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.     

Salary comparisons: New methods for correcting old fallacies

Institutions of higher education place high priority on ensuring that their faculties are adequately paid. Since gauging the adequacy of compensation is a problem of relative measure, this assessment is usually performed by comparing university-level average salaries for a group of peer institutions. Often the potential distortions caused by differences in the rank mix of the faculty are forestalled by comparing average salaries for each of the four ranks. However, few if any institutions take similar steps to prevent the skewing of averages owing to differences in cost of living, differences in distribution of faculty in academic year and twelve-month appointments, and differences in the configuration of the faculty over disciplines. This paper shows that salary comparisons that do not compensate for these variation in staff mix are seriously flawed. A recommended methodology is demonstrated.     

Flexible use of lexical and sublexical information in word recognition

Within the dual‐route framework it is hypothesised that readers exhibit flexibility in their use of lexical and non‐lexical information in word naming. In the present study, participants named high‐ and low‐frequency regular one‐syllable English words embedded within lists of regular or irregular one‐ or two‐syllable English words. A large number of irregular words should bias the reader toward the lexical route, whereas a list consisting exclusively of regular words should allow more efficient use of sublexical information present in the word. Word frequency effects were obtained when the list was dominated by either regular or irregular two‐syllable filler words. Furthermore, there was an interaction between frequency and regularity for the one‐syllable words, indicating that the frequency effect was significantly larger when the fillers were one‐syllable irregular words relative to one‐syllable regular words. These results extend those reported for a shallow orthography, and indicate strategic control over the use of phonological and lexical information in English word recognition.     

Reframing Resources in Engineering Teaching and Learning

Abstract

The notion of “resources” is often framed in an economic sense: money, time, equipment and the like. The authors reconceptualise this notion, situating resources as embedded in curricular frameworks, teacher practice and student experience. This leads them to define resources as “the potential to participate in socio-cultural action” which is illustrated in this article through a series of reflections on the part of the authors, all within the context of engineering education. First, they demonstrate that curriculum can be productively thought of as a route marker for the development of resources that students need in order to enact their role as professional engineers. Thereafter, they show that lecturers bring tacit resources of trust, care, creativity and credibility to the teaching and learning space, and that these are necessary to overcome the inertia that often resists the transformation of teaching and learning practice. Finally, they reflect on how students’ prior learning experiences can be harnessed as a resource for teaching and learning. In so doing, they present resources as tied to sociocultural practices and personal and institutional histories, and encourage others to take up these ideas so as to consider how resources, viewed in the authors’ sense, are valued within (engineering) education.     

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.

Keeping excitement in teaching

It is imperative for those of us who teach to find ways to stay renewed and refreshed over the long term. There are at least five ways through which most of us can find new energy and excitement in our teaching. One is to focus more genuinely on our students and learn more about them as persons and as colleagues- and to listen more carefully to what they say. Another is to inform them of the important objectives of a course and allow students to assume more of the responsibility for learning. A third tip is to use a greater variety of teaching methods, particularly ones that get students more actively involved in learning. Two additional ways to add excitement to teaching are to teach new courses and to plan periodic activities away from teaching. As John Gardner has recently stated so eloquently, it is our faith in human possibilities that is the generative element in such activities as leading, teaching, and guiding students.His office offers a wide range of instructional and professional services for faculty and academic administrators in the University's thirteen schools and colleges.     

An experimental two‐way video teletraining system: Design,development and evaluation

This paper describes the design, development and evaluation of an experimental two‐way video teletraining (VTT) system consisting of two classrooms equipped with audio and video equipment and linked by a land line to enable two‐way audio/video communication. A US Navy instructor delivered instruction in person to the originating classroom and electronically to the remote classroom. Findings were that student attitudes and performance on examinations were comparable in originating and remote classrooms. Students at the remote classroom were more likely to comment on deficiencies of the VTT system and were particularly sensitive to audio problems. Instructor acceptance of the VTT system was high. Cost analysis indicated that using the VTT system was more costly than sending an instructor to a remote site but that VTT can be less cosdy than sending students to the instructor.

The research described was performed under the sponsorship of the Office of Naval Technology. The opinions expressed in this paper are those of the authors, are not official, and do not necessarily reflect the views of the Navy department.     

Effect of Twinning on Chinese and English Vocabulary Knowledge

Vocabulary knowledge was tested in a native (Cantonese-Chinese) and foreign (English) language in 150 twins and 150 singletons aged 6–11 years, matched on age, gender, grade level, nonverbal intelligence, parents’ education, family income, and number of siblings and household members. The singletons clearly outperformed the twins on the native vocabulary, but this “twinning effect” was much less noticeable for the foreign vocabulary. The effect on English vocabulary was further reduced after exposure to English at home was controlled. Given that these participants learned most of their English in school rather than home, the present findings support the notion that the twinning effect is associated with increased competition for family interaction in twins compared with singletons.     

Are athletes on the right track? The effect of availability of an all-weather athletics track on athletics performance

In February of 2011, an earthquake destroyed the only all-weather athletics facility in the city of Christchurch, New Zealand. The all-weather track has yet to be replaced, and so since the loss of the track, local Christchurch athletes have only had a grass track for training and preparation for championship events. The present study considers what effect the loss of the training facility has had on the performance of athletes from Christchurch at national championship events. Not surprisingly, the study finds that there has been a deterioration in the performance in events that are heavily dependent upon the all-weather surface. However, somewhat more surprisingly, the loss of the track appears to have caused a significant improvement in the performance of Christchurch athletes in events that, while on the standard athletics programme, are not heavily dependent on track surface.