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A study of children learning multicolumn addition with microcomputer software support.
Three computer-aided tutoring procedures were devised to teach multicolumn addition according to the standard school algorithm, one procedure to each of three groups of 2nd-grade children. The key differences between groups were the demands placed on short term memory and the amount of conceptual understanding the procedures attempted to teach. Each child solved a sequence of two-digit problems on a computer screen by touching each digit with a light pen in the correct sequence. The control group did not receive on-screen number-fact assistance. One treatment ( assisted ) group did receive on-screen number-fact assistance, testing the hypothesis that the algorithm is learned more effectively when learned first as a sequence of procedural steps alone, without subjects\u27 need to recall number-facts. A second treatment ( simulation ) group received the same on-screen assistance along with an additional display of simulated blocks which, like concrete manipulative materials, represented symbol manipulations. The simulation group tested a second hypothesis that a concurrent display of the meaning of procedural steps contributes to even more effective algorithmic learning. T-tests (one-tailed, 5% level) applied pair-wise to pretest/posttest difference scores indicated support for the first hypothesis but not for the second, an indication that 2nd-grade children learn the addition algorithm more effectively if demand on short term memory is temporarily lifted. A descriptive framework called superposition of frames is proposed to account for anomalies in findings and for the rich diversity of errors generally manifested by children in multidigit addition. Drawing on current concepts in cognitive psychology and mathematics education, this description suggests that children\u27s mathematical knowledge is fragmented into isolated, unstable, and sometimes entrenched frames of knowledge. When a child finds appropriate correspondences between frames and initiates a superposition of frames, the child\u27s procedural and conceptual knowledge, previously in disarray, may then become integrated. Implications for elementary mathematics instruction are discussed
Calabi-Yau 3-folds from 2-folds
We consider type IIA string theory on a Calabi-Yau 2-fold with D6-branes
wrapping 2-cycles in the 2-fold. We find a complete set of conditions on the
supergravity solution for any given wrapped brane configuration in terms of
SU(2) structures. We reduce the problem of finding a supergravity solution for
the wrapped branes to finding a harmonic function on RCY. We then
lift this solution to 11-dimensions as a product of R and a
Calabi-Yau 3-fold. We show how the metric on the 3-fold is determined in terms
of the wrapped brane solution. We write down the distinguished (3,0) form and
the K{\"a}hler form of the 3-fold in terms of structures defined on the base
2-d complex manifold. We discuss the topology of the 3-fold in terms of the
D6-branes and the underlying 2-fold. We show that in addition to the
non-trivial cycles inherited from the underlying 2-fold there are new
2-cycles. We construct closed (1,1) forms corresponding to these new cycles. We
also display some explicit examples. One of our examples is that of D6-branes
wrapping the 2-cycle in an A ALE space, the resulting 3-fold has
, where is the number of D6-branes.Comment: 30 page
NMR relaxation rate in non-centrosymmetric superconductors
The spin-lattice relaxation rate of nuclear magnetic resonance in a clean
superconductor without inversion center is calculated for arbitrary pairing
symmetry and band structure, in the presence of strong spin-orbit coupling.Comment: 4 page
Signature of superconducting states in cubic crystal without inversion symmetry
The effects of absence of inversion symmetry on superconducting states are
investigated theoretically. In particular we focus on the noncentrosymmetric
compounds which have the cubic symmetry like LiPtB. An appropriate
and isotropic spin-orbital interaction is added in the Hamiltonian and it acts
like a magnetic monopole in the momentum space. The consequent pairing
wavefunction has an additional triplet component in the pseudospin space, and a
Zeeman magnetic field can induce a collinear supercurrent
with a coefficient . The effects of anisotropy embedded in the cubic
symmetry and the nodal superconducting gap function on are also
considered. From the macroscopic perspectives, the pair of mutually induced
and magnetization can affect the distribution of magnetic
field in such noncentrosymmetric superconductors, which is studied through
solving the Maxwell equation in the Meissner geometry as well as the case of a
single vortex line. In both cases, magnetic fields perpendicular to the
external ones emerge as a signature of the broken symmetry.Comment: 16 pages in pre-print forma
Let's Twist Again: General Metrics of G(2) Holonomy from Gauged Supergravity
We construct all complete metrics of cohomogeneity one G(2) holonomy with S^3
x S^3 principal orbits from gauged supergravity. Our approach rests on a
generalization of the twisting procedure used in this framework. It corresponds
to a non-trivial embedding of the special Lagrangian three-cycle wrapped by the
D6-branes in the lower dimensional supergravity. There are constraints that
neatly reduce the general ansatz to a six functions one. Within this approach,
the Hitchin system and the flop transformation are nicely realized in eight
dimensional gauged supergravity.Comment: 31 pages, latex; v2: minor changes, references adde
Phases of dual superconductivity and confinement in softly broken N=2 supersymmetric Yang-Mills theories
We study the electric flux tubes that undertake color confinement in N=2
supersymmetric Yang-Mills theories softly broken down to N=1 by perturbing with
the first two Casimir operators. The relevant Abelian Higgs model is not the
standard one due to the presence of an off-diagonal coupling among different
magnetic U(1) factors. We perform a preliminary study of this model at a
qualitative level. BPS vortices are explicitely obtained for particular values
of the soft breaking parameters. Generically however, even in the ultrastrong
scaling limit, vortices are not critical but live in a "hybrid" type II phase.
Also, ratios among string tensions are seen to follow no simple pattern. We
examine the situation at the half Higgsed vacua and find evidence for solutions
with the behaviour of superconducting strings. In some cases they are solutions
to BPS equations.Comment: 15 pages, 1 figure, revtex; v2: typos corrected, final versio
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