Discussion

This study majorly analyzed the difficulty factor of the cube enumeration test and whether using it to teach students was effective or not. The results showed that invisible cube number and integrity of cubes would affect item difficulty. Moreover, using the system of cube enumeration test did improve the students’ ability of spatial orientation, spatial visualization and volume calculation. In the following, we had a further discussion on the above-mentioned results.

The difficulty of the cube enumeration test was first discussed. This study found out that when using a screen to implement the cube enumeration test the item difficulty was lower, and this was different from the result of the literature review. After interviewing the teachers and students, this study discovered that items displayed on a screen were clearer. Items displayed on a screen could provide 3D visual effect for the students more and they were clearer than those in paper. In addition, it was easy for the students to have wrong judgment because items on paper couldn’t be printed larger due to paper limit and there were only black and white colors. Hence, the difficulty parameter obtained from the computerized cube enumeration test was higher than that of a paper-pencil test and thus the accuracy rate was higher also.

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Next, rotation would be discussed. This study found out that if items could be rotated, then they would be simpler and difficulty parameters became higher. This meant that the factor “rotation” greatly reduced the difficulty of items and therefore proved the argument of the above-mentioned literature. One’s cognitive process of the cube enumeration test was to build up spatial orientation among cubes, produce spatial visualization, and further produce a 3D figure of an item in one’s mind. By calculating the number of cubes in the mind, one could answer the question. It indicated that if items of the cube enumeration test could be rotated, then this would help the examinees to develop their spatial orientation and visualization and finally the ability to calculate the total volume (total cube number).

Hence, the researcher regarded that if items could be rotated then they would become simpler and couldn’t measure their spatial ability any longer. However, if we applied this concept in teaching then we could assist students to further develop their spatial ability and ability of volume calculation. Therefore, this study also developed a learning system which could display 3D objects on a 2D screen, allow students to rotate cubes through image processing technology, and further improve their spatial ability.

In addition, integrity of cubes and invisible cube number defined by this study also assisted the researcher to produce the difficulty formula of the cube enumeration test. Besides, the fact that invisible cube number affected difficulty also proved that difficulty of cube enumeration test would be affected by invisible cube number in the literature review. Integrity of cubes defined by this study also verified the argument of cube integrity and that students used layers to calculate in the literature review. This study discovered that invisible cube number didn’t have much effect as integrity of cubes did. After discussing with the teachers, the researcher thought that invisible cube number in this study was less and hence the effect would be less too considering the results of the test. Besides, if we want to study invisible cube number, then we need items with more total cubes. However, if there are too many cubes in the items, human can’t necessarily answer the questions although the system can generate these items automatically.

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In addition, when there were many cubes in items, integrity of cubes would possibly become lower although there were more invisible cubes. For instance: an item with 4 cubes in height, length and width and an item with 4 cubes in height but only 1 cube for the base, they should be consistent. However, their invisible cube numbers were greatly different. This was also different from the above-mentioned literature on invisible cube number. Therefore, for the effect of invisible cube this study will do more investigation in the future.

The VIB produced by this study was verified and its security was also proved.

When we used the VIB to select items, those items wouldn’t have the problem of item exposure. This also proved that the VIB system could calculate the examinees’ spatial ability; test constructors could use the VIB to implement tests and would obtain the examinees’ performance of spatial ability and volume calculation ability.

Besides, this study used the feature of “rotate” to build up a learning system which was used to teach the students cube stacking and calculate them. As what was mentioned before, this study improved the students’ ability of spatial orientation, spatial visualization and volume calculation. This also meant that using a 2D computer screen to display rotatable 3D objects made students better understand the relative position among all cubes and how to calculate volume.

After summarizing the above discussion, this study first invited some experienced teachers to construct cube enumeration tests. These two tests assisted the study to derive the item difficulty formula and proved that “rotate” function could greatly reduce the examinees’ difficulty to do spatial orientation and visualization. Therefore, in addition to using the item difficulty formula to construct a VIB, this study also used

“rotate” function to build up a learning system. There wasn’t any risk of item exposure of the VIB, and the learning system also assisted the students to improve their spatial ability. These results significantly helped the test.

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