Mathematical analysis of the acoustic properties of historical buildings (SOI-TR-610)
Author: Nevin Bilgiçli
School/Organisation: Kırıkkale High School
Below is a summary of the activities, but you can download the full Story of Reuse and its annexes here in Word and PDF.
In this activity, 18 of my students (16-year-old) researched some sacred structures on the UNESCO World Heritage List on the Europeana platform. They gave presentations on the historical and mathematical features used in the architecture of Hagia Sophia, Notre Dame Cathedral and Cologne Cathedral. Then, using the original numerical values of these three structures, they calculated their acoustic properties. They compared and interpreted the acoustic properties of these three structures from a mathematical and geometric perspective.
I was inspired by the Learning Scenario “Mathematical and visual synthesis of historical structures” that I previously created and published on the Teaching with Europeana website. However, because the topic of acoustics was not included in detail in the activity and some parts were quite different (3D visualization instead of 3D printing etc.), I thought it would be more beneficial to create a new activity.
Lesson 1 (40′):
Firstly, I showed students some photos of historical buildings on the smart board. Then, I asked them the following questions:
- 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/ future?
- Do you think great historical sacred structures have impressive acoustics? Has mathematics played a role in this?
Students shared their ideas. Then I opened the Europeana website and introduced what it is. I asked them to conduct research via Europeana and identify three sacred structures, one from our country and two from the rest of Europe. They researched and announced their choices. We made a vote by writing the different options from the students on the board. We decided on Hagia Sophia (Türkiye), Notre Dame Cathedral (France), and Cologne Cathedral (Germany). We named these 3 buildings “reference structures”.
Lesson 2 (40′)
Each group presented the history and mathematical properties of their refence structure to the other groups. Before the presentations, I gave each group a worksheet (Annex 2). While students listened to the presentations, they took notes of the structures’ properties on their worksheets. Following the presentations, students discuss the similarities and differences between the buildings.
Lesson 3 (40′)
I then provided students with information about architectural acoustics. I talked about the influence of mathematics on the acoustic properties of architectural structures.
Next, I asked them to calculate the reverberation times of these three reference structures by using the numerical data they had previously recorded in their worksheets. I gave them two documents to help them with their calculations:
- One with the reverberation time formula, an important formula in architectural acoustics (Annex 3).
- One with the sound absorption coefficients of some materials (Annex 4).
The students made the necessary calculations. Some groups use pen, paper and calculators while others use online digital tools (such Google Sheets or similar). These tools enabled better interpretation of the results using graphs etc.
Each group interpreted the results and compared the three constructs acoustically. They discussed and noted which variables caused the differences.
Lesson 4 (40′)
I gave students some information about the influence of geometry on the acoustic properties of architectural structures. I asked them to examine the geometric elements used within the structures (see Annex 4).
In the first half of the lesson students examined images of the interiors of reference structures. Students analysed the convex, concave, flat, etc. surfaces of the buildings and interpreted them from an acoustic perspective. They then interpreted the results and compared the three constructs acoustically.
In the second half of the lesson each group gave their final presentations about their results, based on their previous calculations on reverberation times (RT values) and geometric analyses.
Lesson 5 (40′)
In this session, I asked the students to create their own structures according to the following conditions:
- It should include construction or decorative materials from each of the reference structures.
- It should include geometric and visual elements from each of the reference structures.
- Its reverberation time (RT) should be within the ideal acoustic range (3-6 seconds).
Also, I told them to use the GeoGebra program (https://www.geogebra.org/?lang=en) and its variable sliders when designing the acoustics (RT) of their buildings. Each group first worked on the numerical values of their own structures. They decided on the volume, surface area, and materials to be used for a suitable RT. Then they determined the geometric-visual elements (materials) they would use in their structures. They decided final design and 3D mathematical model of their own structures. Using some artificial intelligence tools, they created 3D visualizations of their designs.
After school, they worked remotely and prepared some promotional materials. Using some online programs (such as Canva, https://www.canva.com/, QR generator, https://online-qr-generator.com, Narakeet, https://www.narakeet.com/), students created the audio and final promotional posters in both their native language and English. They used the poster in their native languages at the promotional stand and the English poster to disseminate their work on international online education platforms.
Lesson 6 (40′)
The students printed out all the promotional materials they had prepared before the lesson. Then they created a stand for an exhibition at school.
Evaluation
At the end of this STEAM activity, a test (see Annex 5) was given to the students, including questions about the information they obtained from Europeana and other sources about important historical buildings, architectural acoustics, the 3D modelling, as well as the mathematical formulas used in the activity. Additionally, the students’ opinions about this activity were taken with the last two questions in this test.
In addition to this test, throughout the activity, I observed the students’ attitudes towards different parts of the activity and their social interactions. I observed that this activity successfully brought together students with different interests and learning styles.
Outcomes for students
- Science: Students remembered the “optics” topic of physics, while they were studying the propagation of sound on convex and concave surfaces, because sound waves travel like light.
- Technology: Students’ knowledge and skills in technology and ICT improved thanks to their use of Al tools, Geogebra, Google sheets and different Web 2.0 tools (Canva, Kahoot, QR generator, Narakeet…)
- 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 design of their building. They also developed their aesthetic perspectives while preparing promotional visual posters and examining images of reference buildings.
- Mathematics: They reinforced their knowledge of 9th and 10th grade mathematics topics (ratios, proportions, averages) and geometry topics (area and volume of 3D solids, concave and convex shapes, angles) while calculating the acoustic properties of reference structures and their own structure. By brainstorming about the effect of variables on the RT ratio, they improved their problem solving and analytical thinking skills.
Outcomes for the educator
As a teacher, I learned many new things while exploring the resources on the Europeana platform. This increased my motivation to create new learning scenarios and use innovative teaching methods in the future. I believe that creating such activities contributes to the creativity of teachers. Also, implementing these scenarios in the classroom and achieving positive results contributes to my professional development. In addition, getting to know many technological applications more closely with my students has improved my ICT skills.
Public Domain : the featured image used to illustrate this article has been found on Europeana and has been provided by the Rijksmuseum