Beliefs influencing students' mathematical learning and problem solving are structured and intertwined with larger affective and cognitive structures. This theoretical article explores a psychological concept we term an engagement structure, with which beliefs are intertwined. Engagement structures are idealized, hypothetical constructs, analogous in many ways to cognitive structures. They describe complex ''in the moment'' affective and social interactions as students work on conceptually challenging mathematics. We present engagement structures in a self-contained way, paying special attention to their theoretical justification and relation to other constructs. We suggest how beliefs are characteristically woven into their fabric and influence their activation. The research is based on continuing studies of middle school students in inner-city classrooms in the USA.

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The purpose of this study was to evaluate a model for considering general and specific elements of student experience in a gateway course in undergraduate Financial Accounting in a large university on the East Coast, USA. Specifically, the study evaluated a bifactor analytic strategy including a general factor of student classroom experience, conceptualized as student engagement as rooted in flow theory, as well as factors representing specific dimensions of experience. The study further evaluated the association between these general and specific factors and both student classroom practices and educational outcomes. The sample of students (N = 407) in two cohorts of the undergraduate financial accounting course participated in the Experience Sampling Method (ESM) measuring students' classroom practices, perceptions, engagement, and perceived learning throughout the one-semester course. Course grade information was also collected. Results showed that a two-level bifactor model fit the data better than two traditional (i.e., non-bifactor) models and also avoided significant multicollinearity of the traditional models. In addition to student engagement (general factor), specific dimensions of classroom experience in the bifactor model at the within-student level included intrinsic motivation, academic intensity, salience, and classroom self-esteem. At the between-student level, specific aspects included work orientation, learning orientation, classroom self-esteem, and disengagement. Multilevel Structural Equation Modeling (MSEM) demonstrated that sitting in the front of the classroom (compared to the sitting in the back), taking notes, active listening, and working on problems during class had a positive effect on within-student variation in student engagement and attention. Engagement, in turn, predicted perceived learning. With respect to between-student effects, the tendency to sit in front seats had a significant effect on student engagement, which in turn had a significant effect on perceived learning and course grades. A significant indirect relationship of seating and active learning strategies on learning and course grade as mediated by student engagement was found. Support for the general aspect of student classroom experience was interpreted with flow theory and suggested the need for additional research. Findings also suggested that active learning strategies are associated with positive learning outcomes even in educational environments where possibilities for action are relatively constrained.

We studied test preparation activity among fourth-grade math and science teachers in New Jersey, using a survey of almost 300 teachers and observations of and interviews with almost 60. New Jersey uses a mix of open-ended and multiple-choice tests; links few stakes to test results, except for publication of scores; and offers limited professional development to teachers. New Jersey teachers are adopting specific techniques associated with more inquiry-oriented instruction, but their basic approach does not appear to be changing. Teachers do teach the content on the test with a new wrinkle. When more students are passing a test in one subject area than in another, emphasis shifts to the area with low scores. We found more direct instruction in lower socioeconomic districts but not less inquiry-oriented instruction. Finally, principal support has more influence on the test preparation strategies teachers use than does pressure to comply.

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