Students' mental prototypes for functions and graphs

This research study investigates the concept of function developed by students studying English A-level mathematics. It shows that, while students may be able to use functions in their practical mathematics, their grasp of the theoretical nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop prototypes for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, is not stressed and proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sin*, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. Investigations reveal significant misconceptions. For example, threequarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and threequarters thought that a circle is a function. This reveals a wide gulf between the concepts as perceived to be taught and as actually learned by the students

Diagnosing students' difficulties in learning mathematics

This study considers the results of a diagnostic test of student difficulty and contrasts the difference in performance between the lower attaining quartile and the higher quartile. It illustrates a difference in qualitative thinking between those who succeed and those who fail in mathematics, illustrating a theory that those who fail are performing a more difficult type of mathematics (coordinating procedures) than those who succeed (manipulating concepts). Students who have to coordinate or reverse processes in time will encounter far greater difficulty than those who can manipulate symbols in a flexible way. The consequences of such a dichotomy and implications for remediation are then considered

Origin of molecular oxygen in Comet 67P/Churyumov-Gerasimenko

Molecular oxygen has been detected in the coma of comet 67P/Churyumov-Gerasimenko with abundances in the 1-10% range by the ROSINA-DFMS instrument on board the Rosetta spacecraft. Here we find that the radiolysis of icy grains in low-density environments such as the presolar cloud may induce the production of large amounts of molecular oxygen. We also show that molecular oxygen can be efficiently trapped in clathrates formed in the protosolar nebula, and that its incorporation as crystalline ice is highly implausible because this would imply much larger abundances of Ar and N2 than those observed in the coma. Assuming that radiolysis has been the only O2 production mechanism at work, we conclude that the formation of comet 67P/Churyumov-Gerasimenko is possible in a dense and early protosolar nebula in the framework of two extreme scenarios: (1) agglomeration from pristine amorphous icy grains/particles formed in ISM and (2) agglomeration from clathrates that formed during the disk's cooling. The former scenario is found consistent with the strong correlation between O2 and H2O observed in 67P/C-G's coma while the latter scenario requires that clathrates formed from ISM icy grains that crystallized when entering the protosolar nebula.Comment: The Astrophysical Journal Letters, in pres

Egalitarian justice and expected value

According to all-luck egalitarianism, the differential distributive effects of both brute luck, which defines the outcome of risks which are not deliberately taken, and option luck, which defines the outcome of deliberate gambles, are unjust. Exactly how to correct the effects of option luck is, however, a complex issue. This article argues that (a) option luck should be neutralized not just by correcting luck among gamblers, but among the community as a whole, because it would be unfair for gamblers as a group to be disadvantaged relative to non-gamblers by bad option luck; (b) individuals should receive the warranted expected results of their gambles, except insofar as individuals blamelessly lacked the ability to ascertain which expectations were warranted; and (c) where societal resources are insufficient to deliver expected results to gamblers, gamblers should receive a lesser distributive share which is in proportion to the expected results. Where all-luck egalitarianism is understood in this way, it allows risk-takers to impose externalities on non-risk-takers, which seems counterintuitive. This may, however, be an advantage as it provides a luck egalitarian rationale for assisting ‘negligent victims’

Towards a resolution of some outstanding issues in transitive research: an empirical test on middle childhood

Transitive Inference (deduce B > D from B > C and C > D) can help us to understand other areas of sociocognitive development. Across three experiments, learning, memory, and the validity of two transitive paradigms were investigated. In Experiment 1 (N = 121), 7-year-olds completed a three-term nontraining task or a five-term task requiring extensive-training. Performance was superior on the three-term task. Experiment 2 presented 5–10-year-olds with a new five-term task, increasing learning opportunities without lengthening training (N = 71). Inferences improved, suggesting children can learn five-term series rapidly. Regarding memory, the minor (CD) premise was the best predictor of BD-inferential performance in both task-types. However, tasks exhibited different profiles according to associations between the major (BC) premise and BD inference, correlations between the premises, and the role of age. Experiment 3 (N = 227) helped rule out the possible objection that the above findings simply stemmed from three-term tasks with real objects being easier to solve than computer-tasks. It also confirmed that, unlike for five-term task (Experiments 1 & 2), inferences on three-term tasks improve with age, whether the age range is wide (Experiment 3) or narrow (Experiment 2). I conclude that the tasks indexed different routes within a dual-process conception of transitive reasoning: The five-term tasks indexes Type 1 (associative) processing, and the three-term task indexes Type 2 (analytic) processing. As well as demonstrating that both tasks are perfectly valid, these findings open up opportunities to use transitive tasks for educability, to investigate the role of transitivity in other domains of reasoning, and potentially to benefit the lived experiences of persons with developmental issues

Does the cognitive reflection test measure cognitive reflection? A mathematical modeling approach

We used a mathematical modeling approach, based on a sample of 2,019 participants, to better understand what the cognitive reflection test (CRT; Frederick In Journal of Economic Perspectives, 19, 25–42, 2005) measures. This test, which is typically completed in less than 10 min, contains three problems and aims to measure the ability or disposition to resist reporting the response that first comes to mind. However, since the test contains three mathematically based problems, it is possible that the test only measures mathematical abilities, and not cognitive reflection. We found that the models that included an inhibition parameter (i.e., the probability of inhibiting an intuitive response), as well as a mathematical parameter (i.e., the probability of using an adequate mathematical procedure), fitted the data better than a model that only included a mathematical parameter. We also found that the inhibition parameter in males is best explained by both rational thinking ability and the disposition toward actively open-minded thinking, whereas in females this parameter was better explained by rational thinking only. With these findings, this study contributes to the understanding of the processes involved in solving the CRT, and will be particularly useful for researchers who are considering using this test in their research

The link between job satisfaction and organizational commitment:differences between public and private sector employees

Employees in the public and private sectors experience different working conditions and employment relationships. Therefore, it can be assumed that their attitudes toward their job and organizations, and relationships between them, are different. The existing literature has identified the relationship between organizational commitment and job satisfaction as interesting in this context. The present field study examines the satisfaction–commitment link with respect to differences between private and public sector employees. A sample of 617 Greek employees (257 from the private sector and 360 from the public sector) completed standardized questionnaires. Results confirmed the hypothesized relationship differences: Extrinsic satisfaction and intrinsic satisfaction are more strongly related to affective commitment and normative commitment for public sector employees than for private sector ones. The results are discussed, limitations are considered, and directions for future research are proposed

Crossing Eurasia: trans-regional Afghan trading networks in China and beyond

An expanding body of literature in the field of Central Asian studies has brought attention to the problems of considering the region's complex dynamics through the lens of its nation-states. Comparatively less attention has been paid to the role played by trans-regional circulations in connecting parts of Central Asia to the wider world. This paper situates ethnographic work on trans-regional networks of Afghan traders in China, Central Asia, Russia, Ukraine and the UK in relation to the literature on trans-regional connections and circulation societies. Ethnographically it demonstrates the multi-polar nature of these trans-regional networks, and the importance of trading nodes, especially the Chinese city of Yiwu, to their formation and ongoing vitality