Different Types of Subitizing Activity: A Teaching Experiment with Preconservers

Introduction: Subitizing, a quick apprehension of the numerosity of a small set of items, is consistently utilized to support early number understanding. Perceptual subitizing is the innate ability to recognize less than five items without consciously using other mental or mathematical processes. Conceptual subitizing, which requires higher-level abilities, means perceiving the quantities as groups and performing a mental process on them. Research on conceptual and perceptual subitizing indicates some limitations about the activities regarding the children’s early number development. So, MacDonald and Wilkins (2016) developed a framework that explained the types of activities that young children used while subitizing. In this framework, five sets of perceptual subitizing activity were described to explain how young children’s perceptual subitizing activity changed. Besides, two types of conceptual subitizing activities were defined to explain how children’s limited or flexible number understanding related to their subitizing activity. These seven different types of activities characterize the changes in children’s subitizing actions. The study aims to investigate the relationship between children’s number understanding and subitizing activity. Methods: A teaching experiment was conducted with two preschool-aged children to analyze what perceptual and conceptual processes children relied upon when subitizing. The teaching experiment consisted of twenty-six sessions. The interviews were conducted to determine whether children are able to conserve numbers or not, and whether they rely on a variety of different types of subitizing activity or not. After the interviews, 26 teaching sessions were carried out with two preconserver children. Results: In the experimental process, it was observed that the children rely on the color of items, the gap between items, and symmetrical aspects of items when perceptually subitizing. However, they could not manage to transition their subitizing activity from perceptual to conceptual subitizing. The study indicates that children’s subitizing skills were closely related to their number conservation development. Discussion: Based on the findings from this study, for Eren and Beren, subitizing activities were found to be perceptually limited. Specifically, it was found that four types of perceptual subitizing emerged to explain how symmetry, the gap between items, color of items, and canonical patterns promoted strategies that children relied on when constructing number understanding. During the teaching experiment, although these children carried out the activities that required the separating and combining numbers and seeing the relationship between the subgroups and the composite groups, they used perceptual units in this process. The relationship between the number conservation activity and the conceptual subitizing activity requires the coordination of thinking structures related to both ordering and classification. However, it was found that the children could not move from perceptual to the conceptual subitizing. Limitations: As all studies have some limitations, this study has, too. One of the limitations of the study is the sample size/number of participants. But teaching experiments aim to get a deep understanding, studying with a small sample is an obligation. Secondly, this study focused on some compounds of subitizing such as perceptual and conceptual ones. Conclusion: In order to make the transition from perceptual subitizing to conceptual subitizing children should have more experiences with subitizing activities.When designing mathematical games and assessments for young children, being aware of different types of subitizing categories may provide better support children’s number understanding and subitizing.

Unitary unfoldings of a Bose–Hubbard exceptional point with and without particle number conservation

The conventional non-Hermitian but P T -symmetric three-parametric Bose–Hubbard Hamiltonian H ( γ , v , c ) represents a quantum system of N bosons, unitary only for parameters γ , v and c in a domain D . Its boundary ∂ D contains an exceptional point of order K (EPK; K = N + 1) at c = 0 and γ = v , but even at the smallest non-vanishing parameter c ≠ ~0 the spectrum of H ( v , v , c ) ceases to be real, i.e. the system ceases to be observable. In this paper, the question is inverted: all of the stable, unitary and observable Bose–Hubbard quantum systems are sought which would lie close to the phenomenologically most interesting EPK-related dynamical regime. Two different families of such systems are found. Both of them are characterized by the perturbed Hamiltonians H ( λ ) = H ( v , v , 0 ) + λ V for which the unitarity and stability of the system is guaranteed. In the first family the number N of bosons is assumed conserved while in the second family such an assumption is relaxed. Attention is paid mainly to an anisotropy of the physical Hilbert space near the EPK extreme. We show that it is reflected by a specific, operationally realizable structure of perturbations λ V which can be considered small.

Improved tests of muon and electron flavor symmetry in muon processes

I review the motivation for and status of searches for violation of muon and electron number conservation, concentrating on muon initiated processes. I then discuss the progress expected in the next few years, concentrating on a new experiment, E940 at BNL (the MECO experiment). It will improve the experimental sensitivity for the process μ^-Ν —> e^-N to below 10^-16 , roughly 4 orders of magnitude better than the current limit.