Assessing the mathematics knowledge of college developmental students: A case of the slope of a line

Mathematics education has been a national challenge in the United States. Many high school graduates entering college are underprepared for college-level work and must take at least one developmental mathematics course. Mathematics readiness issues have seriously affected colleges' graduation and retention rates and waste sizeable resources. To improve students' college mathematics readiness and support their successful mathematics learning, an important component is to improve math instruction in both the developmental and college-level classrooms. Successful teaching requires teachers having pedagogical content knowledge (PCK) in addition to strong subject matter knowledge. However, there is not much empirical research on developmental mathematics pedagogy or on students' understanding of developmental mathematics. Diversely represented and conceptualized in many different contexts and settings, slope is a fundamental underpinning concept for students to understand most secondary and postsecondary mathematics ideas. To fill in the gaps in the developmental mathematics pedagogy and to seek effective instructional strategies for college developmental mathematics teaching, this study assessed mathematics knowledge of college developmental students specifically regarding the concept of slope.;Building on the previous studies on procedural knowledge, conceptual knowledge, flexibility and multiple representations, this study conceptualized the knowledge flexibility models, considered the primary source register of multiple representations as a possible change mechanism of knowledge flexibility and developed a competence level model of the representations linking. Within these theoretical models and the theoretical framework of validity, this study developed and validated a slope instrument of thirty-five items using Rasch analysis (n = 4287). Results showed the instrument can measure students' slope knowledge reliably and validly. Hierarchical regression analyses and ANOVA demonstrated the significant effects of knowledge complexity and primary source register on item difficulty. The full model explained 42.35% (adjusted R-square = 0.4235; p-value = 0.0018) variance in the observed item difficulties. This study also investigated twelve out of sixteen mathematics translations between and within four different representations systems. The findings revealed college developmental students' strength and weakness in their slope learning and reflected on their overall mathematics background. Implications for developmental mathematics pedagogy, instructional strategies and practices, and the areas of future study are discussed.