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PubMed1995

Reflections of evolution and culture in children's cognition. Implications for mathematical development and instruction.

An evolution-based framework for understanding biological and cultural influences on children's cognitive and academic development is presented. The utility of this framework is illustrated within the mathematical domain and serves as a foundation for examining current approaches to educational reform in the United States. Within this framework, there are two general classes of cognitive ability, biologically primary and biologically secondary. Biologically primary cognitive abilities appear to have evolved largely by means of natural or sexual selection. Biologically secondary cognitive abilities reflect the co-optation of primary abilities for purposes other than the original evolution-based function and appear to develop only in specific cultural contexts. A distinction between these classes of ability has important implications for understanding children's cognitive development and achievement.

Aptitude↗
PubMed1997

Mediators of gender differences in mathematics college entrance test scores: a comparison of spatial skills with internalized beliefs and anxieties.

This study was designed to investigate whether spatial skill, math anxiety, and math self-confidence functioned as mediators of a significant gender difference in the Mathematics Scholastic Aptitude Test (SAT-M) among the top third of a college-bound sample. Using path analytic techniques, the decomposition of the significant gender-SAT-M correlation into direct and indirect effects indicated that there were no direct effects of gender on SAT-M. Mental rotation and math self-confidence showed indirect effects, mediating the gender-SAT-M relationship; math anxiety did not. Of these indirect effects, 36% was mediated by math self-confidence; 64% by mental rotation. For both these variables, most of the mediational effects of the gender-SAT-M relationship did not occur by way of the causal pathway leading through geometry grades. Thus, the mediational effects cannot simply be attributed to the presence of geometry items on the SAT-M or to math self-confidence acquired during prior geometry coursework.

Achievement↗
PubMed1984

Automatic aftereffects in two-choice reaction time: a mathematical representation of some concepts.

A mathematical model is developed to describe sequential effects in two-choice reaction time experiments with a short response-stimulus interval. Evidence is briefly discussed that in conditions with short response-stimulus intervals, automatic aftereffects dominate sequential effects, and the influence of subjective expectancy can be neglected. In these conditions the model premises three components of automatic aftereffects--facilitation, inhibition, and noise, with a common decay factor. Influence of response-stimulus interval and practice on sequential effects are examined and related to parameter changes in the proposed single-decay model. The decrease of automatic aftereffects with increasing response-stimulus interval is primarily ascribed to an increasing decay factor. The parameter representation of the model also clarifies the issue of the disappearance of automatic aftereffects with practice. It shows a gradual fading of inhibition in the initial stages of practice, together with a slower decrease of the facilitation effect. The single-decay model provides a satisfactory explanation for the processes involved in compatible two-choice reaction time with short response-stimulus interval.

Choice Behavior↗
PubMed2000

Score comparability of a state mathematics assessment across students with and without reading accommodations.

This study investigated the factorial invariance of a fourth-grade state mathematics assessment across groups of general education students and students with learning disabilities with and without reading accommodations. Confirmatory factor analysis was used to assess the fit of a 2-factor model to each of the 3 groups. In addition to the overall fit of this model, several levels of constraint were investigated. Invariance across the 3 groups was supported for factor loadings and intercepts. However, invariance of the factor covariances across the general education group and the groups of students with learning disabilities was not supported. Because of the implications for aggregating reported scores, further research is needed into the relationship between the factors in the different groups.

Child↗
PubMed1993

Semiotic aspects of cognitive development: illustrations from early mathematical cognition.

The premise of this article is that cognitive development involves both conceptual and semiotic achievements. From this perspective, the authors emphasize the distinctness of the semiotic issues and develop a differentiated appreciation of semiotic aspects of cognition, particularly in the field of elementary mathematical cognition. The authors provide semiotic analyses of the differences between counting, adding, and multiplying and of the conventional place-value sign system. The authors introduce the concept of the field of reference of a sign, the differentiation of the field into foreground and background, and the dynamics within the field of reference. Finally, the authors relate these ideas to the dynamics between two dimensions of semiotic relations: the sign-referent dimension and the sign-sign dimension.

Child↗
PubMed1994

A turning point in mathematical learning theory.

This target article by Estes (1950) sparked the mathematical learning theory movement, which took seriously the goal of predicting quantitative details of behavioral data from standard learning experiments. The central constructs of Estes's theory were stimulus variability, stimulus sampling, and stimulus-response association by contiguity, all cast within a framework enabling predictions of response probabilities and latencies. The math models enterprise flourished during the period 1950-1975 and provided successful quantitative accounts of data from Pavlovian and instrumental conditioning, probability learning, verbal learning, concept identification, and other standard learning paradigms. The techniques have been assimilated into the armamentarium of theoretical psychology. Stimulus sampling theory has faded away as it has been transformed into modern descendants such as connectionism and information-processing models of cognition.

History, 20th Century↗
PubMed2004

More on the fragility of performance: choking under pressure in mathematical problem solving.

In 3 experiments, the authors examined mathematical problem solving performance under pressure. In Experiment 1, pressure harmed performance on only unpracticed problems with heavy working memory demands. In Experiment 2, such high-demand problems were practiced until their answers were directly retrieved from memory. This eliminated choking under pressure. Experiment 3 dissociated practice on particular problems from practice on the solution algorithm by imposing a high-pressure test on problems practiced 1, 2, or 50 times each. Infrequently practiced high-demand problems were still performed poorly under pressure, whereas problems practiced 50 times each were not. These findings support distraction theories of choking in math, which contrasts with considerable evidence for explicit monitoring theories of choking in sensorimotor skills. This contrast suggests a skill taxonomy based on real-time control structures.

Cognition↗
PubMed2004

Interhemispheric interaction during global-local processing in mathematically gifted adolescents, average-ability youth, and college students.

Interhemispheric interaction in mathematically gifted (MG) adolescents, average-ability (AA) youth, and college students (CS) was examined by presenting hierarchical letter pairs in 3 viewing conditions: (a) unilaterally to the right hemisphere (RH), (b) unilaterally to the left hemisphere (LH), or (c) bilaterally, with 1 member of the pair presented to each hemisphere simultaneously. Participants made global-local, match-no-match judgments. For the AA and CS, the LH was faster for local matches and the RH for global matches. The MG showed no hemispheric differences. Also, AA and CS were slower on cooperative compared with unilateral trials, whereas the MG showed the opposite pattern. These results suggest that enhanced interhemispheric interaction is a unique functional characteristic of the MG brain.

Adolescent↗
PubMed1976

Simple mathematical models with very complicated dynamics.

First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hiearchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.

Diffusion↗
PubMed1990

Using mathematical models to estimate drug resistance and treatment efficacy via CT scan measurements of tumour volume.

A previously described mathematical model designed to evaluate resistance and tumour-kill for individual patients, and to predict changing tumour sizes, has been applied to patients with small cell lung cancer. The model requires tumour volume measurements, and these were obtained via computed tomography scans of the chest. The model fitted the data well, and was able to predict later tumour volumes using earlier ones, as well as suggesting times at which to change or abandon treatment for individual patients. The model gave estimates for resistance and tumour-kill which may provide additional useful outcome measures for clinical trials, and help in the design of future studies.

Drug Resistance↗
PubMed1987

Enhancement of neutrophil-mediated phagocytosis by human granulocyte-macrophage colony-stimulating factor demonstrated using a novel mathematical model.

A mathematical model is presented which may be applied to describe and analyse data from microscopic phagocytosis assays. The method has been used to investigate the phagocytosis of opsonized yeast by peripheral blood neutrophils treated with purified recombinant human granulocyte-macrophage colony-stimulating factor (rH GM-CSF) in vitro. Under limiting conditions of serum opsonization, rH GM-CSF decreased the proportion of non-phagocytic cells and increased the mean number of ingested yeast per cell. Stimulation of phagocytosis was dose-dependent and occurred with concentrations of rH GM-CSF in the range 10-320 units/ml. The effect was dependent on a heat-labile component in serum and was not attributable to endotoxin contamination of the preparation.

Colony-Stimulating Factors↗
PubMed1983

Mathematical and statistical aids to evaluate data from renal patients.

The monitoring of renal patients and the making of many decisions during their management involves consideration of sequences of numerical data. Renal function results after renal transplantation were used as an example of how graphical presentations, simple mathematical transforms, statistical evaluation and adjustments to the data, to take into account other biological and technical sources of error, can all contribute to better understanding. Experience with a statistical technique, the 4-state Kalman filter not often used in the biological sciences, was summarized and its use suggested as a method to quantitate some traditionally subjective decisions about individual patients, for example, the onset of allograft rejection. The method has identified in retrospect and in prospect events after transplantation earlier than did experienced clinicians. Other statistical techniques to set the sensitivity and specificity of monitoring methods, to detect change points and to quantitate rhythmic sequences of clinical data were discussed, with examples, and with increasing access to computers, these can be used more easily by nephrologists, transplant surgeons, and others.

Computers↗
PubMed2004

Behavioral equivalence, but not neural equivalence--neural evidence of alternative strategies in mathematical thinking.

In a functional magnetic resonance imaging study, we investigated how people solve mathematically equivalent problems presented in two alternative formats: verbal, story format or symbolic, equation format. Although representation format had no effect on behavior, anterior prefrontal activation was greater in the story condition and posterior parietal activation was greater in the equation condition. These results show that there exist alternative neural pathways that implement different and yet equally efficient problem-solving strategies.

Behavior↗
PubMed1997

Initial investigation of mathematical morphology for the digital extraction of the skeletal characteristics of trabecular bone.

OBJECTIVE: To develop a filter utilizing mathematical theory to extract the skeletal patterns of trabecular bone. METHODS: Studies of morphology in the extraction of patterns of calcification in mammograms provided the theoretical framework. Using these studies as a basis, a morphological filter was applied to extract skeletal patterns from digital images of trabecular bone. Sequential images (subset) were combined in a structured fashion to create an aggregate (sumset) which compared with the original images, skeleton and line skeleton images. RESULTS: Binary images of the skeletal patterns in continuous, round and mesh-like forms were obtained from the original images by processing with the skeleton operation using a disc-shaped single structuring element. The line skeleton operation using line structuring elements with constant directions allowed the extraction of linear and discontinuous patterns. Both the skeleton and line skeleton operations extracted binary subset images which depicted skeletal patterns correlating with the operation sequence. CONCLUSIONS: Modification of the morphological filter enhanced the extraction of skeletal characteristics of trabecular bone. A morphological filter may be a useful adjunct in computer-aided structural analysis of bone.

Bone Density↗
PubMed2005

Two mathematical models for the correction of carbohydrate and protein interference in the determination of uronic acids by the m-hydroxydiphenyl method.

The most common method in the routine determination of uronic acids, the m-hydroxydiphenyl reaction, recently adapted to rapid microplate analysis, has as a main inconvenience, in any one of their modalities, interferences due to the frequent presence of proteins and neutral carbohydrates in the samples. Corresponding corrections in the literature are unsatisfactory when applied to complex matrices, and further adaptation to the microplate analysis is not free from additional problems. With particular reference to hyaluronic acid, the interactions between the principal reactants and the interfering materials are studied kinetically under realistic conditions, and simple mathematical models are proposed which satisfactorily describe the experimental results and allow adequate corrections to be made.

Animals↗
PubMed1985

Mathematical modelling of the formation of coated vesicles. A theoretical geometrical approach.

The mechanism by which flat hexagonal lattices of clathrin trimers transform into pentagonal/hexagonal spheres remains a mystery. In light of the geometrical nature of this process we have pursued a mathematical approach to the question. Through the geometrical analysis of flat hexagonal lattices we have discovered three possible forms of transformation to introduce curvature into the centre of the lattice: hub-centre transformation; hub-edge transformation; fringe transformation. Hub-edge and fringe transformations are used first to close the lattice while introducing localized curvature at the edges of the lattice. Hub-centre transformation is used after closure to relax the severely localized curvature generated during closure. This scheme not only maximizes the size of the coated vesicle generated, but also minimizes the number of transformations, thus minimizing the energy expended.

Clathrin↗
PubMed2003

Mathematical analysis of atelectasis formation in middle ears with sealed ventilation tubes.

AIM: A mathematical model was developed to identify time periods of atelectasis induction in middle ear (ME) ventilated via ventilating tubes (VT). Atelectatic ears are characterized by a total gas pressure lower than 760 mmHg. METHODS: Ventilating tubes were deliberately sealed and ME gas content changed in the presence of a preset blood gas pressure. Once sealed, CO2 rapidly diffuses out of the blood via lining tissues into the ME cleft. This results in initially a total ME pressure rise followed by a decrease in subatmospheric pressures. Time periods for atelectasis reformation was determined once ME pressure crossed the 760 mmHg value and continued to decline as the atelectasis reached higher grades. RESULTS: Time periods calculated by the model varied from 18 to 125 min in ME cavities ranging in volume from 0.5 to 3.5 mL, respectively. These results were calculated for conditions of venous blood in the lining mucosa blood and are consistent with prior clinical tests that measured an induced return to previous atelectasis state following the closure of the VT in 33 tested ears within 25-120 min (43 min on average). CONCLUSIONS: The model demonstrates that under the above conditions, diffusive gas transfer in relation to blood gas content is the leading mechanism to alterations in ME pressure and volume. It may be used as a tool to determine ME physiological cavity volume of ears with VT.

Carbon Dioxide↗
PubMed1998

Mathematical modelling of morphogenesis in fungi: spatial organization of the gravitropic response in the mushroom stem of Coprinus cinereus.

The purpose of this work was to establish how the distribution of local curvatures changed during the mushroom stem gravitropic reaction and to suggest a suitable mathematical model based on these new data. The gravitropic bending of base- and apex-pinned Coprinus cinereus (Fries) S. F. Gray stems was recorded on videotapes. The images were captured from the tapes after each 10 min, rotated by 45 degrees and transformed into tables of changing co-ordinates of points for each stem. The non-linear regression of these points was performed using Legendre polynomials. From the resulting equations the patterns of changing local curvature for 50 subsections per stem during 400 min of gravitropic reaction were calculated. It was observed that base-pinned stems first bent from the apex, but later the curvature of this part decreased, and in the late stages the apex became nearly completely straight again. Subsections, located about one third of stem length from the base determined the main part of the final curvature. The free basal part of the apex-pinned stems bent upward and after a certain bending time also began to straighten. However, this process started significantly later and was weaker. Bending of the subsections close to the pinned apex did not stop when they reached the vertical position, and the final angle of gravitropic curvature could exceed 180 degrees. Plotting various functions of local bending speed and its derivatives against each other and against local angle indicated that, if the hypothetical signal about reorientation arises in the apex, its propagation towards the base did not follow simple wave or simple diffusion laws. The importance of the local angle of all subsections both for signal origin and transmission was established and a signal transmission equation, involving local angle of each subsection, was derived. After creating a suitable program this partial differential equation was solved numerically. The generated shapes of the bending stem coincided in high degree with experimentally observed images.

Computer Simulation↗