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MATHEMATICS TEACHING RESEARCH JOURNAL 58 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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.
MATHEMATICS TEACHING RESEARCH JOURNAL 59 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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;
MATHEMATICS TEACHING RESEARCH JOURNAL 60 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
– 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
MATHEMATICS TEACHING RESEARCH JOURNAL 61 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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).
MATHEMATICS TEACHING RESEARCH JOURNAL 62 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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).
MATHEMATICS TEACHING RESEARCH JOURNAL 63 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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).
MATHEMATICS TEACHING RESEARCH JOURNAL 64 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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
MATHEMATICS TEACHING RESEARCH JOURNAL 65 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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
MATHEMATICS TEACHING RESEARCH JOURNAL 66 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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).
MATHEMATICS TEACHING RESEARCH JOURNAL 67 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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
MATHEMATICS TEACHING RESEARCH JOURNAL 68 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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
MATHEMATICS TEACHING RESEARCH JOURNAL 69 SPRING 2020 Vol 12, no 1 Vol 12, no 1
Readers are free to copy, display, and distribute this article as long as the work is attributed to the author(s) and Mathematics Teaching-Research Journal Online, it is distributed for non-commercial purposes only, and no alteration or transformation is
made in the work. All other uses must be approved by the author(s) or MTRJ. MTRJ is published by the City University of New York. http://www.hostos.cuny.edu/mtrj/
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