Why and how mathematicians read proofs: further evidence from a survey study

In a recent paper (Weber & Mejia-Ramos, Educational Studies in Mathematics, 76, 329–344, 2011), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians (N?=?118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians.     

Enhancing Students’ Learning Process Through Self-Generated Tests

The use of new technologies in higher education has surprisingly emphasized students’ tendency to adopt a passive behavior in class. Participation and interaction of students are essential to improve academic results. This paper describes an educational experiment aimed at the promotion of students’ autonomous learning by requiring them to generate test type questions related to the contents of the course. The main idea is to make the student feel part of the evaluation process by including students’ questions in the evaluation exams. A set of applications running on our university online learning environment has been developed in order to provide both students and teachers with the necessary tools for a good interaction between them. Questions uploaded by students are visible to every enrolled student as well as to each involved teacher. In this way, we enhance critical analysis skills, by solving and finding possible mistakes in the questions sent by their fellows. The experiment was applied over 769 students from 12 different courses. Results show that the students who have actively participated in the experiment have obtained better academic performance.     

Experimental evidence for improved neuroimaging interpretation using three-dimensional graphic models

Three-dimensional (3D) or volumetric visualization is a useful resource for learning about the anatomy of the human brain. However, the effectiveness of 3D spatial visualization has not yet been assessed systematically. This report analyzes whether 3D volumetric visualization helps learners to identify and locate subcortical structures more precisely than classical cross-sectional images based on a two dimensional (2D) approach. Eighty participants were assigned to each experimental condition: 2D cross-sectional visualization vs. 3D volumetric visualization. Both groups were matched for age, gender, visual-spatial ability, and previous knowledge of neuroanatomy. Accuracy in identifying brain structures, execution time, and level of confidence in the response were taken as outcome measures. Moreover, interactive effects between the experimental conditions (2D vs. 3D) and factors such as level of competence (novice vs. expert), image modality (morphological and functional), and difficulty of the structures were analyzed. The percentage of correct answers (hit rate) and level of confidence in responses were significantly higher in the 3D visualization condition than in the 2D. In addition, the response time was significantly lower for the 3D visualization condition in comparison with the 2D. The interaction between the experimental condition (2D vs. 3D) and difficulty was significant, and the 3D condition facilitated the location of difficult images more than the 2D condition. 3D volumetric visualization helps to identify brain structures such as the hippocampus and amygdala, more accurately and rapidly than conventional 2D visualization. This paper discusses the implications of these results with regards to the learning process involved in neuroimaging interpretation.     

Defending Constructivism in Science Education

After an impressive development throughout the last two decades, supported by a greatamount of research and innovation, science education seemed to be becoming a newscientific domain. This transformation of Science Education into a specific field of researchand knowledge is usually associated with the establishment of what has been called anemergent consensus about constructivist positions. However, some voices have begunto question these constructivist positions and therefore the idea of an advancementtowards a coherent body of knowledge in the field of science education. The goalof this work is to analyse some of the current criticisms of the so-called constructivistorientations and to study their implications for the development of science education asa coherent body of knowledge.