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Attention is drawn to the relationship between information and biological models in the study of problems of human physiology and physiopathology. The biomathematical principles, to which reference must be made for the elaboration of the expert systems, necessary in evaluating biological feedback in particular are analyzed.
BACKGROUND: Understanding arithmetical principles is a key part of a conceptual understanding of mathematics. However, very little attention has been paid to children's understanding of multiplicative, as compared to additive, principles. AIMS: This study investigated (a) children's ability to use commutative and distributive cues to solve multiplication problems, (b) whether their ability to use these cues depends on the problem context, and (c) whether separate mechanisms might underlie children's understanding of commutativity and distributivity. SAMPLE: Twenty-seven 9-year-olds (Year 5) and thirty-two 10-year-olds (Year 6). METHODS: Forty-eight multiplication problems (with a multiple-choice response format) were presented to children. There were four types of problem: Commutative, Distributive, Combined commutative-distributive (all preceded by a cue) and No cue problems. Each type of problem was presented in three different contexts: Isomorphism of measures, Area, and Cartesian product. RESULTS: Children demonstrated a good understanding of commutativity but a very poor understanding of distributivity. A common mistake in the distributive problems was to select the number that was one more, or one less, than the answer in the cue. Children's understanding of distributivity (but not commutativity) seemed to depend on the problem context. Factor analysis suggested that separate factors underlie the ability to solve commutative and distributive problems. CONCLUSIONS: Nine- and 10-year-olds understand commutativity, but are unable to use the distributive principle in multiplication. Their errors suggest that they may confuse some of the principles of multiplication with those of addition. When children do begin to understand the principle of distributivity, they most easily apply it in the context of Isomorphism of measures multiplication problems. The implications for mathematical education are discussed.
Assignment to an appropriate test level is crucial to the validity of standardized test scores. The present study examined the efficiency of the Reading Screening Tests and Mathematics Screening Tests developed by the Center for Assessment and Demographic Studies (CADS) of Gallaudet University for assigning levels of the Stanford Achievement Test, 8th edition (SAT-8), to deaf and hard-of-hearing students. Data were collected from programs participating in the 1990 test standardization with this population. It was found that while the screening procedures provide a credible means of assigning test levels, they are more effective for the SAT-8's Reading Comprehension, Total Language, Mathematics Computation, and Mathematics Applications subtests than for the Spelling and Concepts of Number subtests. This article concludes with recommendations in regard to individual subtest assignments.
The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain information-processing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r2 = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition.
Physician profilers encompass an array of technological products that purport to evaluate individual clinician performance on utilization and other measures. Prior to installation, an organization should be familiar with a number of major concepts that encompass understanding of clinical practice processes, claims data limitations and idiosyncrasies, ethical issues, and mathematical principles. Twelve specific elements are described to assure that these concepts are addressed.
A mathematical model of soluble microbial products (SMPs) formation-degradation was established based on the activated sludge model no. 1 and was applied to the membrane bioreactor process with high concentration of activated sludge under intermittent aerobic operational condition. The simulation results were in good agreement with the experimental data which indicated that the coefficients used in the model could successfully describe the treatment performance. The most advantage of this modified model over the conventional one was that the significant importance of SMP existence was demonstrated and the model provided an reasonable comprehension for SMP concept. The present study demonstrated that SMPs contributed most to the organic matter in the effluent, and the results coincided well with the observations of many other researchers.
A mathematical formalization of cognitive processes based on concepts previously given in Experientia is presented. Cognitive processes are described as a set of exponentially decaying interaction probabilities determined by previous interactions between different elements and an association parameter, the inverse of the cognitive stability. The interactions occur in a 'cognitive string' surrounded by a 'cognitive plasma' which carries the contextual information. The 'complexity' of a cognitive process is proportional to time.
The L-state Potts model for rumor is the N-spin chain describing how a simple rumor transmitted by N recreant rumormongers is aggrandized. The studied rumor is represented mathematically by a simple proposition with the universal quantifier, which again is represented geometrically by a point in a proposition space. During the transmission, such a proposition is changed with the change of the rumor, which has individual number N0 at the beginning of the transmission. Correspondingly, the point expressing the proposition may start from an arbitrary site at the proposition space, and then it shifts in the space. Thus, a spin sum of the Potts model corresponds to a walk of a point in the proposition space and spin configuration's numbers is given by enumerating the corresponding walks. The concept of the lattice path in combinatorial mathematics is introduced and the exact series representation of the configuration's numbers is given. The partition function exhibits the transition of the chain and critical equivalent inverse temperature beta(c) is determined. Moreover, there is a crossover value of the individual number, N00. The model has a first-order transition when N0 N00.
A model based on elementary principles of hydrodynamics and mathematics is proposed for classroom research on concepts related to blood flow physiology. This is an analog model of the vascular system in which blood flow is represented by electrical current flowing in a resistance circuit. The model permits analysis of the change in hemodynamics with local stenosis of both large and peripheral vessels.
Deterministic mathematical equations are derived to describe the pattern of marker excretion in the faeces of ruminants under steady-state conditions when diffusion and viscosity concepts are introduced into a simple two-compartment scheme of the gastrointestinal tract. The basic scheme comprises a pure-mixing pool obeying first-order kinetics and a second compartment exhibiting streamline flow. Introduction of a velocity gradient, longitudinal diffusion or both into the second compartment, even with various simplifying assumptions, yields analytically insoluble equations. The impact of these mechanisms is to be investigated numerically rather than analytically in future work.
Estimates of anaerobic work capacity (AWC) were determined in 26 subjects using the critical power concept. AWC estimates were derived using three mathematically equivalent models: nonlinear power-time, linear power-1/time, and linear work-time. We hypothesized that large variability between estimates generated using the three models might reflect systematic error in the data and that large standard errors of estimate (SEE) might reflect random error. Therefore, subjects were grouped according to the variability in the three AWC estimates and according to the magnitude of the SEE of the parameter estimates. Then AWC estimates were compared to a criterion measure of anaerobic capacity, namely maximal oxygen deficit. When variability in the AWC estimates was low (< 10%), all provided accurate measures of oxygen deficit; also, when SEE were low, estimates were accurate. When variability was high, mean AWC estimates derived using linear models differed from oxygen deficit by over 15%; when SEE were high, mean AWC estimates derived using all models differed from oxygen deficit by over 10%. It is concluded that the accuracy of estimates of anaerobic capacity derived using the critical power concept can best be ensured by accepting values only when all three models provide the same value and/or when the SEE of the AWC estimate is small.
The objectives of this study were to determine the influence of various motivation variables on task-specific mathematics performance and to explore whether these variables change during the first year of middle school (N = 273). Students' task-specific self-efficacy was the only motivation variable to predict performance and did so both at start and end of year. There were no differences in anxiety, self-concept, or self-efficacy for self-regulation between start and end of year, but, by end of year, students described mathematics as less valuable and reported lower effort and persistence. Gifted students had stronger mathematics self-concept beliefs, and they had more accurate and less overconfident self-efficacy beliefs than did regular education students. There were no gender differences in any of the motivation constructs. Copyright 1999 Academic Press.
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In the first part of this article, we describe the basic objectives of the math curriculum in Spain as well as the basic contents, teacher resources, and obstacles perceived in mathematics instruction. Second, we briefly describe the concept of learning disabilities (LD) as they are currently defined in Spain. As stated in the recent educational reform, a student with LD is any student with special educational needs. The emphasis is placed on the educational resources that these students need in order to achieve the curricular objectives that correspond to their age group or grade. Third, we comment specifically on the educational services model and the evaluation and instructional procedures for students with math learning disabilities. Finally, we describe some lines of research that have appeared in the last few years in Spain that have led to the development of new evaluation and intervention procedures for students with LD in computation and problem solving.
Nonlinear dynamics gives us a language within which to express the basic concepts of physiology. Form, qualitative change, oscillation, stability, and other biologically important notions find natural expressions in this new mathematical approach. An introduction to qualitative dynamics is sketched, its relevance to physiology is argued, and several examples of its usefulness in providing models and metaphors for describing brain activity are offered.
"Wolff's law" is a concept that has sometimes been misrepresented, and frequently misunderstood, in the anthropological literature. Although it was originally formulated in a strict mathematical sense that has since been discredited, the more general concept of "bone functional adaptation" to mechanical loading (a designation that should probably replace "Wolff's law") is supported by much experimental and observational data. Objections raised to earlier studies of bone functional adaptation have largely been addressed by more recent and better-controlled studies. While the bone morphological response to mechanical strains is reduced in adults relative to juveniles, claims that adult morphology reflects only juvenile loadings are greatly exaggerated. Similarly, while there are important genetic influences on bone development and on the nature of bone's response to mechanical loading, variations in loadings themselves are equally if not more important in determining variations in morphology, especially in comparisons between closely related individuals or species. The correspondence between bone strain patterns and bone structure is variable, depending on skeletal location and the general mechanical environment (e.g., distal vs. proximal limb elements, cursorial vs. noncursorial animals), so that mechanical/behavioral inferences based on structure alone should be limited to corresponding skeletal regions and animals with similar basic mechanical designs. Within such comparisons, traditional geometric parameters (such as second moments of area and section moduli) still give the best available estimates of in vivo mechanical competence. Thus, when employed with appropriate caution, these features may be used to reconstruct mechanical loadings and behavioral differences within and between past populations.
In Gaussian optics properties such as dioptric power, lateral and angular magnification and thickness are simple scalar concepts. In linear optics, the optics of thick astigmatic systems, however, these concepts generalize to three-dimensional concepts in some cases (the dioptric power of thin systems, for example) and to four-dimensional concepts in general. As a result, the quantitative treatment of these properties in astigmatic systems presents challenges to the researcher in optometry, ophthalmology, and vision science. Considerable progress has been made only in the case of dioptric power. This paper presents a generalized approach to astigmatic optics which allows different physical properties to be treated in the same way: the theory is unified and, in a sense, complete. Mathematical and statistical methods developed for treating one concept become directly applicable to others. The paraxial optical properties of any optical system are completely defined by the 4 x 4 ray transfer matrix, called here the (ray) transference. The transference defines four fundamental properties of an optical system, tentatively called here positional magnification, optical thickness, divergence, and directional magnification. They are the four 2 x 2 submatrices A, B, C, and D of the transference. Each fundamental property is a modification of a familiar concept. Divergence is the negative of dioptric power expressed as the dioptric power matrix F. The four fundamental optical properties A, B, C, and D, and the derived property F, despite being different physically, all have the same underlying mathematical structure. This fact is exploited in developing a unified theory. The theory is complete in the sense that the fundamental properties fully characterize the paraxial optics of any system. The paper presents a general treatment that applies to any of the five properties. The implications are far reaching and extend beyond what can be described in the paper. Dioptric power of thin systems is treated as a particular application of the general theory. The result is the resolution of a number of issues of current interest to the researcher. It is shown, for example, that root-mean-squared (curvital) power, root-mean-squared torsional power, and length of the power vector (or dioptric strength) have a Pythagorean relationship, the power vector being the hypotenuse. Mean-squared curvital and torsional powers are in effect the area enclosed by polar profiles of curvital and torsional power, respectively. The full character of dioptric power cannot be represented by a single vector in the usual sense of the term. Two vectors are required: they are the meridional (vector) power and the orthogonal (vector) power, both of which are associated with the reference meridian. The power along a meridian (often thought of as a scalar or as two scalars) is a vector, the meridional power. This meridional power has components along (the meridional component of the meridional power) and perpendicular to (the orthogonal component of the meridional power) the meridian. In the literature, these components are the curvital power and the negative of the torsional power, respectively. The paper also examines the generalization of these results to the dioptric power of thick systems. Dioptric power is not a fundamental optical property but a derived property. Divergence, the negative of dioptric power, is the corresponding fundamental property. The theory described here is ray-based. The concept of the wavefront is unnecessary. The many formulas and concepts that apply in the context of dioptric power apply directly to the fundamental properties as well. The theory has the potential to provide a complete framework for future studies of astigmatic systems and could systematize the approach to and enhance the knowledge of astigmatism.
Dietary glycemic load, the mathematical product of the glycemic index (GI) of a food and its carbohydrate content, has been proposed as an indicator of the glucose response and insulin demand induced by a serving of food. To validate this concept in vivo, we tested the hypotheses that 1). portions of different foods with the same glycemic load produce similar glycemic responses; and 2). stepwise increases in glycemic load for a range of foods produce proportional increases in glycemia and insulinemia. In the first study, 10 healthy subjects consumed 10 different foods in random order in amounts calculated to have the same glycemic load as one slice of white bread. Capillary blood samples were taken at regular intervals over the next 2 h. The glycemic response as determined by area under the curve was not different from that of white bread for nine foods. However, lentils produced lower than predicted responses (P < 0.05). In the second study, another group of subjects was tested to determine the effects of increasing glycemic load using a balanced 5 x 5 Greco-Latin square design balanced for four variables: subject, dose, food and order. Two sets of five foods were consumed at five different glycemic loads (doses) equivalent to one, two, three, four and six slices of bread. Stepwise increases in glycemic load produced significant and predictable increases in both glycemia (P < 0.001) and insulinemia (P < 0.001). These findings support the concept of dietary glycemic load as a measure of overall glycemic response and insulin demand.