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.

Teaching programming to beginners in a massive open online course

The University of Reading’s first Massive Open Online Course (MOOC) “Begin Programming: Build your first mobile game” (#FLMobiGame) was offered in Autumn 2013 on the FutureLearn platform. This course used a simple Android game framework to present basic programming concepts to complete beginners. The course attracted wide interest from all age groups. The course presented opportunities and challenges to both participants and educators. While some participants had difficulties accessing content some others had trouble grasping the concepts and applying them in a real program. Managing forums was cumbersome with the limited facilities supported by the Beta-platform. A healthy community was formed around the course with the support of social media. The case study reported here is part of an ongoing research programme exploring participants’ MOOC engagement and experience using a grounded, ethnographical approach.

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.

Virtualization for cost-effective teaching of assembly language programming

A virtual system that emulates an ARM-based processor machine has been created to replace a traditional hardware-based system for teaching assembly language. The proposed virtual system integrates, in a single environment, all the development tools necessary to deliver introductory or advanced courses on modern assembly language programming. The virtual system runs a Linux operating system in either a graphical or console mode on a Windows or Linux host machine. No software licenses or extra hardware are required to use the virtual system, thus students are free to carry their own ARM emulator with them on a USB memory stick. Institutions adopting this, or a similar virtual system, can also benefit by reducing capital investment in hardware-based development kits and enable distance learning courses.

Sex differences in the programming effects of prenatal stress on psychopathology and stress responses: an evolutionary perspective

There is strong evidence from animal studies that prenatal stress has different effects on male and female offspring. In general, although not always, prenatal stress increases anxiety, depression and stress responses, both hypothalamic–pituitary–adrenal and cardiovascular, in female offspring rather than in male. Males are more likely to show learning and memory deficits. There have been few studies so far in humans which differentiate effects of prenatal stress on male and female psychopathology. Some studies support the animal models, but the evidence is inconsistent. The mediating mechanisms for any sex specific effects are little understood, but there is evidence that placental function can differ depending on the sex of the fetus. We suggest that there may be an evolutionary reason for any sex differences in the long term effects of prenatal stress. In a stressful environment it may be adaptive for females, who are more likely to stay in one place and look after children, to be more vigilant, alert to danger and thus show more stress responsiveness. This can give rise to a more anxious or depressed phenotype. With males it may be more adaptive to go out and explore new environments, compete with other males, and be more aggressive. For this it may help to be less responsive to external stressors. More research is needed into sex differences in the effects of prenatal stress in humans, to test these ideas.

Evidence for sex differences in fetal programming of physiological stress reactivity in infancy

Associations between low birth weight and prenatal anxiety and later psychopathology may arise from programming effects likely to be adaptive under some, but not other, environmental exposures and modified by sex differences. If physiological reactivity, which also confers vulnerability or resilience in an environment-dependent manner, is associated with birth weight and prenatal anxiety, it will be a candidate to mediate the links with psychopathology. From a general population sample of 1,233 first-time mothers recruited at 20 weeks gestation, a sample of 316 stratified by adversity was assessed at 32 weeks and when their infants were aged 29 weeks (N = 271). Prenatal anxiety was assessed by self-report, birth weight from medical records, and vagal reactivity from respiratory sinus arrhythmia during four nonstressful and one stressful (still-face) procedure. Lower birth weight for gestational age predicted higher vagal reactivity only in girls (interaction term, p = .016), and prenatal maternal anxiety predicted lower vagal reactivity only in boys (interaction term, p = .014). These findings are consistent with sex differences in fetal programming, whereby prenatal risks are associated with increased stress reactivity in females but decreased reactivity in males, with distinctive advantages and penalties for each sex.

Maternal antenatal anxiety, postnatal stroking and emotional problems in children: outcomes predicted from pre and postnatal programming hypotheses

Background Mothers' self-reported stroking of their infants over the first weeks of life modifies the association between prenatal depression and physiological and emotional reactivity at 7 months, consistent with animal studies of the effects of tactile stimulation. We now investigate whether the effects of maternal stroking persist to 2.5 years. Given animal and human evidence for sex differences in the effects of prenatal stress we compare associations in boys and girls. Method From a general population sample of 1233 first-time mothers recruited at 20 weeks gestation we drew a random sample of 316 for assessment at 32 weeks, stratified by reported inter-partner psychological abuse, a risk indicator for child development. Of these mothers, 243 reported at 5 and 9 weeks how often they stroked their infants, and completed the Child Behavior Checklist (CBCL) at 2.5 years post-delivery. Results There was a significant interaction between prenatal anxiety and maternal stroking in the prediction of CBCL internalizing (p = 0.001) and anxious/depressed scores (p < 0.001). The effects were stronger in females than males, and the three-way interaction prenatal anxiety × maternal stroking × sex of infant was significant for internalizing symptoms (p = 0.003). The interactions arose from an association between prenatal anxiety and internalizing symptoms only in the presence of low maternal stroking. Conclusions The findings are consistent with stable epigenetic effects, many sex specific, reported in animal studies. While epigenetic mechanisms may be underlying the associations, it remains to be established whether stroking affects gene expression in humans.

Fetal programming of adipose tissue function: an evolutionary perspective

Obesity is an escalating threat of pandemic proportions and has risen to such unrivaled prominence in such a short period of time that it has come to define a whole generation in many countries around the globe. The burden of obesity, however, is not equally shared among the population, with certain ethnicities being more prone to obesity than others, while some appear to be resistant to obesity altogether. The reasons behind this ethnic basis for obesity resistance and susceptibility, however, have remained largely elusive. In recent years, much evidence has shown that the level of brown adipose tissue thermogenesis, which augments energy expenditure and is negatively associated with obesity in both rodents and humans, varies greatly between ethnicities. Interestingly, the incidence of low birth weight, which is associated with an increased propensity for obesity and cardiovascular disease in later life, has also been shown to vary by ethnic background. This review serves to reconcile ethnic variations in BAT development and function with ethnic differences in birth weight outcomes to argue that the variation in obesity susceptibility between ethnic groups may have its origins in the in utero programming of BAT development and function as a result of evolutionary adaptation to cold environments.

Don’t aim too high for your kids: parental over-aspiration undermines students’ learning in mathematics

Previous research has suggested that parents’ aspirations for their children’s academic attainment can have a positive influence on children’s actual academic performance. Possible negative effects of parental over-aspiration, however, have found little attention in the psychological literature. Employing a dual-change score model with longitudinal data from a representative sample of German schoolchildren and their parents (N = 3,530; grades 5 to 10), we showed that parental aspiration and children’s mathematical achievement were linked by positive reciprocal relations over time. Importantly, we also found that parental aspiration that exceeded their expectation (i.e., over-aspiration) had negative reciprocal relations with children’s mathematical achievement. These results were fairly robust after controlling for a variety of demographic and cognitive variables such as children’s gender, age, intelligence, school type, and family SES. The results were also replicated with an independent sample of US parents and their children. These findings suggest that unrealistically high parental aspiration can be detrimental for children’s achievement.