4. Methods4.1. ParticipantsParticip

4. Methods
4.1. Participants
Participants were 125 undergraduate students (106 female, 19 male) aged 18–23 who attended four mathematics con-tent methods courses for elementary, special education, and/or early childhood teacher education majors at a Midwestern university in the United States. They were selected by convenience sampling; that is, the participants were contacted by the first author and were recruited on the basis of their willingness to participate in the study. Four courses, CI/Math 120,CI/Math 220, CI/Math 321, and CI/Math 322, are the mathematics courses required by the university for the Elementary School. CI/Math 120 and CI/Math 220 are required courses for all undergraduate students in the elementary, special education, and early childhood teacher education programs, whereas CI/Math 321 is a required course for all undergraduate students in only the elementary and special education teacher education programs. Forty-six participants, who were fresh-men and were aged 18–19, came from CI/Math 120; 16 participants, who were sophomores and were aged 19–20, came from CI/Math 220; 48 participants, who were juniors and were aged 20–21, came from CI/Math 321; and 15 participants,who were juniors and were aged 21–22, came from CI/Math 322. All four courses were divided into 16 weeks and met 4 hper week.The four courses (CI/Math 120, 220, 321 and 322) run in a hierarchical system. For example, students who enroll inCI/Math 120 and 220 and do not receive at least a grade of C are not eligible to enroll in CI/Math 321. Moreover, studentsin CI/Math 120 gain introductory mathematical knowledge, including fraction-based knowledge, whole numbers, integers,rational numbers, and the essential arithmetic operations. At the CI/Math 220 level, students concentrate on learning rational and irrational numbers. At the CI/Math 321 level, students are instructed in three-dimensional geometry and are introduced to the fundamental ideas of statistics and probability, which are based on the mastery of fractional knowledge obtained inCI/Math 120 and 220. Students in CI/Math 322 are expected to obtain the knowledge of algebra, and recognize relations and functions, as well as their applications to real-life problems. All four courses, focusing on multiple representations of mathematical concepts, provide not only procedural and conceptual knowledge in mathematics but also pedagogical content knowledge regarding teaching mathematics for the elementary school.With regard to this study, students who enrolled in the four courses were taught about knowledge of fractions for the first six weeks. Especially, many students in CI/Math 321 and 322 transferred from another college with different content curriculum, so this review process associated with rational numbers was necessary. Since this study does not target participants based on age, gender, or other characteristics, such characteristics were not considered in the analysis of the data.4.2. Instrument The Fraction Knowledge Test (see Appendix A), comprised of 32 items that were adapted from Crameretal. (2002) and Ma (1999), was designed to access pre-service elementary mathematics teachers’ procedural and conceptual knowledge off actions. The fractional items specifically designed to measure knowledge of fractions in areas related to eight dimensions:(1) concepts (CON), (2) equivalence (EQU), (3) order (ORD), (4) operation: addition (ADD), (5) operation: subtraction (SUB),(6) operation: multiplication (MUL), (7) operation: division (DIV), and (8) transfer (TRA). These dimensions were selected in order to assess pre-service teachers’ fraction knowledge from Long and DeTemple (1996). Four items were designed for each fractional dimension (CON, EQU, ORD, ADD, SUB, MUL, DIV, and TRA). Within each fractional dimension, the four items were divided into two sets of questions—one was procedural knowledge and the other one was conceptual knowledge. The summary of each item used in the Fraction Knowledge Test is presented in Table 1.