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

Conceptions of basic education teachers about math proof: influence of professional experience

A prova é uma atividade que desempenha um papel fundamental na construção do conhecimento matemático, razão pela qual adquire relevância nos programas escolares de Matemática. Admitindo que as conceções dos professores sobre a prova afetam a forma como ela é tratada em sala de aula, procuramos averiguar as conceções de professores portugueses de Matemática do 3.º ciclo do Ensino Básico (do 7.º ao 9.º ano) sobre diferentes aspetos da prova matemática e a influência que a experiência profissional tem nessas conceções. Adotando uma abordagem metodológica mista, recolhemos os dados através de um questionário, respondido por 72 professores, e de uma entrevista a duas professoras com experiências profissionais diferentes. Os resultados revelam que os professores, sobretudo os que têm menos tempo de docência, consideram que a prova matemática tem uma natureza distinta da de outras disciplinas, é uma atividade essencial para a construção do conhecimento matemático, tem como função verificar e explicar a veracidade de uma afirmação e permite desenvolver o raciocínio e a comunicação matemática. No que respeita à participação dos alunos na atividade de provar, são os professores com mais tempo de docência que mais o destacam, o que permite aos alunos perceberem a natureza desta atividade. Em termos curriculares, são os professores com menos experiência docente que mais concordam com a presença da prova logo nos primeiros anos, embora salientem que esta atividade só faz sentido em alguns tópicos programáticos.Proof is an activity that plays a key role in the construction of mathematical knowledge, which is why it acquires relevance in mathematics programs. Admitting that teachers' conceptions about proof influence how it is handled in the classroom, we tried to investigate the conceptions of Portuguese mathematics teachers from the 3rd cycle of basic education (grade 7 to 9) on different aspects of mathematical proof and the relationship that the teachers’ experience has in these conceptions. Adopting a mixed methodological approach, we collected data through a questionnaire answered by 72 teachers, and an interview with two teachers with different professional experience. The results show that teachers, especially those with less experience, consider that mathematical proof has a distinct nature from other disciplines, it is an essential activity for the construction of mathematical knowledge and its function is to verify and explain the truth of a statement by developing reasoning and mathematical communication. With respect to student participation in the activity of proving, teachers with more teaching experience are the ones that most highlight it, which allows students to realize the nature of this activity. In curriculum terms, teachers with less experience are the ones that most agree with the presence of proof in early grades, while emphasizing that this activity only makes sense in some program topics.Este trabalho contou com o apoio de Fundos Nacionais através da FCT – Fundação para a Ciência e a Tecnologia no âmbito do projeto PEst-OE/CED/UI1661/2014, do CIEdUM e do projeto UID/Multi/04016/2016info:eu-repo/semantics/publishedVersio

Approaching Proof in the Classroom Through the Logic of Inquiry

The paper analyses a basic gap, highlighted by most of the literature concerning the teaching of proofs, namely, the distance between students' argumentative and proving processes. The analysis is developed from both epistemological and cognitive standpoints: it critiques the Toulmin model of reasoning and introduces a new model, the Logic of Inquiry of Hintikka, more suitable for bridging this gap. An example of didactical activity within Dynamic Geometry Environments is sketched in order to present a concrete illustration of this approach and to show the pedagogical effectiveness of the model

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

Perioperative management and anaesthetic considerations in pelvic exenterations using Delphi methodology: Results from the PelvEx Collaborative

Background: The multidisciplinary perioperative and anaesthetic management of patients undergoing pelvic exenteration is essential for good surgical outcomes. No clear guidelines have been established, and there is wide variation in clinical practice internationally. This consensus statement consolidates clinical experience and best practice collectively, and systematically addresses key domains in the perioperative and anaesthetic management. Methods: The modified Delphi methodology was used to achieve consensus from the PelvEx Collaborative. The process included one round of online questionnaire involving controlled feedback and structured participant response, two rounds of editing, and one round of web-based voting. It was held from December 2019 to February 2020. Consensus was defined as more than 80 per cent agreement, whereas less than 80 per cent agreement indicated low consensus. Results: The final consensus document contained 47 voted statements, across six key domains of perioperative and anaesthetic management in pelvic exenteration, comprising preoperative assessment and preparation, anaesthetic considerations, perioperative management, anticipating possible massive haemorrhage, stress response and postoperative critical care, and pain management. Consensus recommendations were developed, based on consensus agreement achieved on 34 statements. Conclusion: The perioperative and anaesthetic management of patients undergoing pelvic exenteration is best accomplished by a dedicated multidisciplinary team with relevant domain expertise in the setting of a specialized tertiary unit. This consensus statement has addressed key domains within the framework of current perioperative and anaesthetic management among patients undergoing pelvic exenteration, with an international perspective, to guide clinical practice, and has outlined areas for future clinical research

Induction chemotherapy followed by chemoradiotherapy versus chemoradiotherapy alone as neoadjuvant treatment for locally recurrent rectal cancer: Study protocol of a multicentre, open-label, parallel-arms, randomized controlled study (PelvEx II)

Background: A resection with clear margins (R0 resection) is the most important prognostic factor in patients with locally recurrent rectal cancer (LRRC). However, this is achieved in only 60 per cent of patients. The aim of this study is to investigate whether the addition of induction chemotherapy to neoadjuvant chemo(re)irradiation improves the R0 resection rate in LRRC. Methods: Thismulticentre, international, open-label, phase III, parallel-arms study will enrol 364 patients with resectable LRRC after previous partial or total mesorectal resection without synchronous distant metastases or recent chemo- and/or radiotherapy treatment. Patients will be randomized to receive either induction chemotherapy (three 3-week cycles of CAPOX (capecitabine, oxaliplatin), four 2- week cycles of FOLFOX (5-fluorouracil, leucovorin, oxaliplatin) or FOLFORI (5-fluorouracil, leucovorin, irinotecan)) followed by neoadjuvant chemoradiotherapy and surgery (experimental arm) or neoadjuvant chemoradiotherapy and surgery alone (control arm). Tumours will be restaged usingMRI and, in the experimental arm, a further cycle of CAPOX or two cycles of FOLFOX/FOLFIRI will be administered before chemoradiotherapy in case of stable or responsive disease. The radiotherapy dose will be 25 × 2.0 Gy or 28 × 1.8Gy in radiotherapy-naive patients, and 15 × 2.0Gy in previously irradiated patients. The concomitant chemotherapy agent will be capecitabine administered twice daily at a dose of 825mg/m2 on radiotherapy days. The primary endpoint of the study is the R0 resection rate. Secondary endpoints are long-termoncological outcomes, radiological and pathological response, toxicity, postoperative complications, costs, and quality of life. Discussion: This trial protocol describes the PelvEx II study. PelvEx II, designed as a multicentre, open-label, phase III, parallel-arms study, is the first randomized study to compare induction chemotherapy followed by neoadjuvant chemo(re)irradiation and surgery with neoadjuvant chemo(re)irradiation and surgery alone in patients with locally recurrent rectal cancer, with the aim of improving the number of R0 resections

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