Mathematical and visual synthesis of historical structures (SoI-TR-594)

Author: Dr. Nevin Bilgiçli / Mathematics teacher

School/Organisation: Kirikkale High School

Introduction

It is a known fact that negative prejudices against mathematics lesson are common. However, there
are scientific studies showing that students’ motivation for the course increases when this course is
supported with examples from daily life. The field of architecture is one of the best examples showing
the relationship of mathematics with daily life. Additionally, there are scientific sources showing that
3D modeling and printing are effective as a method of concretizing mathematics and geometry
lessons.
Based on this fact, we wanted to organize an activity in line with the STEAM approach with my
students. Together with my 20 students aged 15-16, we examined learning scenarios about the
relationship between mathematics and architecture on the “Teaching with Europeana” platform.
We were inspired by the scenarios on this platform, but instead of making major changes to them
and implementing them, we thought it would be more beneficial to create a new learning scenario
and implement it. Therefore, inspired by the Europeana platform, I created a new learning scenario
called “mathematical and visual synthesis of historical structures’ and implemented it with my
students.
In this implementation my students researched important towers, most of which are on the UNESCO
World Heritage List, on the Europeana platform. They made presentations about the historical and
mathematical properties used in the architecture of the Eiffel, Pisa, Big Ban and Galata towers. They
then designed their own unique towers, which were a synthesis of both the visuals and mathematical
properties of these four towers. They also created the story of their own tower. Then, they created a
3D model of their own tower using the Tinkercad program and took a 3D printout. Finally, they
created audio QR codes and promotional materials explaining what they did. They created a
project stand at school and introduced their products to their peers in other classes at school,
teachers, administrators and some parents.
Thus, they both developed academically/socially and gained awareness of how historical buildings
carry the common cultural heritage of humanity from the past to the present.

Learning process

Lesson 1(40′):

First of all, I showed the pictures of some important historical buildings in our country and Europe on
the smart board. Then, I asked the following questions to the students.

  • What is the importance of these structures for humanity?
  • How might mathematics and geometry have been used in the construction of these
    structures?
  • Does mathematics have an impact on these structures survival from past to present?
  • What can be done against the danger of damage to important historical buildings in the
    future?

Students shared their ideas (10′). Then I opened the Europeana website and explained to the
students what this platform does and how to use it. I asked the students to take a look at the
important historical towers from this platform. Students searched the Europeana platform and
researched historical towers in Europe and Türkiye (15′). Then I asked them to identify 4 towers, one
of which was from our country. We made a vote by writing the different options from the students
on the board. We decided on Pisa, Eiffel, Big Ben and Galata towers. We named these 4 towers as
“reference towers” (10′).
At this stage, the students were divided into four groups (5′). Each group was assigned to research
the history of a different refence tower and the mathematical properties used in its construction.
After the lesson, each group worked remotely and prepared presentations about their group’s
tower.

Lesson 2 (40′):

Each group (7-8′) gave information to each other by making presentations about the history and
mathematical properties of a refence towers.
Before the presentations, I gave each group a worksheet. While all students listened to the
presentations, they took notes on the numerical properties of the structures on their worksheets.
After the presentations, students discussed the reflections of historical periods on these buildings,
similar and different aspects of the buildings, remarkable stories of the towers, etc (10′)

Lesson 3 (40′):

I asked the students to design a new tower that had the characteristics of the refence towers, both
visually and mathematically. I told them to start with mathematical synthesis first, that is, to first create
the dimensions of the new tower.
In the first half of the lesson, students discussed how they could create the dimensions of the new
tower by using the mathematical data of the reference towers (Pisa, Eiffel, Big Ban and Galata
towers). We decided to create the dimensions of the new tower as follows;

  • Width: The arithmetic mean of the widths of the refence towers,
  • Height: The geometric mean of the heights of the reference towers,
  • Diameter: The harmonic mean of the diameters of the largest circular part of the reference
    towers,
  • Clock diameter: Big Ben clock diameter

Since a miniature tower was to be created, these dimensions had to be reduced to a certain extent.
As the reduction ratio, the range of the construction years of the reference towers (difference
between the largest year and smallest year) was chosen. In other words, we reduced the mean
values mentioned above by 541 times.
Additionally, the following was decided about the new tower;

  • Number of pieces (floors): The squared mean of the floor number of reference towers
  • Clock: It will show 5.30, which is the inclination angle of Pisa tower.

During the second half of the lesson, students made the necessary calculations. Thus, the length of
the new structure was finalized as; height:19cm, width: 8cm, diameter: 4cm, clock diameter: 3cm
and number of piece: 9.


In this section, we made size-related calculations to put more emphasis on the mathematical
dimension of the STEAM approach. We have created a unique miniature product, remaining
completely faithful to the original dimensions of the 4 reference towers. This is the most unique part
of our activity.

Lesson 4 (40′):

Students worked on the visual design of the new tower in accordance with the mathematical
features determined in the previous lesson. Each group created different tower designs by drawing
on paper. Afterwards, the whole class discussed and decided on the best common design. Thus, the
students created their own tower, which has both the visual and mathematical features of the four
tower. All tower designs were discussed. Then a common tower was decided upon.

In addition, the whole students discussed about the new tower’s name and story. The common story
was determined as:
“In 2025, a natural disaster occurred that affected the whole world and most of the high-rise buildings
were destroyed. UNESCO organized an architectural competition to both forget the damage left by
this sad event and to carry the important buildings on the world heritage list to future generations.
Four young architecture students from Türkiye, France, England and Italy came together and
created a design that is the synthesis of four historical towers that were destroyed in their countries.
This design won first place in the competition. This tower was built in the Bormio, Sandria region of
Italy at a location that is the arithmetic average of the coordinates of the original 4 towers. This tower
was named “Bensafelta“, a combination of the names of Big Ben, Pisa, Eiffel and Galata towers”(20′).

Lesson 5 (40′):

I asked the students “How they could make their designs 3D?. Some students answered “3D
modeling and 3D printing”. We knew that some students had knowledge about this subject. We
asked them to make a 3D modeling of our tower. These students, whom we called the technology
group, first introduced the Tinkercad program to others (10′).
They then began creating a 3D model of the original common tower using the Tinkercad program.
Meanwhile, other students in the class followed the modeling stages on the smart board (30′). The
technology group completed the modeling by doing remote group work after class. Then, they went
to another school and started the 3D printing process, since there was no 3D printer in our school.

Lesson 6 (40′):

The technology group brought the new tower to the classroom, the 3D printing of which was
completed before this lesson. They gave information about how 3D printing processes took place.
They discussed the latest settings made with the Cura program. At this stage, whole students talked
about mathematical calculations. They saw how mathematics is used to determine the location,
size, duration and quantity of material in 3D printing (20′).


The application time does not include the time of printing from the 3D printer. Our printing process
took about ten hours. This time may vary depending on printer, material and design settings.
In the second half of the lesson, groups began working collaboratively to create informative
promotional materials for the new tower (audio QR codes with Vocaroo, collaborative posters with
Canva, and other promotional materials). They prepared a promotional poster with an audio QR
code both in their native language and in English. They used the one in the native language at the
promotion stand, and the one in English to disseminate their work on international online education
platforms. In addition, preparations have begun for a platform surrounded by greenery where the
tower will be exhibited (20′). After class, they completed all the materials and printed them out.

Lesson 7 (40′):

Students created a stand at school with the materials they had prepared before (10′).

They then introduced their work to other students, teachers, administrators, and some parents at
school. We also gave information about STEAM and the “STEM Discovery Campaign” (30′).


In addition, a student with special needs who has learning difficulties in our class did not participate
in some stages of the activity but wanted to make an animation work in her own field of interest. Link
to this student’s work: link . This student, who also had difficulties in terms of social communication,
socialized by showing the animation video she created to the participants during the promotional
activities.


Teacher Note:
Before doing this work, I was inspired by examining some learning scenarios such as “Geometry
Hidden Behind Old Walls” on the “Teaching with Europeana” platform.
But I realized that the scope of the activity I wanted to do had become very different and expanded
from the existing scenarios with the addition of the calculations in lesson 3, 3D modelling, 3D printing,
QR code creation, creating a stand and promoting it to others. There were also differences in terms
of grade levels, mathematics subjects to be reinforced, and duration.
Creating a unique miniature 3D product by remaining completely faithful to the original dimensions
of the 4 reference towers, especially with the mathematical average calculations made in the lesson
3, distinguishes our activity from other learning scenarios. I thought it would be more beneficial to
create a new learning scenario rather than implementing a current scenarios by making major
changes. For these reasons, I created and implemented my own learning scenario.

Evaluations

At the end of this STEAM activity, a Kahoot test (see annex 3) was applied to the students, including
questions about the information they obtained from Europeana and other sources about important
historical towers, the 3D modeling, mathematical formulas used in the application (arithmetic,
geometric, harmonic and square averages, range calculation, reduction ratio, etc.). Additionally,
the students’ opinions about this activity were taken with the last two questions in this test.
When I evaluated both the observations I made during the activity and the test taken at the end of
the activity, it was seen that the students generally (over %70) improved in terms of the achievements
specified in the outcomes section and enjoyed group work. Students expressed that they were willing
to do similar activities, especially in theoretically focused numerical courses.
Additionally, it was seen that the learning scenario could be implemented exactly.

Outcomes

The concrete outcome we obtained as a result of the application was the original tower designed
by the students. The students had the pleasure of seeing this structure, which they designed using
their own creativity, in concrete form. They also took pride in introducing their products to others.
In addition to this concrete product, abstract outputs were also obtained. Since STEAM work was
carried out in this activity, many gains were achieved about the different disciplines such as
mathematics, technology, science, history, geography, visual arts, native language, foreign
language, architecture etc.

  • Science: Students remembered the science topic “center of gravity “, while they were discussing
    how to keep their towers balanced,
  • Technology: Students’ knowledge and skills in technology and ICT improved thanks to their use of
    Tinkercad, 3D printer and different Web 2.0 tools (Canva, Kahoot, Google maps, Vocaroo etc.)
  • Engineering: They discovered the mathematics used in the architecture of buildings and created
    new structures. As they were able to establish a connection between daily life and mathematics
    thanks to architecture, their motivation for the math course increased.
  • Art: They developed their artistic skills and creativity while creating an original visual design of the
    tower.
  • Mathematics: They reinforced their knowledge of 9th grade mathematics curriculum topics:
    arithmetic mean, geometric mean, harmonic mean, square mean and range, while calculating the
    dimensions of their towers. Also, students saw concrete use of mathematics in terms of area, volume,
    time, and location while setting up 3D printing.

In addition, their ability to work in groups improved because they worked in teams, their native
language usage and communication skills improved because they made presentations, and their
foreign language practices improved because they used foreign sources and terms during research
and they prepared posters in written and spoken English. While examining the history of the towers,
they also gained new information in the fields of history. They remembered their geography lesson
knowledge because they used parallels and meridians while calculating the location of their tower.
In summary, students improved in terms of the Steam criteria and other competencies described in
the learning scenario.
Also, as a teacher, I got to know the resources on the Europeana platform more closely. I was inspired
by the learning scenarios available on this platform. While implementing this scenario, many creative
ideas came to my mind and my motivation increased to create new scenarios and use innovative
teaching methods in my lessons.
Finally, my students and I got to know the structures in Europe, most of which are on the UNESCO
World Heritage List. Thus, our awareness about the role of architectural structures in carrying the
common cultural heritage of humanity from the past to the present increased.

Suggestions

  • My colleagues from different countries can do similar activities with different historical towers
    from Europe and their countries. Similar studies can be carried out on different types of
    structures instead of towers.
  • Students can use different mathematical topics (simpler or more complex depending on
    grade level) to create the dimensions of their own unique structures.
  • For situations where 3D printing facilities are not available, only 3D modeling can be done or
    3D pens can be used or cardboard models can be created.
  • Different branch teachers can organize this activity to highlight their own fields. For example,
    instead of making math calculations, students can arbitrarily decide the size of their
    structures. Teachers may ask students to use information about their own lessons in the part
    of creating the story about their structure.

Annex

1. Presentation file containing documents of all work stages of implementation:
Online file:
STEAM Discovery (M&V Design of Structures)
PDF:
https://drive.google.com/file/d/1Tc0G3y0sQYUjy7IAz0CSDx6tYVJBtZCq/view?usp=sharing

2. Work sheet (PDF):
https://drive.google.com/file/d/1FmubtkcL5e1vuwnJziSPF6f3A6Bwm1qn/view?usp=sharing

3. Evaluation test (Kahoot PDF):
https://drive.google.com/file/d/1-PSDW-h72OZybCMExWK8sE7WPpEsb3cc/view?usp=
sharing

4. Learning Scenarios of “Mathematical and visual synthesis of historical structures” (PDF):
https://drive.google.com/file/d/1oDmwiLZMKG69119Hor0EtIoIUiRzsvKt/view?usp=sharing

5. Europeana resources

    6. Teaching with Europeana resources

      7. UNESCO world heritage resources:

        8. Online tools used:

          • Canva
          • Kahoot
          • Tinkercad
          • Cura
          • Vocaroo
          • QR genarator
          • Google maps

          9. Scientific article on the use of 3D printing in mathematics/ STEAM education
          https://journals.sagepub.com/doi/full/10.1177/27527263221129357

            Do you want to discover more stories of implementation? Click here.

            CC0 1.0: the featured image used to illustrate this article has been found on Europeana and has been provided by the Museum of Arts and Crafts, Hamburg.

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