Changing classroom culture, curricula, and instruction for proof and proving: how amenable to scaling up, practicable for curricular integration, and capable of producing long-lasting effects are current interventions?

This paper is a commentary on the classroom interventions on the teaching and learning of proof reported in the seven empirical papers in this special issue. The seven papers show potential to enhance student learning in an area of mathematics that is not only notoriously difficult for students to learn and for teachers to teach, but also critically important to knowing and doing mathematics. Although the seven papers, and the intervention studies they report, vary in many ways—student population, content domain, goals and duration of the intervention, and theoretical perspectives, to name a few—they all provide valuable insight into ways in which classroom experiences might be designed to positively influence students’ learning to prove. In our commentary, we highlight the contributions and promise of the interventions in terms of whether and how they present capacity to change the classroom culture, the curriculum, or instruction. In doing so, we distinguish between works that aim to enhance students’ preparedness for, and competence in, proof and proving and works that explicitly foster appreciation for the need and importance of proof and proving. Finally, we also discuss briefly the interventions along three dimensions: how amenable to scaling up, how practicable for curricular integration, and how capable of producing long-lasting effects these interventions are

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion , a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42932/1/10758_2004_Article_6999.pd

Teachers’ appraisals of adjectives relating to mathematics tasks

Curricular implementations are unlikely to deliver the anticipated benefits for mathematics learners if written guidance to teachers is interpreted and enacted differently from the ways that policymakers and curriculum designers intend. One way in which this could happen is in relation to the mathematics tasks that teachers deploy in the classroom. Teachers and curriculum designers have developed an extensive vocabulary for describing tasks, using adjectives such as ‘rich’, ‘open’, ‘real-life’, ‘engaging’ and so on. But do teachers have a shared understanding of what these adjectives mean when they are applied to mathematics tasks? In Study 1 we investigated teachers’ appraisals of adjectives used to describe mathematics tasks, finding that task appraisals vary on seven dimensions, which we termed engagement, demand, routineness, strangeness, inquiry, context and interactivity. In Study 2, focusing on the five most prominent dimensions, we investigated whether teachers have a shared understanding of the meaning of adjectives when applied to mathematics tasks. We found that there was some agreement about inquiry and context, some disagreement about routineness, and clear disagreement about engagement and demand. We conclude that at least some adjectives commonly used to describe tasks are interpreted very differently by different teachers. Implications for how tasks might be discussed meaningfully by teachers, teacher educators and curriculum designers are highlighted

How Mathematicians Obtain Conviction: Implications for Mathematics Instruction and Research on Epistemic Cognition

This is an Accepted Manuscript of an article published by Taylor & Francis in Educational Psychologist on 16th January 2014, available online: http://wwww.tandfonline.com/10.1080/00461520.2013.865527The received view of mathematical practice is that mathematicians gain certainty in mathematical assertions by deductive evidence rather than empirical or authoritarian evidence. This assumption has influenced mathematics instruction where students are expected to justify assertions with deductive arguments rather than by checking the assertion with specific examples or appealing to authorities. In this paper, we argue that the received view about mathematical practice is too simplistic; some mathematicians sometimes gain high levels of conviction with empirical or authoritarian evidence and sometimes do not gain full conviction from the proofs that they read. We discuss what implications this might have, both for for mathematics instruction and theories of epistemic cognition

Changing outcomes following pelvic exenteration for locally advanced and recurrent rectal cancer

Background Pelvic exenteration for locally advanced rectal cancer (LARC) and locally recurrent rectal cancer (LRRC) is technically challenging but increasingly performed in specialist centres. The aim of this study was to compare outcomes of exenteration over time. Methods This was a multicentre retrospective study of patients who underwent exenteration for LARC and LRRC between 2004 and 2015. Surgical outcomes, including rate of bone resection, flap reconstruction, margin status and transfusion rates, were examined. Outcomes between higher- and lower-volume centres were also evaluated. Results Some 2472 patients underwent pelvic exenteration for LARC and LRRC across 26 institutions. For LARC, rates of bone resection or flap reconstruction increased from 2004 to 2015, from 3.5 to 12.8 per cent, and from 12.0 to 29.4 per cent respectively. Fewer units of intraoperative blood were transfused over this interval (median 4 to 2 units; P = 0.040). Subgroup analysis showed that bone resection and flap reconstruction rates increased in lower- and higher-volume centres. R0 resection rates significantly increased in low-volume centres but not in high-volume centres over time (low-volume: from 62.5 to 80.0 per cent, P = 0.001; high-volume: from 83.5 to 88.4 per cent, P = 0.660). For LRRC, no significant trends over time were observed for bone resection or flap reconstruction rates. The median number of units of intraoperative blood transfused decreased from 5 to 2.5 units (P < 0.001). R0 resection rates did not increase in either low-volume (from 51.7 to 60.4 per cent; P = 0.610) or higher-volume (from 48.6 to 65.5 per cent; P = 0.100) centres. No significant differences in length of hospital stay, 30-day complication, reintervention or mortality rates were observed over time. Conclusion Radical resection, bone resection and flap reconstruction rates were performed more frequently over time, while transfusion requirements decreased

Outcomes of obstructed abdominal wall hernia: results from the UK national small bowel obstruction audit

Background: Abdominal wall hernia is a common surgical condition. Patients may present in an emergency with bowel obstruction, incarceration or strangulation. Small bowel obstruction (SBO) is a serious surgical condition associated with significant morbidity. The aim of this study was to describe current management and outcomes of patients with obstructed hernia in the UK as identified in the National Audit of Small Bowel Obstruction (NASBO). Methods: NASBO collated data on adults treated for SBO at 131 UK hospitals between January and March 2017. Those with obstruction due to abdominal wall hernia were included in this study. Demographics, co-morbidity, imaging, operative treatment, and in-hospital outcomes were recorded. Modelling for factors associated with mortality and complications was undertaken using Cox proportional hazards and multivariable regression modelling. Results: NASBO included 2341 patients, of whom 415 (17·7 per cent) had SBO due to hernia. Surgery was performed in 312 (75·2 per cent) of the 415 patients; small bowel resection was required in 198 (63·5 per cent) of these operations. Non-operative management was reported in 35 (54 per cent) of 65 patients with a parastomal hernia and in 34 (32·1 per cent) of 106 patients with an incisional hernia. The in-hospital mortality rate was 9·4 per cent (39 of 415), and was highest in patients with a groin hernia (11·1 per cent, 17 of 153). Complications were common, including lower respiratory tract infection in 16·3 per cent of patients with a groin hernia. Increased age was associated with an increased risk of death (hazard ratio 1·05, 95 per cent c.i. 1·01 to 1·10; P = 0·009) and complications (odds ratio 1·05, 95 per cent c.i. 1·02 to 1·09; P = 0·001). Conclusion: NASBO has highlighted poor outcomes for patients with SBO due to hernia, highlighting the need for quality improvement initiatives in this group

Contemporary Management of Locally Advanced and Recurrent Rectal Cancer: Views from the PelvEx Collaborative

Pelvic exenteration is a complex operation performed for locally advanced and recurrent pelvic cancers. The goal of surgery is to achieve clear margins, therefore identifying adjacent or involved organs, bone, muscle, nerves and/or vascular structures that may need resection. While these extensive resections are potentially curative, they can be associated with substantial morbidity. Recently, there has been a move to centralize care to specialized units, as this facilitates better multi-disciplinary care input. Advancements in pelvic oncology and surgical innovation have redefined the boundaries of pelvic exenterative surgery. Combined with improved neoadjuvant therapies, advances in diagnostics, and better reconstructive techniques have provided quicker recovery and better quality of life outcomes, with improved survival This article provides highlights of the current management of advanced pelvic cancers in terms of surgical strategy and potential future developments