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This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students’ meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students’ use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving. 

Understanding how students reason with logical implication is essential for supporting students’ construction of increasingly powerful ways of reasoning in proofs-based mathematics courses. We report on the results of an NSF-funded case study with a mathematics major enrolled in an introductory proofs course. We investigate the epistemological obstacles that she experienced and how they might relate to her treatment of logical implications as actions. Evidence shows that an action conception may pose challenges when students transform or quantify implications and may contribute to erroneous assumptions of biconditionality. Our report on available ways of operating with logical implications as actions is a first step in designing instructional tasks that leverage students’ existing reasoning skills to support their continued development. 

INTRODUCTION Balance between excitatory and inhibitory neuron (interneuron) populations in the cortex promotes normal brain function. Interneurons are primarily generated in the medial, caudal, and lateral ganglionic eminences (MGE, CGE, and LGE) of the ventral embryonic forebrain; these subregions give rise to distinct interneuron subpopulations. In rodents, the MGE generates cortical interneurons, the parvalbumin + (PV + ) and somatostatin + (SST + ) subtypes that connect with excitatory neurons to regulate their activity. Defects in interneuron production have been implicated in neurodevelopmental and psychiatric disorders including autism, epilepsy, and schizophrenia. RATIONALE How does the human MGE (hMGE) produce the number of interneurons required to populate the forebrain? The hMGE contains progenitor clusters distinct from what has been observed in the rodent MGE and other germinal zones of the human brain. This cytoarchitecture could be the key to understanding interneuron neurogenesis. We investigated the cellular and molecular properties of different compartments within the developing hMGE, from 14 gestational weeks (GW) to 39 GW (term), to study their contribution to the production of inhibitory interneurons. We developed a xenotransplantation assay to follow the migration and maturation of the human interneurons derived from this germinal region. RESULTS Within the hMGE, densely packed aggregates (nests) of doublecortin + (DCX + ) and LHX6 + cells were surrounded by nestin + progenitor cells and their processes. These DCX + cell–enriched nests (DENs) were observed in the hMGE but not in the adjacent LGE. We found that cells within DENs expressed molecular markers associated with young neurons, such as DCX, and polysialylated neural cell adhesion molecule (PSA-NCAM). A subpopulation also expressed Ki-67, a marker of proliferation; therefore, we refer to these cells as neuroblasts. A fraction of DCX + cells inside DENs expressed SOX2 and E2F1, transcription factors associated with progenitor and proliferative properties. More than 20% of DCX + cells in the hMGE were dividing, specifically within DENs. Proliferating neuroblasts in DENs persisted in the hMGE throughout prenatal human brain development. The division of DCX + cells was confirmed by transmission electron microscopy and time-lapse microscopy. Electron microscopy revealed adhesion contacts between cells within DENs, providing multiple sites to anchor DEN cells together. Neuroblasts within DENs express PCDH19, and nestin + progenitors surrounding DENs express PCDH10; these findings suggest a role for differential cell adhesion in DEN formation and maintenance. When transplanted into the neonatal mouse brain, dissociated hMGE cells reformed DENs containing proliferative DCX + cells, similar to DENs observed in the prenatal human brain. This suggests that DENs are generated by cell-autonomous mechanisms. In addition to forming DENs, transplanted hMGE-derived neuroblasts generated young neurons that migrated extensively into cortical and subcortical regions in the host mouse brain. One year after transplantation, these neuroblasts had differentiated into distinct γ-aminobutyric acid–expressing (GABAergic) interneuron subtypes, including SST + and PV + cells, that showed morphological and functional maturation. CONCLUSION The hMGE harbors DENs, where cells expressing early neuronal markers continue to divide and produce GABAergic interneurons. This MGE-specific arrangement of neuroblasts in the human brain is present until birth, supporting expanded neurogenesis for inhibitory neurons. Given the robust neurogenic output from this region, knowledge of the mechanisms underlying cortical interneuron production in the hMGE will provide insights into the cell types and developmental periods that are most vulnerable to genetic or environmental insults. Nests of DCX + cells in the ventral prenatal brain. Schematic of a coronal view of the embryonic human forebrain showing the medial ganglionic eminence (MGE, green), with nests of DCX + cells (DENs, green). Nestin + progenitor cells (blue) are present within the VZ and iSVZ and are intercalated in the oSVZ (where DENs reside). The initial segment of the oSVZ contains palisades of nestin + progenitors referred to as type I clusters (light blue cells) around DENs. In the outer part of the oSVZ, DENs transition to chains of migrating DCX + cells; surrounding nestin + progenitors are arranged into groups of cells referred to as type II clusters (white cells). In addition to proliferation of nestin + progenitors, cell division is present among DCX + cells within DENs, suggesting multiple progenitor states for the generation of MGE-derived interneurons in the human forebrain. ILLUSTRATION: NOEL SIRIVANSANTI