||This study reanalyzed a part of the national data of the responses of 288 students in grades 6 to 8 on the “One-Variable Linear Equation Virtual Situational Test” collected by Professor Pao-Kuei Wu from August 1, 2001 through July 31, 2003. The analyses were based on the “One-Variable Linear Equation Conceptual Tables”. The results of the analyses are the following.|
I. The use of variables
A. Compared to 7th and 8th graders, 6th graders would first solve the numerical arithmetic and solve the unknown parts next. But if the students could not handle the unknown parts, the 6th graders tended to ignore or even not list the unknown variable in the equations.
B. When encountering the unknown situations, most 6th graders are not accustomed to using symbols to represent unknown variables. Instead, they would observe the numerical components first to try to deduce what the unknown variable would be, and proceed from there. Some students would even set up some constants to represent those unknown variables. These results indicate that the 6th graders’ ability to use symbolic representation is still in the beginning stages.
C. In the unknown virtual situations, the majority of 7th graders were able to use symbolic representations. However, most of them would use pictorial representations such as □, instead of alphabetical representations such as x, y and z. Moreover, many students use the same symbols to represent different variables; this shows that although the 7th graders know to use symbols to represent unknown variables, they still are not able to fully comprehend unknown variables. Hence, the 7th graders’ ability to use symbolic representation is in the transitional stage.
D. When encountering unknown virtual situations, the majority of the 8th graders would able to use the numerical symbols such as x, y and z to represent the unknown variables. The frequency of using pictorial representations such as □ becomes less and less, and the tendency to use the same symbols to represent different variables is decreasing. All these indicate that the 8th graders’ development of the concept of unknown variables is maturing.
II. The concept of problem solving
A. The 6th graders’ ability to use symbolic representation is still in the beginning stages:
1. They only deal with the simple part; for the more complicated part, they chose to ignore.
2. Due to their immature development of symbol representation, when encountering the two variable linear equation problems, they even do not have the ability to write the ‘complete’ equation, not to mention to solve the equations.
B. The 7th graders’ ability to use symbolic representation is in the transitional stage:
1. Compared to the 6th graders, the 7th graders are more able to draw relationships among the different components of the problem.
2. The fact that the substantially decreasing proportion of 7th graders conceiving the unknown variable as a certain numeric compared with 6th graders means that the 7th graders have deeper recognition of unknown variables.
3. When encountering ‘simple’ two-variable linear equation virtual situations, some 7th graders can translate at least one condition into an equation. This result shows that the 7th graders have developed some ability to translate the conditions embedded in the virtual situation into some equations. But when the situation gets more complicated, due to conception immaturity of solving two equations simultaneously, the 7th graders either solve each equation independently, or mess up and tangle the clues of all the conditions together. Moreover, they would use the same symbol to stand for different variables.
C. The 8th graders’ development of the concept of unknown variables is maturing:
1. Most of the 8th graders can use the clues of all the conditions in the virtual situation in a sufficient way.
2. Only a few 8th graders would use the same symbol to stand for different variables during their problem-solving procedure. This result indicates that the ability to use the symbolic way to represent unknown variables is more mature among the 8th grade students.
3. When encountering two-variable linear equation virtual situations, the 8th graders can formulate two independent equations and solve them simultaneously. This result shows that the 8th grade students possess more profound skills to solve two-variable linear equations.
III. Proportion of answering questions correctly:
In general, for simpler virtual problems, there does not exist many differences among grades. Whereas, for the more difficult virtual problems, the 8th graders outperform the 7th graders, and the 7th graders, in turn, outdo the 6th grade students.