Student Teachers' Construction of Mathematical Theorem of Set and Cardinality

Abstract

Constructivists suggest that teachers should guide and facilitate their students in reinventing and or reconstructing mathematics concepts as in reconstructing a theorem. A theorem in mathematics holds a vital role as a fundamental aspect in building the mathematics itself. Hence, students must learn and acquire this ability. Furthermore, by reconstructing a theorem, students not only learn about the theorem but also learn about problem-solving since the theorem that should be reconstructed can be presented as a problem. In which, this skill is needed and promoted in the 21st century. Hence, the teacher has to be able to reconstruct and prove their construction of theorem. However, many studies were focused on investigating students' mathematical proof ability, not on the ability to reconstruct a theorem. This study is aimed to investigate mathematics undergraduate students' ability to reconstruct and prove a theorem. A problem to make a general statement regarding the numbers of all possibility of the cardinality of A B, with A, B is set, and n(A) is a and n(B) is b, is given to 60 undergraduate students who are majoring mathematics education. The statement then is analyzed and categorized into 0 to 5 based on the developed framework. The result shows that most students' answers (29 answers) are categorized as an incomplete or irrelevant theorem (level 0), while only seven answers can be categorized as level 5 (well structured). As for level 2, 3, and 4, there are 13, 8, 3 answers representatively. Also, there is no student answer, which is categorized as level 1. The mistakes occur because students do not fully understand the mathematics notion. Then, the theorem made is different from the one that is asked, the given condition mentioned by the students is not general enough (does not represent all cases); the conclusion made is false. However, no conclusion is unrelated to the condition.