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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↗
PubMed1972

Nonmathematical concepts of selection, evolutionary energy, and levels of evolution.

The place of mathematics in hypotheticodeductive processes and in biological research is discussed. (Natural) Selection is defined and described as differential elimination of performed sets at any level. Sets and acting sets are groups of units (themselves sets of smaller units) at any level that may or do interact. A pseudomathematical equation describes directional change (evolution) in sets at any level. Selection is the ram of evolution; it cannot generate, but can only direct, evolutionary energy. The energy of evolution is derived from molecular or chemical levels, is transmitted upwards through the increasingly complex sets of sets that form living systems, and is turned in directions determined by the sum of selective processes, at different levels, which may either supplement or oppose each other. All evolutionary processes conform to the pseudomathematical equation referred to above, use energy as described above, and have a P/OE (ratio of programming to open-endedness) that cannot be measured, but can be related to other P/OE values. Phylogeny and ontogeny are compared as processes af directional change with set selection. Stages in the evolution of multi-cellular individuals are suggested, and are essentially the same as stages in the evolution of some multi-individual insect societies. Thinking is considered as a part of ontogeny involving an irreversible, nonrepetitive process of set selection in the brain.

Biological Evolution↗
PubMed1994

First principles: physical science concepts as a foundation for advanced studies in physiology.

The importance of mathematics and physical science to physiology is amply evident in both the classic and current literature of this field. Students who have completed typical medical school preparatory programs, however, have been poorly equipped to embark on serious graduate study in physiology. Because undergraduate education in the sciences is typically structured as narrowly defined, disjointed degree programs in separate disciplines, students of biology have had only limited access to the resources of physical science departments. Attempts by physiology faculty to remedy specific deficiencies on a piecemeal basis have been found to be both time consuming and ineffective. We describe an intensive one-semester course as an alternative. Maintaining high levels of generality and concentrating on the development of physical insight were possible only when the mathematical foundation attained by the students was adequate. Although physiological applications were stressed throughout, extended examination of complex problems was limited to a few examples of particular interest. Emphasis was placed on rigor and sophistication rather than proficiency and detail.

Mathematics↗
PubMedDate not supplied

The concepts and terms of mechanics.

Mechanics can be understood in non-mathematical terms. Two important kinds of strength are ultimate or breaking strength, and yield strength--the stress which will cause a permanent deformity (e.g., bending a bone plate). Stress, like pressure, is measured in pounds per square inch. Strain is the percentage "gain in length." The strength of bone is roughly comparable to oak wood. With hyperemia and resorption of bone substance, this strength can fall drastically in a matter of weeks. Bone cement is well named; like cement it has almost no useful tensile strength. Its compressive strength is roughly comparable to bone. Doubling the depth of a beam quadruples its strength

Biomechanical Phenomena↗
PubMed2006

Towards a new model of the homeopathic process based on quantum field theory.

BACKGROUND: Quantum theory's notions of non-locality and entanglement have previously informed attempts to model the therapeutic process. Of these, Weak Quantum Theory (WQT) and Patient- Practitioner-Remedy (PPR) entanglement are developing into mathematically- based models of homeopathy. OBJECTIVE: The present study attempted to identify fundamental concepts within quantum field theory (QFT) that could be used to broaden the scope of PPR entanglement models, prior to constructing a more rigorous mathematical treatment. METHODS: In QFT, particles and forces are considered as fully interacting relativistic quantum matter and force fields, respectively. These interactions are visualized graphically as spacetime Feynman diagrams. Further, these interacting field systems can have ground states with broken symmetry; the so-called Higgs field being responsible for this symmetry breaking. In the new model, patient, practitioner and remedy are imagined as fully interacting quantum-like fields; patients and practitioners in terms of quantum matter-type fields, and remedies and diseases as quantum interaction-type fields. RESULTS: Disease manifestation by the Vital Force (Vf) could be an event similar to spontaneous symmetry breaking in QFT: the curative remedy acting to restore the broken symmetry of the Vf field. Entanglement between patient, practitioner, and remedy might be representable as Feynman-like diagrams. CONCLUSION: QFT demonstrates that quantum properties can be physical without being observable. Thus, an underlying similarity in discourse could exist between homeopathy and quantum theory which could be useful for modelling the homeopathic process. This preliminary investigation also suggested that key elements of previous quantum models of the homeopathic process, may become unified within this new QFT-type approach.

Homeopathy↗
PubMed2005

Modelling of two-stage anaerobic digestion using the IWA Anaerobic Digestion Model No. 1 (ADM1).

The aim of the study presented was to implement a process model to simulate the dynamic behaviour of a pilot-scale process for anaerobic two-stage digestion of sewage sludge. The model implemented was initiated to support experimental investigations of the anaerobic two-stage digestion process. The model concept implemented in the simulation software package MATLAB/Simulink is a derivative of the IWA Anaerobic Digestion Model No.1 (ADM1) that has been developed by the IWA task group for mathematical modelling of anaerobic processes. In the present study the original model concept has been adapted and applied to replicate a two-stage digestion process. Testing procedures, including balance checks and 'benchmarking' tests were carried out to verify the accuracy of the implementation. These combined measures ensured a faultless model implementation without numerical inconsistencies. Parameters for both, the thermophilic and the mesophilic process stage, have been estimated successfully using data from lab-scale experiments described in literature. Due to the high number of parameters in the structured model, it was necessary to develop a customised procedure that limited the range of parameters to be estimated. The accuracy of the optimised parameter sets has been assessed against experimental data from pilot-scale experiments. Under these conditions, the model predicted reasonably well the dynamic behaviour of a two-stage digestion process in pilot scale.

Anaerobiosis↗
PubMed2002

Mathematical modelling in the post-genome era: understanding genome expression and regulation--a system theoretic approach.

This paper introduces a mathematical framework for modelling genome expression and regulation. Starting with a philosophical foundation, causation is identified as the principle of explanation of change in the realm of matter. Causation is, therefore, a relationship, not between components, but between changes of states of a system. We subsequently view genome expression (formerly known as 'gene expression') as a dynamic process and model aspects of it as dynamic systems using methodologies developed within the areas of systems and control theory. We begin with the possibly most abstract but general formulation in the setting of category theory. The class of models realised are state-space models, input--output models, autoregressive models or automata. We find that a number of proposed 'gene network' models are, therefore, included in the framework presented here. The conceptual framework that integrates all of these models defines a dynamic system as a family of expression profiles. It becomes apparent that the concept of a 'gene' is less appropriate when considering mathematical models of genome expression and regulation. The main claim of this paper is that we should treat (model) the organisation and regulation of genetic pathways as what they are: dynamic systems. Microarray technology allows us to generate large sets of time series data and is, therefore, discussed with regard to its use in mathematical modelling of gene expression and regulation.

Gene Expression↗
PubMed1991

Mathematical problem solving by analogy.

We report the results of 2 experiments and a verbal protocol study examining the component processes of solving mathematical word problems by analogy. College students first studied a problem and its solution, which provided a potential source for analogical transfer. Then they attempted to solve several analogous problems. For some problems, subjects received one of a variety of hints designed to reduce or eliminate the difficulty of some of the major processes hypothesized to be involved in analogical transfer. Our studies yielded 4 major findings. First, the process of mapping the features of the source and target problems and the process of adapting the source solution procedure for use in solving the target problem were clearly distinguished: (a) Successful mapping was found to be insufficient for successful transfer and (b) adaptation was found to be a major source of transfer difficulty. Second, we obtained direct evidence that schema induction is a natural consequence of analogical transfer. The schema was found to co-exist with the problems from which it was induced, and both the schema and the individual problems facilitated later transfer. Third, for our multiple-solution problems, the relation between analogical transfer and solution accuracy was mediated by the degree of time pressure exerted for the test problems. Finally, mathematical expertise was a significant predictor of analogical transfer, but general analogical reasoning ability was not. The implications of the results for models of analogical transfer and for instruction were considered.

Adult↗
PubMedDate not supplied

Achieving meaningful mathematics literacy for students with learning disabilities. Cognition and Technology Group at Vanderbilt.

In this article we consider issues relevant to the future of mathematics instruction and achievement for students with learning disabilities. The starting point for envisioning the future is the changing standards for mathematics learning and basic mathematical literacy. We argue that the shift from behaviorist learning theories to constructivist and social constructivist theories (see Rivera, this series) provides an opportunity to develop and implement a hybrid model of mathematics instruction. The hybrid model we propose embeds, or situates, important skill learning in meaningful contexts. We discuss some examples of instructional approaches to complex mathematical problem solving that make use of meaningful contexts. Evaluation data on these approaches have yielded positive and encouraging results for students with learning disabilities as well as general education students. Finally, we discuss various ways in which technology is important for realizing hybrid instructional models in mathematics.

Achievement↗
PubMed2003

The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.

This is the first in a series of four articles developed for the readers of. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease, abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Cardiology↗
PubMed2003

The conceptual basis of mathematics in cardiology IV: statistics and model fitting.

This is the fourth in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Cardiology↗
PubMedDate not supplied

Developmental change and individual differences in children's multiplication.

Age-related change and patterns of individual differences in children's knowledge and skill in multiplication were investigated for students in Grades 4 and 6 (approximately ages 9 and 11, respectively) by examining multiple measures of computational skill, conceptual knowledge, and working memory. Regression analyses revealed that indexes reflecting probability of retrieval and special problem characteristics overshadow other, more general indexes (problem size and frequency of presentation) in predicting solution latencies. Some improvement in the use of conceptual knowledge was evident between Grades 4 and 6, but this change was neither strong nor uniform across tasks. Finally, patterns of individual differences across tasks differed as a function of grade level. The findings have implications for understanding developmental change and individual differences in mathematical cognition.

Child↗
PubMed1987

Blood lactate concentration increases as a continuous function in progressive exercise.

The relationship between arterialized blood lactate concentration [( La-]) and O2 uptake (VO2) was examined during a total of 23 tests by eight subjects. Exercise was on a cycle ergometer with work rate incremented from loadless pedaling to exhaustion as a 50-W/min ramp function. Two different mathematical models were studied. One model employed a log-log transformation of [La-] and VO2 to yield [La-] threshold as proposed by Beaver et al. (J. Appl. Physiol. 59: 1936-1940, 1985). The other model was a continuous exponential plus constant of the form La- = a + b[exp(cVO2)]. In 21 of 23 data sets, the mean square error (MSE) of the continuous model was less than that of the log-log model (P less than 0.001). The MSE was on average 3.5 times greater in the log-log model than in the continuous model. The residuals were randomly distributed about the line of best fit for the continuous model. In contrast, the log-log model showed a nonrandom pattern indicating an inappropriate model. As an index of the position of the [La-]-VO2 continuous model, the VO2 at which the rate of increase of [La-] equaled the rate of increase of VO2 (d[La-]/dVO2 = 1) was determined. This VO2 was 2.241 +/- 0.081 l/min, which averaged 64.6% of maximal VO2. It is proposed that this lactate slope index could be used as a relative indicator of fitness instead of the previously applied threshold concept. The change in [La-] could be better described mathematically by a continuous model rather than the threshold model of Beaver et al.

Adolescent↗
PubMed1979

Transient response of the Geman-Miller respiratory oscillator model.

Recent evidence suggests that there is a persistence of elevated respiratory center activity for many respiratory cycles after the cessation of a neural stimulus. To explore the theoretical possibility that this behavior may be a consequence of the neural oscillator that dictates respiratory rhythm, the transient response of a mathematical model of a medullary respiratory oscillator recently described by Geman and Miller was examined using computer simulation. This concept was motivated by the presence of a persistent transient response behavior inherent in oscillators from mathematical physics. The results indicate that a transient persistence effect is evident in the model behavior under some conditions, and this effect as well as the steady-state amplitude is markedly sensitive to the shape of the saturation function that interconnects the neuron populations. Furthermore, this behavior is initiated by either an abrupt decrease in a tonic input or an abrupt decrease in the synaptic weights connecting subpopulations of neurons.

Animals↗
PubMed2003

The conceptual basis of mathematics in cardiology III: linear systems theory and integral transforms.

This is the third in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas.This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Cardiology↗