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The effects of 3D-representation instruction on composite-solid surface-area learning for elementary school students

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Abstract

Providing instruction on spatial geometry, specifically how to calculate the surface areas of composite solids, challenges many elementary school teachers. Determining the surface areas of composite solids involves complex calculations and advanced spatial concepts. The goals of this study were to build on students’ learning processes for basic and composite solids and employ Google SketchUp, an Internet resource tool, to develop and implement surface-area instructional and learning strategies (SAILS) for composite solids, and then measure its effect on learning achievement and attitudes. The fifth-grade students (N = 111) who were enrolled in this study were divided into an experimental and a control group. The experimental group (N = 56) received SAILS instruction, whereas the control group (N = 55) received traditional instruction. The results indicated that students who received SAILS instruction exhibited better performance on both immediate and maintained surface-area learning achievement tests compared to those who received traditional instruction; furthermore, this effect was more prominent among boys than girls. Low- and moderate-ability students who received SAILS instruction exhibited significantly greater improvement of attitudes toward learning mathematics compared to those receiving traditional instruction with physical teaching aids.

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Acknowledgments

The author appreciate the funding supports from the Ministry of Science and Technology, Taiwan (102-2911-I-003-301; 102-2511-S-003-001-MY3; 101-2511-S-003-058-MY3; 101-2511-S-003-047-MY3; 99-2631-S-003-003).

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Authors and Affiliations

  1. Department of Educational Psychology and Counseling, National Taiwan Normal University, No. 162, Sec. 1, Heping E. Rd., Da’an Dist., Taipei City, 106, Taiwan

    Yao-Ting Sung

  2. Graduate Institute of Information and Computer Education, National Taiwan Normal University, Taipei City, Taiwan

    Pao-Chen Shih & Kuo-En Chang

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  1. Yao-Ting Sung
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  2. Pao-Chen Shih
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Correspondence to Yao-Ting Sung.

Appendices

Appendix A: The procedures, strategies, and SketchUp functions used in the SAILS program

Procedure 1

Establishing a link between 2D and 3D spatial structure representations. Basic solids (e.g., a cuboid) can be illustrated in a 2D plane. However, clues regarding the top, side, and perspective views are lacking, and may thus prevent children from understanding the conversion between plane figures in 2D space and solid objects in 3D space (Battista 1999; Piaget and Inhelder 1956; Piaget et al. 1960). This also increases the difficulty of transforming 3D images into actual 3D mental models. Therefore, the first essential task is to help students convert 2D graphical representations into 3D mental representations.

There are two instructional strategies in the first procedure. The first, Strategy 1A, involves the visualization of 3D models. Here learners are guided to build a 3D object (Fig. A1) based on 2D or physical object images by using the modeling function of Google SketchUp. They could also employ three colors to represent the three-axis planes in order to identify the sides/faces of the cubes. Strategy 1B includes the coordination and structuring of 2D and 3D features. Using functions such as dragging and rotating, hiding planes (Fig. A2), and X-ray (Fig. A3), learners can manipulate the 3D model and perform tasks such as rotation, displaying side and top views, and hiding axis planes. In this way learners establish the correspondence between points, line segments, and side relationships for both 3D and 2D physical objects.

Procedure 2

Decomposing, calculating, and recomposing the area of each side of the basic solid. Determining the surface area of a basic solid involves the calculation of multiple areas. Therefore, children must first decompose the sides/faces of the basic solids and grasp how many different sides/faces there are as well as their differences and similarities. They should then calculate the area of each side/face using Battista’s enumeration method or by applying a general formula, and then sum (re-compose) the areas of multiple sides to arrive at a surface-area value.

There are four instructional strategies in the second procedure. Strategy 2A is unfolding objects and counting the number of sides. Using the frame function options, 3D images can be converted to show top, bottom, front, back, left, right, and isometric views, enabling learners to understand the views of each side of the cube. By unfolding the basic solids, learners can determine the number of sides and side shapes for these solids, as well as the symmetric or corresponding sides (Fig. A4). Strategy 2B is enumeration and calculation. Learners first identify the corresponding or symmetric sides and conduct enumeration or use formulae to calculate the area (Fig. A4). Strategy 2C is recomposing and summing the areas of all sides/faces. Students sum the total surface areas of the six sides of the cube. Finally, strategy 2D is the verification of the calculation results. Learners compared the results they obtained with those provided by Google SketchUp through the calculation functions.

Procedure 3

Restructuring the basic solid into a composite solid to establish a link between basic- and composite-solid representations. Composite solids can be considered as composites of basic solids (Figs. A5, A6, A7), which means that composite solids are deformations resulting from the addition or subtraction of basic solids. However, the similarities and differences between mental models for basic and composite solids can only be understood through unit filling, shifting and conservation.

There are two instructional strategies in the third procedure. The first, strategy 3A, is unitization by cubes. Using unit cubes (in which each side equaled 1 in length, see Fig. A5), basic cuboids are decomposed as composites of unit cubes (Fig. A6). Using the coloring function, learners color the corresponding three pairs of sides red, yellow, and blue. Using the virtual building blocks function, learners form a basic cuboid by copying and pasting unit cubes. Strategy 3B is re-organizing and re-locating. Cube units are added to or subtracted from the basic solid to produce composite solid (Fig. A7). The relationship between the location of the basic and composite solids is also identified.

Procedure 4

Decomposing, calculating, and recomposing the surface area of each side of the composite solid. Similar to Procedure 2, in which the surface of a basic solid should be decomposed into several sides, and then recomposed into the surface area of each side, composite solids must first be decomposed and then converted into a composite of one or more basic-solid to facilitate the calculation of each basic solid surface areas. After the surface area of each piece is calculated, the values are summed to obtain the surface area of the composite solid. In addition to possessing the ability to complete this step, children must also know not to repeat the calculation of shifted or filled-in surface areas.

There are four instructional strategies in the fourth procedure. Strategy 4A is identifying the location and number of the sides/faces. The number of sides of the composite solid is verified by observing the rotational, top, side, and perspective views. The similarities and differences between the shapes of the sides, as well as their relationships, are also verified. Strategy 4B is calculating the surface area of basic-cuboid sides by filling, shifting and integrating. The filling function of Google SketchUp is employed to fill up the holes in the concave side of a concave object and deform the concave into a conglomerate of basic solids (like Fig. A6); the shifting function is employed to remove the protrusion in the convex side of a convex object and dismantle the convex into basic solids (like Figs. A6, A7). The processes and end-results of filling and shifting will form dynamic images which are helpful for students’ understanding of the differences and similarities of the original and deformed solids and their corresponding surfaces. The total surface area of the basic solid is subsequently calculated. For example, after the unit cube in Fig. A7 was shifted, the surface area of the yellow side in Fig. A8 comprised 3 × 3 = 9 units of area.

Strategy 4C is calculating the surface area of non-basic-cuboid sides. In addition to the basic-cuboid sides, the surface areas of the concave or convex sides are calculated. For example, in addition to the top-view of Fig. A6, both the red (Fig. A9) and green (Fig. A10) sides comprised (2 × 3) + 1 = 7 units of area. Strategy 4D is recomposing and summing the surface areas of all sides: The total surface area of the composite solid can be obtained by recomposing the surface area of the basic solid and that of the concave/convex composite. For example, the areas of the three colors in Fig. A7 can be summed as (9 + 7 + 7) + (9 + 7 + 7) = 46. After the calculation, learners verify the results they obtained with those provided by Google SketchUP through the calculation function.

See Table 6.

Table 6 Figures of Google SketchUP functions
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Appendix B

See Table 7.

Table 7 Exemplar items of the composite-solid surface-area achievement test
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Appendix C

See Table 8.

Table 8 Summary table of the four-factor mixed design ANCOVA for the CSSAAT
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Appendix D

See Table 9.

Table 9 Summary table of the four-factor mixed design ANOVA for the MaLAS
Full size table

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Sung, YT., Shih, PC. & Chang, KE. The effects of 3D-representation instruction on composite-solid surface-area learning for elementary school students. Instr Sci 43, 115–145 (2015). https://doi.org/10.1007/s11251-014-9331-8

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