Level of combinatorial thinking in solving mathematical problems

Abstract

Combinatorial thinking is an important reasoning process in building one's knowledge and experience. The purpose of this study is to describe the characteristics of the level of combinatorial thinking in solving mathematical problems. The subjects of the study were 40 students of Elementary Teacher Education Department (PGSD): 20 students of the second semester and the others of the sixth semester. The reason for choosing subjects from these two levels is to meet all levels of combinatorial thinking. All research subjects were given test questions about combinatorial problems. From 40 subjects, five students were selected to be interviewed as they had fulfilled all five levels of combinatorial thinking. The data validity was conducted by triangulation through recording interview results and comparing it with data from students' written test results to ensure the validity and reliability of this research. The results show that there are five levels of combinatorial thinking in solving mathematical problems: investigating “some cases’, systematically checking cases, using the calculation order, systematically generating all cases, and changing the problem into another combinatorial problem. Level one is the identification of the possibility of students’ understanding the questions incorrectly, or vice versa, already can answer the questions with systematic procedures but the results are less precise. Level two is conducting systematic checking about students' understanding of the combination material. Besides, it also concerns about the ability to answer problems systematically using diagram trees. Level three is students are able to apply the calculation orders, which are addition and multiplication. Level four is systematically generating all cases about the ability to calculate possibilities without schematic, drawings, or diagrams. Level five is changing the problem into another combinatorial problem, it is the ability to calculate possibilities with complex problems Based on the research findings, it turns out there is another level of combinatorial thinking, which is using the calculation order and this is found between level two and level three. The researchers recommend further research to explore more on the application of calculation order.

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Hierarchy of Students’ Reflective Thinking Levels in Mathematical Problem Solving

“…These efforts assisted them in drawing the correct conclusions. We conclude that students need time to understand the information provided and the intent of the question (Hidayati et al, 2020), and they do it by working on the provided information in different ways, such as by rereading and redrawing it. These efforts improve their performance significantly (Hidayati et al, 2020;Pradana et al, 2020).…”

“…We conclude that students need time to understand the information provided and the intent of the question (Hidayati et al, 2020), and they do it by working on the provided information in different ways, such as by rereading and redrawing it. These efforts improve their performance significantly (Hidayati et al, 2020;Pradana et al, 2020). When associated with the problem-solving step by Polya (1973), the first level is identic with the first stage of problem solving, i.e., understanding the problem.…”