Students’ Abductive Reasoning in Solving Quadratic Pattern Generalization Problem Based on the Initial Mathematical Ability

Abstract

The purpose of this study was to describe students' abductive reasoning in solving quadratic pattern generalization problem based on initial mathematical ability. Abductive reasoning is a process of drawing conclusions based on certain facts where the conclusion is still an assumption that can be revised based on new information. This type of research is descriptive research with a qualitative approach. The subjects of this study were 6 junior high school students based on the category of students' initial mathematical ability, namely 2 students with high initial mathematical ability, 2 students with moderate initial mathematical ability, and 2 students with low initial mathematical ability. The data collection technique was carried out through an initial mathematical ability test to determine the research subjects, generalization problems of quadratic patterns to identify students' abductive reasoning processes, and interviews. Data analysis was carried out based on the process and indicators of abductive reasoning. The results of the study showed that in the process of (1) realizing the existence of abductive problems, all subjects had never solved generalization problems of quadratic patterns so that the problem became a surprise because it was something new that was obtained, all subjects also found differences in generalization problems of quadratic patterns with number pattern problems that had been encountered before, all subjects carried out this process as an initial process of abductive reasoning; (2) identifying solutions, all subjects explained the discrepancy between the information obtained from the facts of observations and previous knowledge, namely students with high and moderate initial mathematical abilities explained the difference in the given generalization questions on quadratic patterns only presenting the 4th pattern, while the number pattern questions that have been encountered usually contain patterns 1 to 4 in sequence, students with low initial mathematical abilities explained the difference lies in the process of working on it; all subjects mentioned alternative solution guesses that might help to solve the problem, namely all subjects made guesses about different pattern shapes and determined the fixed difference in the number of black circles in each pattern; students with high initial mathematical abilities grouped the black circles into three shapes, using the quadratic formula, the first and second level arithmetic sequence formulas, and obtained two alternative solutions; students with moderate initial mathematical abilities used the first level arithmetic sequence formula and obtained one alternative solution; while students with low initial mathematical abilities did not get an alternative solution; (3) choosing the best solution, students with high and medium initial mathematical ability choose a certain solution from the alternative solutions provided and explain the reasons for choosing this solution as the best solution, while students with low initial mathematical ability do not; students with high initial mathematical ability choose one formula from two formulas obtained because this formula is easier; students with low initial mathematical ability only get one formula and choose this formula because it is easy to use (4) assimilating the chosen solution, students with high and medium initial mathematical ability use the chosen solution to solve the problem, while students with low initial mathematical ability do not; the solution used by students with high initial mathematical ability produces the correct answer, while the solution used by students with medium initial mathematical ability can be the right solution if it is adjusted to the estimated pattern shape that has been made.

Keywords: abductive reasoning, initial mathematical ability, pattern generalization, quadratic pattern generalization problem.