The aim of this study was to integrate 2 theories on academic self-concept-that is, the intemal/extemal frame of reference (I/E) model and the big-fish-little-pond effect (BFLPE)-into a combined model. The rationale for the combined model is that the I/E model actually focuses on internal/dimensional comparison (e.g., a comparison between one's own mathematics and science achievements) and the BFLPE focuses on extemal/social comparison (e.g., a comparison between one's own mathematics achievement and one's schoolmates'). The I/E, BH^^PE, and combined models for mathematics and science were examined using multilevel analyses of 139,174 students in 4,231 schools in 27 countries. The results showed that the I/E model and the BFLPE fitted data well, and the combined model fitted data even better. The I/E model remained unchanged when the BFLPE was combined, suggesting distinctly differential roles of intemal and extemal comparisons in constructing academic self-concepts. In addition, the combined model was examined for each country. The findings revealed that the predictions of the combined model were generally supported for data from most countries.Keywords: academic self-concepts, achievement, the intemal/extemal frame of reference model, the big-fish-little-pond effectThe understanding of psychological processes in constructing academic self-concepts is of paramount importance for educators. This understanding can facilitate suitable interventions provided by teachers who aim to help students obtain appropriate academic self-concepts and desirable learning outcomes, such as adequate allocation of time and effort, sense of control over leaming, and maximizafion of achievements. These desirable outcomes in tum will benefit students in their pursuit of appropriate careers and sense of happiness (Marsh, Craven, & Mclnemey, 2003). As such, appropriate academic self-concepts have become one of the major objecfives of educafion, and the understanding of the process in constructing academic self-concepts can bring both practical and theoretical benefits to education.Two of the major theorefical models in relation to the psychological processes in constructing academic self-concepts to date are the intemal/extemal frame of reference (I/E) model (Marsh, 1986) and the big-fish-little-pond effect (BFLPE; Marsh, 1987a). The basic I/E model highlights two major psychological processes (i.e., intemal and extemal comparisons), while the BFLPE focuses on the process of extemal comparison only. Intemal comparison refers to a psychological process in which individuals compare their own achievement in a particular domain (e.g., mathematics)

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This study investigated the approaches to teaching by three fifth-grade teachers' of creative and non-creative mathematical problems for fractions. The teachers' personal constructs of the two kinds of problems were elicited by interviews through the use of the repertory grid technique. All the teaching was observed and video-recorded. Results revealed that the teachers had slightly distinctive constructs of creative and noncreative problems, and professed a greater preference for creative problems. Based on the teachers' creations of problems in classrooms and related features, the study identified three types of teaching approaches: liberal, reasoning, and skill approaches. The liberal approach appeared to indicate the most appropriate teaching methods for creative problems.Researchers have identified a variety of types of mathematical problems, such as word and time-consuming problems, and word and calculation problems (Vermeer, Boekaerts & Seegers, 2000). The consideration of these problems is more related to the length of time and the language needed for solving a problem than to the use of creativity in solving it. Another typology for mathematical problems is 'routine' and 'non-routine' problems (McLeod, 1988(McLeod, , 1994. This dichotomy, however, relates more to learners' experiences than to problem types. For example, word problems are likely to be non-routine for students used to solving calculation problems. For example, a study was described in which students were often invited by the teachers to create mathematical games and articles, such as Chinese new-year calendars in the cycle of twelve: for these students, these creative tasks were 'routine'.Well-structured and ill-structured problems are another taxonomy, as indicated by Nitko (1996) and Jonassen (1997). Well-structured problems are tasks that are clearly laid out, give students all the information they need, and usually have one correct answer that students can obtain by applying a procedure taught in class. The purpose of well-structured problems is to give students opportunities to rehearse the procedures or algorithms taught in class. In contrast, most authentic problems are illstructured. In order to solve an ill-structured problem, students have to organize, clarify, and obtain information not readily available for understanding the problem. There are likely to be a number of correct answers for an ill-structured problem. Jonassen views well-structured and ill-structured problems as 'a continuum from decontextualized problems with convergent solutions to very contextualized problems with multiple solutions' (p. 67). Jausovec's (1994) study defines well-defined problems as being clearly defined with given states, goal states, and an operator, while ill-defined problems are viewed as having vaguely defined goals, which can only be solved by creative strategies. In the present study, creative problems refer to the extreme end of problems with multiple or divergent (often limitless) solutions, while non-creative problems were those with si...

Past studies have examined the models for the effects of early numeracy activities on children's later mathematics achievement, with parental values as a precondition and socioeconomic status (SES) as an interaction measure with numeracy activities. This study proposed models of (1) the multiple effects of early numeracy activities, conditioned by parental value, on mathematics achievement and affect (e.g. confidence and interest) and (2) the multiple effects to be mediated by SES and early numeracy activities conditioned by parental value and child gender. The proposed models were examined using structural equation modeling with data from Taiwanese parental reports and child grade-4 tests and reports (N = 4,291; 49% girls) of TIMSS 2015. Three major results go beyond those previous research findings and may provide recommendations for educational practices. (1) Early numeracy activities have effects on mathematics confidence and interest in addition to achievement. (2) SES mediates the effects of early numeracy activities on achievement and confidence but not on interest. The mediating effect of SES suggests that high-quality educational provision should be provided during early numeracy activities. (3) Parents provide fewer numeracy activities for girls, which suggests that parents should provide more early numeracy activities to girls.

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The skill-development model contends that achievements have an effect on academic self-confidences, while the self-enhancement model contends that selfconfidences have an effect on achievements. Differential psychological processes underlying the 2 models across the domains of mathematics and science were posited and examined with structural equation modeling using the data of grade 8 students from the Trends in International Mathematics and Science Study across 28 countries (N = 144,069), which generated 2 major findings. In statistical terms, (1) there were negative cross-domain paths leading from achievements to self-confidences, controlling for the positive matching domain paths, as predicted by the skill-development model across domains; (2) there were positive cross-domain paths leading from self-confidences to achievements, controlling for the positive matching domain paths, as predicted by the selfenhancement model across domains. There were, however, qualitative variations in the degrees of support for the 2 models across countries.