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1. ALGEBRAIC SEEDS FOR GRAPHING FUNCTIONS

   López, MA; Strachota, S; Brizuela, BM; Pérez-Martos, MC; Gardiner, AM; Blanton, M (July 2024, PME)

   Evans, T; Marmur, O; Hunter, J; Leach, G; Jhagroo, J (Ed.)

   This case study of one first grade student involves the analysis of three interviews that took place before, during, and after classroom teaching experiments (CTEs). The CTEs were designed to engage children in representing algebraic concepts using graphs. Using a knowledge-in-pieces perspective, our analysis focused on identifying students’ natural intuitions and ways of thinking algebraically about a functional relationship represented using graphs. Findings reveal four seeds, two of which were identified in prior studies, and how the activation and coordination of these seeds results in students' production of function graphs. 

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2. A case study of a seven-year old’s reasoning with and understandings of function graphs

   Liao, N; Strachota, S; López, M; Brizuela, B M; Blanton, M; Gardiner, A (February 2025, Fourteenth Congress of the European Society for Research in Mathematics Education (CERME14))

   This study consisted of Grades 1 and 2 (ages 6-8) classroom teaching experiments (CTE) with a lesson sequence focused on graphical representations of algebraic relationships. We interviewed students before, during, and after the CTE. Here we report on the progression of one Grade 2 (age 7) student’s thinking across the CTE. Prior to the CTE, the student had not previously interacted with representations of algebraic relationships. By the end of the CTE, the student was able to generalize about the functional relationships using graphs. This study illustrates how a learning trajectory modeling students’ understandings of function graphs can be used to characterize one children’s learning and provides evidence that young students are able to reason with function graphs. 

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3. The Carnivoran Adaptive Landscape Reveals Trade-offs among Functional Traits in the Skull, Appendicular, and Axial Skeleton

   https://doi.org/10.1093/iob/obaf001  

   Law, C_J; Hlusko, L_J; Tseng, Z_J (January 2025, Integrative Organismal Biology)

   Synopsis Analyses of form–function relationships are widely used to understand links between morphology, ecology, and adaptation across macroevolutionary scales. However, few have investigated functional trade-offs and covariance within and between the skull, limbs, and vertebral column simultaneously. In this study, we investigated the adaptive landscape of skeletal form and function in carnivorans to test how functional trade-offs among these skeletal regions contribute to ecological adaptations and the topology of the landscape. We found that morphological proxies of function derived from carnivoran skeletal regions exhibit trade-offs and covariation across their performance surfaces, particularly in the appendicular and axial skeletons. These functional trade-offs and covariation correspond as adaptations to different adaptive landscapes when optimized by various factors including phylogeny, dietary ecology, and, in particular, locomotor mode. Lastly, we found that the topologies of the optimized adaptive landscapes and underlying performance surfaces are largely characterized as a single gradual gradient rather than as rugged, multipeak landscapes with distinct zones. Our results suggest that carnivorans may already occupy a broad adaptive zone as part of a larger mammalian adaptive landscape that masks the form and function relationships of skeletal traits. 

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4. HOW TRANSITIONS BETWEEN RELATED ARTIFACTS SUPPORT STUDENTS’ COVARIATIONAL REASONING

   Germia, E.; York, T.; Panorkou, N. (January 2022, Proceedings of the forty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.)

   Lischka, A. E.; Dyer, E. B.; Jones, R. S.; Lovett, J. N..; Strayer, J.; & Drown, S. (Ed.)

   Many studies use instructional designs that include two or more artifacts (digital manipulatives, tables, graphs) to support students’ development of reasoning about covarying quantities. While students’ forms of covariational reasoning and the designs are often the focus of these studies, the way students’ interactions and transitions between artifacts shape their actions and thinking is often neglected. By examining the transitions that students make between artifacts as they construct and reorganize their reasoning, our study aimed to justify claims made by various studies about the nature of the synergy of artifacts. In this paper, we present data from a design experiment with a pair of sixth-grade students to discuss how their transitions between artifacts provided a constructive space for them to reason about covarying quantities in graphs. 

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5. Community context mediates effects of pollinator loss on seed production

   https://doi.org/10.1002/ecs2.4569  

   Arrowsmith, K. C.; Reynolds, Victoria A.; Briggs, Heather M.; Brosi, Berry J. (June 2023, Ecosphere)

   Abstract A critical goal for ecologists is understanding how ongoing local and global species losses will affect ecosystem functions and services. Diversity–functioning relationships, which are well‐characterized in primary producer communities, are much less consistently predictable for ecosystem functions involving two or more trophic levels, particularly in situations where multiple species in one trophic level impact functional outcomes at another trophic level. This is particularly relevant to pollination functioning, given ongoing pollinator declines and the value of understanding pollination functioning for single plant species like crops or threatened plants. We used spatially replicated, controlled single‐pollinator‐species removal experiments to assess how changes in bumble bee species richness impacted the production of fertilized seeds in a perennial herb—Delphinium barbeyi—in the Rocky Mountains of Colorado, USA. To improve predictability, we also assessed how traits and abundances in the plant and bumble bee communities were related toD. barbeyireproductive success. We hypothesized that trait‐matching between pollinator proboscis length andD. barbeyi's nectar spurs would produce a greater number of fertilized seeds, while morphological similarity within the floral community would dilute pollination services. We found that the effects of pollinator removal differed depending on the behavioral patterns of pollinators and compositional features of the plant and pollinator communities. While pollinator floral fidelity generally increasedD. barbeyiseed production, that positive effect was primarily evident when more than half of theBombuscommunity was experimentally removed. Similarly, communities comprising primarily long‐tongued bees were most beneficial toD. barbeyiseed production in tandem with a strong removal. Finally, we observed contrasting effects of morphological similarity in the plant community, with evidence of both competition and facilitation among plants. These results offer an example of the complex dynamics underlying ecosystem function in multitrophic systems and demonstrate that community context can impact diversity–functioning relationships between trophic levels. 

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