Apprehending Mathematical Structure: A Case Study of Coming to Understand a Commutative Ring

Abstract algebra courses tend to take one of two pedagogical routes: from examples of mathematics structures through definitions to general theorems, or directly from definitions to general theorems. The former route seems to be based on the implicit pedagogical intention that students will use their understanding of particular examples of an algebraic structure to get a sense of those properties which form the basis of the fundamental definitions. We will explain the transition from examples to abstract algebra as a series of shifts of attention and in this paper we will use a case study to examine the initial shift, which we will call apprehending a structure, and examine how one student came to apprehend the structure of the commutative ring Z₉₉.     

Exploring the relationship between homework task difficulty,student engagement and performance

This article considers the relationship between differential homework task difficulty, student engagement and performance across four Irish post-primary school sites. A total of 236 participants completed all elements of this study. All participants were junior cycle students, aged between 12 and 16 years. The study employed a standardised test–retest approach, with a one week interval between tests. Between testing, participants were randomly subdivided into three cohorts. Each cohort received homework questions of different difficulty levels to complete for one week. The results of this study found that Cohort A, who received the least difficult homework tasks, completed the most non-compulsory questions. Although Cohort A demonstrated higher levels of engagement, no statistically significant difference was found in the change in performance scores of the three cohorts. The results highlight the negative impact of increased task difficulty on student engagement with independent homework tasks. The findings suggest that in order to support student engagement it is important that teachers afford ample opportunity for student success when designing self-directed tasks such as homework.     

The pedagogical value and the interdisciplinary nature of inductive processes in forming generalizations: Reflections from the classroom

The tendency to generalize from specific experiences leading to new, more abstract concepts is a natural aspect of human thought. Generalizations are the end result of an inductive process that begins with the identification of similarities in seemingly disparate situations. It is the existence of such generalizations that makes it possible for us to understand each other and the world around us. It is pedagogically weak to present generalizations to students and expect them to know how and when to apply them. On the other hand if students experience the inductive process in classrooms and discover generalizations, they are likely to remember and use this process when tackling other problems. The authors illustrate the pedagogical value of such an approach and the interdisciplinary nature of the inductive process by reflecting on teaching practices in English literature and mathematics in a high school classroom. In particular the authors reflect on how the inductive process was applied to four short stories and four problem-solving situations, which resulted in high school students arriving at generalizations that characterized the stories and the problems. A conceptual model that illustrates how inductive processes facilitate generalizations in the classroom is presented.     

Grief – The Expression of Romanian Artistic Spirituality

This paper expresses a personal point of view of the main features of Romanian artistic spirituality based on the writings of Romanian philosopher, Emil Corian, on personal knowledge of Romanian folk art, of the art of the initiated, and on teaching and art-making experience.     

Selling abroad: Are we doing enough?

Despite the dominance of the English language, publishers in the United States had export sales in 1987 of less than 6 percent of their total sales. Adrian Higham analyzes the major overseas markets for U.S. books and suggests ways for publishers to improve their export sales to each. The key to higher sales, he argues, is acquiring “insider status” in each market. Although he is convinced that publishers can do better, he reminds us that there is no simple solution. Adrian Higham brings to his consultancy an extensive background in international publishing. Following his wide career with Longman he served as managing director of John Wiley & Sons’ Company in the United Kingdom and subsequently as senior vice president and general manager of the International Group of John Wiley & Sons in New York.     

Public libraries as reserves of cultural and digital capital: Addressing inequality through digitalization

Public libraries are among key sites for the acquisition of cultural capital, and possibly one of the most potent means through which the state can address inequality. While digitalization of public libraries already attracted significant scholarly attention, the evidence of its contribution to the acquisition of cultural skills and knowledge and social mobility remains limited, as does the conceptual understanding of links between digitalization, cultural capital, and social stratification. This article draws on two disconnected bodies of research, the sociological analysis of cultural capital and stratification and research on digital divides. To bridge these two bodies of research, the interplay of cultural and digital capital in public libraries was investigated. The extensive dataset from the UK Taking Part Survey (2016–17) was analyzed using two-step cluster analysis and multinomial regression models to explore the contrasting profiles of contemporary library users. Results identify four distinct user groups: Traditional, Active, Family, and Tech Access, which possess different degrees of cultural and digital capital, have different demographic profiles, and benefit from digitalized libraries in different ways. If libraries are to fulfil their role in reducing social inequalities, it is important that they tailor their digital services to the specific characteristics of each user group. This approach also provides a useful template for exploring the interplay of digitalization and (in)equality in other cultural institutions.     

Activity of metalloproteinases and adiponectin in obese patients— a possible factor of incisional hernias after bariatric procedures

Purpose

Metalloproteinases are a key component of the pathogenesis of abdominal hernias. Obesity is considered a risk factor in herniogenesis and hernia recurrence. The aim of this study was to evaluate the serum concentrations of metalloproteinase-2 (MMP-2), MMP-9, MMP-13, and adiponectin in morbidly obese and nonoverweight controls.

Materials and methods

The participants were recruited from among patients undergoing bariatric and non-bariatric surgery and divided into two groups: I (body mass index (BMI)≥35 kg/m², n=40) and II (BMI<25 kg/m², n=30). Serum concentrations of MMP-2, MMP-9, MMP-13, and adiponectin were measured using enzyme-linked immunosorbent assay (ELISA).

Results

A statistically significant difference between groups was observed for MMP-2 concentration. The median MMP-9 concentration was higher in the obese group, but the difference was not statistically significant. Median MMP-13 concentrations did not differ between groups. Serum adiponectin concentration was insignificantly higher in the non-obese group.

Conclusions

The elevated serum MMP-2 and MMP-9 concentrations in obese individuals may be related to the higher incidence of incisional hernias in this population.

     

Conditional inference and advanced mathematical study

Many mathematicians and curriculum bodies have argued in favour of the theory of formal discipline: that studying advanced mathematics develops one’s ability to reason logically. In this paper we explore this view by directly comparing the inferences drawn from abstract conditional statements by advanced mathematics students and well-educated arts students. The mathematics students in the study were found to endorse fewer invalid conditional inferences than the arts students, but they did not endorse significantly more valid inferences. We establish that both groups tended to endorse more inferences which led to negated conclusions than inferences which led to affirmative conclusions (a phenomenon known as the negative conclusion effect). In contrast, however, we demonstrate that, unlike the arts students, the mathematics students did not exhibit the affirmative premise effect: the tendency to endorse more inferences with affirmative premises than with negated premises. We speculate that this latter result may be due to an increased ability for successful mathematics students to be able to ‘see through’ opaque representations. Overall, our data are consistent with a version of the formal discipline view. However, there are important caveats; in particular, we demonstrate that there is no simplistic relationship between the study of advanced mathematics and conditional inference behaviour.