smoothly enough on some surfaces, the huge pine cones with right and left spirals, the puzzle about the Pythagorean theorem ... 'mathematik begreifen' is a wonderful exhibition, a real adventure. "It would be the perfect addition to the junior campus mainz program," concludes Kroll.

He then closes up the crates, puts the blanket over the map, and switches off the light. But he knows that as soon as a suitable venue has been found, the exhibits will be rebuilt and put on display again.