The purpose of the study was threefold: (1) to determine the factor structure of mathematical thinking at the within-classroom and at the between-classroom level / (2) to investigate the extent of variation in the relationships among different mathematical thinking constructs at the within- and between-classroom levels / and (3) to examine the cross-level interactions among different types of mathematical thinking. Previous research was extended by investigating the factor structure of mathematical thinking in derivative at the within- and between-classroom levels, and further examining the direct, indirect, and cross-level relations among different types of mathematical thinking. Multilevel analyses of a cross-sectional dataset containing two independent samples of undergraduate students nested within classrooms showed that the within-structure of mathematical thinking includes enactive, iconic, algorithmic, algebraic, formal, and axiomatic thinking, whereas the between-structure contains formal-axiomatic, proceptual-symbolic, and conceptual-embodied thinking. Major findings from the two-level mathematical thinking model revealed that: (1) enactive, iconic, algebraic, and axiomatic thinking varied primarily as a function of formal and algorithmic thinking / (2) the strongest direct effect of formal-axiomatic thinking was on proceptual-symbolic thinking / (3) the nature of the relationships was cyclic at the between-classroom level / (4) the within-classroom mathematical thinking constructs significantly moderate the relationships among conceptual-embodied, proceptual-symbolic, and formal-axiomatic thinking / and (5) the between-classroom mathematical thinking constructs moderate the relationships among enactive, iconic, algorithmic, algebraic, formal, and axiomatic thinking. The challenges when using multilevel exploratory factor analysis, multilevel confirmatory factor analysis, and multilevel structural equation modeling with categorical variables are emphasized. Methodological and educational implications of findings are discussed.

This study aimed to provide evidence of reliability and validity for the 42-item Perceptions of Practices Survey. The scale was designed to assess educators’ perceptions of the extent to which their schools were implementing multi-tiered system of supports (MTSS) practices. The survey was initially given as part of a larger evaluation project of a 3-year, statewide initiative designed to evaluate MTSS implementation. Elementary educators (Level-1 n = 2,109, Level-2 n = 62) completed the survey in September/October of 2007, September/October of 2008 (Level-1 n = 1,940, Level-2 n = 61), and January/February of 2010 (Level-1 n = 2,058, Level-2 n = 60). Multilevel exploratory and confirmatory factor analysis procedures were used to examine the construct validity and reliability of the instrument. Results supported a correlated four-factor model: Tiers I & II Problem Solving, Tier III Problem Identification, Tier III Problem Analysis & Intervention Procedures, and Tier III Evaluation of Response to Intervention. Composite reliability estimates for all factors across the three years approximated or exceeded .84. Additionally, relationships were found between the Perceptions of Practices Survey factors and another measure of MTSS implementation, the Tiers I & II Critical Components Checklist. Implications for future research regarding the psychometric properties of the survey and for its use in schools are discussed.