Diversity in mathematics assessment equals diversity in opinion?

We undertook a study to investigate the views of both students and staff in our department towards assessment in mathematics, as a precursor to considering increasing the diversity of assessment types. In a survey and focus group there was reasonable agreement amongst the students with regards major themes such as mode of assessment. However, this level of agreement was not seen amongst the staff, where discussions regarding diversity in mathematics assessment definitely revealed a difference of opinion. As a consequence, we feel that the greatest barriers to increasing diversity may be with staff, and so more efforts are needed to communicate to staff the advantages and disadvantages, in order to give them greater confidence in trying a range of assessment types.

Relationship between classroom authority and epistemological beliefs as espoused by primary school mathematics teachers from the very high and very low socio-economic regions in Thailand

This article presents findings of a larger single-country comparative study which set out to better understand primary school teachers’ mathematics education-related beliefs in Thailand. By combining the interview and observation data collected in the initial stage of this study with data gathered from the relevant literature, the 8-belief / 22-item ‘Thai Teachers’ Mathematics Education-related Beliefs’ (TTMEB) Scale was developed. The results of the Mann-Whitney U Test showed that Thai teachers in the two examined socio-economic regions espouse statistically different beliefs concerning the source and stability of mathematical knowledge, as well as classroom authority. Further, these three beliefs are found to be significantly and positively correlated.

A molecular, morphological and cytological investigation of the identity of non-native Ludwigia (Onagraceae) populations in Britain

A combined molecular, morphological and cytological analysis was used to study the identity and number of species of Ludwigia section Oligospermum occurring in British waterways. Only one taxon was identified for which the name L. grandiflora subsp. hexapetala (Hook. & Arn.) G.L. Nesom & Kartesz is preferred. A chromosome count of 2n = 80 was made for all plants tested and DNA evidence demonstrates that at least two clones are present in Britain. Morphological characters to differentiate L.grandiflora subsp. hexapetala and L. peploides subsp. montevidensis (Spreng.) P.H. Raven are provided. Though the production of fruit in Britain by apparently isolated populations is noted, repeated introduction into the wild from gardens is judged to be primarily responsible for the British distribution of the taxon. Legislative implications are considered and an amendment to Schedule 9 of the UK Wildlife and Countryside Act (1981) is strongly advocated.

The time-course of morphological constraints: Evidence from eye-movements during reading

Lexical compounds in English are constrained in that the non-head noun can be an irregular but not a regular plural (e.g. mice eater vs. *rats eater), a contrast that has been argued to derive from a morphological constraint on modifiers inside compounds. In addition, bare nouns are preferred over plural forms inside compounds (e.g. mouse eater vs. mice eater), a contrast that has been ascribed to the semantics of compounds. Measuring eyemovements during reading, this study examined how morphological and semantic information become available over time during the processing of a compound. We found that the morphological constraint affected both early and late eye-movement measures, whereas the semantic constraint for singular non-heads only affected late measures of processing. These results indicate that morphological information becomes available earlier than semantic information during the processing of compounds.

The time-course of morphological constraints: a study of plurals inside derived words

The avoidance of regular but not irregular plurals inside compounds (e.g. *rats eater vs. mice eater) has been one of the most widely studied morphological phenomena in the psycholinguistics literature. To examine whether the constraints that are responsible for this contrast have any general significance beyond compounding, we investigated derived word forms containing regular and irregular plurals in two experiments. Experiment 1 was an offline acceptability judgment task, and experiment 2 measured eye movements during reading derived words containing regular and irregular plurals and uninflected base nouns. The results from both experiments show that the constraint against regular plurals inside compounds generalizes to derived words. We argue that this constraint cannot be reduced to phonological properties, but is instead morphological in nature. The eye-movement data provide detailed information on the time-course of processing derived word forms indicating that early stages of processing are affected by a general constraint that disallows inflected words from feeding derivational processes, and that the more specific constraint against regular plurals comes in at a subsequent later stage of processing. We argue that these results are consistent with stage-based models of language processing.

The time-course of morphological processing in a second language

We report findings from psycholinguistic experiments investigating the detailed timing of processing morphologically complex words by proficient adult second (L2) language learners of English in comparison to adult native (L1) speakers of English. The first study employed the masked priming technique to investigate -ed forms with a group of advanced Arabic-speaking learners of English. The results replicate previously found L1/L2 differences in morphological priming, even though in the present experiment an extra temporal delay was offered after the presentation of the prime words. The second study examined the timing of constraints against inflected forms inside derived words in English using the eye-movement monitoring technique and an additional acceptability judgment task with highly advanced Dutch L2 learners of English in comparison to adult L1 English controls. Whilst offline the L2 learners performed native-like, the eye-movement data showed that their online processing was not affected by the morphological constraint against regular plurals inside derived words in the same way as in native speakers. Taken together, these findings indicate that L2 learners are not just slower than native speakers in processing morphologically complex words, but that the L2 comprehension system employs real-time grammatical analysis (in this case, morphological information) less than the L1 system.

Flow-induced morphological instability of a mushy layer

A morphological instability of a mushy layer due to a forced flow in the melt is analysed. The instability is caused by flow induced in the mushy layer by Bernoulli suction at the crests of a sinusoidally perturbed mush–melt interface. The flow in the mushy layer advects heat away from crests which promotes solidification. Two linear stability analyses are presented: the fundamental mechanism for instability is elucidated by considering the case of uniform flow of an inviscid melt; a more complete analysis is then presented for the case of a parallel shear flow of a viscous melt. The novel instability mechanism we analyse here is contrasted with that investigated by Gilpin et al. (1980) and is found to be more potent for the case of newly forming sea ice.

Palaeobiology of the medieval population of Albano (Rome, Italy): a combined morphological and biomolecular approach

Recent years have seen increased interest in skeletal populations from the Imperial Roman Age in Italy, but much less is known about diet and standards of living in the subsequent medieval period. To fill this gap, we conducted a morphological analysis of human remains from Albano, an Italian town near Rome, as well as a stable isotope analysis of bone collagen to reconstruct diet. The sample was recovered from a Medieval cemetery (1040–1220 cal. yr. BP) located in the gardens of the historical Palazzo Doria Pamphili in Albano. A minimum number of 40 individuals (31 adults and 9 sub-adults) were examined using standard methods. Though the general health status of the population was good, signs of cribra orbitalia and diffuse enthesopathies were noted during the morphological examination. Stable carbon and nitrogen isotope analyses of the bone collagen from 24 adult humans and three faunal bones indicate that the diet of the population may be described as predominantly terrestrial and C3-plant based although the data for some of the individuals suggest a modest consumption of C4-(millet) based or aquatic proteins. No evidence of significant dietary differences between the sexes was found. The comparison of the isotope data from Albano with those from populations recovered in the same region is consistent with a shift from a terrestrial, possibly plant foods-dominated subsistence in the Early Middle Ages to a diet with a higher contribution from animal proteins, both terrestrial and aquatic, in the Later Middle Ages.

Morphological decomposition in Broca's aphasia

Background: Few studies have investigated how individuals diagnosed with post-stroke Broca’s aphasia decompose words into their constituent morphemes in real-time processing. Previous research has focused on morphologically complex words in non-time-constrained settings or in syntactic frames, but not in the lexicon. Aims: We examined real-time processing of morphologically complex words in a group of five Greek-speaking individuals with Broca’s aphasia to determine: (1) whether their morphological decomposition mechanisms are sensitive to lexical (orthography and frequency) vs. morphological (stem-suffix combinatory features) factors during visual word recognition, (2) whether these mechanisms are different in inflected vs. derived forms during lexical access, and (3) whether there is a preferred unit of lexical access (syllables vs. morphemes) for inflected vs. derived forms. Methods & Procedures: The study included two real-time experiments. The first was a semantic judgment task necessitating participants’ categorical judgments for high- and low-frequency inflected real words and pseudohomophones of the real words created by either an orthographic error at the stem or a homophonous (but incorrect) inflectional suffix. The second experiment was a letter-priming task at the syllabic or morphemic boundary of morphologically transparent inflected and derived words whose stems and suffixes were matched for length, lemma and surface frequency. Outcomes & Results: The majority of the individuals with Broca’s aphasia were sensitive to lexical frequency and stem orthography, while ignoring the morphological combinatory information encoded in the inflectional suffix that control participants were sensitive to. The letter-priming task, on the other hand, showed that individuals with aphasia—in contrast to controls—showed preferences with regard to the unit of lexical access, i.e., they were overall faster on syllabically than morphemically parsed words and their morphological decomposition mechanisms for inflected and derived forms were modulated by the unit of lexical access. Conclusions: Our results show that in morphological processing, Greek-speaking persons with aphasia rely mainly on stem access and thus are only sensitive to orthographic violations of the stem morphemes, but not to illegal morphological combinations of stems and suffixes. This possibly indicates an intact orthographic lexicon but deficient morphological decomposition mechanisms, possibly stemming from an underspecification of inflectional suffixes in the participants’ grammar. Syllabic information, however, appears to facilitate lexical access and elicits repair mechanisms that compensate for deviant morphological parsing procedures.

Inclusive approaches to teaching and learning mathematics

This chapter explores the role of mentors in supporting pre-service teachers to include all children in mathematics teaching, no matter what their individual needs.

fMRI evidence for the involvement of the procedural memory system in morphological processing of a second language

Behavioural evidence suggests that English regular past tense forms are automatically decomposed into their stem and affix (played  = play+ed) based on an implicit linguistic rule, which does not apply to the idiosyncratically formed irregular forms (kept). Additionally, regular, but not irregular inflections, are thought to be processed through the procedural memory system (left inferior frontal gyrus, basal ganglia, cerebellum). It has been suggested that this distinction does not to apply to second language (L2) learners of English; however, this has not been tested at the brain level. This fMRI study used a masked-priming task with regular and irregular prime-target pairs (played-play/kept-keep) to investigate morphological processing in native and highly proficient late L2 English speakers. No between-groups differences were revealed. Compared to irregular pairs, regular pairs activated the pars opercularis, bilateral caudate nucleus and the right cerebellum, which are part of the procedural memory network and have been connected with the processing of morphologically complex forms. Our study is the first to provide evidence for native-like involvement of the procedural memory system in processing of regular past tense by late L2 learners of English.

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.