Making connections in mathematics

Math textbooks, which usually represent the mathematics curriculum, seem to be linked to the poor math performance of U.S. students. The major shortcomings of math textbooks are described in this article; then an alternative perspective is offered (the sameness analysis), along with research conducted with students with learning disabilities and at-risk students. The article then presents a detailed illustration of the sameness analysis--how to teach the addition-subtraction and multiplication-division relationships and their interrelationships in the context of solving word problems in mathematics.     

Making connections: the logical structuring of hypertext documents

The results from an experiment analysing hypertextual semantic linking is reported. In the study participants are asked to describe logical relationships between related, semi-related and unrelated paragraphs of text. It is concluded that existence and themes of relationships are (as expected) fairly consistently recognised, however, specific relationships do not readily fall into natural predicate like statements. Purely thematic relationships, or a palette of available relationships might be the more practical approach. Further work on dynamic linking in hypertext documents is also reported. Dynamic linking eliminates some of the problems associated with a mismatch between the mental models of authors and readers and allows the readers to construct their own hypertexts from linear documents. This approach offers useful data on logical connectivity in hypertext documents but also represents an alternative to conventional hypertext authoring and reading.     

Making global connections: The knowledge,understanding and motivation of trainee teachers

This article begins by noting some of the global concerns that young people have today and highlights the crucial role of global education in responding to such concerns. It then considers the importance of teacher training in such matters and reports on a preliminary study relating to the initial training of teachers in England. In particular this raises questions about the knowledge and understanding that such students bring to their courses. It concludes by identifying some of the consequent dilemmas faced by teacher trainers working in this context.     

Making the right connections: Differential effects of reading intervention for subgroups of comprehenders

The purpose of this study was to compare the effects of different types of questioning interventions on students' reading comprehension. Fourth-grade students (n = 246) were identified as struggling, average, or good readers and assigned randomly within school to one of three questioning interventions: two inferential conditions (Causal or General) or one literal condition (“Who, What, Where, When” or W-questioning). Teachers delivered the interventions for 20–30 min, 2–4 times per week, for 8–10 weeks. All readers made reliable pre- to posttest comprehension gains as measured by story recall (ps < .001 to .04). Differential effects for intervention were found between two subgroups of struggling comprehenders—elaborators and paraphrasers. Elaborators benefited more than paraphrasers from Causal questioning (d = .86) whereas paraphrasers benefited more than elaborators from General questioning (d = 1.46). These findings suggest that identifying subgroups is important in developing and evaluating the effectiveness of reading comprehension interventions.     

Making connections: Using skill theory to recognize how students build and rebuild understanding

In this companion to Marc Schwartz and Kurt Fischer's article, Patricia King and JoNes VanHecke describe how student affairs educators can help students become sophisticated thinkers.     

Place,matters of concern,and pedagogy: Making impactful connections with our planet

Neoliberalism, capitalist ideas, and the disastrous human-induced state of the environment are evidence of the lack of connection between humans and the earth, calling for a rethinking of the relationships between humans and the planet. As early childhood educators, we wonder about our role in rethinking these relationships and in particular, the relationship between children and planet. In this article, we articulate the actions and positioning of teachers in three different research studies and various contexts (Victoria, AUS; Oregon, USA; California, USA) utilizing Latour’s (2004) ‘matters of concern’ as a framework to rethink engagement and relationality. Each project considers how place can be a provocation that makes visible the entanglement of children, families, teachers, and the more-than-human as a way to rethink pedagogy. The limitations, tensions, and possibilities that occur within and across these entanglements are explored, highlighting how the enacted practices could disrupt dominant early childhood discourses and practices.     

Making connections: a conference approach to developing transformative environmental and sustainability education within initial teacher education

Abstract

Children today face significant challenges in response to living in a globalised world and the predicted environmental threats to the planet. However, implementing environmental and sustainability education (ESE) within schools can be complex and is often ignored within initial teacher education (ITE). In response, this project aimed to evaluate pedagogies for developing transformative learning for ESE within the context of university-based ITE. We ran a conference for trainee teachers informed by critical, participatory and arts-based pedagogic approaches. Alongside this, we undertook an interpretive case study, collecting data through questionnaires, focus groups and in-conference spoken reflection. Findings suggest trainee teachers gained a more nuanced understanding of the value and nature of ESE, as well as more affective and informed engagement with it, across the conference. Implications for ESE within ITE focus around the potential of pedagogies which provide affective experiences, in particular participatory and arts-based pedagogical approaches, for inspiring and empowering trainee teachers to enter into schools and develop their own ESE practice as classroom teachers.     

Making connections among multiple graphical representations of fractions: sense-making competencies enhance perceptual fluency,but not vice versa

Prior research shows that representational competencies that enable students to use graphical representations to reason and solve tasks is key to learning in many science, technology, engineering, and mathematics domains. We focus on two types of representational competencies: (1) sense making of connections by verbally explaining how different representations map to one another, and (2) perceptual fluency that allows students to fast and effortlessly use perceptual features to make connections among representations. Because these different competencies are acquired via different types of learning processes, they require different types of instructional support: sense-making activities and fluency-building activities. In a prior experiment, we showed benefits for combining sense-making activities and fluency-building activities. In the current work, we test how to combine these two forms of instructional support, specifically, whether students should first work on sense-making activities or on fluency-building activities. This comparison allows us to investigate whether sense-making competencies enhance students’ acquisition of perceptual fluency (sense-making-first hypothesis) or whether perceptual fluency enhances students’ acquisition of sense-making competencies (fluency-first hypothesis). We conducted a lab experiment with 74 students from grades 3–5 working with an intelligent tutoring system for fractions. We assessed learning processes and learning outcomes related to representational competencies and domain knowledge. Overall, our results support the sense-making-first hypothesis, but not the fluency-first hypothesis.     

Making connections: reflections on over three decades of EU initiatives in Early Childhood Education and Care (ECEC)

Recent anniversaries remind us of the lengthy history of the European Union’s involvement in ECEC, from the early work of the European Commission’s Childcare Network, from 1986–1996, to the European Commission’s publication in 2015 of an indicative Roadmap outlining potential new initiatives. European-funded research and policy reviews are numerous, frequently highlighting the significance of integrated ECEC systems in developing services and other areas of provision. This article looks back on three decades of EU policy initiatives and suggests that whilst key EC reviews have advocated integrated systems for member states, the Commission’s own fragmented approach has impeded the development of an effective framework at an EU level. It is argued that a tougher approach to improving ECEC, as suggested by recent developments, requires an integrated policy framework, unified targets and a strong single lead able to initiate and not just support actions at an EU level.     

Reliability of power connections

Despite the use of various preventive maintenance measures, there are still a number of problem areas that can ad- versely affect system reliability. Also, economical constraints have pushed the designs of power connections closer to the limits allowed by the existing standards. The major parameters influencing the reliability and life of Al-Al and Al-Cu connections are identified. The effectiveness of various palliative measures is determined and the misconceptions about their effectiveness are dealt in detail.     

Making connections and making friends: social interactions between two children labelled with special educational needs and their peers in a nursery setting

This paper reports on a pilot study of the social interactions between two children labelled with special educational needs and their peers in an early years setting. Data from play observations and staff interviews are used to examine the dynamics of friendship groups that the two children have developed and the way that they attempt to make new connections with other children. Comparisons between the two children’s interactions with their recurrent playmates and less familiar peers are drawn and the significance of their agency in making decisions about developing relationships is highlighted. A theme that also emerges is that staff facilitating rather than directing interactions between the two children and their peers has the most positive impact. Areas for further investigation are suggested including the range of relationships that children identified with special educational needs establish with their peers and the nature of adult support that most effectively supports friendships between all children.     

Real-world connections in secondary mathematics teaching

The mathematics-education community stresses the importance of real-world connections in teaching. The extant literature suggests that in actual classrooms this practice is infrequent and cursory, but few studies have specifically examined whether, how, and why teachers connect mathematics to the real world. In this study, I surveyed 62 secondary mathematics teachers about their understanding and use of real-world connections, their purposes for making connections in teaching, and factors that support and constrain this practice. I also observed 5 teachers making real-world connections in their classrooms and I conducted follow-up interviews; these qualitative data are used to illuminate findings from the survey data. The results offer an initial portrayal of the use of real-world connections in secondary mathematics classes and raise critical issues for more targeted research, particularly in the area of teacher beliefs about how to help different kinds of students learn mathematics.

Of connections and fields - I

We describe some instances of the appearance of Chern’s mathematical ideas in physics. By means of simple examples, we bring out the geometric and topological ideas which have found application in describing the physical world. These applications range from magnetic monopoles in electrodynamics to instantons in quantum chromodynamics to the geometric phase of quantum mechanics. The first part of this article is elementary and addressed to a general reader. The second part is somewhat more demanding and is addressed to advanced students of mathematics and physics.     

Of connections and fields—II

In the first part of this article¹ we gave an elementary introduction to Chern’s ideas and their impact on modern physics. In this concluding article we describe some more advanced applications of Chern’s ideas. This second part is some-what more demanding than the first part and is addressed to students with some background in mathematics and physics. Joseph Samuel is a theoretical physicist and by natural inclination a classical mechanic. Over the years he has strayed into other fields like optics, general relativity and very recently DNA elasticity. A unifying theme in his work is differential geometry and topology in physics. He keeps moderately fit by raising and lowering indices and relaxes by playing semiclassical guitar.