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Mathematical abilities of children with specific language impairment: a 2-year follow-up.

A 2-year follow-up of the mathematical abilities of young children with specific language impairment (SLI) is reported. To detect the nature of the difficulties children with SLI exhibited in mathematics, the first- and second-grade children's performance was compared to mental age and language age comparison groups of typically developing children on a series of tasks that examined conceptual, procedural, and declarative knowledge of mathematics. Despite displaying knowledge of many conceptual aspects of mathematics such as counting plates of cookies to decide which plate had "more," children with SLI displayed marked difficulty with declarative mathematical knowledge that required an immediate response such as rote counting to fifty, counting by 10's, reciting numerals backwards from 20, and addition facts such as 2 + 2 =?. Moreover, children with SLI performed similarly to their cognitive peers on mathematical tasks that allowed children to use actual objects to count and on math problems that did not require them to exceed the sequence of numbers that they knew well. These findings offer further evidence that storage and/or retrieval of rote sequential material is particularly cumbersome for children with SLI.

Child↗

Visual-sequential and visuo-spatial skills in dyslexia: variations according to language comprehension and mathematics skills.

This study focused on visual-sequential and visuo-spatial functions in a group of 39 heavily dyslexic children, compared to a Control group. Mean age was 12.72 (SD 1.71). The dyslexia group was divided into three subgroups by language comprehension and mathematics skills. Only on a visual-sequential task was no difference seen between the groups. The main differences occurred between the two dyslexic subgroups with no language comprehension impairment, but with varying mathematics skills. Whereas the subgroup with good mathematics skills scored within the upper ranges, the mathematics-impaired subgroup showed significantly lower scores. The third dyslexic subgroup, with both language comprehension and mathematics impairments, performed within the norm. The study indicates a dissociation between language comprehension and visuo-spatial skills in dyslexia, which has implications for how variations in dyslexia should be understood. The results also show that the visuo-spatial impairments seen in one of the dyslexia subgroups lead to two ways of understanding mathematics impairment when it co-occurs with dyslexia: (1) as a visuo-spatial problem; (2) as a linguistic problem. These distinctions should imply different intervention strategies in dyslexia.

Adolescent↗

Skolem and pessimism about proof in mathematics.

Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege's first formalization, to the debates over intuitionism and other schools, through Hilbert's program and the decisive blow of the Gödel Incompleteness Theorem. A critical role is played by the Skolem-Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert's belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert's assumption. The vast majority of questions even in elementary number theory, of reasonable complexity, are beyond the reach of any such reasoning. Of course this cannot be proved and we present only plausibility arguments. The great success of mathematics comes from considering 'natural problems', those which are related to previous work and offer a good chance of being solved. The great glories of human reasoning, beginning with the Greek discovery of geometry, are in no way diminished by this pessimistic view. We end by wishing good health to present-day mathematics and the mathematics of many centuries to come.

Algorithms↗

The mathematics of hair restoration surgery.

BACKGROUND: Mathematics is prevalent in almost all aspects of life, and so it is with hair restoration surgery. This article explores some of the mathematical relationships that exist in this quickly growing subspecialty of cosmetic surgery. OBJECTIVE: The objective of this article is to demonstrate a few of the mathematical correlations that are present in hair restoration surgery. METHODS: Several subdivisions of this subspecialty are analyzed from a mathematical perspective. RESULTS: Mathematics pervades many of the aspects of hair restoration, from the donor site, to the recipient site, to scalp reduction, to scalp extension, and the list goes on. CONCLUSION: To be able to perform hair restoration surgery in a very precise and accurate way, one should have a good grasp of the mathematical relationships that exist in this field.

Hair↗

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↗

The effects of metacognitive training versus worked-out examples on students' mathematical reasoning.

BACKGROUND: The present study is rooted in a cognitive-metacognitive approach. The study examines two ways to structure group interaction: one is based on worked-out examples (WE) and the other on metacognitive training (MT). Both methods were implemented in cooperative settings, and both guided students to focus on the problem's essential parts and on appropriate problem-solving strategies. AIMS: The aim of the present study is twofold: (a) to investigate the effects of metacognitive training versus worked-out examples on students' mathematical reasoning and mathematical communication; and (b) to compare the long-term effects of the two methods on students' mathematical achievement. SAMPLE: The study was conducted in two academic years. Participants for the first year of the study were 122 eighth-grade Israeli students who studied algebra in five heterogeneous classrooms with no tracking. In addition, problem-solving behaviours of eight groups (N = 32) were videotaped and analysed. A year later, when these participants were ninth graders, they were re-examined using the same test as the one administered in eighth grade. METHOD: Three measures were used to assess students' mathematical achievement: a pretest, an immediate post-test, and a delayed post-test. ANOVA was carried out on the post-test scores with respect to the following criteria: verbal explanations, algebraic representations and algebraic solution. In addition, chi-square and Mann-Whitney procedures were used to analyse cooperative, cognitive, and metacognitive behaviours. RESULTS: Within cooperative settings, students who were exposed to metacognitive training outperformed students who were exposed to worked-out examples on both the immediate and delayed post-tests. In particular, the differences between the two conditions were observed on students' ability to explain their mathematical reasoning during the discourse and in writing. Lower achievers gained more under the MT than under WE condition. CONCLUSIONS: The findings indicate that the kind of task and the way group interaction is structured are two important variables in implementing cooperative learning, each of which is likely to have different effects on mathematical communication and achievement outcomes.

Adolescent↗

Evaluating change in attitude towards mathematics using the 'then-now' procedure in a cooperative learning programme.

BACKGROUND: Tertiary students' attitudes to mathematics are frequently negative and resistant to change, reflecting low self-efficacy. Some educators believe that greater use should be made of small group, collaborative teaching. However, the results of such interventions should be subject to assessments of bias caused by a shift in the frame of reference used by students in reporting their attitudes. AIMS: This study was designed to assess whether traditional pretest-post-test procedures would indicate positive changes in mathematics attitude during a programme of cooperative learning, and whether an examination of any attitudinal change using the 'then-now' procedure would indicate bias in the results due to a shift in the internal standards for expressing attitude. SAMPLE: Participants were 141 undergraduate students enrolled in a 12-week statistics and research design component of a course in educational psychology. METHOD: Using multivariate procedures, pretest, post-test, and then-test measures of mathematics self-concept and anxiety were examined in conjunction with a cooperative learning approach to teaching. RESULTS: Significant positive changes between pretest and post-test were found for both mathematics self-concept and mathematics anxiety. There were no significant differences between the actual pretest and retrospective pretest measures of attitude. The results were not moderated by prior level of mathematics study. CONCLUSION: Conclusions about the apparent effectiveness of a cooperative learning programme were strengthened by the use of the retrospective pretest procedure.

Anxiety↗

A causal model of mathematics performance in early adolescence: the role of sex.

Using path analysis, the present investigation was done to clarify possible causal linkages among general scholastic aptitude, academic achievement in mathematics, self-concept of ability, and performance on a mathematics examination. Subjects were 122 eighth-grade students who completed a mathematics examination as well as a measure of self-concept of ability. Aptitude and achievement measures were obtained from school records. Analysis showed sex differences in prediction of performance on the mathematics examination. For boys, this performance could be predicted from scholastic aptitude and previous achievement in mathematics. For girls, performance only could be predicted from previous achievement in mathematics. These results indicate that the direction, strength, and magnitude of relations among these variables differed for boys and girls, while mean levels of performance did not.

Achievement↗

Frames of reference for self-evaluation of ability in mathematics.

Measures of eight frame-specific self-evaluations of ability in mathematics were used to predict general mathematics self-concept and self-efficacy. Participants were 900 Norwegian students in Grade 6 (n = 277), Grade 9 (n = 236), Grade 11 (n = 263), and adult students attending senior high school (n = 124). Four items measured frame-specific self-evaluation of achievement based on external frames of reference whereas four items measured frame-specific self-evaluation based on internal frames of reference. Regression analyses were used to test relations between the frame-specific self-evaluations and general mathematics self-concept and self-efficacy. The analyses indicated that self-evaluation based on comparison with other students in class (an external frame of reference) and on comparison of mathematics achievement with achievement in other school subjects (an internal frame of reference) were robust predictors of both mathematics self-concept and self-efficacy. The analyses also indicated that students are using multiple frames of reference when evaluating their mathematics ability. Implications of the result for the internal-external frame of reference model are discussed.

Achievement↗

[Mathematical protocol for radiofrequency ablation of liver tumors and its clinical application].

OBJECTIVE: To develop a preoperative protocol for ultrasonography-guided percutaneous radiofrequency ablation (RFA) on liver tumors larger than 3.5 cm in diameter based on mathematical models and clinical experience, and to evaluate its ablated effect compared to the previous non-math method. METHODS: One hundred and twenty-five patients with 80 primary and 55 secondary liver tumors (4.7 +/- 0.9 cm in diameter, ranged from 3.6 - 7.0 cm) were enrolled in this study, of which the first 22 patients (23 tumors) had been treated empirically before the mathematical model was set up and were referred as the non-math group. The rest 103 patients (112 tumors) were treated based on the mathematical model and referred as the math group. Based on principle of overlapping spheres, a mathematical analysis was performed to investigate how multiple ablation spheres could overlap and cover larger tumors most efficiently. Some mathematical models such as regular prism and regular polyhedron model were chosen to estimate the mathematical protocol which included least ablation (sphere) number and optimal overlapping mode required to adequately ablate a large and spherical target lesion. The target volume consisted of the tumor plus a 0.5 - 1 cm tumor-free margin. The operation method for electrode placement was also described. RESULTS: The procedure success rate for the math group was 88.4% (99/112), local recurrence rate and estimated mean time until local recurrence were 25.9% (29/112) and 17.5 months, respectively. While for the non-math group the results were 52.2% (12/23) (P < 0.01), 56.5% (13/23) (P < 0.05) and 11.9 months (P < 0.05), respectively. The therapy results in the math group were much better than in the non-math group. CONCLUSION: This study provides theoretic basis and clinical guidance for RFA therapy for liver tumors larger than 3.5 cm. These results could be used to reduce local recurrence rate and improve treatment response.

Adult↗

[New trends in the evaluation of mathematics learning disabilities. The role of metacognition].

INTRODUCTION: The current trends in the evaluation of mathematics learning disabilities (MLD), based on cognitive and empirical models, are oriented towards combining procedures involving the criteria and the evaluation of cognitive and metacognitive processes, associated to performance in mathematical tasks. AIMS: The objective of this study is to analyse the metacognitive skills of prediction and evaluation in performing maths tasks and to compare metacognitive performance among pupils with MLD and younger pupils without MLD, who have the same level of mathematical performance. Likewise, we analyse these pupils' desire to learn. Subjects and methods. We compare a total of 44 pupils from the second cycle of primary education (8-10 years old) with and without mathematics learning disabilities. RESULTS: Significant differences are observed between pupils with and without mathematics learning disabilities in their capacity to predict and assess all of the tasks evaluated. As regards their 'desire to learn', no significant differences were found between pupils with and without MLD, which indicated that those with MLD assess their chances of successfully performing maths tasks in the same way as those without MLD. Finally, the findings reveal a similar metacognitive profile in pupils with MLD and the younger pupils with no mathematics learning disabilities. CONCLUSIONS: In future studies we consider it important to analyse the influence of the socio-affective belief system in the use of metacognitive skills.

Learning Disabilities↗

Mathematics reflecting sensorimotor organization.

This review combines short presentations of several mathematical approaches that conceptualize issues in sensorimotor neuroscience from different perspectives and levels of analysis. The intricate organization of neural structures and sensorimotor performance calls for characterization using a variety of mathematical approaches. This review points out the prospects for mathematical neuroscience: in addition to computational approaches, there is a wide variety of mathematical approaches that provide insight into the organization of neural systems. By starting from the perspective that provides the greatest clarity, a mathematical approach avoids specificity that is inaccurate in characterizing the inherent biological organization. Approaches presented include the mathematics of ordered structures, motion-phase space, subject-coincident coordinates, equivalence classes, topological biodynamics, rhythm space metric, and conditional dynamics. Issues considered in this paper include unification of levels of analysis, response equivalence, convergence, relationship of physics to motor control, support of rhythms, state transitions, and focussing on low-dimensional subspaces of a high-dimensional sensorimotor space.

Animals↗

Mathematical modeling of cancer: the future of prognosis and treatment.

BACKGROUND: Cancer research has undergone radical changes in the past few years. Producing information both at the basic and clinical levels is no longer the issue. Rather, how to handle this information has become the major obstacle to progress. Intuitive approaches are no longer feasible. The next big step will be to implement mathematical modeling approaches to interrogate the enormous amount of data being produced and extract useful answers (a "top-down" approach to biology and medicine). METHODS: Quantitative simulation of clinically relevant cancer situations-based on experimentally validated mathematical modeling-provides an opportunity for the researcher, and eventually the clinician, to address data and information in the context of well-formulated questions and "what if" scenarios. RESULTS AND CONCLUSIONS: At the Vanderbilt Integrative Cancer Biology Center (VICBC), we are integrating cancer researchers, oncologists, chemical and biological engineers, computational biologists, computer modelers, theoretical and applied mathematicians, and imaging scientists, in order to implement a vision for a combined web site and computational server that will be a home for our mathematical modeling of cancer invasion. The web site (www.vanderbilt.edu/VICBC/) will serve as a portal to our code, which simulates tumor growth by calculating the dynamics of individual cancer cells (an experimental "bottom-up" approach to complement the top-down model). Eventually, cancer researchers outside of Vanderbilt will be able to initiate a simulation based on providing individual cell data through a web page. We envision placing the web site and computer cluster directly in the hands of biological researchers involved in data mining and mathematical modeling. Furthermore, the web site will also contain teaching props for a new generation of biomedical researchers fluent in both mathematics and biology. This is unconventional bioinformatics: We will be incorporating biological data and functional information into a unified community-based mathematical framework. The result will be a tool for cancer modeling that will ultimately have basic research, therapeutic and educational value.

Animals↗

An application of Lacker's mathematical model for the prediction of ovarian response to superstimulation.

Introduction. A mathematical model of ovarian follicular growth is applied to the problem of predicting ovarian response in a superstimulation protocol. Methods. Fifty-four women enrolled in an ovarian superstimulation program of therapy for the amelioration of idiopathic infertility had their ovarian cycles synchronized by taking Demulen 30 for two weeks prior to the study. Daily ultrasonographic imaging, measurements of serum estradiol and doses of hMG began on day 5 after the patients stopped taking Demulen. The diameters of individual follicles were measured and followed daily. When the largest follicle attained a diameter of 19 mm, hCG was given to induce ovulation. Individual follicle growth data were fit to a mathematical model of ovarian follicle maturation and the resulting parameters were used to classify patients into low and high ovarian response groups. Results. The parameters computed from the mathematical model fit were found to be predictive of ovarian response with a sensitivity of 71% and a specificity of 70%. The parameters were also meaningful within the context of the original mathematical model and have value for determining how doses of hMG may be adjusted during the course therapy to increase the ovarian response in individuals. Conclusion. Mathematical modeling of ultrasonographically derived follicular growth data has significant potential for clinical application in ovarian superstimulation protocols. The method of fitting follicular growth data to a mathematical follicle maturation surface furthermore provides a straightforward approach for the characterization of ovarian follicular dynamics in general.

Adult↗

Mathematical modeling of biofiltration in activated pine-bark charge of a biofilter.

AIM, SCOPE AND BACKGROUND: Human economic activities cause emissions of various pollutants of an organic nature: butanol, butyl acetate, methanol, formaldehyde, phenol, benzene, toluene, xylene, etc. These compounds are emitted to atmosphere by various enterprises of food, chemistry, wood processing industries, from transportation means, agricultural enterprises, etc. Therefore, when purifying air from these pollutants, it is necessary to apply efficient and inexpensive air purification methods. In this dimension, the biological air purification is chosen from all possible air cleaning methods. An experimental biofilter with the activated charge of pine bark was developed at the Department of Environment Protection of the Vilnius Gediminas Technical University. In the course of the experimental investigation, it was determined that this air purification method is efficient. Filter efficiency, when purifying air of volatile organic compounds (butanol, butyl acetate and xylene), to a great extent, depending on the nature and concentrations (up to 100 mg/m3) of pollutants injected, might go up to 70-98%. The mathematical model of the biofilter was developed based on the research results and fully taking into consideration physical, chemical, and biological processes going on during its operations. MAIN FEATURES: The aim of this article is to determine biodegradation constant alpha, absorption capacity beta, and half empiric expressions of filter efficiency. Knowing this, it is possible to find out the dependence of the filter efficiency on the operational parameters of the filter (i.e. on the concentrations and the height of biocharge of the initial pollutants (butanol, butyl acetate, xylene) fed through it). CONCLUSIONS: With the help of mathematical modeling, the biodegradation constants and absorption capability of volatile organic compounds (butanol, butyl acetate, and xylene) fed into the biofilter charged with the activated pine bark and used for the cleaning of volatile organic compounds, as well as the efficiency of the biofilter in half empiric expression, have been established. It has been discovered that the constant of pollutant biodegradation alpha is a value inverse to the time during which the amount of pollutants in the filter becomes n times higher. It is rather complicated to carry out theoretical calculation of the biodegradation constant at a molecular level, therefore this constant has been established based on the results obtained in the course of research. The equations describing pollutant dynamics in the filter charge and the air cleaning processes going on in it have been derived from diffusion equations in a mobile medium. The modeling helped to find out the absorption capacity beta of the examined pollutants, which by its numeric value is equal to the volume unit of the absorbed gas amount. The latter factor, the same as the biodegradation constant, was determined basing on the experimental results. Mathematical modeling brought a range of formulas expressing dependences of each pollutant's efficiency on its initial concentrations and filter charge height. RECOMMENDATIONS/OUTLOOK: Based on the experimental data, a mathematical model has been developed which will allow the measuring of the filter efficiency not only with regard to the absorption and biodegradation of the pollutants under examination, but also with regard to other pollutants and their compounds, etc., having an impact on the filter performance. The results of the mathematical modeling have revealed that the modeling of processes going on in the filter is much simpler than isthe performance of long and costly experiments. The developed mathematical model makes it possible to measure the filter efficiency at the present moment.

Air Pollutants↗

Mathematical modeling of pharmacy systems.

Mathematical modeling and its potential applications in pharmacy are discussed. A model is a simplified representation of the real world. As an experimental approach, modeling minimizes expense, risk, and disruption, but its validity can be hard to ascertain. Mathematical models describe numerically the relationships among elements of a system and are a powerful tool in making decisions affecting that system. There are two types of mathematical models: analytical models, which directly describe the relationships between system inputs and outputs using mathematical equations (such as pharmacokinetic models), and simulation models, which involve the replication, usually with a computer, of events as they occur in the real world. Analytical models are easier to develop but are not appropriate for describing highly complex systems. In continuous-time simulation, the system is represented as an uninterrupted flow of material; in discrete-event simulation, it is assumed that events occur only at distinct times. Various simulation programs are commercially available. The stages of a mathematical modeling study are (1) formulate the problem, (2) determine the model's structure, (3) collect and analyze initial data, (4) develop the model further, (5) validate the model, (6) experiment using the model, and (7) use the results. There have been many applications of modeling in health care, but relatively few have involved the study of pharmacy systems. Mathematical modeling offers pharmacists a low-risk, low-cost tool for aiding decisions about pharmacy systems by predicting alternative futures.

Models, Organizational↗

Mathematics word problem solving for deaf students: a survey of practices in grades 6-12.

One hundred and thirty-three mathematics teachers of deaf students from grades 6-12 responded to a survey on mathematics word problem-solving practices. Half the respondents were teachers from center schools and the other half from mainstream programs. The latter group represented both integrated and self-contained classes. The findings clearly show that regardless of instructional setting, deaf students are not being sufficiently engaged in cognitively challenging word problem situations. Overall, teachers were found to focus more on practice exercises than on true problem-solving situations. They also emphasize problem features, possibly related to concerns about language and reading skills of their students, rather than analytical and thinking strategies. Consistent with these emphases, teachers gave more instructional attention to concrete visualizing strategies than to analytical strategies. Based on the results of this study, it appears that in two of the three types of educational settings, the majority of instructors teaching mathematics and word problem solving to deaf students lack adequate preparation and certification in mathematics to teach these skills. The responses of the certified mathematics teachers support the notion that preparation and certification in mathematics makes a difference in the kinds of word problem-solving challenges provided to deaf students.

Journal Article↗

[Mathematical modelling in medicine and biology. Theoretical basis and fundamentals].

Mathematical modelling is currently a common tool in the study of physiological and biochemical systems. Its basis and fundaments are not, however, well known by the non-specialist. Its aims are to describe, explain and predict physiological and biochemical phenomena. Mathematical models provide a concise and objective description of complex dynamic processes by defining, through mathematical equations, the relationships between quantitative measurements; they indicate, also, ways to improve experimental designs, and allow the testing of hypotheses about physiological or biochemical phenomena. Mathematical models can be developed from simple non-compartmental representations to large scale multi-compartmental models. The basic steps in the formulation of a model include conceptualization, realization and solution of the model. Each step has to be verified and validated. In the case of compartmental models, mass-balance equations are used to represent each compartment. A brief review of the theory of system's analysis and the general aims of mathematical modelling is presented here. The modelling process is usually started with a definition of the problem and a parameter identification followed by the setting up of a clear conceptual model of the system. The model consists of the description of the principal flows of material (in and out) and of the main components which store, convert or transmit these flows. A selection of the class of mathematical representation follows, i.e. linear or non-linear, in order to formulate the equations relating the input and output flows of material for each individual component of the system.(ABSTRACT TRUNCATED AT 250 WORDS)

Models, Biological↗