Transreal limits expose category errors in IEEE 754 floating-point arithmetic and in mathematics

The IEEE 754 standard for oating-point arithmetic is widely used in computing. It is based on real arithmetic and is made total by adding both a positive and a negative infinity, a negative zero, and many Not-a-Number (NaN) states. The IEEE infinities are said to have the behaviour of limits. Transreal arithmetic is total. It also has a positive and a negative infinity but no negative zero, and it has a single, unordered number, nullity. We elucidate the transreal tangent and extend real limits to transreal limits. Arguing from this firm foundation, we maintain that there are three category errors in the IEEE 754 standard. Firstly the claim that IEEE infinities are limits of real arithmetic confuses limiting processes with arithmetic. Secondly a defence of IEEE negative zero confuses the limit of a function with the value of a function. Thirdly the definition of IEEE NaNs confuses undefined with unordered. Furthermore we prove that the tangent function, with the infinities given by geometrical con- struction, has a period of an entire rotation, not half a rotation as is commonly understood. This illustrates a category error, confusing the limit with the value of a function, in an important area of applied mathe- matics { trigonometry. We brie y consider the wider implications of this category error. Another paper proposes transreal arithmetic as a basis for floating- point arithmetic; here we take the profound step of proposing transreal arithmetic as a replacement for real arithmetic to remove the possibility of certain category errors in mathematics. Thus we propose both theo- retical and practical advantages of transmathematics. In particular we argue that implementing transreal analysis in trans- floating-point arith- metic would extend the coverage, accuracy and reliability of almost all computer programs that exploit real analysis { essentially all programs in science and engineering and many in finance, medicine and other socially beneficial applications.

MA TEFL programmes in Iran: change in a globalised era

This chapter is a modest attempt to investigate how MA TEFL programmes in Iran are changing in a globalised world. Our previous research in this area (Hasrati & Tavakoli, in print; Tavakoli & Hasrati, in preparation) has shown how MAs in English Language Teaching programmes are developing in Anglophone countries, but little or no research has been conducted to study changes in MA TEFL programmes in Iran. In what follows, we will first introduce MA TEFL programmes in Iran, before presenting and discussing different definitions of globalisation. We will then explain how we collected the data for this study and report our findings, making comparisons with the other contexts when appropriate. We will conclude by elaborating on possible extensions of this study in similar contexts.

Morphological awareness and reading development in bilingual English-Chinese children in Singapore

Addressing two aspects of morphological awareness – derivational and compound, this study investigates the relationships between morphological awareness and vocabulary and reading comprehension in English-Chinese bilingual primary 3 children in Singapore (N = 76). Comparable tasks in Chinese and English were administered to examine the children’s morphological awareness, vocabulary, and reading comprehension. The results show that morphological awareness is highly related to vocabulary and reading comprehension, with higher correlations between morphological awareness and reading comprehension than between morphological awareness and vocabulary. This indicates that morphological awareness may have direct influence on reading comprehension beyond the mediating effect of vocabulary. Furthermore, the results indicate that children displayed more compound than derivational morphological awareness for Chinese due to the dominance of compound morphology in Chinese. However the children also displayed similar levels of derivational and compound morphological awareness for English despite far more derivatives than compounds in English. The robust cross-linguistic correlations suggest that Chinese compound morphological knowledge plays a facilitating role not only in learning English compounds, but also in learning transparently derived words that do not involve phonological or orthographic shifts.

Production of definite and indefinite articles in typically developing English-speaking children and children with SLI

The present paper examines the production of definite and indefinite articles in English-speaking typically developing (TD) children and children with Specific Language Impairment (SLI). Twenty four English-speaking children with SLI (mean age: 7;5), twenty nine TD age-matched (TD-AM) children (mean age: 7;5) and eleven younger (mean age: 5;5) TD vocabulary-matched (TD-VM) children participated in a production task involving short stories without picture props based on Schafer and de Villiers (2000). Article production was examined in two different semantic contexts for the definite article, namely in the anaphoric and the bridging context. In the anaphoric condition, definiteness is established via linguistic means, whereas in the bridging condition via shared world knowledge. Indefinite article production was examined in the referential specific, non-referential predicational, and non-referential instrumental contexts. The referential specific context involves [+speaker, −hearer] knowledge and the non-referential predicational and instrumental [−speaker, −hearer] knowledge. Results showed that in the definite article contexts, all three groups performed better on the bridging compared with the anaphoric condition; in the indefinite article contexts, they had better performance on the non-referential predicational vs. the referential specific and the non-referential instrumental conditions. In terms of errors, the TD-VM children and the children with SLI produced significantly more substitutions than the TD-AM children in the definite article contexts. In the indefinite article contexts, the three groups did not differ in terms of accuracy or error patterns. The present results point towards problems in the discourse integration of entities that are part of the speaker's and hearer's knowledge in children with SLI and TD-VM controls, especially in definite articles. These problems are accentuated in the children with SLI due to their grammatical impairment and suggest that children with SLI exhibit a delayed acquisition profile.

Internationalizing the MFA in Creative Writing

is article explores the prospects for internationalizing the Master of Fine Arts (MFA) in Creative Writing, a degree that has gained considerable popularity in the United States in the past half century but has yet to gain much of a foothold in other countries. As part of this exploration, we describe the experiences of estab- lishing the first low-residency Master of Fine Arts in Creative Writing in Asia at City University of Hong Kong, explaining the justification for setting up such a program with reference to the history of teaching creative writing and the current conditions for literary writing in English in Asia and globally. We also reflect upon the processes of planning, curriculum design, and administrative negotiation and that went into setting up the program and report on feedback from the first cohort of students. e experience of setting up this program is used as the basis for raising a number of more general issues regarding the teaching of creative writing in English in interna- tional contexts.

Children's preconceptions of science: how these can be used in teaching

This paper focuses on young children’s initial ideas about science prior to any teaching and discusses teachers’ identification of these ‘preconceptions’ when teaching science in the early years. The research focuses on early years teaching in public and private kindergartens with children from three to five. The area of the children’s preconceptions has been extensively investigated by other researchers in the past. However, research focusing on children’s preconceptions and how teachers work with these in the early years is still limited in comparison, especially within Cypriot context. A case study was employed which facilitated in-depth investigation employing different methods of data collection including interviews and observations. The results indicate that teachers tend to avoid identification of the children’s preconceptions when planning and teaching science. This suggests a lack of appreciation of the children’s preconceptions and the consequences when they are not acknowledged. To help teachers respond to the children’s preconceptions, this paper provides a number of suggestion on how to identify children’s preconceptions.