Algebraic proofs over noncommutative formulas

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003). We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitin?s formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991).The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.     

Symbols and the bifurcation between procedural and conceptual thinking

Abstract:

Symbols occupy a pivotal position between processes to be carried out and concepts to be thought about. They allow us both to do mathematical problems and to think about mathematical relationships. In this article we consider the discontinuities that occur in the learning path taken by different students, leading to a divergence between conceptual and procedural thinking. Evidence will be given from several different contexts in the development of symbols through arithmetic, algebra, and calculus, then on to the formalism of axiomatic mathematics. This evidence is taken from a number of research studies recently conducted for doctoral dissertations at the University of Warwick by students from the United States, Malaysia, Cyprus, and Brazil, with data collected in the United States, Malaysia, and the United Kingdom. All the studies form part of a broad investigation into why some students succeed, while others fail.     

Human factors in continuous time-concerned cooperative systems represented by NΣ-labeled calculus

NΣ-labeled calculus is a generalization of N-labeled calculus so as to describe time-concerned recognition, knowledge, belief and decision of humans or computer programs together with related external physical or logical phenomena. N-labeled calculus is the smallest formal system applicable to verification and analysis of cooperative real-timing systems on natural number time introduced as an adaptation of tense arithmetic (TA). A merging problem of vehicles with misunderstanding or incorrect recognition is discussed as an example of cooperating systems controlling continuously changing objects including human factor. Euler’s approximation is introduced in order to represent the continuously changing objects. Through this example, relationship among artificial intelligence, external environment and human factors is investigated.     

A Survey of World Wide Web Use by Freshman English Students

ABSTRACT

Freshman English Bibliographic Instruction (BI) students were surveyed about their usage of the World Wide Web (WWW). The survey sought to determine the WWW use patterns of the students, the perceived expertise level of the users, their views on the utility of the WWW, and the students' general satisfaction with it. The results indicated that students lack the searching and critical thinking skills needed to use the WWW effectively for research. Therefore BI programs should adopt a multi-layered approach in order to teach these skills to students.     

Formal development of industrial-scale systems in <Emphasis Type="BoldItalic">Circus</Emphasis>

Circus is a new notation that may be used to specify both data and behavioural aspects of a system, and has an associated refinement calculus. In this work, we present rules to translate Circus programs to Java programs that use JCSP, a library that implements Communicating Sequential Processes constructs. These rules can be used as a complement to the Circus algebraic refinement technique, or as a guideline for implementation. They are a link between the results on refinement in the context of Circus and a practical programming language in current use. The rules can also be used as the basis for a tool that mechanises the translation. Although a few case studies are already available in the literature, the industrial fire control system, whose refinement and implementation is discussed in this paper, is, as far as we know, the largest case study on the Circus refinement strategy.     

On Transitions Between Representations: The Role of Contextual Reasoning in Calculus Problem Solving

This article argues for a shift in how researchers discuss and examine students' uses and understandings of multiple representations within a calculus context. An extension of Zazkis, Dubinsky, and Dautermann's (1996) visualization/analysis framework to include contextual reasoning is proposed. Several examples that detail transitions between modes of reasoning and how these transitions inform students' reasoning in a calculus context are discussed. These examples are used to provide evidence for the usefulness of the model for unpacking student reasoning.     

Applying fuzzy arithmetic to the system dynamics for the customer–producer–employment model

This paper presents a system dynamics analysis based on the application of fuzzy arithmetic. Traditional crisp system dynamics observe that some variables/parameters may belong to the uncertain factors. It is necessary to extend the system dynamics to treat also the vague variables/parameters. The evaluation of fuzzy system dynamics may provide the decision-maker information regarding the system's behavior uncertainties. In this paper, the customer–producer–employment model is examined with the fuzzy system dynamics in two types of fuzzy arithmetic, α-cut fuzzy arithmetic and T_ω weakest t-norm operator. Symmetrical and nonsymmetrical triangular fuzzy number (TFN), varied amount of fuzzy inputs’ fuzziness, and length of the system time delay are examined with useful results provided. Particularly, it is revealed that (1) both types of fuzzy arithmetic can provide the steady-state analysis of the system's variables as their counterpart, the crisp arithmetic analysis. (2) The α-cut arithmetic realizes the fuzziness of the model interactive variables fuzzier than that of the T_ω fuzzy arithmetic due to the accumulating phenomenon of fuzziness of the α-cut arithmetic. The fuzzier the inputs, the higher the level and/or oscillation of the cyclically steady or stable pattern of the stability of these variables exhibit with the α-cut arithmetic. (3) The T_ω arithmetic gives a smaller fuzziness and defuzzified values due to the concept that it takes only the maximal fuzziness encountered and calculated in the operation. In this case, the T_ω arithmetic provides more stable (or conversely less sensitive) results to the amount of fuzziness and nonsymmetricity (fuzziness) of input.     

How does the smartphone usage of college students affect academic performance?

The present study aims to investigate the effects of smartphone use by college students on their perceived academic performance. Using five hypotheses derived from the literature related to smartphone use, the initial model was set up for path analysis to reveal the relationships among variables regarding college students' smartphone use in the academic setting. Moreover, multiple group analyses were additionally conducted to verify whether students exhibited different relationships in the hypothesized model depending on their majors. The results from path analyses showed that all path coefficients were positive and statistically significant, which indicated that all five hypothesized paths were supported. Above all, the results from multiple group analysis showed that one path (from Behavioural Intention to Use Smartphone to Academic Performance) differed significantly across groups. The findings from the current study provide educational policymakers and educators with information on how smartphone utilization in learning activities influences students' academic performance.     

Can structured representation enhance students' thinking skills for better understanding of E-learning content?

This paper proposes an e-learning model to assess the effects of online structured representation of content on learners' understanding. We designed a structured representation based on the theory of distributed cognition that provides seven segments for reading research articles. The study hypothesized that motivation, attention, and interactivity are essential factors that affect students' thinking skills for understanding e-learning content. To investigate and confirm the effect of these factors on the students' thinking skills, we designed a survey and analyzed the responses of 210 university students concerning the proposed structured representation. The results revealed that motivation, attention, and interactivity did contribute to the students' thinking skills. They also demonstrated that the structured representation helped students achieve an adequate level of thinking skills as they read research articles, which had a positive effect on their understanding. This finding demonstrates that structured representation has significant potential as a learning tool and that structure-based e-learning can influence students' metacognitive activities and facilitate their understanding.     

Usability,expectation, confirmation,and continuance intentions to use electronic textbooks

A trend in education is the use of electronic textbooks (e-textbook, [Stone, R. W., and L. Baker-Eveleth. 2013. “Students’ Expectation, Confirmation, and Continuance Intention to Use Electronic Textbooks.” Computers in Human Behavior 29 (3): 984–990.]). A variety of research has examined the motivations for e-textbook adoption as well as continuance intentions of e-textbooks after an initial adoption. This research extends the continuance intentions literature using the expectation–confirmation model ([Bhattacherjee, A. 2001. “Understanding Information Systems Continuance: An Expectation–Confirmation Model.” MIS Quarterly 25 (3): 351–370.]) and including the constructs of e-textbook usability and its dimensions. The data were responses to an online questionnaire by students at a medium-sized (USA) university. A total of 1434 responses were received for a 12.52% response rate. These responses were partitioned to include only complete responses from students who had previously used an e-textbook. This produced a sample of 639 questionnaire responses for empirical analysis. The analysis was performed using structural equation modelling (i.e. Procedure Calis in PC SAS version 9.2). The results showed that students' continuance intentions of e-textbooks are driven by students' satisfaction with e-textbooks and perceived usefulness. Additionally, students' expectation–confirmation and e-textbook usability positively influence both students' satisfaction and perceived usefulness and hence the intentions for continued e-textbook adoptions. Finally, the usability dimensions of efficiency and effectiveness significantly influence e-textbook usability.     

Is There Social Capital in a Social Network Site?: Facebook Use and College Students' Life Satisfaction,Trust, and Participation1

This study examines if Facebook, one of the most popular social network sites among college students in the U.S., is related to attitudes and behaviors that enhance individuals' social capital. Using data from a random web survey of college students across Texas (n = 2,603), we find positive relationships between intensity of Facebook use and students' life satisfaction, social trust, civic engagement, and political participation. While these findings should ease the concerns of those who fear that Facebook has mostly negative effects on young adults, the positive and significant associations between Facebook variables and social capital were small, suggesting that online social networks are not the most effective solution for youth disengagement from civic duty and democracy.     

Motivation factors and barriers to the continuous use of blended learning approach using Moodle: students' perceptions and individual differences

ABSTRACT

Practitioners of blended learning encounter challenges on how to cater students individual needs and to increase their motivational involvement with the system. Therefore, this study aims to identify and evaluate the motivation factors and barriers that influence students' decisions to continue using blended learning based on Moodle platform. The study proposes a research model to gain a better understanding of students' behavioural attitudes, motivations, and barriers to the continuous use of blended learning. In addition, a multi-group analysis was conducted to examine individual differences between students from three different universities in Palestine. The findings provide a comprehensive understanding of students' behaviours and the investigated factors of the proposed model were truly significant in forming students' perceptions and reflecting their decisions to continue using such learning approach. Features of blended learning system were found to be the most significant motivational antecedents in relation to influencing students' perceptions and involvements in the learning environment, while perceived behavioural control and subjective norms were found to be the primary factors driving students' continuous use. Finally, the findings shed light on the impact of individual differences on students' perceptions and usage behaviours and suggest that these differences could be leveraged to provide adaptive usage decisions.     

Towards an analysis framework for investigating students' engagement and learning in educational blogs

Educational blogs are currently gaining in popularity in schools and higher education institutions, and they are widely promoted as collaborative tools supporting students' active learning. However, literature review on educational blogging revealed a lack of a complete and consistent framework for studying and assessing students' engagement and the impact of blogging on students' learning. This study reports on the application of an analysis framework for evaluating blogging learning activities, based on two well‐documented models, those of Community of Inquiry (CoI) and Social Network Analysis. The framework proposed is examined through an empirical study involving 21 K‐9 students, coming from two classes, in Greece. This investigation shed light into the different ways of students' engagement in a blog‐based project, namely their social and cognitive presence that supported the development of a CoI and learning. The results suggest that the students, through their different roles in the blog, achieved higher thinking and cognitive levels.     

Efficiency of learning environment using GeoGebra when calculus contents are learned in collaborative groups

In this paper we present a modern approach of teaching mathematics based on the computer supported collaborative learning (CSCL) of calculus contents. The collaborative learning was used in calculus course at the University of Novi Sad, Serbia, for examining functions and drawing their graphs. In 2012 the authors decided to improve the collaborative learning introducing GeoGebra application. Small four member groups were formed by using Kagan's (1994) principles. Two groups of students, the experimental, and the control one were observed. The students in the experimental group learned with the help of GeoGebra, and the students in the control group learned without using GeoGebra.Comparison between those two groups of the first year calculus students, regarding their way of learning and the results achieved, is described below. Before the students' collaborative learning, they were tested with a pre-test and their knowledge necessary for examining functions was verified. The pre-test showed that there was no significant statistical difference between the experimental and the control group. The experimental group worked with the help of the computer and the control one without it. After the collaborative learning, the students were tested with a test (colloquium) and the results of the experimental group were significantly better than the results of students in the control group. At the end of the course the students did their exams (post-test), and the results of the experimental group were significantly better than the results of students in the control group.Some students from the experimental group had to answer questions in an interview related to the use of GeoGebra during their collaborative learning. In order to see the students' difficulties in solving problems, students in the experimental group were asked to cross out incorrect parts of solutions, not to erase them. The teachers reviewed the students' tasks done during the collaborative learning and after that the students who had corrected their mistakes were invited for an interview about using GeoGebra for overcoming their difficulties. Based on the students' results in the tests, answers in the questionnaire and in the interview, it can be concluded that GeoGebra has enabled an easier learning of this material. The GeoGebra package enables the students to check whether each step in the process of solving a task was correctly done or not. The results of our research show that GeoGebra can help those students having insufficient knowledge (necessary for solving those tasks) to improve it.We can say that our research shows that the students' learning achievement in examining functions and drawing their graphs is better when they use GeoGebra, working in collaborative groups than without using it. Also, GeoGebra enables creation of effective learning environment for examining functions and drawing their graphs.     

Which way of design programming activities is more effective to promote K-12 students' computational thinking skills? A meta-analysis

Promoting the cultivation of computational thinking (CT) skills in programming activities has become a key issue in the K-12 curriculum, however, there is no unified conclusion on how to design programming activities to promote the acquisition of CT skills more effectively. The purpose of this study is to determine the effectiveness of using programming to enhance K-12 students' CT skills and explore the influence of various programming instructional design factors on the acquisition of CT skills. This study presents a quantitative meta-analysis. A systematic search of randomized controlled studies on the influence of programming on CT skills. A total of 86 empirical studies with 114 effect sizes met the study selection criterion. The results showed that programming improved K-12 students' CT skills in general (Hedges' g = 0.601, 95% CI [0.505, 0.697], p < 0.001). No publication bias was detected. Besides, we found that the interdisciplinary integration of programming, the duration of programming intervention within 1 week to 1 month, the class size of less than 50 students, and a reasonable selection of programming instruments and CT assessment types may be more conducive to promoting students' CT skills. Based on these findings, we suggest that programming instructional design should be planned reasonably to make the best use of technology towards developing students' CT skills. This will be of great significance to programming teaching and CT education in K-12.     

Être à l’écoute des élèves : un incontournable dans la classe de mathématiques

Abstract

This article examines a unique professional development initiative in mathematics teaching that focuses on the design and field testing of pedagogical resources for primary teachers. We examine a particular aspect of the project—namely, helping teachers move from a model centred on explanation and repetitive exercises to an approach centred on listening and responding to students' thinking. In particular, the initiative emphasizes the important role played by students’ reasoning in the construction of their own computational algorithms. This analysis provides evidence that professional development that aims to deepen understanding of students’ mathematical thinking, that provides resources to support the implementation of new teaching strategies, and that strives to establish a collaborative environment enables educators to gain confidence and to develop expertise in their interactions with students in the context of problem solving.     

Arithmetic sequence as a bridge between conceptual fields

Abstract

Arithmetic sequence is used in this study as a means to explore pre‐service elementary school teachers’ connections between additive and multiplicative structures as well as several concepts related to introductory number theory. Vergnaud's theory of conceptual fields is used and refined to analyze students’ attempts to test membership of given numbers and to generate elements that are members of a given infinite arithmetic sequence. Our results indicate that participants made a strong distinction between two types of arithmetic sequences: sequences of multiples (e.g., 7, 14, 21, 28,…) and sequences of ‘non‐multiples,’ (e.g., 8, 15, 22, 29,…). Students were more successful in recognizing the underlying structure of elements in sequences of multiples, whereas for sequences of non‐multiples students often preferred algebraic computations and were mostly unaware of the invariant structure linking the two types. We examine the development of students’ schemes as they identify differences and similarities in the situations presented to them.     

STEM learning attitude predicts computational thinking skills among primary school students

Computational thinking (CT) plays a vital role in the fields of science, technology, engineering and mathematics (STEM). However, whether students' learning attitude towards STEM is related to their CT skills remains unknown. Two studies were conducted to address this knowledge gap. In Study 1, we validated a newly developed STEM learning attitude scale among a sample of Chinese primary school students (N = 489). Exploratory and confirmatory factor analysis results revealed that the scale which consisted of three factors (mathematics, science and information technology) could sufficiently measure primary school students' STEM learning attitude in the current sample. In Study 2, we explored the association between students' STEM learning attitude and their CT skills. Evidence revealed that learning attitude towards STEM significantly predicted CT skills. We also found that girls had a more positive learning attitude towards STEM than boys, and the fourth grade might be the key period for the development of CT skills. Implications for promoting primary school students' STEM learning and CT skills were also discussed.     

Expectation versus variation: Students’ decision making in a sampling environment

Abstract:

A task adapted from one used by Tversky and Kahneman (1971) was used in an interview or questionnaire context with 122 students from Grade 3 to Grade 13. Two questions assessed student understanding of the relationship of a sample to a population and of the expected value of the arithmetic mean, with and without information on a single value from the sample. Combining responses to the two questions, increasingly complex hierarchical sequences were identified in the observed responses in relation to the expected value of the means, an expectation variable, and in relation to the degree that ideas about variation were used to support judgements, a variation variable. Using data from another task performed by 68 of the students, responses were associated with the observed development of understanding of sampling more generally, a basic sampling variable. Overall, the associations of levels of response among the variables was not strong, suggesting more explicit discussion of sampling issues is required in classrooms.