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A variational principle is introduced which minimizes an action formulated for configurations of vacuum Dirac seas. The action is analyzed in position and momentum space. We relate the corresponding Euler-Lagrange equations to the notion of state stability. Examples of numerical minimizers are constructed and discussed.

Research has identified two core difficulties many students have with fractions: first, they often struggle with processing fraction magnitudes, and second, they rely on natural number concepts in fraction problems ["Natural Number Bias" (NNB)]. Yet, the relation between these two difficulties is not well-understood. Moreover, while most studies of the NNB relied on analyses of whole samples, there is empirical evidence that the occurrence of the NNB differs between student subgroups. In the present study, we investigate individual students' profiles of the occurrence of the NNB and their ability to process fraction magnitude, using a dynamic assessment that utilizes continuous diagrams on touchscreen devices. We analyze data of 234 low-achieving 6th-grade students from Germany who completed a symbolic fraction comparison task, and a fraction magnitude estimation task with continuous circle and tape diagrams. A cluster analysis on the comparison task revealed three distinct clusters: a Typical Bias cluster (better performance on symbolic fraction comparison items congruent to natural number-based reasoning), a Reverse Bias cluster (better performance on items incongruent to natural number-based reasoning), and a No Bias cluster (similar performance on congruent and incongruent items). Only students in the No Bias cluster but not students in the other clusters demonstrated a distance effect in symbolic fraction comparison, suggesting fraction magnitude processing. Linear mixed models on the percent absolute error in the magnitude estimation task revealed significantly lower percent absolute error for students in the No Bias cluster compared to students in the other two clusters. Students in the No Bias cluster were significantly slower to solve both fraction comparison and fraction magnitude estimation tasks than students in the other clusters. The results of this study suggest that the occurrence of the natural number bias and the ability to process fraction magnitude are closely related. The continuous representations used in our digital assessment tools appeared to be suitable for assessing both the natural number bias and fraction magnitude processing.

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The transition from classical to electronic textbooks seems to be a logical step in the digitization advancing worldwide. However, developing an e-book ought to be more than digitizing text: key features of computer-based learning environments such as interactive exercises, adaptive demands, or automatic feedback should be integrated to take advantage of the digitization. The "ALICE:fractions" project aims at designing and evaluating an interactive mathematics textbook for introducing fractions in the classroom. It is based on research in mathematics education and includes the elements just mentioned. This paper describes the electronic textbook's implementation and its theoretical background. Moreover, it introduces aspects that allow further information on learning processes to be gleaned. As an example, time on task is regarded in a study with 6th graders who used the electronic textbook in the classroom. Linear mixed models revealed a negative effect of time on task on task success. The effect was moderated by exercise difficulty and slightly by student competence: the effect was less pronounced in difficult exercises and for low-achieving students, whereas for high-achieving students or in easier exercises, the effect intensified.