Background: Identification and retention of effective teachers in STEM education play cardinal roles in teacher recruitment exercises worldwide. Studies on factors that characterize effective teachers have therefore gained popularity in recent times. Teacher self-efficacy, job satisfaction and school climate are among other factors that have attracted global attention. Thus, proper understanding of the relations between these factors is equally important. The purpose of this study is to validate and cross-validate a model of direct/indirect effects of school climate and teacher self-efficacy on job satisfaction. Results:The data used for the current study are extracted from a publicly available data of Teaching and Learning International Survey (TALIS) 2018 survey. Structural equation modeling approach was used in the analyses coupled with robust maximum likelihood to ensure accurate estimations in the models. The results of the validated models show a strong direct impact of school climate on job satisfaction, a direct impact of teacher self-efficacy on job satisfaction and a mediating effect of teacher self-efficacy between school climate and job satisfaction. This model exhibits structural invariance in factor loadings, intercepts and regression weights across two independent samples from a population of 3951 lower secondary school teachers in Norway. Conclusion: The findings of this study do provide empirical evidence for the relations between teacher selfefficacy, job satisfaction and school climate among Norwegian lower secondary school teachers. The crossvalidation of these relations was also established using an independent sample to enhance generalization of the findings. Two methodological observations concerning recoding of some items as well as an addition of item cross-loading in the measurement model of the job satisfaction scales are raised and addressed. It is therefore recommended that researchers who will be using TALIS 2018 data should take note of these observations.

The revised two-factor study process questionnaire (R-SPQ-2F) has widely been considered valid and reliable in many contexts for measuring students’ learning approaches. However, its cultural specificity has generated considerable discussion, with inconclusive results when translated to different languages. This paper provides more insights into the construct validity of a Norwegian version of this instrument. The R-SPQ-2F is composed of ten items designed to expose deep learning approaches and 10 items designed to expose surface learning approaches. A survey research design involving a sample of 253 first year university students in a mathematics course was adopted. Ten hypothesized models were compared using a series of confirmatory factor analyses following the model proposals reported in the literature. A weighted least square mean and variance adjusted (WLSMV) estimator was used to enhance model parameter estimations under multiple violations of assumptions inherent in ordinal data. The results favored a two first-order factor model with ten items measuring the deep approach and nine items measuring the surface approach including a deletion of one item from this instrument. The findings of this study provide empirical evidence for the cultural specificity of the instrument that is consistent with the literature. The R-SPQ-2F is therefore recommended to assess students’ approaches to learning, and further studies into its cultural specificity are recommended.

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This study was framed within a quantitative research methodology to develop a concise measure of calculus self-efficacy with high psychometric properties. A survey research design was adopted in which 234 engineering and economics students rated their confidence in solving year-one calculus tasks on a 15-item inventory. The results of a series of exploratory factor analyses using minimum rank factor analysis for factor extraction, oblique promin rotation, and parallel analysis for retaining extracted factors revealed a one-factor solution of the model. The final 13-item inventory was unidimensional with all eigenvalues greater than 0.42, an average communality of 0.74, and a 62.55% variance of the items being accounted for by the latent factor, i.e., calculus self-efficacy. The inventory was found to be reliable with an ordinal coefficient alpha of 0.90. Using Spearman’ rank coefficient, a significant positive correlation ρ ( 95 ) =   0.27 ,   p <   0.05 (2-tailed) was found between the deep approach to learning and calculus self-efficacy, and a negative correlation ρ ( 95 ) =   − 0.26 ,   p <   0.05 (2-tailed) was found between the surface approach to learning and calculus self-efficacy. These suggest that students who adopt the deep approach to learning are confident in dealing with calculus exam problems while those who adopt the surface approach to learning are less confident in solving calculus exam problems.

Theories of self-efficacy and approaches to learning are wellestablished in the psychology of learning. However, studies on relationships between the primary constructs on which these theories are developed are rarely reported in mathematics education research. Thus, the purpose of the current study is to provide empirical evidence for a potential causal relationship between perceived self-efficacy and approaches to learning. The present study adopts a cross-sectional survey research design that includes 195 engineering students enrolled on a first-year introductory calculus course. The data are collected using two well-developed and validated instruments with established high psychometric properties. Two hypotheses are formulated and tested using a structural equation modelling approach coupled with a weighted least square mean and variance adjusted estimator. The findings show that a high sense of perceived self-efficacy has a strong tendency to induce a deep approach to learning mathematics. In contrast, a low sense of perceived selfefficacy induces a surface approach to learning mathematics with a strong effect. This study represents a shift from the usual correlational studies that characterize quantitative research in mathematics education literature to causal relation research. Therein, causal assumptions are made and tested against the collected data, and some recommendations are made for future studies.

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Let (Γ, * ) be a finite group and S a possibly empty subset of Γ containing its non-self-invertible elements. In this paper, we introduce the inverse graph associated with Γ whose set of vertices coincides with Γ such that two distinct vertices u and v are adjacent if and only if either u * v ∈ S or v * u ∈ S. We then investigate its algebraic and combinatorial structures.