Supervision of teachers based on adjusted arithmetic learning in special education

This article reports on 20 children's learning in arithmetic after teaching was adjusted to their conceptual development. The report covers periods from three months up to three terms in an ongoing intervention study of teachers and children in schools for the intellectually disabled and of remedial teaching in regular schools. The researcher classified each child's current counting scheme before and after each term. Recurrent supervision, aiming to facilitate the teachers’ modelling of their children's various conceptual levels and needs of learning, was conducted by the researcher. The teaching content in harmony with each child's ability was discussed with the teachers. This approach gives the teachers the opportunity to experience the children's own operational ways of solving problems. At the supervision meetings, the teachers theorized their practice together with the researcher, ending up with consistent models of the arithmetic of the child. So far, the children's and the teachers’ learning patterns are promising.     

Arithmetical thinking in children attending special schools for the intellectually disabled

This article focuses on spontaneous and progressive knowledge building in “the arithmetic of the child.” The aim is to investigate variations in the behavior patterns of eight pupils attending a school for the intellectually disabled. The study is based on the epistemology of radical constructivism and the methodology of multiple clinical interviews. Theoretical models elucidate behavior patterns and the corresponding mental structures underlying them. The individual interviews of the pupils were video recorded. The results show that the activated behavior patterns, which are responses to well-adapted contexts presented by the researcher, are compatible with findings in Swedish compulsory schools. Six of the pupils’ mental structures in the study are numerical. A substantial implication for special education is the harmonization of the content in teaching with the children's own ways of operating, which implies a triadic teaching process.     

Asymptotic stability in the pth moment for stochastic differential equations with Lévy noise

This paper is devoted to study a class of stochastic differential equations with Lévy noise. In comparison to the standard Gaussian noise, Lévy noise is more versatile and interesting with a wider range of applications. However, Lévy noise makes the analysis more difficult owing to the discontinuity of its sample paths. In this paper, we attempt to overcome this difficulty. We propose several sufficient conditions under which we investigate the long-time behavior of the solution including the asymptotic stability in the pth moment and almost sure stability. Also, we discuss two types of continuity of the solution: continuous in probability and continuous in the pth moment. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.     

Operator identities involving the bivariate Rogers-Szegö polynomials and their applications to the multiple q-series identities

In this paper, we first give several operator identities involving the bivariate Rogers-Szegö polynomials. By applying the technique of parameter augmentation to the multiple q-binomial theorems given by Milne [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187], we obtain several new multiple q-series identities involving the bivariate Rogers-Szegö polynomials. These include multiple extensions of Mehler's formula and Rogers's formula. Our U(n+1) generalizations are quite natural as they are also a direct and immediate consequence of their (often classical) known one-variable cases and Milne's fundamental theorem for An or U(n+1) basic hypergeometric series in Theorem 1.49 of [S.C. Milne, An elementary proof of the Macdonald identities for , Adv. Math. 57 (1985) 34-70], as rewritten in Lemma 7.3 on p. 163 of [S.C. Milne, Balanced summation theorems for U(n) basic hypergeometric series, Adv. Math. 131 (1997) 93-187] or Corollary 4.4 on pp. 768-769 of [S.C. Milne, M. Schlosser, A new An extension of Ramanujan's summation with applications to multilateral An series, Rocky Mountain J. Math. 32 (2002) 759-792].