Introduction 

Makerspaces, varying in shape and size, serve as spaces where students convene to create, invent, tinker, explore, and discover using a diverse range of tools and materials. Makerspaces are natural places for developing mathematical arguments and mathematical models, where geometric assumptions are supported and real-world complexities arise. Drawing inspiration from an eighth grade makerspace project called Kinetic Sculpture, I illustrate the ways in which construction within makerspaces offer mathematics teachers ample opportunities to foster students’ geometric thinking across various domains. This article presents tasks centered around the construction of kinetic sculptures, shedding light on the geometric properties and assumptions that can be explored through students’ physical creations. Exploring the geometric properties inherent in the design of kinetic sculptures presents valuable opportunities to not only justify the correctness of Euclidean construction (SLO 8), but also encourage the investigation of geometric properties spanning multiple dimensions and adapt to diverse opportunities and constraints (e.g., materials, mobility, and assembling) often encountered in makerspaces. 

Makerspaces 

Constructivism speaks to the value of the learner constructing their own knowledge based on experiences and interactions with the real world (Bringuier & Piaget, 1980). The learning happens where the learner is positioned to actively construct meaningful, physical objects for the purposes of sharing playful and useful creations with the world (Hansen et al., 2019; Papert & Harel, 1991). In makerspaces, the focus is not solely on the types of tools and materials but rather on the act of making itself. Unlike traditional approaches in geometric constructions that often rely on straightedge and compass constructions, makerspaces offer a broader range of possibilities. These spaces provide a platform for learners to explore and express their creativity, utilizing various tools and materials to bring their mathematical constructions to life, thus fostering a deeper understanding of mathematical concepts through tangible and relatable constructions. 

Makerspaces naturally allow students to translate between the physical world and its representation using geometric models through drawing or design. At the same time, students learn that adding a strand of wire to a sculpture cannot be contained within the strictest geometrical definitions of a line, an infinitely long object with no depth or curvature. No physical circle made by nature or human can be as perfect as its definition and set of written properties, such as having a circumference consisting of equidistant points from the center. Students realize that constructions from the 2D to 3D spaces involve different mathematical processes. The exploration in a makerspace can allow students to appreciate the limitations and variations that exist between the precise geometric definitions and the physical manifestations of objects, such as a clock, which may deviate from their ideal form that students are used to in school mathematics. 

Bridging the Gap between Mathematics and Makerspaces

Despite the potential benefits, there is a lack of support for students in recognizing how the construction of artifacts in makerspaces can enhance their learning of mathematics. Collaboration between makerspace or design teachers and mathematics teachers is often limited. One challenge seems to be the unpreparedness of mathematics teachers in effectively planning and communicating ways for their students to do mathematics in the makerspace. Makerspace teachers often support other disciplinary teachers in integrating their subjects into the makerspace environment. In conversation with the makerspace teacher who developed the Kinetic Sculpture project, it was surprising to find out that among all disciplinary teachers, mathematics teachers were found to be the least likely to make use of the makerspace. It is interesting to note that in makerspace or design projects involving mathematical construction, the responsibility of communicating geometric properties to students often falls on the makerspace teacher. Additionally, the makerspace teacher revealed that opportunities for students to reflect on their physical constructions tend to be missed or overlooked. This presents a significant opportunity for mathematics teachers to design impactful learning experiences that integrate geometry and physical artifacts in makerspaces. By incorporating geometric constructions and properties into the context of a Kinetic Sculpture makerspace project, I aim to demonstrate the potential for mathematics teachers to seize these opportunities and facilitate meaningful geometry learning for their students. 

Kinetic Sculpture Construction 

In the eighth grade makerspace project titled “Kinetic Sculpture,” students engaged in constructing sculptures that express themselves through the use of ratios. Students strove to convey a harmonious, balanced, and aesthetically appealing composition while expressing their individual messages. Taking inspiration from the renowned American sculptor Alexander Calder, students were taught how his mobile  kinetic sculptures achieved balance by distributing force evenly through their linkages, remaining sometimes still but never static. The goal of this project was for students to successfully create a kinetic sculpture capable of spinning. The tasks to reach this goal were: 

1. Use precise measurement and proportions to design a kinetic sculpture that is balanced in its structure (e.g., symmetry, equality, congruence). 

2. Compare the geometric properties underlying the 2D construction and identify any properties that are not upheld in the 3D construction. How do these properties contribute to the sculpture’s construction and spinning motion? 

In these tasks, students applied mathematical principles related to measurement, proportion, balance, symmetry, and geometric properties to create a functioning kinetic sculpture that spins. They critically assessed the alignment between their design intentions and the realized 

construction, highlighting any geometric properties that may impact the sculpture’s performance. In particular, when the students constructed the 3D sculpture, they put together various layers of the sculpture (e.g., Figure 2a) to connect them at the center or axle, allowing for the spinning motion. The completed kinetic sculptures below (Figures 2b, 3, 4, 5a, 6, 7) were developed by students who participated in the Kinetic Sculpture project. Presenting these geometric constructions in the mathematics classroom allowed opportunities for students to identify geometric properties in their construction and notice the different properties that exist in the 2D and 3D construction and occur when the sculpture is mobile. In this article, I discuss how the completed kinetic sculptures can be brought back to the mathematics classroom for the mathematics teacher to verify these geometric properties work. 

A significant number of kinetic sculptures prominently feature circular elements. Some of these sculptures incorporate circular components characterized by a central point and curved spokes that extend outward in various directions, converging at a centralized location within the circle (Figures 3 and 4). In Figure 4, the design of each layer is made up of five curved spokes, and these spokes are designed to be congruent. The mathematics teacher could ask the students how they might figure out mathematically whether they are congruent. These curved spokes are constrained within the circle, and their congruence can be described using various geometric properties, such as the arc or sector of a circle. Students might notice that these curved spokes are like arcs of a circle. In Figure 3, the design of the purple layer has six petals while the white layer has eight petals. If a hexagon or octagon is drawn where the end of each congruent petal is the vertice of the polygon, are these polygons regular? In particular, in a regular hexagon, each petal is made of the arcs of congruent circles whose centers are the vertices of the hexagon. All the arcs intersect at the center of the original circle or hexagon. The same property does not hold for a regular octagon but it would be interesting to see what other properties can be discovered. 

Figures 6 and 7 also incorporate spoke-like shapes that converge at the center and extend outwards but do not reach a rim. In Figure 7, describing the jagged lines is not a straightforward task, as they possess unique characteristics. These characteristics include increasing line segments separated by jagged points. An intriguing investigation involves characterizing each line segment based on its length, direction, and angle that maximize the jaggedness without causing intersections with the other spokes. The kinetic sculptures can also be described with proportions and ratios. In particular, the concept of the Golden Ratio can be used to describe the completed sculptures. Many patterns observed in nature, such as the arrangement of leaves or petals around a stem or center, follow spiral structures that can be described using the Golden Ratio. The number of spirals, the count of shapes, or measurements of neighboring line segments in one direction often align with Fibonacci numbers (a sequence in which each number is the sum of the two preceding ones). Determining whether the features of the kinetic sculptures

adhere to the Golden Ratio would justify the balance of geometric shapes. For instance, it may provide a rationale for the dimensions of the distorted quadrilaterals in Figure 6, further enhancing our understanding of the sculptures’ geometric composition. 

Up to this point, I have explored geometric properties that describe each layer of a kinetic sculpture. However, the completed kinetic sculpture allows for reflection on what needs to be considered when transitioning from the 2D design to the 3D construction, delving into real-world applications. In makerspaces and professional contexts like architecture, carpentry, or quilting, factors such as material properties, construction techniques, measurement errors, and tool limitations introduce variations that no longer prioritize the idealized geometric properties defined by Euclidean geometry. In the context of the kinetic sculpture project, understanding and utilizing geometric properties become crucial for ensuring proper alignment and functionality, particularly when dealing with a concentric axle and multiple layers. 

One key consideration is the placement of the axle, which greatly impacts the functionality of the sculpture. Achieving concentricity, where the sculpture’s layers connect at the center, is essential for promoting balanced rotation and minimizing vibrations. When the layers within a kinetic sculpture are identical (Figure 3) or mirror images of each other (Figures 4, 6, 7), establishing concentricity is relatively straightforward. However, when different layers are involved, additional considerations arise. For example, in the construction of the kinetic sculpture depicted in Figure 5b, two distinct layers are utilized (Figure 5a). Finding the center of the oval becomes essential for determining the placement of the first layer (left image, Figure 5a), while locating the center of the circle becomes crucial for positioning the second layer (right image, Figure 5b). This example shows the practical aspects of working with non-ideal shapes and the need to adapt geometric concepts to accommodate these variations. 

Figure 2b presents another important and practical consideration when assembling the layers for the 3D construction. It is crucial to ensure that the larger circular layer (right image, Figure 2a) has a circumference with a width sufficient to accommodate axles that align with the centers of the smaller circular layers (left images, Figure 2a). This consideration highlights the geometric relationship between the layers and the need for precise alignment to maintain the structural integrity and functionality of the kinetic sculpture. By ensuring that the axles fit with the centers of the smaller circular layers, the rotational movement of the sculpture is secure and balanced. The weight of the sculpture is an additional dimension in the construction process. In Figure 5b, for example, the stars extend outward from the more stable center of the kinetic sculpture. Therefore, when constructing the sculpture, it is essential to consider the materials’ properties to ensure that the structure is robust enough to withstand the intended spinning movement. By carefully selecting materials with appropriate strength and durability, the construction can support the weight of the sculpture and maintain its structure. 

Finally, spinning the kinetic sculptures offers a unique perspective to explore and appreciate their geometric properties. In addition to the constructed sculptures, students took videos of their kinetic sculpture spinning and what these videos capture are invaluable for geometric analysis. The dynamic nature of spinning allows for the observation of patterns,

shapes, and symmetries that may not be readily apparent when the objects are stationary. The kinetic sculpture may exhibit rotational symmetry as the kinetic sculpture repeatedly displays the same pattern or shape at regular intervals. This symmetry becomes more pronounced during rotation, revealing the underlying structure and balance within the object. In addition, the concept of kinetic symmetry can arise, where an object appears symmetrical when in motion but reveals asymmetry when at rest. This dynamic interplay between symmetry and asymmetry adds depth and complexity to the understanding of geometric properties. For example, spinning Figure 4 reveals moments when the curved spokes create flower-like petal shapes, providing a visual confirmation of the design intention. Spinning Figure 2b provides a captivating opportunity to witness the intersections between the smaller circles and the circumference of the larger circle at multiple points. By adjusting the spinning parameters of the larger circle, such as its rotational speed (which can be simulated by increasing or decreasing the speed of the video) or direction (which can be simulated by fast-forwarding or rewinding the video), students can observe how these factors might cause the smaller circles to appear more tightly clustered or spread out along the circumference. This could lead to different observable patterns, such as spirals or complex curvatures, as the smaller circles interact with the rotating larger circle. As such, the construction of 3D kinetic sculptures that spin can offer interesting insights into geometric properties. While there may be deviations from the idealized or intended geometric decisions, the real and dynamic nature of these sculptures enriches our understanding of geometry in valuable ways. 

Conclusion and Implications

In conclusion, the geometric constructions in makerspaces involve a delicate balance between idealized geometry and practical considerations. While deviations from idealized concepts are inevitable, the fundamental principles of geometry guide a successful construction. Engaging with the construction in the various domains, such as 2D, 3D, static, and mobile, provides valuable insights into the adaptability and versatility of geometry. Practical considerations highlight the importance of incorporating real-world constraints and variations into geometric construction processes. By working with the variations introduced by material properties, construction techniques, and measurement limitations, students engaging in the Kinetic Sculpture project gain a deeper understanding of the role geometry plays in real-world applications and the need to adapt geometric principles to achieve functional, balanced, and aesthetically pleasing end products. 

Given the current lack of collaboration between mathematics and makerspaces, there is a need to enhance the preparation of future mathematics teachers, providing them with the necessary skills to develop meaningful geometric tasks that seamlessly integrate geometry with students’ physical creations. The tasks described in this article can be undertaken by future mathematics teachers, enabling them to explore geometric constructions beyond the confines of the mathematics classroom and encouraging the utilization of available resources or artifacts at school, which they may have been hesitant to employ otherwise. Even without access to makerspace resources, future teachers can leverage images or video recordings of kinetic sculptures, which offer an engaging method to convey geometric concepts and identify geometric properties. By embracing an interdisciplinary approach to teaching, future mathematics teachers will become better equipped to establish connections between various fields like architecture, engineering, and prototyping, thereby developing a more holistic understanding of constructions rooted in geometry.

By incorporating hypothetical mathematical tasks that simulate the construction process, mathematics teachers can significantly enhance students’ understanding of geometry within various learning contexts. These tasks provide avenues for exploring the intricacies of geometric construction in makerspaces, where physical limitations may constrain actual construction. The constructions in geometry require a rigorous understanding and application of geometric principles. Success lies not only in the physical construction but also in the ability to articulate the geometric properties that exist within the constructions. Engaging with such tasks provides a deeper connection with geometry, fostering analytical, creative, and systematic thinking among students. As students navigate the construction process, transitioning from 2D to 3D and from static to mobile structures, they gain an appreciation for the importance of precision, adaptability, and the interplay between idealized geometry and practical considerations. This preparation aims to equip students for future endeavors where they may encounter complex geometric problems, requiring them to make informed decisions based on both mathematical principles and real-world opportunities.

Resources

Bringuier, J. C., & Piaget, J. (1980). Conversations with jean piaget. University of Chicago Press. 

Hansen, A. K., McBeath, J. K., & Harlow, D. B. (2019). No bones about it: How digital fabrication changes student perceptions of their role in the classroom. Journal of Pre-College Engineering Education Research (J-PEER), 9(1), 6. 

IdeelArt. (2016, August 8). Alexander Calder mobile art and its many forms. IdeelArt. https://www.ideelart.com/magazine/alexander-calder-mobile

MoMA. (2023). Alexander Calder. MoMA. https://www.moma.org/artists/922 

Papert, S., & Harel, I. (1991). Situating constructionism. constructionism, 36(2), 1-11.