Exploring argumentation, objectivity, and bias: The case of mathematical infinity

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece (e.g., Dubinsky et al., 2005). Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise (e.g., Brown et al., 2010; Fischbein et al., 1979, Tsamir, 1999). This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation as manifested in different presentations and normative interpretations or resolutions of well-known paradoxes of infinity. Paradoxes have been described as occasioning major epistemological reconstructions (e.g., Quine, 1966), and I highlight such occasions as they emerged for both novices and experts with connection to current conceptualisations of objectivity (e.g., Daston, 1992). Of interest is the perception that one single objective truth about “actual” mathematical infinity exists – indeed, this is brought to question at an axiomatic level with both theoretical and empirical research implications

Polysemy of symbols: Signs of ambiguity

This article explores instances of symbol polysemy within mathematics as it manifests in different areas within the mathematics register. In particular, it illustrates how even basic symbols, such as ‘+’ and ‘1’, may carry with them meaning in ‘new’ contexts that is inconsistent with their use in ‘familiar’ contexts. This article illustrates that knowledge of mathematics includes learning a meaning of a symbol, learning more than one meaning, and learning how to choose the contextually supported meaning of that symbol

Population Projections for Forty-Four European Countries: The Ongoing Population Ageing

In the current paper we present the population projections for forty-four European countries. The dynamics of migration might prevent some European countries from experiencing population decline in the near future, although fertility has been below replacement for some time. Similarly, the EU-27 population is projected to further increase. Our results confirm that population ageing is underway all around Europe, albeit with clear differences across countries. According to the traditional measures of population age structure, the countries with the oldest populations are expected to be found within the EU. However, these measures do not take into account the longevity change: a man of 65 living in a country with a higher life expectancy might be at a different stage of the life cycle in comparison to a man of 65 living in a country with lower life expectancy. Relying on three alternative measures of age which consider explicitly changes in the remaining life expectancy, we find evidence that ageing will continue, but (1) it might be more severe in some countries where population is shrinking towards older ages but life expectancy is still rather low; (2) it might not be as fast as it appears when not adjusting for the longevity change. As an example, the former Soviet Union states and some countries of the Balkan region show the highest proportion of population with a remaining life expectancy of 15 years or less; in Italy and Germany the prospective median age and the proportion of population with a remaining life expectancy of 15 years or less might increase at a slower pace than the corresponding conventional measures, i.e. the population median age and the proportion of people aged 65+

Population Projections for non-EU / non-EFTA Countries in Europe

In this study we develop population projections for the non EU/non EFTA countries in Europe. The natural population dynamics recently observed in these countries are moving in a similar direction as in the rest of Europe, even though there is significant, country-specific heterogeneity regarding the intensity and timing of these changes. Contrary to other European countries, the majority of these countries will see a favourable period in terms of the characteristics of their population age profiles in the near future. With a low share of young and elderly populations on the one hand, and a prominent working-age population on the other hand, this demographic window could trigger socio-economic development. Yet this situation will only prevail during a short period, until the dependency ratio once more increases as the ageing process advances, which also seems to be an item on the future demographic agenda of these countries

Risks Worth Taking? Social Risks and the Mathematics Teacher

In this article, we explore notions of risk as perceived or experienced by individuals involved in mathematical education. We present this exploration in the form of vignettes, each illustrating a form of risk: a parent’s reaction to classroom “propaganda”; a teacher trying to do justice by her students; a teacher confronted by his administration; and a college professor who believes university policy to be unjust. Each vignette sheds light on areas in which teacher education may offer additional support in fostering the mathematical knowledge, pedagogical sensitivity, and social awareness required to foster, what are in our view, much needed risks in the mathematical (and otherwise) education of pupils. Following the vignettes, we offer a discussion of factors that contributed to the risks perceived or experienced by teachers: neoliberal discourses, and the powerful cultural scripts that leave teachers feeling that they must hold all control, authority, and knowledge

European Demographic Datasheet 2016

The European Demographic Datasheet 2016 shows key demographic data, population trends and projections until 2050. It covers fertility, mortality, migration and population structure, including population ageing, and their changes. The datasheet combines data for all countries of Europe and for broader European regions, as well as maps, population pyramids, tables, graphs and featured thematic boxes. It pays special attention to the importance of migration for the current and future population changes across the continent and to the alternative indicators of population ageing. In comparison to the Data Sheet Poster, the online version provides expanded data coverage, additional maps and population pyramids, ranking charts and details about data sources and definitions. It also allows users to download all data and images

Full STEAM Ahead: Building Preservice Teachers’ Capacity in Makerspace Pedagogies

This paper explores teacher candidates’ understandings of 1) makerspace/constructionist pedagogies; 2) the issue of bullying; and, 3) working with at-risk youth, as they evolved over the course of a six-month partnership. The partnership included researchers and teacher candidates at a Faculty of Education and the teacher librarian at a local elementary school who were participating in a larger Social Sciences and Humanities Research Council of Canada (SSHRC)- funded project that focuses on building, implementing and evaluating an effective model for a school improvement program that increases teachers’ capacity, experience and specific fluency and expertise with technologies supporting STEAM learning and digital literacies. In this paper, we discuss qualitative ethnographic case study research, which examines in depth the experiences of five teacher candidates as they worked with 20 students in a grade 6 class in a high needs school on makerspace activities related to bullying prevention in their school community. Qualitative research documentation includes digital video and audio recordings, on the-ground field notes and observational notes, pre and post interviews with participants and focus group sessions. Results from this study contribute new knowledge in the areas of preservice teacher development and digitally-enhanced learning environments for K-6 learners

The Influence of Task Demands on Manual Asymmetries for Reaching Movements to Tools

In this dissertation, three experiments were conducted that examined the influence of task demands on manual asymmetries for the performance of reaching movements to tools. In all three experiments, the difference between the hands (in terms of preference for Experiment 1 and performance for Experiments 2 and 3) was studied in response to varying task demands for grasping movements to tools. In the first experiment, 82 right-handed and 60 left-handed university students performed reaching movements to tools and dowels at five positions within working space. Differences in the reaching patterns of the left and right hands to the tools and dowels were examined, as well as the effect of task demands (lift, use) and type of object (tool, dowel) on the reaching patterns. Dowels were used in order to examine if participants would treat a neutral object as if it were a tool in terms of their reaching patterns in working space. Results confirmed and extended prior research on the influence of task demands on reaching patterns within working space. Overall, there were more similarities in the general reaching patterns of left- and right-handed participants than differences. However key differences between the handedness groups emerged in the treatment of the dowel and the frequency of switches (reaching to lift the object with the non-preferred hand and transferring it to the preferred hand to use). Results also showed that tools enjoy a privileged association with the preferred hand, and that the intent of the movement has a very real goal on movement planning. The first experiment examined patterns of hand use across working space in response to differing task demands. In the next experiments performance differences between the hands in terms of movement planning and initiation were examined through the use of reaction time and movement time. In these experiments, reaction time represented the time from the presentation of a go signal to when the participant first lifted their hand, and movement time was the time between lifting the hand to lifting a tool off a sensor. Movement time represented the time to pick up the tool, and did not include the time to use the tool to perform a particular task and complete the reaching movement. In the second experiment, reaction time and movement time to tools placed at the midline position were examined under varying degrees of advance information using a precue paradigm. Three precue conditions were used which presented advance information on the hand to use to perform the movement (left or right) and/or the task (lift, use, or pantomime) to be performed: (1) both hand and task were cued in advance (Both precue); (2) task only was cued in advance (Task precue); and (3) neither hand nor task were cued in advance (No precue). Twenty-four right-handed university students performed reaching movements to tools under the three different precue conditions. The results of Experiment 2 showed that reaction time was sensitive to the amount of advance information presented in the precue. For reaction time manual asymmetries were observed in one condition only – a right hand advantage was present in the No precue condition. In contrast manual asymmetries in favor of the right hand were clearly observed with the movement time results. Experiment 2 was the first experiment reported in the literature to systematically examine reaction time for reaching and grasping movements to tools. In order to further explore these results, in Experiment 3 a fourth precue condition (in which the hand to be used was cued in advance; the Hand precue) was added to the precue paradigm used in Experiment 2. An additional variable called replacement time, which represented the time spent interacting with the tool, was also examined. Forty-two right-handed university students participated in Experiment 3. The results of Experiment 3 largely replicated the findings of Experiment 2, and indicated that both the amount and type of precue information had an effect on reaction time. The addition of the Hand precue condition suggested that having advance knowledge of the hand to be used to perform the task was of greater importance for movement planning than was advance knowledge of the task to be performed. Regarding the movement time results, Experiment 3 was one of the first experiments to show the influence of task demands on the magnitude of manual asymmetries. The lack of differences between the hands for the replacement time results also suggested that the initial execution of the movement (represented by movement time) was most sensitive to manual asymmetries. Overall, these experiments provided further insight into manual asymmetries for the performance of reaching movements, and illustrated how simple manipulations of task demands led to differences between the hands in measures of both preference and performance when interacting with tools