An Interview With Albert W. Tucker

The mathematical career of Albert W. Tucker, Professor Emeritus at Princeton University, spans more than 50 years. Best known today for his work in mathematical programming and the theory of games (e.g., the Kuhn-Tucker theorem, Tucker tableaux, and the Prisoner\u27s Dilemma), he was also in his earlier years prominent in topology. Outstanding teacher, administrator and leader, he has been President of the MAA, Chairman of the Princeton Mathematics Department, and course instructor, thesis advisor or general mentor to scores of active mathematicians. He is also known for his views on mathematics education and the proper interplay between teaching and research. Tucker took an active interest in this interview, helping with both the planning and the editing. The interviewer, Professor Maurer, received his Ph.D. under Tucker in 1972 and teaches at Swarthmore College

Suppression of the cottony cushion scale in Louisiana

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An Exploratory Study of Attributes, Affordances, Abilities, and Distance in Children\u27s Use of Mathematics Virtual Manipulative iPad Apps

This exploratory qualitative study investigated the presence of and relationships among constructs that contribute to children\u27s interactions with educational technology, leading to the development of the modification of attributes, affordances, abilities, and distance (MAAAD) for Learning framework. For this study, each of 10 fifth-grade children participated in one individual video-recorded semistructured interview session, during which they interacted with two mathematics virtual manipulative iPad apps and responded to follow-up questions. Video recordings and observation field notes were analyzed for evidence of attributes, affordance-ability relationships, distance, and relationships among these constructs. Constant comparative data analysis using memoing and eclectic coding provided evidence of the presence of each focus construct. Further analysis and interpretation, including quantization of qualitative data for visualization using novel rhombus plots, also led to the identification of emergent themes related to each construct and revealed relationships among the constructs. Emergent themes included categorization, alignment, and modification of attributes, variations and interrelationships among affordance-ability relationships, and the identification of and interactions among mathematical and technological distance. Furthermore, each construct related to each other construct. The evidence and interpretations led to the development of the MAAAD for Learning framework. The results of the study suggest that the MAAAD for Learning framework models relationships among attributes, affordance-ability relationships, and distance in the context of user-app interactions. the framework could serve as a tool for app developers designing apps, educators using apps to support children\u27s learning, and researchers characterizing user-app interactions and the outcomes of those interactions. The constructs, relationships, and framework identified in this study advance the literature on children\u27s interactions with educational technology tools, in particular literature concerning children\u27s interactions with mathematics virtual manipulative iPad apps

The Luminosity Function of Galaxies in the Las Campanas Redshift Survey

We present the $R$-band luminosity function for a sample of 18678 galaxies, with average redshift $z = 0.1$, from the Las Campanas Redshift Survey. The luminosity function may be fit by a Schechter function with $M^* = -20.29 \pm 0.02 + 5 \log h$, $\alpha = -0.70 \pm 0.05$, and $\phi^* = 0.019 \pm 0.001 \ h^3$~Mpc$^{-3}$, for absolute magnitudes$-23.0 \leq M - 5 \log h \leq -17.5$. We compare our luminosity function to that from other redshift surveys; in particular our normalization is consistent with that of the Stromlo-APM survey, and is therefore a factor of two below that implied by the$b_J \approx 20$bright galaxy counts. Our normalization thus indicates that much more evolution is needed to match the faint galaxy count data, compared to minimal evolution models which normalize at$b_J \approx 20$. Also, we show that our faint-end slope$\alpha = -0.7$, though ``shallower'' than typical previous values$\alpha = -1$, results primarily from fitting the detailed shape of the LCRS luminosity function, rather than from any absence of intrinsically faint galaxies from our survey. Finally, using [OII] 3727 equivalent width$W_{\lambda} = 5$~\AA \ as the dividing line, we find significant differences in the luminosity functions of emission and non-emission galaxies, particularly in their$\alpha$values. Emission galaxies have Schechter parameters$M^* = -20.03 \pm 0.03 + 5 \log h$and$\alpha = -0.9 \pm 0.1$, while non-emission galaxies are described by$M^* = -20.22 \pm 0.02 + 5 \log h$and$\alpha = -0.3 \pm 0.1$. (abridged abstract)Comment: 41 pages, including 13 postscript figures, uses AASTEX v4.0 style files. Important clarification of R-band definition, plus correction of luminosity densities and updated references. Main conclusions unchanged. Final version to appear in Ap

Living and Teaching in Two Worlds: Professional Identity Development in Transnational Dual Language Immersion Teachers

This study explores how transnational teachers working in dual language immersion schools in the United States negotiate their professional identity to pursue a career in teaching. Researchers pay attention to tensions between transnational teachers’ individual agency and the sociocultural influence of the workplace. An important issue is how chainging teacher’s professional identity impacts the ways in which they implement curriulum, particularly related to accountability. Framed by third space theory, we explore their heterogeneous stories and socio-culturally contextualized teaching experiences via a qualitative multiple case study. The three teachers wer all teaching Spanish or Chinese in urban public elementary schools in the Intermountain West region of the United States. Data sources include a semester long classroom observation, semi-structured teacher interviews, informal conversations with the teachers, teacher journal entries, and artifacts. Our results indicate that these teachers’ professional identity development processes were diverse, complex, and ongoing. All three displayed multicultural awareness regarding the codes of their new educational and cultural settings and exercised their agency through strong self-concept and frequent reflection within their situated contexts. Their prior cross-cultural teaching and schooling experiences served as a springboard that enabled them to gain a better understanding of their culturally and linguistically diverse students and to teach their target languages by incorporating elements of their students’ sociocultural backgrounds. This study supports a rich sociocultural appreciation of the processes and contexts of transnational teachers’ professional identity development. Keywords: Professional identity development, teacher identity, transnational teacher, third space theory, dual language immersio

Developing number sense with Fingu: a preschooler’s embodied mathematics during interactions with a multi-touch digital game

Early number sense, including subitizing and composition, is a foundation for mathematics, and bodies, especially fingers, are integral to number sense. Multi-touch technology offers innovative opportunities for developing and studying number sense, especially using conceptually congruent gestures that match the mathematics. However, there have been few investigations of the development of early number sense, particularly in embodied forms. Therefore, this mixed-methods study explores a preschooler’s development of early number sense during a month of interactions with the multi-touch digital mathematics game Fingu. Key findings related to the development of early number sense include relevance of configuration and quantity, relationships among gestures and quantities, and development of estimation and precision. This research adds new perspectives to our understandings of early number sense research and practice, calling for consideration of embodiment and conceptually congruent gestures