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.     

Examining the literature to reveal the nature of community EE/ESD programs and research

Interest in community environmental education (EE) and community education for sustainable development (ESD) is increasing, as evidenced by the increase in studies examining community EE/ESD approaches and NAAEE’s current development of the Community EE Guidelines for Excellence. Thus, the purpose of this paper is to: (1) provide a review of research which identifies as examining community EE/ESD issues from 1994 to 2013, (2) examine themes specific to community-oriented program implementation and outcomes, and (3) examine theoretical trends within this work. While no single framework emerged from the diverse studies on community EE/ESD, themes suggested greater success for programs that were rooted in community issues, involved multiple community partners, were oriented towards collaborative and civic action, and incorporated reflection on social institutions and power dynamics. These characteristics reflect an emancipatory approach to education which seeks to enhance human development through public participation and engagement. To this end, future researchers and practitioners should consider: (1) the resources needed to understand community values and concerns, (2) skills and knowledge required to build relationships and attend to community culture, values and democratic processes, (3) theories that account for the socio-historical contexts of participants and their pathways for transformation and empowerment, and (4) a process of action that continually reflects on power structures and dynamics.     

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.