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MATHEMATICS TEACHING RESEARCH JOURNAL 58 SPRING 2020 Vol 12, no 1 Vol 12, no 1

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Students make interactive exhibition “Experimental Mathematics” for

the Museum of Entertaining Sciences Maria Pavlova, Maria Shabanova

Northern (Arctic) Federal University named after M. V. Lomonosov,

Moscow Center for Educational Quality

Abstract: Museums of entertaining sciences is one of the most interesting forms of popularization

of scientific knowledge and scientific activity. The first museum of entertaining sciences was

opened in 1906 in Germany. Today, there are numerous similar museums in many countries. The

main advantage of such museums is the interactive nature of exhibits: everyone may touch the

exhibits and experiment with them. We suggest going further by giving students an opportunity to

create an exhibition themselves. The purpose of this article is to present our experience of

realization of this idea. Students of universities and secondary schools have made a holistic

interactive exposition “Experimental Mathematics” for a museum of entertaining sciences in

Archangelsk. The students wanted to present mathematics in a new, unusual perspective of

“experimental science”, to tell them about the role of experiments in mathematical discoveries,

and to make them feel themselves like real researchers and experimental mathematicians.

Keywords: museum of entertaining sciences, experimental mathematics, mathematics education

MUSEUMS OF ENTERTEINING SCIENCES OF POPULARIZATION OF SCIENTIFIC

KNOWLEDGE AND SCIENTIFIC ACTIVITY

Scientific education and popularization of scientific knowledge were at all times important areas

of work for scientists and teachers. They can help create an atmosphere of positive attitude to

scientific achievements and research work and increase both learning motivation of the younger

generation and their interest in scientific activities. The significance of this direction for the

development of mathematical education in our country is emphasized in the Concept of the

development of mathematical education (2013). It says that in addition to traditional forms of

popularization of mathematical knowledge in Russia, such as mathematical competitions and

clubs, we should develop new forms including interactive museums of mathematics.

However, it is not correct to call this form a new one. The very idea of museums as contributors

to creation of new values in addition to their traditional functions (e.g. preserving treasures of the

past) was expressed at the beginning of the 20th century at a conference in Mannheim called

“Museums as educational institutions” (1903). In 1905, a professional musicological journal Neue

Museumskunde (New Museology) appeared, which published articles on interaction of museums

and public education.




The development of a new scientific direction began, which Freudenthal (1931) called “Museum

pedagogy.” Museum pedagogy is a scientific discipline at the junction of museology, pedagogy

and educational psychology. It focuses on not only the use of museum objects, expositions, or

museum spaces for educational purposes, but also the creation of a special kind of museums, the

so-called children's museums. The children's museums apply a specific principle of collecting

museum objects: “Objects may not necessarily have intrinsic value to science, history, art, or

culture, and can include constructed activity pieces and exhibit components» (Standards for

Professional Practice in Children’s Museums, 2012). All expositions of such museums are

interactive.

The first interactive museum of entertaining sciences in Russia was opened on October 15, 1935

in Leningrad thanks to the efforts of a famous science popularizer, mathematician, physicist,

journalist, and teacher Yakov Perelman. This museum was generally known as a House of

Entertaining Sciences, but one of the main parts of its exposition was devoted to mathematics.

Unfortunately, the Second World War had hampered its work, and the museum was closed.

For a long time, creation of such museums was a work of enthusiastic scientists. Thus, in 1969 in

San Francisco (USA), a group of enthusiasts headed by famous physicist Frank Oppenheimer

created the Exploratorium. Up to now, it exhibits devices made by their own hands. Widespread

occurrence of such museums and exhibitions began in the late 20th – early 21st century. This was

facilitated by development of museum pedagogy as an independent branch of science and by

further understanding the role of museums, their mission, and forms of museum communication.

Let's turn to a review of museum expositions devoted to mathematics.

OVERVIEW OF EXHIBITS INTENTED FOR POPULARIZATION OF

MATHEMATICS IN MUSEUMS OF ENTERTAINING SCIENCES IN RUSSIA

The ideas of expositions of the House of Entertaining Sciences created by Yakov Perelman formed

the basis of modern museums of entertaining sciences, which began to appear in many cities of

Russia at the beginning of the 21st century. These include Experimentarium Science Museum in

Moscow, Einstein Museum of Entertaining Sciences in Yaroslavl, ExperimentUm in Abakan,

republic of Khakassia, LabyrinthUm in St. Petersburg, Newton Park in Krasnoyarsk, Museum of

Entertaining Sciences NARFU named after M.V. Lomonosov in Arkhangelsk, Museum of Science

and Technology SB RAS in Novosibirsk, Park of Scientific Entertainment in Perm.

In this regard, we would like to begin the review with a story about the heritage of Yakov Perelman.

He has published many popular science books on mathematics for schoolchildren, where he

outlined the idea of creating interactive expositions. He tried to pick up the exhibits based on the

following principles of popularization:

– the exhibits should cause surprise and interest, attract attention of visitors with unusualness, and

shouldn’t leave them indifferent;




– each exhibit should be not only entertaining, but also instructive. It should help discover the

information that was previously unknown;

– the exhibits should be accessible, i.e. visitors can touch them, view from all sides, explore how

they work, clearly see their design and intelligently use them;

– each exhibit should be somehow related to the content of school programs in mathematics.

Bogomolov (2002), in his article, comprehensively described the exposition of a mathematics

room called a Numeral Chamber in honor of Magnitsky. The door of the room was designed as

the cover of “Arithmetic, or Numeral Science”, a famous book written by Magnitsky (1703). The

ceiling gave a visual representation of a “million” as a million yellow luminous circles (“stars”)

depicted on a dark blue background. To capture the imagination of visitors entering the math room,

the true number of stars visible with a naked eye on the hemisphere of the sky was enclosed in a

white circle. Many exhibits were presented in the form of colorful posters and panel pictures such

as "Panel of Indian problems" and “Pi Number”. This information was supplemented by large-

scale three-dimensional models and interactive equipment. For example, the “Pi” panel came with

the equipment for an experiment carried out by Georges Buffon, a famous French naturalist of the

17th century. This experiment can be used to collect data for calculating the approximate value of

the Pi number. Students tossed short needles on squared sheets of cardboard lying on the floor,

performing this procedure dozens of times. To calculate the Pi number, they counted the number

of intersections of the needles with the lines on the cardboard sheets and then divided this value

by the number of tosses. Math lovers could also see the development of proportion between the

number of intersections and the length of the needle.

Mathematical tricks with guessing numbers were presented in an uncommon manner: a reviving bird

called “The Wise Owl”, “The Tale of Scheherazade about Magic Number 1001” — a large book made

of plywood sheets, and many other interesting things. Dozens of math games, puzzles and devices

were collected in the math room.

Overview of mathematical expositions of Russian and foreign museums of entertaining science

shows that there are not so many exhibits that represent scientific facts from the school course of

mathematics (e.g. Pythagoras theorem, the theorem about equal-area polygons, the properties of

regular polygons) in an entertaining interactive form. Most of these expositions contain interactive

models of mathematical objects not studied at school, such as Reuleaux triangle, Pascal’s triangle,

Fractals, Möbius loop. In order to engage the audience in vigorous activity with the models of

mathematical objects, museums usually offer various mathematical games and puzzles. To create

interactive exhibits, the foreign museums of entertaining sciences widely use capabilities of

computer technology and special software for visualization of mathematical models.

Today, museums of entertaining sciences offer their visitors not only thematic excursions, but also

many interesting educational events: educational quests, show programs, design workshops, and




museum lessons. Implementation of museum lessons requires adding the exhibits that support the

school curriculum. To solve this problem, we do not need a “new Perelman” (Romanovsky, 2002).

Going further with this idea, we believe that schoolchildren and students can do this work

themselves.

Creating an interactive exhibition “Mathematical Experiment” by schoolchildren and students

The idea of creating a holistic exposition “Experiments in Mathematics” arose as a part of

implementation of a similarly named educational project supported by the Dynasty Foundation in

2014. Initially, it was a small mobile exhibition “The History of Experiments in Mathematics”

prepared by students of NArFU named after M.V. Lomonosov. It was several times presented at

university events. Today, students from the "Experimental Mathematics" club have also joined in

the work on creating the exposition. A joint team of students and the club leader is engaged in

creating a large-scale interactive exposition that will include several zones: “Experimental

Mathematics,” “Mathematical Game Library,” “Mathematics and Art,” and “Japanese

Mathematical Courtyard”.

In the Experimental Mathematics zone, the students planned to collect the exhibits for visitors to

learn about the history of scientific discoveries where computer experiment is now replacing

mental experimentation and experiments with material models: the Four Color Theorem, the

Plateau problem of finding minimal surfaces bounded by a framework, etc.

Experimental mathematics is an area where computers are widely used. Therefore, the exhibits in

this zone present the most important results obtained by computer experiments. Here, we will show

the peculiar features of this zone using the example of an exhibit for the Four Color Theorem.

The Four Color Theorem is a well-known problem solved with the help of computer. This theorem

states that any geographical map can be properly colored using only four different colors. Proper

coloring means that no two adjacent countries have the same color.

Students collected historical information about how this problem was set and solved. They

presented this data as a poster (Fig. 1).

http://itprojects.narfu.ru/kruzhok-exp-mat




Figure 1. The poster of the Four Color Theorem

They also created a computer visualization of the theorem for the visitors. A game called

“Artist and Mathematician” is an interesting addition to the exhibit. The rules of the game are as

follows.

This game is for two players. One player represents an Artist, and his adversary represents a

Mathematician. The Artist has four paints of different colors. The Mathematician has only a pen.

The goal of Mathematician is to create a graph model of a map so that the Artist cannot color it

properly with the four available colors. The Artist’s goal is to distribute the four colors in a way

that enables proper coloring of the map. The graph model of the map consists of circles (vertices

of the graph) and lines (edges of the graph). Vertices of the graph indicate countries. The edges of

the graph indicate the borders between them. In one move, the Mathematician can add only one

country and draw lines to indicate the borders with previously mapped countries. The Artist can

fill only one circle in one move. The game continues until the Artist has enough colors to fill the

circles.

Anyone who knows the Four Color Theorem may think that this game is endless. We show that

this is not so. There is a winning strategy available for the Mathematician (See Table 1).

https://www.geogebra.org/m/mnrww4us




No. Mathematician Artist

1

2

3

4

5

6

The game is over

Table 1. Steps of winning strategy in “Artist and Mathematician”

This zone of the exhibition contains posters depicting the history of setting and solving the

problems together with some equipment for experimenting with material models and dynamic

models for computer experiments (Table 2).




Table 2. Exhibits of Experimental Mathematics zone

Mathematical Game Library is an environment where visitors can test their intellectual abilities by

tackling mathematical tasks (Fig. 2). The students have collected various tasks that can be solved

with the aid of experiments. They presented the tasks on colorful posters, such as on Figure 3, and

supplemented them with experimental equipment: dominoes; sticks; color cardboard sheets;

scissors, etc. Visitors may solve these tasks and share their results with others.

Dynamic model of helicoid

Dynamic models for demonstrating the

properties of arbelos: Inscribed circles; Pappus

chain; Area of arbelos; Rectangle

https://www.geogebra.org/m/dcynsqju

https://www.geogebra.org/m/pwgrbdgp

https://www.geogebra.org/m/dwtzysue

https://www.geogebra.org/m/dwtzysue

https://www.geogebra.org/m/rpz4x7yx

https://www.geogebra.org/m/qeamxftw




Figure 2. Mathematical Game Library Figure 3. Рroblems with matches

In very difficult cases, they can find ready-made answers presented as animations. Here are some

examples of such tasks:

You have a paper sheet of A4 size (210 mm × 297 mm) and a pair of scissors. How many cuts you

need to do in order to obtain a rhombus with the largest possible area? Find minimum number of

cuts required to achieve this. Find the area of the rhombus (cm2). The best result for this task is 1

cut (Fig. 4) and the area of the rhombus is approximately 467.76 cm2.

Figure 4. Folding paper to solve the task 1

2. There are 36 square plates in a box. Each plate has a red and a blue side. If a plate lies with its

red side up, it is considered a “living” cell. If it lies with a blue side up, the cell is “dead”

(Fig. 5).

Figure 5. Initial view Figure 6. Final view

https://www.geogebra.org/m/p8yftznz




If a dead cell has more than two common sides with living cells, it becomes alive. If a living cell

has more than two common sides with dead ones, it dies. In other cases, the cells retain their initial

state.

How many transitions from one state to another are required to obtain the view shown in Fig. 6?

If you want to check your answer, go to computer animation option.

Being inspired by Mathematical Etudes web site, the students created and exhibited their own

interactive puzzles related to Pythagoras' theorem (Fig. 7)

3. Please move and rotate triangles in the dynamic model to obtain a large square.

Figure 7. Proving Pythagoras’ theorem with a puzzle

“Mathematics and Art” is a zone where visitors can see how mathematics is applied to art. Here we

will present this zone by two exhibits. The students had found it interesting to show mathematical

paradoxes of M. Escher’s drawings. They collected nine of his drawings on one poster and prepared

a story to tell the visitors about mathematical principles underlying these famous artworks (Fig. 8).

https://www.geogebra.org/m/majme3rq

https://www.etudes.ru/ru/etudes/pythagorean-theorem/

https://www.etudes.ru/ru/etudes/pythagorean-theorem/




Figure 8. Math. paradoxes of Escher’s drawings Figure 9. A visitor playing pi number music

The visitors will learn about relationship between symmetry and rotations through the example of

“Drawing Hands,” a well-known lithograph by Escher. The picture “Hand with a Reflecting

Sphere” is used to tell about inversion. At the end, the speaker suggests the visitors to draw their

own pictures using geometric transformations. The next exhibit in this zone is for music lovers.

They will find everything for playing or listening pi number music: sheet music, a piano, a music

player and headphones (Fig. 9)

“Japanese Mathematical Courtyard” is an exposition presented as a model of Japanese temple

decorated with sangaku tablets, origami, and models of stones. This exposition will help visitors

to plunge into the world of Japanese mathematics (Fig. 10).

Figure 10. Scale model of “Japanese Mathematical Courtyard” created by the students

This zone will present the problem of mathematical reconstruction of the heritage of Japanese

temple geometry, sangaku. For this exhibition, the students prepared a poster (Fig. 11) and a

https://drive.google.com/drive/u/0/folders/19OEiTlJJ30U4Qr1c0BKDVV-hie5g3OcW




collection if their mathematical reconstructions of sangaku. They not only tell about their results

but also invite visitors to join a network project.

Figure 11. Sangaku poster Figure 12. Japanese Garden of 15 Stones

The important part of this zone is “Japanese Garden of 15 Stones.” Its special feature is that from

any point you can see no more than 14 of the 15 stones (Fig. 12). A dynamic model helps visitors

to verify this and discover other properties of the garden. Visitors can also create their own gardens

with the same property. There is also a table with origami and supply of paper for constructing

models. During these activities, the students show visitors how to formulate mathematics tasks

with these models and how to use paper-folding method for task solving.

CONCLUSIONS

The experience presented here can be useful both for children's museums that have their own

workshops for creating exhibits and for museums without such workshops. The former can use

their workshops to organize new museum activities for pupils and students, while the latter one

can use crowdsourcing opportunities for to replenish their collections.

In our view, creation of exhibits and holistic interactive expositions for the museum of entertaining

sciences is a good result of students’ individual research activity. At this stage, the pupils and

students work together in temporary creative groups, jointly discuss the ways of presenting

scientific information in interesting and accessible manner, implement their ideas in the form of

https://www.geogebra.org/m/yacah6hm

https://sites.google.com/view/sangaku-project/

https://www.geogebra.org/m/cwv38w58




interactive exhibits, and prepare to play the role of guides. Such work is likely to be more

interesting to students than a simple visit to a museum of entertaining sciences created by adults.

When students create their own exhibits instead of only looking at them, they begin to recognize

the importance of scientific knowledge and scientific activity.

REFERENCES

Bogomolov, N. (2003). The House of entertaining science, Magazine “Neva”, 5 (pp. 276-282).

SPb. (in Russian)

Romanovsky, I.V. (2002) Museums of entertaining science. Computer tools in education, 5 (pp.

86-88). SPb: SPU “Informatization of Education” (in Russian)

Concepts of development of mathematical education in the Russian Federation (2013). Approved

by Order of the Government of the Russian Federation of 24.12.2013, №2506-r. Retrieved from

http://www.firo.ru/wp-content/uploads/2014/12/Concept_mathematika.pdf. (in Russian)

Project “Experimental mathematics” Retrieved from http://itprojects.narfu.ru/kruzhok-exp-mat.

Freudenthal H. (1931) Museum – Volksbildung – Schule – Erfurt (in German).

Standards for Professional Practice in Children’s Museums (2012). Association of Children’s

Museums. Retrieved from https://childrensmuseums.org/members/publications

http://www.firo.ru/wp-content/uploads/2014/12/Concept_mathematika.pdf

http://itprojects.narfu.ru/kruzhok-exp-mat

https://childrensmuseums.org/members/publications

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