Why and how mathematicians read proofs: an exploratory study

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.     

Conditional inference and advanced mathematical study: further evidence

In this paper, we examine the support given for the ‘theory of formal discipline’ by Inglis and Simpson (Educational Studies Mathematics 67:187–204, 2008). This theory, which is widely accepted by mathematicians and curriculum bodies, suggests that the study of advanced mathematics develops general thinking skills and, in particular, conditional reasoning skills. We further examine the idea that the differences between the conditional reasoning behaviour of mathematics and arts undergraduates reported by Inglis and Simpson may be put down to different levels of general intelligence in the two groups. The studies reported in this paper call into question this suggestion, but they also cast doubt on a straightforward version of the theory of formal discipline itself (at least with respect to university study). The paper concludes by suggesting that either a pre-university formal discipline effect or a filtering effect on ‘thinking dispositions’ may give a better account for the findings.     

Teaching and learning how to write proofs in Concepts of Geometry

This paper concerns my experience teaching Concepts of Geometry, an inquiry-based course that emphasizes discovery learning, analytical thinking, and individual creativity. The author discusses how to guide the students to recognize the connections among different mathematical ideas; to select the types of reasoning and methods of proof that apply to the problem; to organize their mathematical thinking through communication; to make their own mathematical conjectures; and to analyze and evaluate each other＇s mathematical arguments and proofs. In particular, the comments and the feedback from students are included.     

Epistemic cognition when students read multiple documents containing conflicting scientific evidence: A think-aloud study

This study used think-aloud methodology to investigate 51 Norwegian undergraduates’ topic-specific epistemic cognition while working with six documents presenting conflicting views on the issue of cell phones and potential health risks. Results showed that students’ epistemic cognition was represented by one dimension concerning the certainty and simplicity of knowledge and three dimensions concerning the justification for knowing by different sources. Moreover, components of a mechanism of change, in particular epistemic doubt and resolution strategies, were identified in the think-aloud protocols. Finally, these mechanism of change components seemed to operate within distinct dimensions of epistemic cognition. Three case studies were used to elaborate on and illustrate how epistemic cognition may be differently involved in the reading of multiple conflicting documents over the course of reading.     

Multimedia resources designed to support learning from written proofs: an eye-movement study

This paper presents two studies of an intervention designed to help undergraduates comprehend mathematical proofs. The intervention used multimedia resources that presented proofs with audio commentary and visual animations designed to focus attention on logical relationships. In study 1, students studied an e-Proof or a standard written proof and their comprehension was assessed in both immediate and delayed tests; the groups performed similarly at immediate test, but the e-Proof group exhibited poorer retention. Study 2 accounted for this unexpected result by using eye-movement analyses to demonstrate that participants who studied an e-Proof exhibited less processing effort when not listening to the audio commentary. We suggest that the extra support offered by e-Proofs disrupts the processes by which students organise information, and thus restricts the extent to which their new understanding is integrated with existing knowledge. We discuss the implications of these results for evaluating teaching innovations and for supporting proof comprehension.     

‘Studies have shown that …’ how to read a statistical study

This article provides some background for a classroom discussion of statistical research studies. The history of cigarettes and warning labels in the USA is reviewed here in order to demonstrate the challenges of statistical research and communication.     

The eye-mind of processing written feedback: Unraveling how students read and use feedback for revision

This research aims to promote our understanding of feedback engagement processes in writing tasks using a combination of online and offline measures, including eye-tracking, thinking-aloud, and text-analyses. Study 1 explored how sixteen students read, evaluate, and use feedback for revision. Results revealed three feedback processing strategies: (1) superficial processing (n = 6), which is characterized by reading feedback in a linear way, without critically rereading or revising the text, (2) local processing (n = 6) in which students switched between reading the comments and the commented text, and (3) deep processing (n = 4) in which students integrated the feedback with both commented and uncommented parts of the text and made more substantial revisions. In Study 2, we investigated the local and deep feedback reading strategy in more detail with 41 students using a within-subject design with different types of feedback. Results demonstrated the same strategies among students, but also that the focus of feedback affected students' revision behavior, above and beyond an individual feedback processing strategy. This finding is in line with previous research that emphasized the effects of feedback characteristics on students’ use of feedback. By triangulating various process measures, this research is one of the first that provides empirical evidence for different feedback processing strategies among students. These novel insights in individual feedback engagement processing can be used to extend and refine current theories on how, when, and why feedback works and for whom.     

Morphological analysis,phonological analysis and learning to read French: a longitudinal study

This paper presents a longitudinal study, from kindergarten to secondgrade, which aims to examine the relationship between morphologicalanalysis, phonological analysis and learning to read. Three phonologicalawareness tasks, five derivational and four inflectional subtests wereadministered to fifty children at each of the three levels. Evolution ofperformance was analyzed through the three years. Data showed that withthe exception of two subtests, performance increased from kindergarten tofirst grade and from first grade to second grade, without reaching ceilingperformance in second grade, at least for morphological subtests. Linksbetween morphological and phonological analyses were very strong: inparticular, syllable segmentation was highly correlated with themorphological subtests in kindergarten while phonemic segmentation wascorrelated with morphological subtests in first and second grade. Therewere also strong links between morphological analysis and reading.Regression analyses showed that while phonological awarenessexplained a major part of variance in first grade, both phonologicaland morphological scores explained significant part of variance of bothdecoding and comprehension reading scores in second grade. Thus, thislongitudinal study contributes to the evidence of a link between bothphonological and morphological analysis and learning to read French.     

Special education teachers under stress: evidence from a Greek national study

The present study examines the sources of job stress in a national sample of 373 Greek special education teachers, using a self‐report questionnaire assessing job stress, its perceived sources, and its demographic and professional characteristics. Although the participants in general considered their job moderately stressful, they felt stressed by issues concerning the special needs child, such as the child's progress, safety, and social development. More than half of the teachers indicated that teaching children with autism poses major stress to them, followed by teaching students with behavioural and emotional difficulties. Regression analysis showed that the implementation of the special educational curriculum was the most important predictor of job stress, followed by the social and academic progress of children. The implications of these findings for in‐service and pre‐service teacher preparation, support, and awareness‐raising regarding job stress are discussed.     

Social class and participation in further education: evidence from the Youth Cohort Study of England and Wales

This paper examines the class distribution of young people, aged 16–17 years, in colleges of further education (FE) using data from the Youth Cohort Study. It finds that, contrary to popular perceptions of FE colleges as being for ‘other people’s children’, middle‐class students as well as working‐class students are well represented. However, this does not imply that FE colleges are institutions of choice; middle‐class representation is often related to lower achievement and, for low‐achieving working‐class students, leaving education entirely is more likely than entry to FE. These findings are explored using notions of habitus and field. Their relationship with studies of the education of middle‐class children is also discussed, and the paper suggests that research on class in FE colleges must come to terms with middle‐class presence.     

Why and how people of limited intelligence become calendrical calculators

Abstract

Calendrical calculation is the rare talent of naming the days of the week for dates in the past and future. Calendrical savants are people with low measured intelligence who have this talent. This paper reviews evidence and speculation about why people become calendrical savants and how they answer date questions. Most savants are known to have intensively studied the calendar and show superior memory for calendrical information. As a result they may answer date questions either from recalling calendars or by using strategies that exploit calendrical regularities. While people of average or superior intelligence may become calendrical calculators through internalizing formulae, the arithmetical demands of the formulae make them unlikely as bases for the talents of calendrical savants. We attempt to identify the methods used by a sample of 10 savants. None rely on an internalised formula. Some use strategies based on calendrical regularities probably in conjunction with memory for a range of years. For the rest a decision between use of regularities and recall of calendars cannot be made.     

Negotiating the complementarian gender ideology of an evangelical student subculture: further evidence from women's narratives

This study is based on a longitudinal, qualitative investigation of a burgeoning evangelical student organisation on a university campus in the USA. In addition to four months of observation, in‐depth interviews were conducted with students in their first and third years of college to understand the gender climate and ideology that characterised the organisation, as well as how students' gender ideologies were altered or reinforced after three years of exposure to the organisation. The findings from the initial study revealed that the evangelical student community was steeped in a complementarian gender ideology; that is, the culture embraced normative masculinity, essential gender differences, and separate roles and expectations for men and women with respect to leadership, modesty, and dating/marriage. A narrative analysis of the longitudinal interview data revealed the diverse ways in which four women negotiated the gender ideology of the evangelical organisation during their college years.     

Pathways to literacy: a study of invented spelling and its role in learning to read

This intervention study tested whether invented spelling plays a causal role in learning to read. Three groups of kindergarten children (mean age = 5 years 7 months) participated in a 4-week intervention. Children in the invented-spelling group spelled words as best they could and received developmentally appropriate feedback. Children in the 2 comparison groups were trained in phonological awareness or drew pictures. Invented-spelling training benefited phonological and orthographic awareness and reading of words used in the intervention. Importantly, the invented-spelling group learned to read more words in a learn-to-read task than the other groups. The finding are in accord with the view that invented spelling coupled with feedback encourages an analytical approach and facilitates the integration of phonological and orthographic knowledge, hence facilitating the acquisition of reading.