Pupils' mistakes, if suitably analysed, may give useful suggestions for improving the teaching/learning process of mathematics. We present here the main issues of an investigation on a population of 732 Italian pupils (9-12 years old), addressed to determine the typology of errors in performing written subtraction. We compared our results with those emerging from a study carried out in Brazil on pupils of the same age, concerning the errors in performing the usual algorithm of written subtraction. In our study, one group of Italian pupil utilized the usual algorithm and another group utilized an algorithm also called 'Austrian subtraction', which was commonly taught in the primary schools of Trieste (Italy) up to the sixties and is still taught in Austria. We attempted to assess whether certain error patterns depend on different ethnic-cultural situations, teaching methods and algorithm used, and to what extent. We briefly present results from the experimental research and subsequent developments.

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In this paper we describe a method for computing derivatives of the polar representations of the boundary of a convex body, from generalized chord functions at two points. INTRODUCTIONThe subject of this paper is related to the large range of questions arising from Hammer's X-ray problems [5].Let us consider the problem of reconstructing a planar convex body K, when, for an integer n, generalized n-chord functions from two distinct points P1 and P2, not lying on its boundary, are assigned. In the context of Hammer's X-ray problems, such points may represent two X-ray sources. Results about uniqueness of solutions, in the cases of 'sources' both interior or both exterior to K, can be found in some papers of Falconer ([1], [2]), Gardner ([4]), and Vol~i~ ([7]). The n-chord functions have been considered by Falconer ([1], [2]) and, in a generalized form, by Gardner ([4]); we use Gardner's definition, adapted to the case of angles between half-lines. We suppose that the straight line joining the 'sources' intersects the boundary of K at two distinct points. Methods for computing these intersection points are produced by Falconer ([1], [2]) for generalized n-chord functions, with n ~> 1. These techniques can be adapted also to the case n < 1. We observe that only for n = 1 does a global study of the given n-chord functions appear to be essential, otherwise these points can be obtained by solving a system of algebraic equations.We continue, in some sense, Falconer's program: we provide methods to evaluate the jth derivatives of the polar representations of the boundary, at the points Z1, Z z where OK intersects the line P1P2. Our main result is the following: if the boundary is sufficiently smooth and the sources are both interior or both exterior to K, then thejth derivatives are uniquely determined from two given n-chord functions, except forj = 1 -n (ifn ~< 0). We also study the case in which one and only one 'source' is interior to K and we obtain similar results, except when the 'sources' and the intersection points ZI, Z2 form a harmonic group.The described process is, at least theoretically, constructive; it provides an alternative proof of a theorem of Falconer about uniqueness ([1, Th. 1]), if the boundary is supposed analytic and n > 0.Furthermore, these results show that some assumptions of a theorem of Gardner ([4, Th. 4]) are redundant.

A study of the interactions between mathematics and cognitive science, carried out within a historical perspective, is important for a better understanding of mathematics education in the present. This is evident when analysing the contribution made by the epistemological theories of Ernst Mach. On the basis of such theories, a didactic method was developed, which was used in the teaching of mathematics in Austria at the beginning of the twentieth century and applied to different subjects ranging from simple operations in arithmetic to calculus. Besides the relevance of this method—also named the ‘‘Jacob method’’ after Josef Jacob who proposed it—to teaching practice, it could also be considered interesting in a wider context with reference to the mind-body problem. In particular, the importance that Jacob gives to ‘‘muscular activity’’ in the process of forming and elaborating mathematical concepts, derived from Mach, resounds in the current debate on embodied cognition, where cognitive processes are understood not as expressions of an abstract and merely computational mind but as based on our physicality as human beings,\ud equipped not just with a brain but also a (whole) body. This model has been applied to mathematics in the ‘‘theory of embodied mathematics’’, the objective of which is to study, with the methods and apparatus of embodied cognitive science, the cognitive mechanisms used in the human creation and conceptualisation of mathematics. The present article shows that the ‘‘Jacob method’’ may be considered a historical example of didactical application of analogous ideas