Bridging knowing and proving in mathematics An essay from a didactical perspective

Text of a talk at the conference "Explanation and Proof in Mathematics: Philosophical and Educational Perspective" held in Essen in November 2006International audienceThe learning of mathematics starts early but remains far from any theoretical considerations: pupils' mathematical knowledge is first rooted in pragmatic evidence or conforms to procedures taught. However, learners develop a knowledge which they can apply in significant problem situations, and which is amenable to falsification and argumentation. They can validate what they claim to be true but using means generally not conforming to mathematical standards. Here, I analyze how this situation underlies the epistemological and didactical complexities of teaching mathematical proof. I show that the evolution of the learners' understanding of what counts as proof in mathematics implies an evolution of their knowing of mathematical concepts. The key didactical point is not to persuade learners to accept a new formalism but to have them understand how mathematical proof and statements are tightly related within a common framework; that is, a mathematical theory. I address this aim by modeling the learners' way of knowing in terms of a dynamic, homeostatic system. I discuss the roles of different semiotic systems, of the types of actions the learners perform and of the controls they implement in constructing or validating knowledge. Particularly with modern technological aids, this model provides a basis designing didactical situations to help learners bridge the gap between pragmatics and theory

Justifications-on-demand as a device to promote shifts of attention associated with relational thinking in elementary arithmetic

Student responses to arithmetical questions that can be solved by using arithmetical structure can serve to reveal the extent and nature of relational, as opposed to computational thinking. Here, student responses to probes which require them to justify-on-demand are analysed using a conceptual framework which highlights distinctions between different forms of attention. We analyse a number of actions observed in students in terms of forms of attention and shifts between them: in the short-term (in the moment), medium-term (over several tasks), and long-term (over a year). The main factors conditioning students´ attention and its movement are identified and some didactical consequences are proposed

Biblioteca Patrimonial Digital de la Universitat de Barcelona

La reproducció digital del document forma part de la col·lecció Química de la Biblioteca Patrimonial Digital de la Universitat de Barcelon

Personal radio use and cancer risks among 48,518 British police officers and staff from the Airwave Health Monitoring Study

Background: Radiofrequency electromagnetic fields (RF-EMF) from mobile phones have been classified as potentially carcinogenic. No study has investigated use of Terrestrial Trunked Radio (TETRA), a source of RF-EMF with wide occupational use, and cancer risks. Methods: We investigated association of monthly personal radio use and risk of cancer using Cox proportional hazards regression among 48,518 police officers and staff of the Airwave Health Monitoring Study in Great Britain. Results: During median follow-up of 5.9 years, 716 incident cancer cases were identified. Among users, the median of the average monthly duration of use in the year prior to enrolment was 30.5 min (inter-quartile range 8.1, 68.1). Overall, there was no association between personal radio use and risk of all cancers (hazard ratio [HR] = 0.98, 95% confidence interval [CI]: 0.93, 1.03). For head and neck cancers HR = 0.72 (95% CI: 0.30, 1.70) among personal radio users vs non-users, and among users it was 1.06 (95% CI: 0.91, 1.23) per doubling of minutes of personal radio use. Conclusions: With the limited follow-up to date, we found no evidence of association of personal radio use with cancer risk. Continued follow-up of the cohort is warranted