The thinking styles of university mathematics students

In this paper, we focus on the relationship between studying university mathematics and the ‘thinking styles’ of both undergraduate and postgraduate mathematics students. A crosssectional quantitative study (N = 238) was conducted in a large Greek university, identifying the thinking styles of second, third and fourth year undergraduates, as well as those of students following a postgraduate degree in mathematics. The analysis revealed that the more experienced undergraduates and the postgraduates showed a stronger preference for originality and freedom in thinking, low degrees of structure, high levels of freedom and more complex information processing, combined with non-prioritised thinking. However, the postgraduates combine these preferences with a stronger preference for implementing rules and instructions. A discussion on these findings and on the factors that may account for them is presented

One-step 'change' and 'compare' word problems: Focusing on eye-movements

Introduction: In this study, we focus on the relationship between the students' mathematical thinking and their non-mechanically identified eye-movements with the purpose to gain deeper understanding about the students' reasoning processes and to investigate the feasibility of incorporating eye-movement information in everyday pedagogy. Method: This is a quantitative study. Thirty-eight (N=38) Grade 1 (6 years old) students were verbally presented with six word arithmetic problem: three problems of 'change' and three problems of 'comparison'. The problems were chosen to be within the students' mathematical abilities. Results: The findings of this study appeared to validate the non-mechanical data collection technique.Furthermore, differentiations were found in the students' rightwards eye-movements (suggesting the activity of the left hemisphere) and the eye-movements spread (suggesting the activity of both hemispheres) in line with the literature. The morechallenging problems were found to be linked with a wider spread of eye-movements and to be more time consuming. Moreover, though boys appeared to be faster than the girls in the easier tasks, no statistical differences were found in the more challenging tasks. Discussion: The adopted 'softer' technique was sensitive enough to be in accordance with the existing literature and at the same time helped in gaining deeper understanding in the stu-dents' reasoning about 'change and "compare' problems. For example, the reported gender differences -in line with broader research evidence from education and neurophysiology-are hypothesised to indicate of qualitatively different thinking processes involved, which may be related to different gender-related thinking dispositions. Consequently, it is posited that the adopted technique was is in line with the relevant research evidence, offering at the same time insight in the complex processes involved, thus allowing for further research to be conducted in order to investigate the pedagogical benefits of the incorporation of such 'softer 'eye-movement identification techniques in everyday pedagogical practices. © 2015 Education & Psychology I+D+i and Ilustre Colegio Oficial de Psicólogos de Andalucía Oriantal (Spain)

The objectification of the right-angled triangle in the teaching of the Pythagorean Theorem: An empirical investigation

In this paper, we present the design and the results of a teaching experiment carried out to investigate the hypothesis that it is feasible to facilitate the students' possibility for experiencing the reactivation of the objectification of the right-angled triangle. For this purpose, a teaching design of the Pythagorean Theorem was developed and taught to an experimental class of 14-year old students. The results of our teaching were compared with a control class with the employment of a questionnaire and semi-structured interviews. The quantitative and qualitative analyses supported our hypothesis that the students of the experimental class would develop qualitatively different understandings of the theorem than the control class, thus suggesting their possibility for experiencing the reactivation of the objectification of the right-angled triangle. © 2013 Springer Science+Business Media Dordrecht

Endoscopic Treatment of Posterior Ankle Impingement Secondary to Os Trigonum in Recreational Athletes

Background: The os trigonum (OT)—the most common accessory bone of the foot—although usually asymptomatic, may cause posterior ankle impingement syndrome (PAIS), which may be a severely debilitating problem for recreational or competitive athletes. The aim of the present study was to evaluate effectiveness of posterior ankle arthroscopy and to assess the outcome in the treatment of PAIS secondary to OT impingement or OT fractures within a group of young athletes and their return to previous sports level. Methods: From 2011 to 2018, a retrospective review of 81 recreational athletes of mean age 27.8 years was performed. All patients were diagnosed with PAIS due to OT pathology and were operated on endoscopically with resection of the OT. Pre- and postoperative clinical evaluation were performed at 3 months, 1 year, and 2 years based on visual analog scale (VAS), ankle range of motion (ROM), American Orthopaedic Foot & Ankle Society (AOFAS) hindfoot score, and the Foot & Ankle Disability Index (FADI) scores, in a follow-up of at least 2 years. Results: VAS score was significantly improved from an average of 7.5 (5-9) preoperatively to 1.9 (1-3) at 3 months postoperatively and to 0.6 (0-2) and 0.3 (0 -1) at 1 and 2 years postoperatively. Ankle ROM was significantly improved from an average of 24.8 (10-35) preoperatively to 58.0 (50-65) at 3 months postoperatively and to 64.0 (50-65) at 1 year and 64.7 (60-65) at 2 years postoperatively. AOFAS and FADI scores were significantly improved from 39.4 (18-55) and 49.7 (42.3-62.5) preoperatively to 85.2 (74-89) and 87.3 (81.7-88.5) postoperatively at 3 months to 97.7 (85-100) and 97.9 (93.3-100) postoperatively at 1 year, respectively (P <.001). Only 5 patients dropped to a lower activity level. There were 5 complications (4 transient). Conclusion: Endoscopic treatment of PAIS due to OT pathology demonstrated excellent results. Posterior ankle arthroscopy was an effective treatment and allowed for a prompt return to a high activity level of their athletic performance. Level of Evidence: Level IV, therapeutic study / retrospective case series. © The Author(s) 2020

Introduction to Thematic Working Group 1: Argumentation and Proof

International audienc

Chapter 11. Classical and Digital Technologies for the Pythagorean Theorem

This paper aims to discuss the use of material tools, called mathematical machines, and digital tools in approaching the Pythagorean theorem. These mathematical machines are related to different proofs of the theorem. Teaching experiments with 13-year old students were carried out within the laboratory approach developed from the theoretical frameworks of the Theory of Semiotic Mediation and Instrumental approach in mathematics education. Their analysis shows that behind the kinesthetic experience with the machines, there are important cognitive processes such as the identification of invariants, relationships between the components and usage schemes. It also shows the only manipulation of the first machine does not imply the emergence of the mathematical meanings embedded in the materials tools and the crucial role of the teacher with his different instrumental orchestrations in that process