Learning events in the life of a community of mathematics educators

The paper examines some characteristics of learning events of a community of mathematics educators. Participation in the community entailed gaining familiarity with agreed upon conventions, goals, and forms of communication. The case discussed herein is an attempt to convey the complexities underlying learning in such a community through (re)negotiation of practices and goals. The notion ofreflective discourse is borrowed to describe a group discussion involving collective reflection that constituted an occasion for meaningful learning.     

Relations between Types of Reasoning and Computational Representations

This paper examines the idea that particular representations differentially support and enhance different cognitive processes, in particular different types of reasoning. Five case studies were conducted consisting of detailed observations of pairs of middle-school students interacting with a computer-based learning environment. The software environment, called NumberSpeed, deals with kinematics concepts by having students construct various motion scenarios by adjusting numerical motion parameters: position, velocity and acceleration. NumberSpeed provides feedback about the student-specified motion using two representations: the motion representation and the number-lists representation. Two distinct types of reasoning were recognized in students’ learning while interacting with NumberSpeed: (1) model-based reasoning and (2) constraint-based reasoning. These two types of reasoning are characterized in detail and their roles in problem-solving are analyzed. A cross-analysis between the types of reasoning and the use of particular NumberSpeed representations reveals a correlation between type of reasoning and representational choice. These findings are explained by analyzing the representations’ characteristics and the ways they may differentially support and enhance particular types of reasoning.     

Students' Verification Strategies for Combinatorial Problems

We focus on a major difficulty in solving combinatorial problems, namely, on the verification of a solution. Our study aimed at identifying undergraduate students' tendencies to verify their solutions, and the verification strategies that they employ when solving these problems. In addition, an attempt was made to evaluate the level of efficiency of the students' various verification strategies in terms of their contribution to reaching a correct solution. 14 undergraduate students, who had taken at least 1 course in combinatorics, participated in the study. None of the students had prior direct learning experience with combinatorial verification strategies. Data were collected through interviews with individual or pairs of participants as they solved, 1 by 1, 10 combinatorial problems. 5 types of verification strategies were identified, 2 of which were more frequent and more helpful than others. Students' verifications proved most efficient in terms of reaching a correct solution when they were informed that their solution was incorrect. Implications for teaching and learning combinatorics are discussed.     

Evaluating the Influence of Warmup on Singing Voice Quality Using Acoustic Measures

Vocal warmup is generally accepted as vital for singing performance. However, only a limited number of studies have evaluated this effect quantitatively. In this study, we evaluated the effect of vocal warmup on voice production, among young female singers, using a set of acoustic parameters. Warmup reduced frequency-perturbation (p < 0.001) and amplitude-perturbation values (p < 0.05). In addition, warmup increased singer's formant amplitude (p < 0.05) and improved noise-to-harmonic ratio (p < 0.05). Tone-matching accuracy, however, was not affected by warmup. The effect of vocal warmup on frequency-perturbation parameters was more evident among mezzo-soprano singers than it was among soprano singers. It was also more evident in the low pitch-range than in the higher pitch-ranges (p < 0.05). The results of this study provide valid support for the advantageous effect of vocal warmup on voice quality and present acoustic analysis as a valuable and sensitive tool for quantifying this effect.     

Applying the Voice Handicap Index (VHI) to Dysphonic and Nondysphonic Hebrew Speakers

OBJECTIVE: To evaluate a translated version of the Voice Handicap Index (VHI) as a diagnostic tool for people with and without a laryngeal pathology, among Hebrew speakers. STUDY DESIGN: Parallel group design. METHODS: The VHI was translated and adapted to Hebrew. The translated version was, then, administered to a group of 182 patients with various laryngeal pathologies and a control group of 171 people with no laryngeal pathology. Based on the participants' responses to the VHI, statistical analyses were, initially, performed to assess validity and reliability, and then to evaluate group differences between the pathological and control groups and among the different pathological groups included in the study. RESULTS: Statistical analyses showed high reliability values of the Hebrew version of the VHI (overall Cronbach's alpha r = 0.976). Participants' scores were not affected by their age (P = 0.156) or gender (P = 0.261). The participants in the control group obtained significantly lower scores on the overall VHI score, as well as on all three subscale scores, in comparison with the pathological group (P < 0.001). In addition, within the pathological group, patients with neurogenic pathologies received higher scores than all other pathological groups, whereas patients with laryngeal inflammation received lower scores than all other pathological groups (P < 0.05). CONCLUSION: The VHI is a powerful tool for quantifying patients' perceptions of their voice handicaps, and it maintained its power across translation. The VHI was shown to be valuable for the assessment of speakers with, as well as without laryngeal pathologies.     

Students' understanding of the notion of function in dynamic geometry environments

To make optimal use of computational environments, one must understand how students interact with the environments and how students' mathematical thinking is reflected and affected by their use of the environments. Similarly, to make sense of research on students' thinking and learning, one must understand how the environments and contexts used in the research may affect the conclusions one derives.The research on students' learning of functions has approached the topic in terms of symbols and graphs (see, for example, Leinhardt et al. (1990) for a review of work up to that point; Harel and Dubinsky (1992) for a collection of research; and Dugdale et. al. (1995), for some recent thinking about implications for curriculum reform using technology). Dynamic geometry environments (DGEs) like Cabri Geometry or Geometer's Sketchpad, offer us an opportunity to get a new perspective on these old and important issues. DGEs let students build geometrical constructions and then drag certain objects around the screen in a continuous manner while observing how the entire construction responds dynamically. In this way DGEs model functional relationships that are not specified by symbols or represented by graphs.Based on interviews with undergraduate mathematics majors, this paper presents preliminary observations that confirm some old results and raise some new questions about students' notions of function.     

A fast flow tube study of gas phase H/D exchange of multiply protonated ubiquitin

An electrospray ionization (ESI)/fast-flow technique has been applied to the study of gas phase hydrogen/deuterium (H/D) exchange kinetics. Multiply charged ubiquitin ions [ubiquitin + nH](n)(+), in charge states n = 7-13, were reacted with ND(3). The behavior of ND(3) as exchange reagent is different from that of the previously studied reagents, D(2)O and CH(3)OD. Contrary to those, the maximum number of exchanged hydrogen atoms and the overall exchange rate were observed to increase with increasing charge state of the ubiquitin ions. The results are reagent-dependent because the exchange mechanisms are different for the different reagents. This observation is in agreement with a recent conclusion by Beauchamp and co-workers that contrary to the assumption often expressed in earlier studies, H/D exchange kinetics may not directly reflect ion structures. The results for all three reagents are, however, consistent with observations of previous ion mobility experiments that with increasing charge state the conformers change from more compact, partially folded structures to elongated nearly linear ones. H/D exchange of (ubiquitin + 13H)(13+) with ND(3) leads to two separated ion populations reflecting the possible existence of two conformers with different exchange rates. The ions (ubiquitin + 8H)(8+) and (ubiquitin + 11H)(11+) represent a partially folded structure and an unfolded structure, respectively, and were studied in greater detail. The relative abundances of ions were measured in steps of 0.5 m/z (mass-to-charge ratio), as a function of the ND(3) flow rate. The experimental results were simulated by computer fitted curves based on a recently developed algorithm. The algorithm allows the extraction of sets of grouped rate constants. Eight rate constant groups were deduced for each of the two ions. These rate constants correspond to 32 and 44 H/D exchanges for the 8+ and 11+ charged ions, respectively. The results indicate higher individual rates for most of the exchanged atoms in the 11+ ion compared to the 8+ ion.     

The Elekes–Szabó Theorem in four dimensions

Let F ∈ C[x, y, s, t] be an irreducible constant-degree polynomial, and let A,B,C,D ? C be finite sets of size n. We show that F vanishes on at most O(n^8/3) points of the Cartesian product A × B × C × D, unless F has a special group-related form. A similar statement holds for A,B,C,D of unequal sizes, with a suitably modified bound on the number of zeros. This is a four-dimensional extension of our recent improved analysis of the original Elekes–Szabó theorem in three dimensions. We give three applications: an expansion bound for three-variable real polynomials that do not have a special form, a bound on the number of coplanar quadruples on a space curve that is neither planar nor quartic, and a bound on the number of four-point circles on a plane curve that has degree at least five.     

Porphyrin Nanocage‐Embedded Single‐Molecular Nanoparticles for Cancer Nanotheranostics

Single molecular nanoparticles (SMNPs) integrating imaging and therapeutic capabilities exhibit unparalleled advantages in cancer theranostics, ranging from excellent biocompatibility, high stability, prolonged blood lifetime to abundant tumor accumulation. Herein, we synthesize a sophisticated porphyrin nanocage that is further functionalized with twelve polyethylene glycol arms to prepare SMNPs ( porSMNPs ). The porphyrin nanocage embedded in porSMNPs can be utilized as a theranostic platform. PET imaging allows dynamic observation of the bio‐distribution of porSMNPs , confirming their excellent circulation time and preferential accumulation at the tumor site, which is attributed to the enhanced permeability and retention effect. Moreover, the cage structure significantly promotes the photosensitizing effect of porSMNs by inhibiting the π–π stacking interactions of the photosensitizers, ablating of the tumors without relapse by taking advantage of photodynamic therapy.