Cube enumeration

This section was the discussion on literature related to the abilities evaluated by the cube enumeration test. Besides, there was also a simple investigation on the factors that would affect item difficulty of the cube test expecting that those would assist the evaluation of item difficulty in this study.

Cube, or regular hexahedron cube, can be used to introduce the concept of volume as the basic unit for volume. Previous studies show that enumerating cube arrays are used to measure or build the mental image and manipulation capability of 3D geometric objects.

In the past, most studies related to cube enumeration test used 2-D cuboid figure stacked by cubes, and those researchers mostly explained the examinees’ cognitive process by the perspective of spatial orientation. Ben-Haim (1985) regarded that the examinees’ major difficulties were whether they understood these 3D figures on 2D planes and whether they could consider invisible cubes or not when solving the questions. This cognitive process involved with their ability to understand relative spatial relationship among all cubes and to create mental images appropriately, and this was categorized as spatial orientation ability.

The serial studies of Battista and Clements (Battista, 1999; Battista& Clements, 1996, 1998) pointed out how one coordinated and integrated messages coming from different viewing angles and then formed the mental image of a 3D object in one’s mind when structuralizing an object’s position. This mental image could be stored as an object that could be manipulated mentally in the future. Generally speaking, the cube enumeration process that Battista and Clements investigated still belonged to spatial orientation. How to understand 3D figures represented by 2D ones, form mental image of 3D objects, and perform spatial orientation on each cube were the spatial abilities that a cube enumeration test tried to measure.

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In addition, some scholars indicated that factors affecting the sixth graders’ cube enumeration were their abilities to form correct mental image of invisible cubes and group scattered cubes by manipulating and moving them and finally calculate these groups by the volume formula and add them. The former - the production of mental image - was spatial orientation and the latter one – psychological manipulation - was spatial visualization, and calculation ability was related to mathematical ability of volume calculation. Besides, the fifth graders’ mathematical ability of volume calculation was measured by a cube enumeration test in Taiwan’s elementary school curriculum and the teachers also used cube enumeration to teach the concepts of volume. This also demonstrated that the teaching of cube test could be used in the teaching material of mathematical ability of volume calculation. After summarizing the above literature, it was found out that the abilities measured by a cube enumeration test were closely related to the examinees’ spatial orientation, spatial visualization and mathematical ability of volume calculation.

Next this study described the factors affecting the difficulty that the examinees considered and their strategies when enumerating the cubes expecting that those could provide important reference for difficulty evaluation in this study. Students often consider that cube enumeration test is difficult (Battista & Clements, 1996). For example, in a research done by National Assessment of Educational Progress (NAEP), the hit rate for cube enumeration in cuboids was lower than 50% for students at the age of 9, 13 and 17. The results of this research indicated that cube enumeration tasks, even in regular cuboids, are difficult even for older students (Ben-Haim , Lappan &

Houang, 1985).

In order to design a spatial visualization ability test and examine the effects of instruction intervention, Ben-Haim (1985) referred to the option design of cube enumeration item of Michigan Educational Assessment Program in 1983. Each item was composed of a 2D cuboid stacked by cubes, and the test asked students to select the cube amount that was needed to form the cuboid. The designing principle of the other four distracters were (1) visible surface amount, (2) multiply the visible surface amount by 2, (3) visible cube amount, (4) multiply the visible cube amount by 2.

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Ben-Haim et al. (1985) regarded that there were two major difficulties for the students; one was that whether they could realize 2D figures by a 3D way, and options (3) and (4) were cube amounts and students selecting these two options could realize that 2D figures represented 3D objects while students selecting (1) and (2) couldn’t.

The other difficulty was that whether the students could consider invisible cubes;

options (2) and (4) were to multiply visible surface or cube amount by 2, and therefore the sixth graders selecting these two options could consider invisible cubes while the sixth graders selecting (1) and (3) couldn’t. That study conducted the instruction experiment on the fifth to eighth graders in 3 areas (there were more than one hundred to 4 hundred students in each grade); before the instruction, these students’ correctness rates respectively were 25%, 40%, 45%, 50% and the result generally agreed with the result of NAEP(1977~1978) (47%), the survey on 127,420 seventh graders. Their selecting rates of the wrong answers from (1) to (4) respectively were 17%, 22%, 8%, 8%.

Battista and Clements had conducted a series of cube enumeration study and proposed that spatial structuring, mental model and scheme were related to the reasoning of cube enumeration (Battista, 1999; Battista & Clements, 1996). Spatial structuring was the process of constructing a 3D object, and the process determined the shape or nature of an object by identifying and integrating each component. This process included constructing units, constructing the relations among units (such as relative position) and cognizing that correct repetition of units could produce the whole;

for instance, repeating “row” could produce the whole cuboid. Mental model was the nonverbal and lively mental image activated by physical situations and the image could be used by an individual to explain or reason in the process of doing or thinking.

Scheme was a set of organized sequence of action or operation, abstracted from past experiences, and it enabled people to do the same response in similar environments.

Battista and Clements (1996) pointed out that there were 4 development stages for the students to construct a 3D cuboid in their minds. Stage 1 was medley of viewpoints which meant that the students only paid attention to a surface at one time and then

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pieced viewpoints together to perceive a 3D structure; the spatial organization was partial.

In stage 2 composite units, the students could form composite units and then processed them psychologically but their composite units weren’t necessarily appropriate; it would be more appropriate for students to organize by small 3D units than by 2D units. However, even though some students could tell the difference of

“square” that they saw and “cube” that they need to enumerate, the complicated process of cube surface enumeration always made them forget to pay attention to the difference particularly when they enumerated cube surfaces near the edges because different surfaces represented the same cube, and therefore they needed the ability of stage 3 coordination to coordinate different angles caused by the cuboid surfaces such as the coordination of right angles of the front and the right side and the top and the left side; they would do the enumeration right when they understood that the cube surfaces on the edges represented the same cube.

In stage 4 integration, students had to collect all separate viewpoints to form the whole picture of the object and this stage included two psychological mechanisms. The first process was recollection which enabled an individual to activate the mental models in the past experiences. Facing an object, an individual first compared the different viewpoints of the realistic object and the existing mental models to activate the appropriate mental model. If there was no suitable mental model, then the individual would start to perform a series of integration and transformation to produce new psychological objects through the second psychological mechanism – scheme.

Battista and Clements (1996, 1998) had interviewed with 45 third graders and 78 fifth graders for their enumeration performances; the materials were figures of cuboids stacked by cubes and they found out 5 different strategies. In strategy 1, the students thought by the unit of horizontal or vertical “layer” of the cuboid and the possible methods of enumerating “layer” were doing enumeration one by one or repetitive accumulation or multiplication, and students using this strategy had the highest correctness rate. In strategy 2, they saw the cuboid as a filled-up space and would try

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to enumerate the cube amount inside and outside the cuboid; they may use “row” as the unit to enumerate cubes or they may use unsystematic and random method to do it, but usually the error rate was more than 50%. In strategy 3, students could only enumerate the cubes on the visible surfaces of the cuboid and would ignore the cubes inside the cuboid, and they often would doubly enumerate the cubes on the edges of the cuboid, and hence the error rate was high. In strategy 4, although they could use the volume formula L by W by H to calculate, some of them couldn’t explain why they had to multiply these 3 numbers and hence this didn’t indicate that they understood its meaning. Strategy 5 referred to the strategies other than the above four ones including misusing the volume formula; for instance, they used the wrong numbers of all sides of the cuboid for the multiplication.

Battista and Clements (1996) found out that the students who could do enumeration by “layer” had a higher correctness rate - about 60% of the fifth graders and lower than 20% of the third graders. For those 3 items, there was only 29% and 7% of them who could use this strategy to correctly answer the question for the fifth and third graders. This showed that it still would be difficult for the fifth graders to do enumeration by “layer”. On the contrary, about 60% of the third graders and 20% of the fifth graders used strategy 3 - only calculated the cubes on the cuboid surfaces. It was quite common for them to do repetitive enumeration; 64% of the third graders and 21% of the fifth graders did it at least once, and 33% of the third graders and 6% of the fifth graders did it in all items.

Battista and Clements (1996, 1998), Ben-Haim et al. (1985) and Olkun and Knaupp (2000) used cuboids stacked by cubes or frameworks formed by length, width, and height as the materials, and their purposes were to investigate how individuals consider invisible cubes. Nevertheless, more items were cuboids stacked by cubes

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irregularly in IQ tests (Chiang, 1984; Luh, 1999; Eliot, 1980) or math courses related to cube enumeration. In Taiwan’s math courses, students had already learned the contents mentioned in the study of Battista and Clements before the sixth grade in order to build up the ability of volume concept and measurement and hence they could know the meaning and calculation of cuboid volume by the stacked cubes. According to the studies of Battista (1999) and Olkun and Knaupp (2000), this type of course could assist most students to develop the ability of spatial orientation (or spatial structuring), that is, the ability to tell the difference of cube and square on surfaces, transform 2D figures to construct 3D objects and identify the relative positions of each cube.

The above paragraphs were literature review related to measurement theories, CBT, AIG, VIB and cube enumeration. Ideally, what this study had hoped to conduct was applying IRT theory to estimate item difficulty of VIB and develop a VIB-based computer adaptive cube enumeration test (VIB-based CAT) system. However, practical experimental results indicated that the computing load needed in dynamically estimating IRT-based item difficulty was too heavy to develop an operable VIB-based CAT system. Therefore, this study used CTT to estimate item difficulty and developed a VIB-based CBT instead. However, the experiment results of implementing IRT to estimate item were still provided in the Chapter four.

This study would invite some experienced elementary school teachers to assist the tool construction process and interview with some students in order to prepare for the development of the research tool. For the mechanism of generating CBT item, the researcher would also refer to the distractor generation mechanism from the studies related to AIG and generate the distracters suitable for a cube test. Besides, this study also referred to the distractor generation mechanism of VIB to avoid generating hard-to-answer options. In addition, this study also referred to the studies related to figural tests of AIG and their methods of designing and system construction to construct a test system that would be used in this study. Finally, this study would refer to the studies related to factors affecting cube enumeration difficulty including the number of invisible cube and item type … etc. and then analyze the factors affecting difficulty of

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the cube enumeration test and construct the difficulty formula of the cube enumeration test. Then, the study used this difficulty formula to build up a VIB and construct a research tool that this study needed by referring to the system generated from AIG.

After analyzing literature related to AIG, it was found out that using AIG to generate a learning system had a good performance. Thus, this study also would consider learning system, analyze the characteristics of cube enumeration, and research how to use cube enumeration as a learning tool. Additionally, this study would design an appropriate research tool in order to find out whether using this learning tool had a good learning performance or not by referring to the abilities that cube enumeration could measure and elementary school students’ learning activities.

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