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Student focusing and noticing, which drive reasoning, are important but under researched aspects of student learning. Quadratic functions representations are perceptually and conceptually complex and thus, offer much for students to focus on and notice. Our study compared a teacher’s goals for student focusing and noticing during quadratic functions instruction with what students actually focused on and noticed. Qualitative analysis revealed some alignment but also informative ways that the teacher’s goals and student outcomes for focusing and noticing were misaligned. These results will further the field’s understanding of how students learn about quadratic functions and may have implications for student focusing and noticing of other mathematics topics as well. 

The topic of study in this report is student focusing and noticing. Specifically, we examined a teacher’s goals for student focusing and noticing and the student outcomes for focusing and noticing. The mathematics context for this research was quadratic functions and covariational reasoning. Two whole-class discussion episodes were analyzed. Results showed ways that the teacher’s goals and student outcomes were aligned and three ways that they were misaligned. These results could inform how quadratic functions are taught and how teachers can improve the alignment between their goals for student focusing and noticing and student outcomes for focusing and noticing. 

The first measurements of proton emission accompanied by neutron emission in the electromagnetic dissociation (EMD) of ${}^{208}\mathrm{Pb}$nuclei in the ALICE experiment at the Large Hadron Collider are presented. The EMD protons and neutrons emitted at very forward rapidities are detected by the proton and neutron zero degree calorimeters of the ALICE experiment. The emission cross sections of zero, one, two, and three protons accompanied by at least one neutron were measured in ultraperipheral ${}^{208}\mathrm{Pb}\text{−}{}^{208}\mathrm{Pb}$collisions at a center-of-mass energy per nucleon pair $\sqrt{s_{NN}}=5.02\mathrm{TeV}$. The 0p and 3p cross sections are described by the RELDIS model within their measurement uncertainties, while the 1p and 2p cross sections are underestimated by the model by 17–25%. According to this model, these 0p, 1p, 2p, and 3p cross sections are associated, respectively, with the production of various isotopes of Pb, Tl, Hg, and Au in the EMD of ${}^{208}\mathrm{Pb}$. The cross sections of the emission of a single proton accompanied by the emission of one, two, or three neutrons in EMD were also measured. The data are significantly overestimated by the RELDIS model, which predicts that the (1p,1n), (1p,2n), and (1p,3n) cross sections are very similar to the cross sections for the production of the thallium isotopes ${}^{206,205,204}\mathrm{Tl}$. ©2025 CERN, for the ALICE Collaboration2025CERN