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.     

Reactivity of Singlet Oxygen Toward Proteins: The Effect of Structure in Basic Pancreatic Trypsin Inhibitor and in Ribonuclease A

The reactions of singlet oxygen, ¹O₂, with large peptides have been described previously. It was found that even in these systems, which in their native form are generally not supposed to possess a stable structure in solution, the polypeptide does impede the access of ¹O₂ to the amino acids that react readily with ¹O₂. Here we describe the ¹0₂ reaction with two proteins of well-defined structure. The quenching of ¹O₂ by bovine pancreatic trypsin inhibitor (BPTI) and by ribonuclease A (RNase A) was compared to that of a solution at the same concentration as those of its constituent amino acids that react readily with ¹O₂. The proteins were studied in their native form, when partly denatured by splitting their S-S bonds and when fully denatured. It was found that while in the native form the quenching rate constant was seven times lower in BPTI (2.2 vs 15.2 times 10⁷WM^-1 s^-1) and three times lower in RNase A (11.0 vs 32 times 10⁷M^-l s^-1) than in a mixture of its constituent amino acid residues, it increased upon denaturation reaching in the fully denatured state the value of the corresponding amino acid mixture. More striking is the effect of the protein structure when comparing the fraction of the encounters between ¹O₂ and protein, which cause damage to the protein, as reflected in the decrease of its biological activity. This decrease is assumed to be due to the chemical (oxidative) reactions of ¹O₂ in the protein. In the exceptionally stable BPTI the fraction of such encounters was 0.05 and in RNase A it was 0.2, whereas for the amino acid tryptophan in solution, 0.7 of the collisions with ¹O₂ led to a chemical reaction.     

REACTIVITY OF SINGLET OXYGEN TOWARD LARGE PEPTIDES

Abstract— The reactions of singlet oxygen, ¹O₂, with amino acids and their derivatives have been studied previously. It was found that only five amino acid residues interact readily with ¹O₂. Here we describe its reactions with the large peptides melittin, neuropeptide Y (NPY) and insulin in their native and in their denatured forms. The singlet oxygen quenching by a polypeptide was compared with that of a solution at the same concentration as those of its constituent amino acids, which are known to react efficiently with ¹O₂. It was found that the quenching rate by such a mixture exceeded that of the polypeptides in their native form. The ratio of the rate constants for NPY to that of the corresponding amino acid mixture in solution was 0.75. For melittin in its monomeric form it was 0.83 and for a tetramer of melittin (at high ionic strength) it was 0.70. For native insulin the ratio of the rate constants was 0.55. For oxidized insulin with its -S-S- bridges opened the figure became 0.80. However, the quenching by all the polypeptides in their fully denatured form (in the presence of 6 M urea) equalled that of the corresponding amino acid mixtures. Although polypeptides are generally supposed not to possess a stable secondary structure in solution the effects are explained by shielding of some of the reactive amino acid residues in the chain by temporary folding or incipient secondary structures of the native polypeptide.
It is shown that the kinetics for a homogeneous solution of quenchers applies also to measurements in a polypeptide solution where the quenchers are localized along the polypeptide backbone and thus form clusters in solution.     

Microspheres of Mixed Proteins

This paper describes the synthesis of mixed proteinaceous microspheres (MPMs) by the sonochemical method. The current fundamental research follows the research of Suslick and co‐workers who have developed a method by which high‐intensity ultrasound is used to make aqueous suspensions of proteinaceous microcapsules filled with water‐insoluble liquids. 1 By using high‐intensity ultrasound, we have synthesized microspheres made of a few different proteins. The three proteins used in the current experiments are bovine serum albumin (BSA), green fluorescent protein (GFP), and cyan fluorescent protein–glucose binding protein–yellow fluorescent fused protein (CFP‐GBP‐YFP). The two synthesized microspheres made of mixed proteins are BSA‐GFP and BSA‐(CFP‐GBP‐YFP). This paper presents the characterization of the sonochemically produced microspheres of mixed proteins. It also provides an estimate of the efficiency of the sonochemical process in converting the native proteins to microspheres.     

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.