Exploring the combinatorial reasoning of high school students with reflective and impulsive cognitive style in solving problems

Abstract

Combinatorial reasoning was a basic competence that every student must have for solving mathematical problems, as it highly related to providing argumentation or strategy in solving mathematical problems. It was the process of creating complex constructs out of a set of given elements that satisfy the conditions explicitly given or inferred from the situation. Considering this issue, this study aimed to explore the combinatorial reasoning of high school students with cognitive-reflective andimpulsive styles in solving problems. More specifically, It correlated combinatorial reasoning with tempo cognitive style, since it applied time-based problem solving in which the speed of responding and the frequency of either correct or wrong answer might affect students' mental action in solving problems. This study was a qualitative research. It used High school students in eleventh grade as the research subject through matching familiar figure test. The researchers distributed a task containing several problems that had similar concept for each and then organized an interview to explore the students' combinatorial reasoning in solving the given problems. Cognitivereflective subject decided to use two strategies-formula and filling slot-for the sake of her affirmation, while cognitive-impulsive subject decided to only use one strategy-formula. The cognitive-reflective subject tended to be more accurate and careful in solving the problems. Otherwise, the cognitive-impulsive ones tended to be careless and less accurate, given that the subject decided to do spontaneous mental action. The result of this study found some similarities and differences on the combinatorial reasoning of both reflective and impulsive students. The similarities referred to their ways in explaining the notation in the formula they used and generalizing their strategies. The differences referred to the process of investigating various factors, considering any probabilities that might reveal, and evaluation.

The Development of Students’ Worksheets Based on Combinatoric Reasons in Elementary School

“…So, students find it not too difficult to solve math problems, especially story problems. This is in accordance with previous research which states that combinatorial reasoning is the process of looking for several alternative strategies for solving a problem and combinatorial reasoning has a strong relationship with the ease of solving problems (Nurul Aini et al, 2020;Ammamiarihta et al, 2017;Uripno et al, 2017;Uripno et al, 2020;., 2023;Medová et al, 2020) Figure 4 shows that, among 40 students who participated in the test, 37 students (92,5%) were considered to complete the MMC, while the other 3 students (7,5%) were under the rate of MMC. This incompleteness was probably due to several factors related to students' problem-solving skills.…”

“…It corresponded to the basic concept of combinatoric reason, which referred to assisting students to solve problems, and thus, combinatoric reason fits for narrative tasks. It was in line with some research suggesting that combinatoric reason could drill students to have reasonable and sustainable thinking for problem-solving (N. Aini et al, 2019;Nurul Aini et al, 2020;Ammamiarihta et al, 2017;Cahyanti et al, 2020). Hence, students could proceed and find some alternative strategies for solution of problems, in addition to thinking in more flexibly and creatively.…”

“…Thus, students' worksheets based on combinatoric reason was considered very practical. The students could easily complete the narrative task as the indicators of combinatoric reason were systematic, and it was consistent with previous studies suggesting that combinatoric reason could help students complete the given narrative tasks (N Aini et al, 2018;Nurul Aini et al, 2020). Furthermore, they could already think combinatorically although in a simple way, which was listing.…”

“…The first reason why we need to do this current study was that combinatoric reason by elementary graders still has less attention although it may help students to solve problems. This is due to the fact that combinatoric reason is a process of finding some alternative strategies for solutions to problems (Nurul Aini et al, 2020;Ammamiarihta et al, 2017;Uripno et al, 2023). So it can be said that combinatorial reasoning has a strong relationship with ease in problem solving (Medová et al, 2020).…”

“…In this study, students' worksheets based on combinatoric reason referred to tasks that make student do a reasoning process and problem-solving experience based on the indicators of combinatoric reason through which they could achieve the learning objectives, as they got used to thinking combinatoric. The indicators of combinatoric reason included investigating various factors, considering any possibilities, and evaluating and generalizing the use of strategy (Nurul Aini et al, 2020). Each indicator of combinatoric reason was spread out in the form of questions that led students to combinatoric reason.…”