AUBH Students Use 3D Printing to Minimize Material Waste

Education_BH: Innovation and Excellence celebrates the remarkable innovations implemented in Bahrain’s schools and universities. We learned about these innovations, the process, the challenges, and the lessons learned from some of the leading educational institutions in the Kingdom.

Read all about the American University of Bahrain‘s innovation below.

Employing 3D printing to enforce students’ geometric perception in Calculus II

Students at the American University of Bahrain designed and 3D-printed mugs using calculus concepts, combining abstract mathematics with hands-on learning whilst promoting sustainability through efficient design that maximises capacity with minimal material.

Please share a brief background of your innovation.

Over the past few decades, the Calculus II curriculum in higher education has undergone a steady process of standardisation, reflecting a broader effort to unify mathematical instruction across institutions. Among other topics, it includes the calculation of volumes and surface areas of solids of revolution. This offers instructors and students the challenge of dealing with three-dimensional geometry. One reason this topic is facing inherent difficulties is that traditional teaching methods (textbooks, 2D diagrams, and classroom boards) are often insufficient means to convey 3D spatial relations.

During the last five years, 3D printing has emerged as an educational tool, employed by educators to bridge the gap between abstract concepts and tangible understanding. At the American University of Bahrain (AUBH), Course Learning Outcomes for Calculus II include application of definite integrals in a geometric context as well as computation of integrals using a variety of integration techniques, numerical approximations and software. Students come from Engineering or Computer Science majors. 

All of them have completed Calculus I and can exhibit some competence in soft skills. Approximately one-third of the students have also been exposed in their classes to software allowing manipulation of solid figures, like  SolidWorks. According to its mission, AUBH is committed to transformative, flexible, and innovative teaching. 

There is a 3D printing Lab on campus (D LAB) manned by highly qualified personnel. Students are encouraged to use this facility and work hands-on to complete their projects. Furthermore, a glimpse into some of the recent advancements in the use of 3D printing within mathematics education is provided, for instance, which offer insightful examples of its growing pedagogical impact

How was the innovation planned?

The project focuses on volumes and surface area of solids of revolution. Students need to be able to visualise how different choices of a 2D graph, when revolved about an axis, will affect the shape of the resulting solid. Furthermore, they must provide formulas, following the theory taught in class, to compute the surface area and its volume. This calculation can be carried out,  whenever possible, by hand. If it is beyond the student’s reach, then it can be done by numerical methods and software, such as online integrators. Upon consultation by the personnel of the D LAB, the project consisted of students having to design and 3D print a mug. 

There were upper limits in the lengths of the height and the base/top diameters of the mug for cost reasons. Furthermore, it was required that the mug hold water. But the essential goal was that the ratio of volume to surface area should be as large as possible. This condition is in accordance with the sustainability principle, which encourages humans to satisfy their needs with the minimum amount of resources. Sustainability was an annual academic theme adopted by AUBH. 

Students were divided into groups, taking into account some of their social preferences.  For the designing part, they had to use software like GeoGebra or Mathematica to 2D graph a curve. Then they were supposed to use the rotating facilities of the software to produce 3D graphs and verify that the resulting solid (mug) is within their given specifications. In order to get started with their experimentation, they were given the hint to use a parabola and adjust its parameters to their satisfaction. 

Then they applied theoretical tools established in the course to produce integrals for the calculation of volume and surface area. In the case of volume, they had to employ both disk/washer and shell methods. The calculation of those integrals was carried out using numerical methods and tools acquired in the course. Finally, every group produced a ratio of volume to surface area. 

To increase the desire for scientific inquiry, the group with the highest ratio was given an extra grade bonus. Upon completion of the experimentation, groups went separately to the D LAB to have their mug 3D printed. Lab personnel required the 2D graph to be handed in a specific format, and printing took 6 – 8 hrs. When groups presented their physical mugs, the instructor filled them to the top with water and put them on a scale to make sure that the volume calculation agreed with the net content of water. 

Finally, every group submitted a diary of their experimentation, the theoretical background of measurements of solids of revolution, a record of their calculations and their thoughts about how the whole process helped them to understand geometric motion in 3D. Every group gave a short presentation of their efforts and final results. The project counted for 10% of the total grade.

Give us a brief assessment of your results.

Assessment of the projects was based on a comprehensive rubric encompassing theoretical understanding, applied knowledge, computational proficiency, and soft skills. A key component of the evaluation involved verifying the consistency between the calculated volume of each mug – calculated using integration techniques – and the actual amount of water it could hold. To further encourage excellence and innovation, bonus points were awarded to the group achieving the highest volume-to-surface-area ratio, as well as to those selected to present their work at the AUBH Research Day event. To evaluate the impact of the project on student learning, a diagnostic test was administered twice: first, at the conclusion of the relevant sections, before the project’s announcement, and again during the final week of classes, following project completion. 

The test consisted of questions of basic to moderate difficulty, focusing on volumes and surface areas of solids of revolution, which formed the core conceptual foundation of the project.  Statistical analysis indicates an overall improvement in class performance on the second test.  However, it is noteworthy that for some students, scores remained relatively unchanged, which may be attributed to their limited engagement or peripheral roles within project groups.

In hindsight, what were the most valuable lessons learned while implementing the innovation? Could things have been done differently?

Beyond the quantitative outcomes, this educational activity yielded valuable qualitative insights and experiential learning. Notably, several student groups became intellectually engaged when confronted with discrepancies arising from different methods of calculating volume or surface area. These moments of confusion prompted meaningful discussions—both among peers and with the instructor, serving as effective opportunities to address and resolve misconceptions in calculus and geometric reasoning.

Another example is that although initial guidance suggested employing parabolas, many students independently explored a broader range of mathematical functions to achieve aesthetically and structurally satisfying results. Interestingly, while the project was situated in a three-dimensional context, it also contributed to the refinement of students’ two-dimensional graphing skills.

From a mathematical standpoint, the optimal ratio of volume to surface area is attained by the sphere. Some students identified this principle during their theoretical exploration and attempted to design mugs that approximated a spherical shape, while remaining within the constraints of the project specifications. This demonstrates a commendable level of analytical thinking and an ability to translate abstract mathematical concepts into practical design considerations.

A couple of groups shared their observations on the project: “This project really improved our 3D understanding, as it gave us a means to visualise the volumes and areas of solids of rotation, see our 3D object in the design software, and finally see it come to life as a real 3D object. This process made us understand the transformation from a simple function to a 3D model, and to an actual 3D object. We think this project was really insightful and was really helpful in developing our 3D visualisation skills.”

“This project has honestly been so fun. It allowed us to work with Solid Works for the very first time. Not to mention, that it was the perfect way to use the D-LAB since we had never used it before. Witnessing something being 3-D printed is simply cool; having been the people who designed it makes it even cooler. We really enjoyed this project and were enthusiastic throughout the entire process of creating the mug.”

The project instructor further added: “I would like to continue incorporating 3D printing activities, especially in Calculus II & III classes, since they caption student’s imagination and creativity. From the organisational point of view, I believe that the projects should be announced as early as possible. Furthermore, a small part of the grade should correspond to initial designing, thus pushing students to start working early and encouraging them to improve the quality of the outcome. And last, students should provide individual reports to make sure that they participate in the development of the project.”