Background
Five high school students were asked to participate in this assessment. All participants wished to remain anonymous, but all of them had common characteristics: they were aged over 16 years. Furthermore, they were not currently enrolled in Mathematics programs and had never taken the Mathematics course at Chaminade University. Participants were invited to complete three simple problem-solving tasks focused on the topic of area and perimeter.
Reflecting on the Task Involving a Composite Figure
In the first task, students were required to find the area and perimeter of a composite plane figure consisting of two rectangles. Although the problem was simple, not all participants arrived at the correct answer. In particular, Participant 4 broke down the task into three sections, each with separate calculations, and arrived at the wrong solution for the perimeter length. Furthermore, Participants 4 and 5 completed only half of the task since they did not calculate the area of the figure.
As for those participants who produced the correct result, they used similar approaches to problem-solving. For example, Participants 1 and 2 calculated the perimeter using the formula for the rectangle: P = 2L + 2W, in which W is the width and L is the length. The third and fifth students used a similar approach but supported it by explaining that the opposite sides have the same size because the figure has right angles.
Among those participants who found the area of the shape, all used a uniform approach of dividing the composite figure into two simple figures. The difference is only in the amount of visual aid needed for each student. For example, Participant 2 seemed to divide the figure in mind because the assignment sheet did not show any supporting drawings. In contrast, the first participant drew two rectangles separately and noted the lengths of their sides on the drawing, while the third student drew a dashed line that separated one simple shape from the other.
Reflecting on the Task Involving a Triangle
Further, students were asked to find the perimeter and area of the acute-angled triangle. The lengths of all three sides and the altitude of the triangle were known. All participants gave correct results of the perimeter and used the same approach to arrive at the solution: they added the lengths of all three sides of the triangle. As for the area of the figure, only the first three students managed to complete the task, while Participant 4 did not find the area, and Participant 5 arrived at the wrong answer. Notably, the fifth student could have received the correct result because the formula for calculating the area was right. However, the student misunderstood the drawing and thought that the base length was 15 instead of 12, which led to the incorrect outcome. Eventually, this participant concluded that the base could not be 15 by checking this assumption with the Pythagorean theorem. Despite this, the student did not try to find the correct value of the triangle’s area.
Those students who successfully accomplished the task utilized a similar approach to problem-solving. They created a rectangle, the length of which was equal to the triangle’s base, and the width was identical to the triangle’s altitude. After that, they found the area of this rectangle and divided it by two, arriving at the correct answer. The only difference between the participants’ approaches is that the first and the third students actually drew the rectangle on paper to assist themselves in calculations, while the second participant did this in mind.
Reflecting on the Task of Finding the Side Length
In the third task, students were asked to find the length of the missing side of the rectangle, given that the length of one side and the perimeter of the figure were known. This task was successfully completed by four of the five students. Participant 4 seemed to confuse the concepts of the perimeter and the area because this student divided the rectangle into 20 parts while, according to the assignment, 20 units were the perimeter of the rectangle.
Further, the first and the fifth participants used the perimeter formula to formulate an equation and find the solution to the problem. After arriving at the result, both students completed a check to make sure that their solution made sense. The third student used the same approach, but this participant’s explanation was less suggestive of the student’s understanding that the rectangle’s opposite sides have the same length. This is because this respondent marked the missing sides as S3 and S4 rather than with a single letter, as Participants 1 and 5 did. Finally, the second student did not formulate an equation but solved the task using two operations. First, the student subtracted the sum of the lengths of the two known sides from the perimeter. Second, the participant divided the result by two, arriving at the correct answer.
General Comparison
The students who successfully completed the tasks showed certain differences in their approaches to mathematical problem-solving. Participants 1 and 2 gave the most detailed explanations of their solutions. Yet, the first student relied on additional drawings to better understand the task, while the second participant did these supporting operations in mind. The third student also drew additional lines, shapes, and numbers to facilitate finding the solution, like the first respondent. The fourth participant had the weakest mathematical skills and could not apply the necessary formulas to achieve correct results. Finally, the fifth respondent seemed to have solid mathematical knowledge but needed guidance to understand the task correctly. Although students relied on similar formulas to arrive at the right answers, their approaches to tasks varied, and some demonstrated a need for additional visual aid when solving geometric problems.