Statistical models for cores decomposition of an undirected random graph

The $k$-core decomposition is a widely studied summary statistic that describes a graph's global connectivity structure. In this paper, we move beyond using $k$-core decomposition as a tool to summarize a graph and propose using $k$-core decomposition as a tool to model random graphs. We propose using the shell distribution vector, a way of summarizing the decomposition, as a sufficient statistic for a family of exponential random graph models. We study the properties and behavior of the model family, implement a Markov chain Monte Carlo algorithm for simulating graphs from the model, implement a direct sampler from the set of graphs with a given shell distribution, and explore the sampling distributions of some of the commonly used complementary statistics as good candidates for heuristic model fitting. These algorithms provide first fundamental steps necessary for solving the following problems: parameter estimation in this ERGM, extending the model to its Bayesian relative, and developing a rigorous methodology for testing goodness of fit of the model and model selection. The methods are applied to a synthetic network as well as the well-known Sampson monks dataset.Comment: Subsection 3.1 is new: `Sample space restriction and degeneracy of real-world networks'. Several clarifying comments have been added. Discussion now mentions 2 additional specific open problems. Bibliography updated. 25 pages (including appendix), ~10 figure

Pre-service and in-service teachers’ perceptions on the integration of children’s literature in mathematics teaching and learning in Ireland

The beneficial role that children’s literature plays in facilitating the meaningful integration and advancement of literacy and numeracy in the primary mathematics classroom has been well validated by research findings internationally. In Ireland, supporting the development of literacy and numeracy is a key educational priority. Consequently, a myriad of policy initiatives such as the Literacy and Numeracy for Learning and Life strategy have been introduced. All aim to address concerns about young people’s lack of basic literacy and numeracy skills and to consider new teaching and learning modalities to enhance same. Despite this, no official emphasis is given to incorporating literature in the Irish primary school mathematics curriculum. Therefore, it is pertinent and timely that this study seeks to ascertain pre-service and in-service teachers’ views on the use of literature to support mathematics teaching and learning and to investigate perceived barriers to and enablers for the integration of children’ literature in the mathematics classroom in Ireland. The analysis of the findings will be framed using Ajzen (1991)’s Theory of Planned Behavior (TPB) model. This research is part of a large international research collaboration (see www.mathsthroughstories.org), in which the beliefs of teachers with respect to children’s literature are investigated

Maltese teachers’ beliefs concerning the integration of children’s literature in mathematics teaching and learning

This exploratory mixed-methods study set out to explore Maltese primary school teachers’ perceived barriers to, and enablers for, the integration of children’s literature in mathematics teaching. Data were collected by means of an online questionnaire and semi-structured interviews, and analysed thematically using Ajzen’s Theory of Planned Behaviour. The responses given by the participants showed that integration of mathematics and stories was not a common practice. The perceived barriers were categorised as Resource Constraint, Time Constraint, Lack of Pedagogical Knowledge and Confidence, Doubts about Outcome Expectancy, and Inhibiting Social Norms while the three perceived enablers identified were Pedagogical Benefits, Love of Stories, and Enabling Social Norms. Given that the majority of the participating teachers acknowledged the potential benefits of the approach and expressed a wish for training, one key recommendation of the study is for teaching mathematics through stories to be explicitly included in pre-service and in-service professional development programmes

Parallel Reflections: The Interdisciplinary Process of Co-Constructing Meaning

This paper reports the results of a study conducted by four teacher educators who represent four disciplines in education. The purpose was to understand how pre-service teacher reflections influenced teacher educator reflections within a course. Data collection methods included narrative and document review. Data analysis methods included multiple readings and discussions of individual portraits in search of answers to research questions. Findings include a parallel reflection model that illustrates a dynamic process that occurs within the dialogic space as student and professor thoughts and discourse are interwoven to make theory to practice connections, co-construct new knowledge and begin challenging existing beliefs and dispositions through their reflections

Parallel Reflections: The Interdisciplinary Process of Co-Constructing Meaning

This paper reports the results of a study conducted by four teacher educators who represent four disciplines in education. The purpose was to understand how pre-service teacher reflections influenced teacher educator reflections within a course. Data collection methods included narrative and document review. Data analysis methods included multiple readings and discussions of individual portraits in search of answers to research questions. Findings include a parallel reflection model that illustrates a dynamic process that occurs within the dialogic space as student and professor thoughts and discourse are interwoven to make theory to practice connections, co-construct new knowledge and begin challenging existing beliefs and dispositions through their reflections