Functional motifs: a novel perspective on burst synchronization and regularization of neurons coupled via delayed inhibitory synapses

For one of the most common network motifs, an inhibitory neuron pair, we perform an extensive study of burst synchronization and the related phenomena applying the model of Rulkov maps coupled via delayed synapses. Instigated by the phase-plane analysis, that has the neuron switching between the noninteracting and the interacting map, it is demonstrated how the system evolution may be interpreted by means of the dynamical configurations of the motif, each represented by an extracted subgraph. Under the variation of the synaptic parameters, the probability of finding synchronized neurons in a given configuration is seen to reflect the way in which the anti-phase synchronization is eventually superseded by the synchronization in phase. Such an approach also provides a novel insight into regularization, characterizing the neuron bursting in either of these regimes. Looking into correlation of the two neurons’ bursting cycles we acquire a deeper understanding of the more sophisticated mechanisms by which the regularity in the time series is maintained. Further, it is examined whether introducing heterogeneity in the neuron or the synaptic parameters may prove advantageous over the homogeneous case with respect to burst synchronization.     

Science Talk: Preservice Teachers Facilitating Science Learning in Diverse Afterschool Environments

The purpose of this study was to assess the impact a community‐based service learning program might have on preservice teachers' science instruction during student teaching. Designed to promote science inquiry, preservice teachers learned how to offer students more opportunities to develop their own ways of thinking through utilization of an afterschool science program that provided them extended opportunities to practice their science teaching skills. Three preservice teachers were followed to examine and evaluate the transfer of this experience to their student teaching classroom. Investigation methods included field observations and semi‐structured, individual interviews. Findings indicate that preservice teachers expanded their ideas of science inquiry instruction to include multiple modes of formative assessment, while also struggling with the desire to give students the correct answer. While the participants' experiences are few in number, the potential of afterschool teaching experience serving as an effective learning experience in preservice teacher preparation is significant. With the constraints of high‐stakes testing, community‐based service learning teaching opportunities for elementary and middle‐school preservice teachers can support both the development and refinement of inquiry instruction skills.     

Distinguishing between mathematics classrooms in Australia, China, Japan, Korea and the USA through the lens of the distribution of responsibility for knowledge generation: public oral interactivity and mathematical orality

The research reported in this paper examined spoken mathematics in particular well-taught classrooms in Australia, China (both Shanghai and Hong Kong), Japan, Korea and the USA from the perspective of the distribution of responsibility for knowledge generation in order to identify similarities and differences in classroom practice and the implicit pedagogical principles that underlie those practices. The methodology of the Learner’s Perspective Study documented the voicing of mathematical ideas in public discussion and in teacher–student conversations and the relative priority accorded by different teachers to student oral contributions to classroom activity. Significant differences were identified among the classrooms studied, challenging simplistic characterisations of ‘the Asian classroom’ as enacting a single pedagogy, and suggesting that, irrespective of cultural similarities, local pedagogies reflect very different assumptions about learning and instruction. We have employed spoken mathematical terms as a form of surrogate variable, possibly indicative of the location of the agency for knowledge generation in the various classrooms studied (but also of interest in itself). The analysis distinguished one classroom from another on the basis of “public oral interactivity” (the number of utterances in whole class and teacher–student interactions in each lesson) and “mathematical orality” (the frequency of occurrence of key mathematical terms in each lesson). Classrooms characterized by high public oral interactivity were not necessarily sites of high mathematical orality. In particular, the results suggest that one characteristic that might be identified with a national norm of practice could be the level of mathematical orality: relatively high mathematical orality characterising the mathematics classes in Shanghai with some consistency, while lessons studied in Seoul and Hong Kong consistently involved much less frequent spoken mathematical terms. The relative contributions of teacher and students to this spoken mathematics provided an indication of how the responsibility for knowledge generation was shared between teacher and student in those classrooms. Specific analysis of the patterns of interaction by which key mathematical terms were introduced or solicited revealed significant differences. It is suggested that the empirical investigation of mathematical orality and its likely connection to the distribution of the responsibility for knowledge generation and to student learning ourcomes are central to the development of any theory of mathematics instruction and learning.     

Progressive, classical or balanced— A look at mathematical learning environments in Swiss-German lower-secondary schools

In a national supplement to TIMSS, lower-secondary school teachers (N=102) and their students (N=975) reported on mathematics instruction by means of a teacher questionnaire (teaching-learning methods, instructional sub-goals, facilitated student activities, achievement assessment, teacher role) and a student questionnaire (teachers' instructional proficiency, classroom climate). A cluster analysis performed on the ratings of teaching-learning methods yielded a solution with three clusters referred to as progressive, classical, and balanced learning environment. Cluster-related differences in facilitated student activities, achievement evaluation and preferred teacher role were found but not in instructional sub-goals. Students from different learning environments equally approved teachers' instructional proficiency and classroom climate and also had similar TIMSS mathematics scores. The results of this study provide empirical evidence that in addition to classical teacher-centered learning environments there seem to exist more diversified and studentcentered learning environments that address the needs for students to direct their own learning, communicate and work with others, and develop ways of dealing with complex problems. In line with the research literature it was also found that high mathematics achievement is not restricted to a certain type of learning environment.     

Stock market forecasting by using a hybrid model of exponential fuzzy time series

The initial aim of this study is to propose a hybrid method based on exponential fuzzy time series and learning automata based optimization for stock market forecasting. For doing so, a two-phase approach is introduced. In the first phase, the optimal lengths of intervals are obtained by applying a conventional fuzzy time series together with learning automata swarm intelligence algorithm to tune the length of intervals properly. Subsequently, the obtained optimal lengths are applied to generate a new fuzzy time series, proposed in this study, named exponential fuzzy time series. In this final phase, due to the nature of exponential fuzzy time series, another round of optimization is required to estimate certain method parameters. Finally, this model is used for future forecasts. In order to validate the proposed hybrid method, forty-six case studies from five stock index databases are employed and the findings are compared with well-known fuzzy time series models and classic methods for time series. The proposed model has outperformed its counterparts in terms of accuracy.     

Making practice studyable

A common way to situate professional learning in practice is to use representations of teaching, such as videos of classroom instruction or samples of student work. Using representations of teaching, however, does not automatically lead to teacher learning. Learning in and from practice also requires supports that make such practice studyable. The authors introduce and explore the work of “making practice studyable” by analyzing a case of practice-based professional development in which the professional development designers deliberately tried to mediate participants’ learning in and from practice. From this analysis, the authors identified five categories of work that can help make practice studyable: (1) engaging the content, (2) providing insight into student thinking, (3) orienting to the instructional context, (4) providing lenses for viewing, and (5) developing a disposition of inquiry. These categories are then applied to the use of a representation of mathematics teaching in a course for preservice elementary teachers.     

Making mathematics and science integration happen: key aspects of practice

The integration of mathematics and science teaching and learning facilitates student learning, engagement, motivation, problem-solving, criticality and real-life application. However, the actual implementation of an integrative approach to the teaching and learning of both subjects at classroom level, with in-service teachers working collaboratively, at second-level education, is under-researched due to the complexities of school-based research. This study reports on a year-long case study on the implementation of an integrated unit of learning on distance, speed and time, within three second-level schools in Ireland. This study employed a qualitative approach and examined the key aspects of practice that impact on the integration of mathematics and science teaching and learning. We argue that teacher perspective, teacher knowledge of the ‘other subject’ and of technological pedagogical content knowledge (TPACK), and teacher collaboration and support all impact on the implementation of an integrative approach to mathematics and science education.     

Principal and coach as partners

This paper explores the roles and responsibilities of mathematics coaches and principals. It suggests that principals need to equilibrate power by treating the coach as a partner, even if the coach is subordinate in the educational hierarchy. It posits that coaching would benefit all parties if they worked together across roles to enhance the capacity of educators at every level in the school hierarchy—teacher, coach, principal—to design, analyze, and implement mathematics instruction that results in deeper student learning. It briefly describes the model of Content Coaching. It questions some of the prevailing norms of interaction among educators, and it offers coaching scenarios that demonstrate that the initial design and implementation of coaching initiatives can lead to resistance and misunderstanding of the power and benefits of coaching for all educators and administrators. It goes on to challenge some of our assumptions regarding potential barriers (e.g. confidentiality) in relationships among coaches, teachers, and principals, and suggests that teacher learning goals be public and collaboratively engaged. It posits that positional authority is not a particularly strong lever for improving teacher practice, particularly if the principal lacks facility or interest in mathematics. It argues that if we want to enhance mathematics education in this country (as evidenced by robust student understanding of mathematics and its application in the world), there is great power in working collaboratively and inclusively, using coaching to support educators at every level.     

Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics

One of the manifestations of learning is the student’s ability to come up with original solutions to new problems. This ability is one of the criteria by which the teacher may assess whether the student has grasped the taught mathematics. Obviously, a teacher can never teach the ability to invent new solutions (at least not directly): he/she can ask for it, expect it, encourage it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Guy Brousseau modelled in one of the best-known concepts in didactics of mathematics: the didactical contract. The teacher cannot be confident that the student will learn exactly what the teacher intended to teach and hence the student must re-create it on the basis of what he/she already knows. In this paper, the importance and the role of situations affording mathematical creativity (in the sense of production of original solutions to unusual situations) are demonstrated. The authors present an experiment (with 9–10-year-old children) that makes it possible to show how certain situations are more favourable (for all children) to express some characteristics of mathematical creativity.     

Capitalizing on Emerging Technologies: A Case Study of Classroom Blogging

The challenge many teachers face is how to incorporate new technology into their classrooms that strengthens classroom learning by capitalizing on students’ media literacies. Blogs, a new and innovative technological tool, can be used in math and science classrooms to support student learning by capitalizing on students’ interests and familiarity with on‐line communication. This study explores the emerging blogging practices of one high school mathematics teacher and his class to explore issues of intent, use, and perceived value. Data sources for this case included one year's worth of blog content, an interview with the facilitating teacher, and students ‘perceptions of classroom blogging practices. Findings indicate that (1) teachers’ intentions focused on creating additional forms of participation as well as increasing student exposure time with content; (2) blogs were used in a wide variety of ways that likely afforded particular benefits; and (3) both teacher and students perceived the greater investment to be worthwhile. The findings are used to critically consider claims made in the literature about the potential of blogging to effectively support classroom learning.     

Algebra Teachers' Utilization of Problems Requiring Transfer Between Algebraic,Numeric, and Graphic Representations

For students to develop an understanding of functions, they must have opportunities to solve problems that require them to transfer between algebraic, numeric, and graphic representations (transfer problems). Research has confirmed student difficulties with certain types of transfer problems and has suggested instructional factors as a possible cause. Algebra teachers (n= 28) were surveyed to determine the amount of class time they devote to different types of transfer problems and how many times these problems appear on their teacher‐made assessments. Results suggest that teachers dedicate less class time to graphic to numeric transfer problems than to any other type of transfer problem and that these problems appear less frequently on assessments. These are exactly the types of transfer problems that pose the most difficulty for students. It is conjectured that teachers' familiarity with these problems, combined with assumed student mastery, contribute to this mismatch.     

Rural teacher leadership in science and mathematics

This study adds to our understanding of science and mathematics teacher leadership in rural schools. Through In Vivo and Concept coding of teacher interviews, we investigated 20 secondary science and mathematics teachers' perceptions of rural teacher leadership during their participation in a three-year professional development program. As the teachers developed as teacher leaders, they broadened their focus from improving their own students' learning to sharing new knowledge learned through the program with other teachers both informally and formally. We compared our program components to the Teacher Leader Model Standards and added an emphasis on the importance of disciplinary content knowledge. We also identified patterns in science and mathematics teacher leadership that are contextually connected to teachers' instruction in rural high poverty schools. Rural teacher leadership included the importance of building strong teacher–student relationships, providing new academic opportunities for students, encouraging students' success, and building community connections.     

Expanding the Role of K‐5 Science Instruction in Educational Reform: Implications of an Interdisciplinary Model for Integrating Science and Reading

Addressed is the current practice in educational reform of reducing time for science instruction in favor of traditional reading/language arts instruction. In contrast, presented is an evidence‐based rationale for increasing instructional time for K‐5 science instruction as an educational reform initiative. Overviewed are consensus interdisciplinary research and complementary multi‐year findings of the Science IDEAS model demonstrating the effectiveness of integrating conceptually‐relevant reading within science instruction in improving student achievement in both science and reading comprehension. Based on research summarized, increasing time for integrated K‐5 science is advocated as a meaningful reform‐based approach to science learning and reading comprehension proficiency that, in turn, better prepares students for subsequent success in science and content‐area reading comprehension across upper elementary and middle school grades (3–8).     

Functional reproducing kernel Hilbert spaces for non-point-evaluation functional data

Motivated by the need of processing non-point-evaluation functional data, we introduce the notion of functional reproducing kernel Hilbert spaces (FRKHSs). This space admits a unique functional reproducing kernel which reproduces a family of continuous linear functionals on the space. The theory of FRKHSs and the associated functional reproducing kernels are established. A special class of FRKHSs, which we call the perfect FRKHSs, are studied, which reproduce the family of the standard point-evaluation functionals and at the same time another different family of continuous linear (non-point-evaluation) functionals. The perfect FRKHSs are characterized in terms of features, especially for those with respect to integral functionals. In particular, several specific examples of the perfect FRKHSs are presented. We apply the theory of FRKHSs to sampling and regularized learning, where non-point-evaluation functional data are used. Specifically, a general complete reconstruction formula from linear functional values is established in the framework of FRKHSs. The average sampling and the reconstruction of vector-valued functions are considered in specific FRKHSs. We also investigate in the FRKHS setting the regularized learning schemes, which learn a target element from non-point-evaluation functional data. The desired representer theorems of the learning problems are established to demonstrate the key roles played by the FRKHSs and the functional reproducing kernels in machine learning from non-point-evaluation functional data. We finally illustrate that the continuity of linear functionals, used to obtain the non-point-evaluation functional data, on an FRKHS is necessary for the stability of the numerical reconstruction algorithm using the data.     

On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.     

Teaching Mathematics and Science in Urban Elementary Schools

Elementary teachers from a large urban school district with a 74 percent minority student population were surveyed to assess their capacity to provide quality mathematics and science instruction. Forty-nine percent of the surveys distributed to a random sample of teachers were returned. Both strengths and barriers to capacity building were identified. Strengths included use of collaborative student work, manipulatives, informal learning environments, and alternative assessment practices; availability of calculators and computers; and high expectations for student learning. Barriers included lack of professional development, infrequent science instruction, limited calculator use, lack of planning time, inadequate resources, and the perception that science was not valued as highly as mathematics.     

Relationships Between Inquiry‐Based Teaching and Physical Science Standardized Test Scores

This exploratory case study investigates relationships between use of an inquiry‐based instructional style and student scores on standardized multiple‐choice tests. The study takes the form of a case study of physical science classes taught by one of the authors over a span of four school years. The first 2 years were taught using traditional instruction with low levels of inquiry (non‐inquiry group), and the last 2 years of classes were taught by inquiry methods. Students' physical science test scores, achievement data, and attendance data were examined and compared across both instructional styles. Results suggest that for this teacher the use of an inquiry‐based teaching style did not dramatically alter students' overall achievement, as measured by North Carolina's standardized test in physical science. However, inquiry‐based instruction had other positive effects, such as a dramatic improvement in student participation and higher classroom grades earned by students. In additional inquiry‐based instruction resulted in more uniform achievement than did traditional instruction, both in classroom measures and in more objective standardized test measures.     

Possibilities of introducing different functional circuits on top of a structural neuron triplet: Where do the gains lie?

Instigated by the research on clusterization phenomena in complex neural networks, we study a triplet of bursting Rulkov map neurons connected via inhibitory synapses with delay. It is demonstrated how on a background of structural motif one can build different types of functional circuits. The approach is based on utilizing the properties of the chemical synapses, whose gating is modeled by the fast threshold modulation, in conjunction with the phase plane analysis, allowing the system state to be represented in terms of maps the neurons reside on. For both the dynamical configurations, monitoring the layout of active neurons, and the functional motifs, following the maps where the synchronized neurons lie, we establish a one-on-one correspondence between sequences in the time series and the triads, making up the subgraphs of the original graph. By introducing the appropriate sets of quantities, one obtains not only the distributions of triads as a function of synaptic parameters, but is also able to identify a distinct triad whose presence may be viewed as a signature of the burst synchronization process. In another setup, the regularization of burst cycles for an arbitrary neuron is explained by classifying all the bursts as long or short, with their fractions linked to the abundances of triads under variation of synaptic parameters.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.