Students' solution of arrangement problems and their connection to Cartesian product problems

• Published in: Mathematical thinking and learning Vol. 22; no. 1; pp. 23 - 55
• Main Author: Tillema, Erik S.
• Format: Journal Article
• Language: English
• Published: Philadelphia Routledge 02.01.2020; Taylor & Francis Ltd
• Subjects: Cartesian coordinates; Combinatorial analysis; Combinatorial reasoning; Grade 6; Intermediate Grades; Learning; Magnet Schools; Mathematical Concepts; Mathematical problems; Mathematics education; Mathematics Instruction; middle grades; Middle School Students; Multiplication; multiplicative reasoning; Problem Solving; second order models; Student Attitudes; Students; Teaching Methods
• ISSN: 1098-6065; 1532-7833
• DOI: 10.1080/10986065.2019.1608618

Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11-12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented a balanced mix of students operating with each of three different multiplicative concepts that have been identified in prior research. This paper reports on the 11 students who were using the first or second multiplicative concept. Students operating with different multiplicative concepts all experienced similar perturbing elements in their solution of APs relative to their solution of CPPs, but they operated differently to resolve these perturbing elements. These differences are identified and their significance discussed in relation to other research findings on students' combinatorial and multiplicative reasoning.