Keywords

1 About 3D Wooden Puzzles

Three-dimensional (3D) wooden puzzles are produced worldwide. Most are solid shapes based on cuboids and cubes. Since wood shrinks as it dries, the dimensions begin to differ from the designer’s original intention as the months and years pass after the user’s purchase. Above all, when fitting pieces of a wooden puzzle together, parts can be interlocked by employing an approach similar to configuring traditional wooden joints. However, while interlocking wood into complex wooden joint configurations may be appropriate for buildings not intended to be taken apart, it is not suitable for a 3D wooden puzzle intended as a toy. Therefore, wooden puzzles are assembled with simple wooden joints, allowing them to be disassembled repeatedly, and many are designed so that they can be easily put back together. However, as seen in Figs. 1, 2 and 3, most wooden puzzles are designed with simple wooden joints. Although some wooden puzzles have fewer parts, many are difficult to solve [1–3]. The author noted this fact during his experimental observation in 2012 [4]. One reason assembly is difficult is that while wooden puzzles possess the functions of a puzzle, they are often used as objets d’art.

Fig. 1.
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3D wooden puzzle designed by Gregory Benedetti

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Fig. 2.
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3D wooden puzzle designed by Hirokazu Iwasawa

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Fig. 3.
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3D wooden puzzle designed by Bill Cutler

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There are 3D wooden puzzles, such as the one in Fig. 4, that blend the functions of a toy with those of art and design education. This puzzle is made of resin, and magnets hold the parts together, without wooden joints [5]. Painted in three colors, it is devised such that children can easily assemble it according to the shape of the pieces and their color. With this puzzle, children use its given form. As mentioned above, puzzles are mainly “3D wooden puzzles intended for adults” and “3D wooden puzzles incorporating educational functions intended for children.” This study aims to develop 3D wooden puzzles so that children can learn to calculate volume while having fun, utilizing shapes of wooden joints mostly built with straight surfaces such as cuboids and cubes, which are characteristic of 3D wooden puzzles. This paper includes basic observations in order to gain necessary information to effectively use wooden puzzles as teaching materials.

Fig. 4.
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3D resin wooden puzzle for educational purposes

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2 Teaching Children to Calculate Volume: The Current Circumstances

In Japanese schools, the two methods of instruction for teaching children to measure volume are: “Instruction measuring volume with an instrument and expressing the volume” and “Instruction determining volume based on calculations.” Furthermore, there are two subcategories under the latter: (1) learning formulas to determine the volume of a solid (such as cuboids, cubes, cylinders, pyramids and spheres); and (2) calculating the volume of complex figures that represent a combination of basic shapes (cuboids and cubes).

Students learn to calculate volume in fifth grade (elementary school) when they learn the concepts and units for measuring volume. During this time, they learn the concept of 1 cm3, and quantify volume based on the number of 1 cm3 cubes, which leads to the formulas for the volume of cuboids and cubes. In fifth grade, when students recall what they previously learned about area, they discover that volume can also be expressed in terms of the number of universal units. As a result, they are able to grasp this area of learning and compute volume in an integrated way. During instruction on volume, exercises are devised to find out how many 1 cm sided cubes it takes to fill a cuboid or cube. Through these exercises, students learn formulas such as the following: “volume of a cuboid = length × width × height” and “volume of a cube = side × side × side.” In 2001, the National Institute for Educational Policy Research conducted a survey presenting problems to sixth grade students, asking whether they could “express the volume of the solid as a formula.”

The percentage of correct answers was 79.5 %. The report of the Curriculum Implementation Survey for elementary and junior high schools pointed out that, “One example is actively adopting operational and experiential arithmetic activities, such as ones where solids of the size of a unit are prepared; students construct various cubes and cuboids by actually stacking and arranging them, then find their volumes. Students can understand the meaning of units and how to measure volume. Creating instruction that enhances students’ feel for the size of volume is important.”

The question on volume in the elementary school arithmetic section for the upcoming 2014 national survey is as follows. While the question is different than the one from 2001, which asked about units and measurement, the percentage of correct answers was 81.3 %. We can see that approximately 20 % of students were still unable to understand how to quantify volume. While the preceding manuscript mentions the current situation, whereby 20 % of students do not understand how to calculate volume, we believe this is due to inadequate teaching materials for calculating the volume of 3D objects. Textbooks are two-dimensional (2D), but in practice, students have to imagine 3D objects for calculations. While it is important to also learn problems from the textbook that are simply made 3D, in the next chapter “Observations,” we decided to see what types of teaching materials were possible by combining the aspects of “easy to understand models” and “fun to assemble” for learning how to measure volume.

3 Observations

In order to understand the manner in which wooden puzzles balance the combination of the two aforementioned elements (“easy to understand models” and “fun to assemble”), we carried out basic observations on February 18 and 19, 2014.

We used three types of wooden puzzles for the observation, as shown in Fig. 5; they were made of ABS resin with a 3D printer and designed by us. Puzzle A was a wooden puzzle with pictures printed on the ABS resin. Puzzle B was painted such that the surfaces of pieces that come into contact with each other at the joints were the same color. Puzzle C was made of white ABS resin with a 3D printer and lacked any images or colors.

Fig. 5.
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The 3D wooden puzzles used during observations

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However, the number of pieces and assembly methods of the three types of puzzles were designed based on the author’s previous experience of the puzzle having nearly the same degree of difficulty. A professional designer performed the data design, based on 3D CAD. However, as seen in the comments in Tables 1, 2, 3 and 4, some students perceived difficulty due to size.

Table 1. February 18, 2014, first observation
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Table 2. February 18, second observation
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Table 3. February 19, first observation
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Table 4. February 19, second observation
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Over two days, third and sixth grade students were each given 15 min to put the puzzles together, for a total of two times.

Time and other details were as follows. We observed different students on both days. While third grade students have not yet learned how to measure volume, sixth grade students already have.

February 18, 2014:

  • “10:15–10:35 3 third grade students”

  • “12:55–13:10 3 sixth grade students”

February 19, 2014:

  • “10:15–10:35 3 third grade students”

  • “12:55–13:10 3 sixth grade students”

The three students each assembled all of the 3D puzzles (i.e., A, B, and C). We measured the time required for assembly. Upon completing the 3D puzzles, we interviewed the three students and asked them which puzzle was the easiest, which was the most difficult, and their reasons.

As C was clearly difficult, we stopped the students once 5 min had elapsed and then had them assemble A or B.

As a result of the observations, we found that Puzzle A, with the printed pictures, had a low degree of difficulty, except during the first observation on February 18, 2014. Additionally, we noted that even when presented with a complex structure, if students were able to recognize the rule that the surfaces of pieces were the same color where they meet, they were able to assemble the puzzle easily. However, since students found assembly difficult for shapes with deep joints (described by students as being “rugged,” i.e., uneven, in shape), we believe that the color provided a helpful hint when selecting joints.

4 Future Developments

Unlike planar materials, the size of teaching materials for measuring volume is important to students (i.e., when holding the object in one’s hand). Thus, the size of wooden puzzles must be determined from an alternative ergonomic perspective. We plan to create models resembling the same 3D objects found in textbooks, and perform an experiment on the differences between students who use 3D teaching materials versus those who learn by using traditional 2D textbooks.

The reason behind having professional designers devise the data of the 3D CAD was to understand under which types of processes modeling would occur. According to the designer, shapes were constructed by combining cubes or cuboids with each other, depending on the shape’s degree of difficulty. Another method was to take a large cube or cuboid and subtract cubes and/or cuboids from it. These processes are the same as the basis for calculating volume that children learn in textbooks, as shown in Fig. 6 [6]. This figure shows that in textbooks, there is not just one method for calculating volume; rather, there are several. In addition, the 3D CAD data have all been recorded. In the future, we believe the 3D-CAD can be put to practical use as an application on tablets; for example, as shown in Fig. 7.

Fig. 6.
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Methods for solving volume calculation as shown in textbooks

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Fig. 7.
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Example of an application of volume calculation

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