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

Disaggregating political authority: what's wrong with Rawlsian civil disobedience?

Contemporary philosophical and theoretical discussions of civil disobedience hope to contribute to significant political debates around when and in which forms political dissent, protest and resistance is appropriate. In doing so, they often focus on and criticize John Rawls' work on civil disobedience. However, ignoring the frame in which Rawls discusses civil disobedience has led critics to wrongly attack his theory for being too restrictive when it is more likely to be too permissive. That permissiveness depends on treating any political order which does not come close to fulfilling his theory of justice as absolutely illegitimate. In this sense, Rawls’ theory of political authority is binary and demanding. The problems his theory shares with most others, including his critics’, show that political authority needs to be disaggregated to make sense of the conditions under which different forms of protest and resistance are appropriate

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’

Introduction to the special issue: after trust

“Trust” has long been seen as critical to the success and durability of trading networks, and conceptualised as a positive moral sentiment that is embedded in shared kinship, ethnicity or friendship, or in shared frameworks of morality. Other recent studies of business communities suggest that the ability to work in settings characterised by pervasive mistrust is a key factor in the development of commercial acumen and determining success. This Special Issue argues that the focus on trust and mistrust, and the underlying concern with ethics and morality, obscure equally critical aspects that inform the durability of trading networks. It offers ethnographic accounts of different inter-Asian trading networks active in the city of Yiwu. One of China’s most dynamic and diverse ‘international trade cities’, Yiwu is home to the largest wholesale market of small commodities in the world, and attracts traders and merchants from across the planet; over 12,000 foreign traders are also resident in the city. The collected articles analyse the durability of these networks in relation to broader geopolitical processes and contexts, arguing that success often depends on the ability to negotiate geopolitical shifts and the faultlines of political identity. The articles trace traders’ efforts to create institutions that allow them to withstand geopolitical transformations. They also document the ability of trading networks to operate flexibly across different social fields, showing that resilience often depends on the ability to navigate and profit from shifting relations between economic, political and familial domains

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