# 3D Visualization through the Hologram for the Learning of Area and Volume Concepts

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. The Teaching of Geometry in Secondary Education

#### 2.2. The Importance of Visualization and the Use of Technologies in the Geometry Learning Process

#### 2.3. The Use of the Hologram in Geometry

## 3. Methodology

#### 3.1. Objective and Hypothesis

#### 3.2. Sample

#### 3.3. Research Design

#### 3.4. Information Collection Tools

- Lateral and total areas of polyhedrons
- Lateral and total areas of bodies in revolution
- Volumes of polyhedrons
- Volumes of bodies of revolution

#### 3.5. Procedure

#### 3.6. Data Analysis

## 4. Results and Discussion

#### 4.1. Pre-Test Results

#### 4.2. Post-Test Results

#### 4.3. User Experience Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Pre-Test

1. The polygons are flat figures (T/F) |

2. The quadrilaterals have: |

a. Four equal sides |

b. Four equal sides two to two |

c. Four equal angles |

d. Four right angles |

3. A triangle with two equal sides and a different one is called: |

a. Equilateral |

b. Isosceles |

c. Scalene |

d. Acute |

4. In a triangle with an angle greater than 90º it is called: |

a. Scalene |

b. Acute |

c. Obtuse |

d. Rectangle |

5. The apothem of a regular polygon is: |

a. The side of the polygon |

b. The sum of the sides of the polygon |

c. The distance from the center of the polygon to the vertex |

d. The distance of the polygon to the middle of the side |

6. What is the perimeter of a flat figure: |

a. The sum of its angles |

b. The sum of its sides |

c. The line that forms its outline |

d. The space that collects |

7. The circumference of a circle is: |

a. πr^{2} |

b. 2πr^{2} |

c. 2πr |

d. πr |

8. A piece of circumference is called: |

a. Arc |

b. Sector |

c. Radio |

d. Angle |

9. To which flat figure corresponds the formula for calculating area axb, where a and b are the sides: |

a. Rectangle |

b. Quadrilateral |

c. Triangle |

d. Hexagon |

10. The formula of the area of the triangle is: |

a. Base x height/3 |

b. Base x height/2 |

c. Base x height |

d. Side by side |

11. How is the area of a hexagon calculated? |

a. Decomposing in equilateral triangles |

b. Decomposing in isosceles triangles |

c. Decomposing in triangles |

d. Decomposing in scalene triangles |

12. The area of a circle is: |

a. πr^{2} |

b. 2πr^{2} |

c. 2πr |

d. πr |

13. A polyhedron is: |

a. A flat figure |

b. A 3D figure |

c. A 3D figure whose faces are polygons |

d. A 3D figure whose lateral surface is curved |

14. A orthohedron is shaped like a shoe box (T/F) |

15. What is a polyhedron: |

a. Cube |

b. Prism |

c. Orthopedic |

d. pyramid |

e. Everyone |

16. A body in revolution is: |

a. A flat figure |

b. A 3D figure |

c. A 3D figure whose faces are polygons |

d. A 3D figure whose lateral surface is curved |

17. What is a body in revolution? |

a. Cylinder |

b. Cone |

c. Sphere |

d. Everyone |

18. The word edge refers to: |

a. The side of a polyhedron |

b. The vertex of a polyhedron |

c. The height of a polyhedron |

d. The face of a polyhedron |

19. The total surface area of the polyhedrons is calculated as: |

a. Area of the bases plus the lateral area |

b. Area of a base by height |

20. The surface area of the bodies in revolution is calculated as the sum of the lateral area plus the areas of the base, except that of the sphere |

21. Point out the false statement: |

a. The volume of a cube, orthohedron, prism and cylinder is calculated as the area of the base by height |

b. The volume of a polyhedron is calculated as the area of the base by the height |

c. The volume of a cone is one-third of the volume of the cylinder |

d. The volume of a square base pyramid is one-third of the volume of a cube |

## Appendix B. Post-Test

1. The surface area of a 5 cm long, 2 cm wide and 3 cm high orthohedron is: |

a. 30 cm^{2} |

b. 30 cm^{3} |

c. 62 cm^{2} |

d. 10 cm^{2} |

2. The total surface area of a prism with right triangle base whose legs measure 4 cm and 3 cm and the height of the prism is 6 cm is: |

a. 72 cm^{2} |

b. 84 cm^{2} |

c. 78 cm^{2} |

d. 36 cm^{2} |

3. The total surface area of a hexagonal prism of 7 cm high whose base is formed by a hexagon of 6 cm side and whose apothem is 4 cm is: |

a. 396 cm^{2} |

b. 66 cm^{2} |

c. 276 cm^{2} |

d. 324 cm^{2} |

4. The lateral area of a square-based pyramid whose faces are formed by isosceles triangles in which the equal sides have 5 cm and the other 6 cm is: |

a. 12 cm^{2} |

b. 48 cm^{2} |

c. 84 cm^{2} |

d. 24 cm^{2} |

e. 60 cm^{2} |

5. The lateral area of a trunk squared-base pyramid in which the faces are trapezoids with a great base of 7 cm, a small base of 4 cm and an apothem of 5 cm is: |

a. 70cm^{2} |

b. 280 cm^{2} |

c. 110 cm^{2} |

d. 88 cm^{2} |

6. Label the correct statement/s: |

a. A cylinder is a body of revolution obtained when rectangle or a square rotates on one of its sides so its lateral surface area is calculated by multiplying the length of the circumference it forms by the height |

b. A cone is a body of revolution obtained when a right triangle rotates by one of its sides and the lateral surface area is obtained by multiplying the length of the circumference that it forms by the height |

c. A truncated cone is a body of revolution obtained by cutting a cone or rotating a trapezoid on its straight side and its lateral surface area is πg (R + r) |

d. A sphere is a body of revolution obtained when a semicircle or a circle rotates around its diameter and its surface area is the sum of the area of 4 circles. |

7. Label the correct statement/s: |

a. The volume of a cube and an octohedron is always calculated as the product of the area of the base by the height |

b. The volume of a hexagonal prism is calculated as the product of the area of its base by its height |

c. The volume of a cylinder is calculated as the product of area of the base by height |

d. a and b are correct |

e. All are correct |

8. Point out the correct affirmation or affirmations |

a. Three triangular-base pyramids form a cube |

b. Three triangular-base pyramids form a tetrahedron |

c. Three square-base pyramids form a cube |

d. Three square-base pyramids form a tetrahedron |

9. If a cube has a volume of 27 cm^{3}, what will be the volume of the three equal pyramids it can form? |

a. 3 cm^{3} |

b. 9 cm^{3} |

c. 13.5 cm^{3} |

d. 6.75 cm^{3} |

10. If a squared-base pyramid of side 7 cm and height 6 cm is cut at a height from the ground of 4 cm so that the side of the square of the smaller base is 2 cm, the volume of the pyramid trunk is: |

a. 86 cm^{3} |

b. 95.3 cm^{3} |

c. 6 cm^{3} |

d. 12 cm^{3} |

11. The volume of a cone of diameter 6 cm and height 5 cm is: |

a. 47.1 cm^{3} |

b. 141.3 cm^{3} |

c. 188.4 cm^{3} |

d. 70.65 cm^{3} |

12. If a cone in which the base has a radius of 3 cm and 5 cm in height is cut at a height from the base of 3 cm so that the smallest radius of the circle that forms is 1.2 cm, the volume of the trunk of the cone is: |

a. 47.1 cm^{3} |

b. 44.1 cm^{3} |

c. 42.58 cm^{3} |

d. 3.01 cm^{3} |

13. Label the false statement in relation to the volume of the sphere |

a. It is calculated as one third of the product of the surface area by the radius |

b. It can be calculated by adding the volumes of all the square-base pyramids that constitute it |

c. A sphere can be filled with two cones if the height and diameter of the cone are equal to the diameter of the sphere |

d. Three cylinders can be filled with two spheres if the height and diameter of the cylinder is equal to the diameter of the sphere |

14. Label the false statement/s |

a. With three cones you can fill three cylinders if they have the same height and same base area |

b. With three cones you can fill a cylinder if they have the same height and same base area |

c. With a cone, a semi-sphere can be filled if the height and diameter of the cone are equal to the diameter of the semi-sphere |

d. With three spheres two cylinders can be filled if the height and diameter of the cylinder are equal to the diameter of the sphere |

e. With two spheres three cylinders can be filled if the height and diameter of the cylinder are equal to the diameter of the sphere |

## Appendix C. User Experience Test

Likert-type questions: |

1. My learning results have increased |

2. I had fun learning |

3. I have learned more autonomously |

4. I have managed to increase the logical capacity |

5. My creativity has increased |

6. My motivation has increased |

7. I have been able to self-evaluate my learning process |

8. I liked this methodology more than the traditional one |

9. The learning has been more active on my part |

10. The use of this methodology has encouraged cooperative learning with my colleagues |

11. I have been able to learn at my own rhythm |

12. I believe that I will be able to retain better the contents that I have learned thanks to the use of this methodology |

Open questions: |

1. The physical phenomenon that has taken place in the use of the hologram is: |

2. What I liked most about the use of hologram is: |

3. In what subject/s do you think it could be used in addition to mathematics? |

4. Would you recommend the hologram as a teaching tool? |

5. Is there something you want to comment? |

## References

- Blanco, L.J.; Barrantes, M. Concepciones de los estudiantes para maestro en España sobre la Geometría escolar y su enseñanza-aprendizaje. Relime
**2003**, 6, 107–132. [Google Scholar] - Gómez-Chacón, I.M. Matemática Emocional. Los Efectos en el Aprendizaje Matemático; Narcea: Madrid, Spain, 2000. [Google Scholar]
- Villella, J. Uno, Dos, Tres… Geometría Otra Vez; Aique: Buenos Aires, Argentina, 2001. [Google Scholar]
- Serra, R.; Vega, G.; Ferrat, Á; Lunazzi, J.J.; Magalhães, D.S.F. El holograma y su utilización como un medio de enseñanza de la física en Ingeniería. Revista Brasileira de Ensino de Física
**2009**, 31, 1401. [Google Scholar] [CrossRef] - Orcos, L.; Magreñán, Á.A. The hologram as a teaching medium for the acquisition of STEM contents. Int. J. Learn. Technol.
**2018**, 13, 163–177. [Google Scholar] [CrossRef] - Fabres, R. Estrategias metodológicas para la enseñanza y el aprendizaje de la Geometría, utilizadas por docentes de segundo ciclo, con la finalidad de generar una propuesta metodológica atingente a los contenidos. Estudios Pedagógicos
**2016**, 42, 87–105. [Google Scholar] [CrossRef] - Fujita, T.; Jones, K. The bridge between practical and deductive geometry: Developing the geometrical eye. In Proceedings of the 26th PME International Conference, Norwich, UK, 21–26 July 2002; Cockburn, A.D., Nardi, E., Eds.; University of East Anglia: Norwich, UK, 2002; Volume 2, pp. 384–391. [Google Scholar]
- Jaramillo López, C.; Duarte, P. Enseñanza y aprendizaje de las estructuras matemáticas a partir del modelo de Van Hiele. Revista Educación y Pedagogía
**2009**, 18, 109–118. [Google Scholar] - Schonberger, A.K. The relationship between visual spatial abilities and mathematical problem solving are there sex-related differences? In Proceedings of the 3rd PME International Conference, Coventry, UK, 9–14 July 1979; Volume 1, pp. 179–185. [Google Scholar]
- Gutiérrez, Á. Las representaciones planas de los cuerpos 3-dimensioanles en la enseñanza de la Geometría espacial. Rev. EMA
**1998**, 3, 193–220. [Google Scholar] - Owens, K.; Outhred, L. The Complexity of learning geometry and measurement. In Handbook of Research on the Psychology of Mathematics Education: Past, Present and Future; Gutierrez, A., Boero, P., Eds.; Sense Publishers: Rotterdam, The Netherlands, 2006; pp. 83–115. [Google Scholar]
- Bishop, A.J. Spatial abilities and mathematics education—A review. Educ. Stud. Math.
**1980**, 11, 257–269. [Google Scholar] [CrossRef] - Suh, J.; Seshaiyer, P. The Role of Information Technology in Engaging Elementary Students in Mathematical Modeling. In Proceedings of the Society for Information Technology & Teacher Education International Conference, Kuala Lumpur, Malaysia, 24–25 March 2016; pp. 2576–2583. [Google Scholar]
- Moyer-Packenham, P.; Suh, J. Learning mathematics with technology: The influence of virtual manipulatives on different achievement groups. J. Comput. Math. Sci. Teach.
**2012**, 31, 39–59. [Google Scholar] - Clements, D.H.; Sarama, J. Effects of a preschool mathematics curriculum: Summative research on the Building Blocks project. J. Res. Math. Educ.
**2007**, 38, 136–163. [Google Scholar] - Adams Becker, S.; Cummins, M.; Davis, A.; Freeman, A.; Hall Giesinger, C.; Ananthanarayanan, V. NMC Horizon Report: 2017 Higher Education Edition; The New Media Consortium: Austin, TX, USA, 2017. [Google Scholar]
- Liu, Y.Z.; Pang, X.N.; Jiang, S.; Dong, J.W. Viewing-angle enlargement in holographic augmented reality using time division and spatial tiling. Opt. Express
**2013**, 21, 12068–12076. [Google Scholar] [CrossRef] [PubMed] - Gabor, D. Holography, 1948–1971. Science
**1972**, 177, 299–313. [Google Scholar] [CrossRef] [PubMed] - Lee, H. 3D Holographic Technology and Its Educational Potential. Teach Trends
**2013**, 57, 34–39. [Google Scholar] [CrossRef] - Walker, R.A. Holograms as teaching agents. In Journal of Physics: Conference Series; IOP Publishing: London, UK, 2013; Volume 415, p. 012076. [Google Scholar]
- Ohlmann, O.M. 3D and Education. In Journal of Physics: Conference Series; IOP Publishing: London, UK, 2013; Volume 415, p. 012066. [Google Scholar]
- Suh, J.M.; Moyer-Packenham, P.S. The application of dual coding theory in multi-representational virtual mathematics environments. In Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Seoul, Korea, 8–13 July 2007; Woo, J.-H., Lew, H.-C., Park, K.-S., Seo, Y., Eds.; PME: Seoul, Korea, 2007; Volume 4, pp. 209–216. [Google Scholar]
- Moyer, P.S.; Bolyard, J.J.; Spikell, M.A. What are virtual manipulatives? Teach. Child. Math.
**2002**, 8, 372–377. [Google Scholar] - Moyer-Packenham, P.S.; Lommatsch, C.W.; Litster, K.; Ashby, J.; Bullock, E.K.; Roxburgh, A.L.; Shumway, J.F.; Speed, E.; Covington, B.; Hartmann, C.; et al. How design features in digital math games support learning and mathematics connections. Comput. Hum. Behav.
**2019**, 91, 316–332. [Google Scholar] [CrossRef] - Melhuish, K.; Falloon, G. Looking to the future: M-learning with the iPad. Comput. N. Z. Sch.
**2010**, 22, 1–16. [Google Scholar] - Martín-Gutiérrez, J.; Saorín, J.L.; Contero, M.; Alcañiz, M.; Pérez-López, D.C.; Ortega, M. Design and Validation of an Augmented Reality for Spatial Abilities Development in Engineering Students. Comput. Graph.
**2010**, 34, 7–91. [Google Scholar] [CrossRef] - Kesim, M.; Ozarsla, Y. Augmented Reality in Education: Current Technologies and the Potential for Education. Procedia Soc. Behav. Sci.
**2012**, 47, 297–302. [Google Scholar] [CrossRef] - Heinrich, P. The iPad as a Tool for Education: A Study of the Introduction of iPads at Longfield Academy; The ICT Association: Nottingham, UK, 2012; Available online: http://www.naace.co.uk/publications/longfieldipadresearch (accessed on 10 December 2012).
- Clements, D.H.; Battista, M.T. Geometry and spatial reasoning. In Handbook of Research on Mathematics Teaching and Learning; Grouws, D.A., Ed.; MacMillan: New York, NY, USA, 1992; pp. 420–464. [Google Scholar]
- De la Torre, J.; Martín-Dorta, N.; Soarín, J.L.; Carbonel, C.; Contero, M. Entorno de aprendizaje ubicuo con realidad aumentada y tabletas para estimular la compresión del espacio tridimensional. Revista de Educación a Distancia
**2013**, 37. Available online: http://www.um.es/ead/red/37/ (accessed on 9 March 2019). - Grouws, D.A. Handbook of Research on Mathematics Teaching and Learning. A Project of the National Council of Teachers of Mathematics; MacMillan: New York, NY, USA, 1992; pp. 420–464. [Google Scholar]
- Pozo, J.I.; Monereo, C. Introducción: La nueva cultura del aprendizaje universitario o por qué cambiar nuestras formas de enseñar y aprender. In Psicología del Aprendizaje Universitario: La Formación en Competencias; Pozo, J.I., Pérez, M.P., Eds.; Morata: Madrid, Spain, 2009; pp. 9–28. [Google Scholar]
- LeTendre, G.; McGinnis, E.; Mitra, D.; Montgomery, R.; Pendola, A. American Journal of Education: Retos y oportunidades en las ciencias translacionales y la zona gris de la publicacion academica|The American Journal of Education: Challenges and opportunities in translational science and the grey area of academic. Revista Espanola de Pedagogía
**2018**, 76, 413–435. [Google Scholar] [CrossRef]

**Figure 1.**Image taken from the video of the volume calculation of the cylinder to be projected with the hologram.

**Figure 4.**Box plots of the mean qualifictions. (

**a**) Box plots of the mean qualifications in the post-test in the control group; (

**b**) Box plots of the mean qualifications in the post-test in the experimental group.

**Figure 5.**Comparison of the percentage of correct answers in the post-test of both control and experimental groups.

**Figure 6.**Results of the percentages of responses of students of the experimental (

**top**) and control (

**bottom**) groups in questions 6, 8, and 14.

**Figure 7.**Results of the user experience. (

**a**) Results of the user experience on knowledge of the physical phenomenon on which the hologram is based; (

**b**) Results of the user experience on what they liked most; (

**c**) Results of the user experience on its possible application to other areas; (

**d**) Results of the user experience on and its recommendation as a teaching medium.

**Figure 8.**Average results of the assessment of Likert-type scale items of the user experience in the experimental group.

Independent Samples Test | |||||||
---|---|---|---|---|---|---|---|

Levene Proof | t-Test of Equality of Means | ||||||

F | Sig. | t-Value | gl | Sig. | Mean Differences | Standard Error Differences | |

Assuming equal variances | 3.184 | 0.078 | −15.983 | 76 | 0.000 | −3.9021 | 0.2441 |

Non assuming equal variances | - | - | −16.766 | 70.777 | 0.000 | −3.9021 | 0.2327 |

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## Share and Cite

**MDPI and ACS Style**

Orcos, L.; Jordán, C.; Magreñán, A.
3D Visualization through the Hologram for the Learning of Area and Volume Concepts. *Mathematics* **2019**, *7*, 247.
https://doi.org/10.3390/math7030247

**AMA Style**

Orcos L, Jordán C, Magreñán A.
3D Visualization through the Hologram for the Learning of Area and Volume Concepts. *Mathematics*. 2019; 7(3):247.
https://doi.org/10.3390/math7030247

**Chicago/Turabian Style**

Orcos, Lara, Cristina Jordán, and Alberto Magreñán.
2019. "3D Visualization through the Hologram for the Learning of Area and Volume Concepts" *Mathematics* 7, no. 3: 247.
https://doi.org/10.3390/math7030247