Zig-zagging in geometrical reasoning in technological collaborative environments: a Mathematical Working Space-framed study concerning cognition and affect

This study highlights the importance of cognition-affect interaction pathways in the construction of mathematical knowledge. Scientific output demands further research on the conceptual structure underlying such interaction aimed at coping with the high complexity of its interpretation. The paper discusses the effectiveness of using a dynamic model such as that outlined in the Mathematical Working Spaces (MWS) framework, in order to describe the interplay between cognition and affect in the transitions from instrumental to discursive geneses in geometrical reasoning. The results based on empirical data from a teaching experiment at a middle school show that the use of dynamic geometry software favours students’ attitudinal and volitional dimensions and helps them to maintain productive affective pathways, affording greater intellectual independence in mathematical work and interaction with the context that impact learning opportunities in geometric proofs. The reflective and heuristic dimensions of teacher mediation in students’ learning is crucial in the transition from instrumental to discursive genesis and working stability in the Instrumental-Discursive plane of MWS.

On continuous families of geometric Seifert conemanifold structures

In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space (Formula presented.) and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot (Formula presented.). As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on (Formula presented.), the result of (Formula presented.) surgery in the left-handed trefoil knot (Formula presented.), J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.