THE ANALYSIS OF SELF-EFFICACY FOR STUDENTS ENROLLED IN A CALCULUS I COURSE AT A COMMUNITY COLLEGE

It is generally accepted that learning implies a multitude of factors meant to prepare the children for life and its challenges. Some of these factors are directly related to the level of knowledge of subject matter, but others are based on individual feelings, relationships, or capabilities of developing a sense of belonging and personal worth, confidence, or attitude toward a certain content area. All these elements together form the foundation of student’s future success. On many occasions, certain factors such as the teaching approaches, encouragement from family members and school personnel, or past experiences in learning mathematics are important in creating a positive view of mathematics. From basic arithmetic to the more advanced calculus courses in first years of college, students build knowledge and shape continuously their views on mathematics. These factors and experiences can help form the foundations of a positive attitude toward mathematics and may influence students’ self-efficacy with respect to the mathematics. While literature provides examples where students’ self-efficacy was a determining factor in pursuing and solving mathematics problems, the influence of college mathematics courses on developing higher or lower levels of self-efficacy was a scarcely explored area. The purpose of the current study was to identify the impact of a Calculus and Analytic Geometry I course on college freshman students’ mathematics self-efficacy, and suggest ways of using the results for future teaching and learning. The study used an explanatory sequential mixed methods research design, where the quantitative portion of the study helped determine participants for qualitative portion of the study. The results revealed that the Calculus and Analytic Geometry I course improved general students’ self-efficacy in some areas of mathematics related to daily mathematical tasks. Also, it was determined that students’ background in previous mathematics courses had a strong impact in maintaining a high level of self-efficacy, and class interaction had a particular role in increasing most students’ self-efficacy. The majority of students reported that interactivity in mathematics classes in high school was less evident than the interactivity observed in these college mathematics classes. Not surprisingly, some students reported a preference to follow algorithms and step-by-step methods when solving mathematics problems rather than rely on deeper understanding. Although the results of the study cannot be generalized, it was suggested that instructors should insist on constructing a strong mathematical background of students early in students’ mathematics classes. While peer work and collaboration should be encouraged, the instructor should also identify those students who prefer different approaches in studying mathematics. The same observation about teaching and learning approaches is valid in the case of using algorithms and step-by-step problem solving methods. Keywords: self-efficacy, interactivity, mathematical background, algorith

New Solutions to the Firing Squad Synchronization Problems for Neural and Hyperdag P Systems

We propose two uniform solutions to an open question: the Firing Squad Synchronization Problem (FSSP), for hyperdag and symmetric neural P systems, with anonymous cells. Our solutions take e_c+5 and 6e_c+7 steps, respectively, where e_c is the eccentricity of the commander cell of the dag or digraph underlying these P systems. The first and fast solution is based on a novel proposal, which dynamically extends P systems with mobile channels. The second solution is substantially longer, but is solely based on classical rules and static channels. In contrast to the previous solutions, which work for tree-based P systems, our solutions synchronize to any subset of the underlying digraph; and do not require membrane polarizations or conditional rules, but require states, as typically used in hyperdag and neural P systems

Edge- and Node-Disjoint Paths in P Systems

In this paper, we continue our development of algorithms used for topological network discovery. We present native P system versions of two fundamental problems in graph theory: finding the maximum number of edge- and node-disjoint paths between a source node and target node. We start from the standard depth-first-search maximum flow algorithms, but our approach is totally distributed, when initially no structural information is available and each P system cell has to even learn its immediate neighbors. For the node-disjoint version, our P system rules are designed to enforce node weight capacities (of one), in addition to edge capacities (of one), which are not readily available in the standard network flow algorithms.Comment: In Proceedings MeCBIC 2010, arXiv:1011.005

Sublinear P system solutions to NP-complete problems

Many membrane systems (e.g. P System), including cP systems (P Systems with compound terms), have been used to solve efficiently many NP-hard problems, often in linear time. However, these solutions have been independent of each other and have not utilised the theory of reductions. This work presents a sublinear solution to k-SAT and demonstrates that k-colouring can be reduced to k-SAT in constant time. This work demonstrates that traditional reductions are efficient in cP systems and that they can sometimes produce more efficient solutions than the previous problem-specific solutions

Structured Modeling with Hyperdag P Systems: Part A

P systems provide a computational model based on the structure and interaction of living cells. A P system consists of a hierarchical nesting of cell-like membranes, which can be visualized as a rooted tree. Although the P systems are computationally complete, many real world models, e.g., from socio-economic systems, databases, operating systems, distributed systems, seem to require more expressive power than provided by tree structures. Many such systems have a primary tree-like structure completed with shared or secondary communication channels. Modeling these as tree-based systems, while theoretically possible, is not very appealing, because it typically needs artificial extensions that introduce additional complexities, nonexistent in the originals. In this paper we propose and define a new model that combines structure and flexibility, called hyperdag P systems, in short, hP systems, which extend the definition of conventional P systems, by allowing dags, interpreted as hypergraphs, instead of trees, as models for the membrane structure. We investigate the relation between our hP systems and neural P systems. Despite using an apparently less powerful structure, i.e., a dag instead of a general graph, we argue that hP systems have essentially the same computational power as tissue and neural P systems. We argue that hP systems offer a structured approach to membrane-based modeling that is often closer to the behavior and underlying structure of the modeled objects. Additionally, we enable dynamical changes of the rewriting modes (e.g., to alternate between determinism and parallelism) and of the transfer modes (e.g., the switch between unicast or broadcast). In contrast, classical P systems, both tree and graph based P systems, seem to focus on a statical approach. We support our view with a simple but realistic example, inspired from computer networking, modeled as a hP system with a shared communication line (broadcast channel). In Part B of this paper we will explore this model further and support it with a more extensive set of examples

A sublinear Sudoku solution in cP Systems and its formal verification

Sudoku is known as a NP-complete combinatorial number-placement puzzle. In this study, we propose the first cP system solution to generalised Sudoku puzzles with m×m cells grouped in m blocks. By using a fixed constant number of rules, our cP system can solve all Sudoku puzzles in sublinear steps. We evaluate the cP system and discuss its formal verification

Membrane computing with water

We introduce water tank systems as a new class of membrane systems inspired by a decentrally controlled circulation of water or other liquids throughout cells called tanks and capillaries called pipes. To our best knowledge, this is the first proposal addressing the behavioural principle of floating and stored water for modelling of information processing in terms of membrane computing. The volume of water within a tank stands for a non-negative rational value when acting in an analogue computation or it can be interpreted in a binary manner by distinction of “(nearly) full” or “(nearly) empty”. Water tanks might be interconnected by pipes for directed transport of water. Each pipe can be equipped with valves which in turn either fully open or fully close the hosting pipe according to permanent measurements whether the filling level in a dedicated water tank exceeds a certain threshold or not. We demonstrate dedicated water tank systems together with simulation case studies: a ring oscillator for generation of clock signals and for iteratively making available amounts of water in a cyclic scheme, analogue arithmetics by implementation of addition, non-negative subtraction, division, and multiplication complemented by systems in binary mode for implementation of selected logic gates

Logarithmic SAT Solution with Membrane Computing

P systems have been known to provide efficient polynomial (often linear) deterministic solutions to hard problems. In particular, cP systems have been shown to provide very crisp and efficient solutions to such problems, which are typically linear with small coefficients. Building on a recent result by Henderson et al., which solves SAT in square-root-sublinear time, this paper proposes an orders-of-magnitude-faster solution, running in logarithmic time, and using a small fixed-sized alphabet and ruleset (25 rules). To the best of our knowledge, this is the fastest deterministic solution across all extant P system variants. Like all other cP solutions, it is a complete solution that is not a member of a uniform family (and thus does not require any preprocessing). Consequently, according to another reduction result by Henderson et al., cP systems can also solve k-colouring and several other NP-complete problems in logarithmic time

Towards Structured Modelling with Hyperdag P Systems

Although P systems are computationally complete, many real world models, such as socio-economic systems, databases, operating systems and distributed systems, seem to require more expressive power than provided by tree structures. Many such systems have a primary tree-like structure augmented with shared or secondary communication channels. Modelling these as tree-based systems, while theoretically possible, is not very appealing, because it typically needs artificial extensions that introduce additional complexities, inexistent in the originals. In this paper, we propose and define a new model called hyperdag P systems, in short, hP systems, which extend the definition of conventional P systems, by allowing dags, interpreted as hypergraphs, instead of trees, as models for the membrane structure. We investigate the relation between our hP systems and neural P systems. Despite using an apparently restricted structure, i.e., a dag instead of a general graph, we argue that hP systems have essentially the same computational power as tissue and neural P systems. We argue that hP systems offer a structured approach to membranebased modelling that is often closer to the behavior and underlying structure of the modelled objects