Micro productive failure and the acquisition of algebraic procedural knowledge

Productive failure has shown positive effects on conceptual and transfer measures, but no clear effects on procedural measures. It is therefore an open question whether, and to what extent, productive failure methods may be used to enhance the learning of procedural skills. A typical productive failure study focuses on a single, complex concept; in contrast, procedural knowledge generally consists of a series of less-complex procedural steps. In this study, failure occasions were adapted to specifically fit procedural knowledge by introducing procedural problems prior to the formal instruction of relevant principles. These procedural problems offered brief but multiple occasions for failure, which we call micro productive failure. A total of 85 sixth-graders were introduced to algebraic expression simplification by providing problem-solving prior to instruction (PS-I condition), compared to providing problem-solving after instruction (I-PS condition). Findings reveal a stable effect of offering micro productive failure occasions for procedural learning; however, as anticipated, there were no effects on conceptual or transfer measures.

On Schrödinger–Poisson systems involving concave–convex nonlinearities via a novel constraint approach

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: [Formula: see text] where [Formula: see text] are two parameters, [Formula: see text], [Formula: see text] is a potential well, [Formula: see text] and [Formula: see text]. Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on [Formula: see text] due to [Formula: see text]. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.

Existence results for semilinear problems in the two-dimensional hyperbolic space involving critical growth

We consider semilinear elliptic problems on two-dimensional hyperbolic space. A model problem of our study iswhere H1(𝔹2) denotes the Sobolev space on the disc model of the hyperbolic space and f(x, t) denotes the function of critical growth in dimension 2. We first establish the Palais–Smale (PS) condition for the functional corresponding to the above equation, and using the PS condition we obtain existence of solutions. In addition, using a concentration argument, we also explore existence of infinitely many sign-changing solutions.

Verbal mediation of cognition in children with specific language impairment

Private speech (PS) and inner speech (IS) are thought to be functionally important for children's and adults’ cognition, but they have not been studied systematically in children with specific language impairment (SLI). Participants were 21 children with SLI (7–11 years, expressive or receptive verbal IQ ≤ 75, nonverbal IQ ≥ 84) and 21 age- and nonverbal IQ-matched controls. Participants completed three sets of Tower of London problems: one with no dual task (PS condition), one with articulatory suppression, and one while foot tapping (control condition). Participants also completed a digit span task. There was no group difference in the susceptibility of Tower of London performance to articulatory suppression, but the PS of the SLI group was less internalized than that of the controls on both tasks. The findings suggest that children with SLI experience a significant delay in the development of PS/IS, but that their PS/IS is effective for Tower of London performance in middle childhood. Findings are discussed with reference to the interpretation of the nonlinguistic deficits associated with SLI, and in terms of clinical implications.

Effect of lumbar total disc arthroplasty on the segmental motion and intradiscal pressure at the adjacent level: an in vitro biomechanical study

Object The artificial disc has been proposed as an alternative to spinal fusion for degenerative disc disease. The primary aim of this biomechanical study was to compare motion and intradiscal pressure (IDP) in a ball-and-socket artificial disc–implanted cadaveric lumbar spine, at the operative and adjacent levels, using a displacement-controlled setup. A secondary comparison involved a “salvage” construct, consisting of pedicle screws (PSs) added in supplementation to the artificial disc construct. Methods Ten human cadaveric lumbosacral spines (L2–S1) were potted at L-2 and S-1. All measurements were initially made in the intact spine, followed by implantation of the artificial disc, and finally by the salvage PS condition. For the artificial disc condition, a Maverick ball-and-socket artificial disc was implanted at L4–5. For the PS condition, CD Horizon PSs were placed at L4–5, and the artificial disc was left in place. A displacement-controlled, custom-designed testing apparatus was used to impart motion in the sagittal and coronal planes. Motion at both the implanted level (L4–5) and immediately adjacent levels (L3–4 and L5–S1) was measured. Intradiscal pressure at the rostral adjacent level (L3–4) was also measured. The Tukey test was used for statistical analysis (p < 0.05). Results In flexion, no significant difference was noted between the artificial disc and the intact spine with regard to motion at the operative level, motion at adjacent levels, or IDP. In lateral bending, while the artificial disc significantly decreased operative-level motion (p < 0.05), no significant difference was noted in adjacent-level motion or IDP. With regard to extension, the artificial disc significantly increased operative level motion and decreased the rostral adjacent level (L3–4) motion and IDP (p < 0.05). Caudal adjacent-level (L5–S1) motion was not significantly different. In flexion and lateral bending, the addition of PSs significantly decreased motion at the implanted level when compared with the intact spine and the artificial disc (p < 0.05). This decrease in motion at the index level was associated with a compensatory increase in motion at both adjacent levels in flexion only (p < 0.05), but not in lateral bending (p > 0.05). The IDP was significantly increased in lateral bending but not in flexion. With regard to extension, the significant decrease in IDP that was noted with the artificial disc persisted despite the addition of PSs (p < 0.05). Conclusions The artificial disc either maintains or reduces adjacent-level motion and pressure, compared with the intact spine. The addition of PSs to the artificial disc construct leads to significantly increased motion at adjacent levels in flexion and significantly increased IDP in lateral bending. At the operative level, the artificial disc is associated with hypermobility in extension, which is restored to the intact state after the addition of supplementary PSs.

The Use of Cerami Sequences in Critical Point Theory

The concept of linking was developed to produce Palais-Smale (PS) sequencesG(uk)→a,G'(uk)→0forC1functionalsGthat separate linking sets. These sequences produce critical points if they have convergent subsequences (i.e., ifGsatisfies the PS condition). In the past, we have shown that PS sequences can be obtained even when linking does not exist. We now show that such situations produce more useful sequences. They not only produce PS sequences, but also Cerami sequences satisfyingG(uk)→a,(1+||uk||)G'(uk)→ 0as well. A Cerami sequence can produce a critical point even when a PS sequence does not. In this situation, it is no longer necessary to show thatGsatisfies the PS condition, but only that it satisfies the easier Cerami condition (i.e., that Cerami sequences have convergent subsequences). We provide examples and applications. We also give generalizations to situations when the separating criterion is violated.