History of the Infinitely Small and the Infinitely Large in Calculus

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17^th through the 20^th centuries. We will also present `didactic observations' at relevant places in the historical account. This revised version was published online in July 2006 with corrections to the Cover Date.     

Flavor preferences conditioned with starch in rats

Rats were trained to associate artificial cherry or grape flavors with 1% starch suspensions. Conditioning was assessed by offering the rats a choice of the cherry versus grape flavors without starch. Conditioned preferences were moderately strong and persistent; 3 days of conditioning produced a preference that did not fully extinguish within 18 days. Food deprivation substantially increased intake of 1% starch suspension, However, the degree of preference conditioned was not influenced by the availability of food during the conditioning period; the rats that had been food deprived during training acquired as strong a preference as did those fed freely during training. The degree of preference obtained was similar in the rats given reinforcing and nonreinforcing fluids simultaneously or sequentially. Starch conditioned a slightly stronger preference than did the same concentration of glucose, even though these substances contain the same amount of calories. A statistically significant, but weak, preference was conditioned by 0.5%, but not by 0.25%, starch. It is proposed that the flavor of starch, independent of its calories, is reinforcing to rats.     

Discriminative control of interspecific killing in hungry and satiated rats

Hungry and satiated killer Long-Evans rats were exposed to two species of prey, mice and frogs. Experiment 1 demonstrated that the rats learned to discriminate between prey when attacks upon one of the prey were punished by electric shocks and attacks upon the alternative were not punished. Thus, killing of the “dangerous” prey was suppressed, while killing of the alternative, “safe” prey continued. However, in Experiment 2—in which the consequences of killing the prey differed in that one was allowed to be eaten following a kill but the other was not—no evidence of discriminative attack resulted. Hungry and satiated rats did not differ in their responsiveness toward prey as objects of attack, but hungry killers were more responsive to prey as food, thus demonstrating a dissociation of killing and eating control.     

Carbohydrate-induced stimulation of saccharin intake: Yoked controls

Dilute, intragastric infusions of carbohydrate stimulate saccharin intake. To determine whether this effect is due to an associative process, a yoked control was employed. One group of rats was trained in the conventional fashion: they received carbohydrate (maltodextrin) infusions whenever they drank saccharin. Whenever these rats were infused with carbohydrate, yoked rats were infused with the same amount of carbohydrate. In an experiment in which saccharin was available continuously, carbohydrate infusions stimulated saccharin intake in conventionally trained rats but not in yoked rats. In a subsequent experiment, in which rats were offered saccharin for only 2.5 h/day, carbohydrate infusions stimulated intake in conventionally trained and in yoked rats. However, when these rats were switched to continuous saccharin availability, saccharin intake declined in yoked rats but remained elevated in conventionally trained rats. Thus, carbohydrate infusions stimulate saccharin intake via an associative process.     

Systems,transfer, and fields: Evaluating a new approach to energy instruction

Energy is a central concept in science in every discipline and also an essential player in many of the issues facing people everywhere on the globe. However, studies have shown that by the end of K-12 schooling, most students do not reach the level of understanding required to be able to use energy to make sense of a wide range of phenomena. Many researchers have questioned whether the conceptual foundations of traditional approaches to energy instruction may be responsible for students' difficulties. In response to these concerns, we developed and tested a novel approach to middle school physical science energy instruction that was informed by the recommendations of the Framework for K-12 Science Education (National Research Council, 2012a) and the Next Generation Science Standards (NGSS) (NGSS Lead States, 2013). This new approach differs substantially from more traditional approaches to energy instruction in that it does not require energy forms and it emphasizes connections between energy, systems, and fields that mediate interaction-at-a-distance. We investigated student learning during this novel approach and contrasted it with student learning within a comparable unit based on a more traditional approach to energy instruction. Our findings indicate that students who learned in the new approach outperformed students who learned in the traditional approach in every quantitative and qualitative aspect considered in this study, irrespective of their prior knowledge of energy. They developed more parsimonious knowledge networks in relation to energy that focused primarily around the concept of energy transfer. This study warrants further investigation into the value of this new approach to energy instruction in both middle and high school.     

Storage and retrieval considerations of binary data bases

A binary approach to data storage and retrieval is introduced. It views the data base as a two-dimensional matrix that relates entities to all possible values the attributes of these entities may take. As such, it provides a unified solution to the two conflicting types of data base transactions—operational and managerial. An analytical investigation of the feasibility of binary storage and a compression method for reducing meaningless areas of the matrix are presented. Storage efficiencies of binary and conventional inverted file methods are compared and evaluated. An analysis of retrieval considerations associated with the binary matrix is given, particularly the issue of going from high to low orders of compression. Results of these analyses indicate that the binary data base's efficiency increases with increases in query complexity. Future research directions are sited and discussed.     

Beyond teaching practice: Exploring individual determinants of student performance on a research skills module

The purpose of this study was to identify the extent to which intrinsic student factors such as cognitive learning style, learning strategies, motivation and statistics anxiety, as well as certain demographic factors, predict performance on a research design and analysis course (RDAIIA) in the Department of Psychology at the University of the Witwatersrand, South Africa, with a view to developing future interventions for improving student performance. Using a sample of eighty students, it was established that the factors most predictive of RDA performance were secondary school performance (p < 0.0001) and age (p = 0.0106). Excluding demographic factors, self-efficacy (p < 0.0001), help-seeking (p = 0.0531) and a reflective learning style (p = 0.0071) were also found to be predictive. Performance also differed between students who had completed Mathematics on the higher grade in secondary school and those who had completed standard grade (p = 0.016). These findings present both possible directions and substantial challenges for future performance-intervention strategies.     

Computed tomography‐enhanced anatomy course using enterprise visualization

Rapid changes in medical knowledge are forcing continuous adaptation of the basic science courses in medical schools. This article discusses a three‐year experience developing a new Computed Tomography (CT)‐based anatomy curriculum at the Sackler School of Medicine, Tel Aviv University, including describing the motivations and reasoning for the new curriculum, the CT‐based learning system itself, practical examples of visual dissections, and student assessments of the new curriculum. At the heart of this new curriculum is the emphasis on studying anatomy by navigating inside the bodies of various living individuals utilizing a CT viewer. To assess the students’ experience with the new CT‐based learning method, an anonymous questionnaire was administered at the end of the course for three consecutive academic years: 2008/2009, 2009/2010, 2010/2011. Based upon the results, modifications were made to the curriculum in the summers of 2009 and 2010. Results showed that: (1) during these three years the number of students extensively using the CT system quadrupled (from 11% to 46%); (2) students' satisfaction from radiologists involvement increased by 150%; and (3) student appreciation of the CT‐based learning method significantly increased (from 13% to 68%). It was concluded that discouraging results (mainly negative feedback from students) during the first years and a priori opposition from the teaching staff should not weaken efforts to develop new teaching methods in the field of anatomy. Incorporating a new curriculum requires time and patience. Student and staff satisfaction, along with utilization of the new system, will increase with the improvement of impeding factors. Anat Sci Educ 6: 332–341. © 2013 American Association of Anatomists.     

Time Comparison between the Cross-Over and Jab-Step Starts

Abstract

The purpose of this investigation was to evaluate and quantify physiological differences among groups of distance runners. The subjects included 20 elite distance runners (8 marathon, 12 middle-long distance) and 8 good runners. Working capacity and cardiorespiratory function were determined by submaximal and maximal treadmill tests, and body composition by hydrostatic weighing. The variables studied were maximum oxygen uptake ([Vdot]O₂ max), [Vdot]O₂ submax, lactic acid submax, lean body weight, and fat weight. MANOVA showed that the good runners differed from the elite runners (p < 0.01) and the elite marathon runners differed from the elite middle-long distance runners (p < 0.05). Discriminant analysis showed that both functions were significant. The first was a general physiological efficiency factor that separated the good and elite runners. The second separated the elite marathon and middle-long distance groups. The second function showed that the marathon runners had lower lactic acid submax values. The middle-long distance runners had higher [Vdot]O₂ max values. Classification analysis was used to evaluate the accuracy of the discriminant analysis; 80% of the elite runners were correctly classified as marathon or middle-long distance runners. The discriminant functions were used to develop a multivariate scaling model for evaluating distance runners. Two premier runners, one marathoner (F. Shorter) and one middle-long distance runner (S. Prefontaine), were found to be at the extremes of the scale. The data showed that the discriminant functions provided a valid model for evaluating differences among elite distance runners.