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Abstraction in mathematics.

Some current interpretations of abstraction in mathematical settings are examined from different perspectives, including history and learning. It is argued that abstraction is a complex concept and that it cannot be reduced to generalization or decontextualization only. In particular, the links between abstraction processes and the emergence of new objects are shown. The role that representations have in abstraction is discussed, taking into account both the historical and the educational perspectives. As languages play a major role in mathematics, some ideas from functional linguistics are applied to explain to what extent mathematical notations are to be considered abstract. Finally, abstraction is examined from the perspective of mathematics education, to show that the teaching ideas resulting from one-dimensional interpretations of abstraction have proved utterly unsuccessful.

Concept Formation↗

Brain potentials to mathematical syntax problems.

Anterior negativities obtained during sentence processing have never been unambiguously reported in the mathematical domain. The reason for this might be that the tasks explored in the mathematical domain have been far from resembling those typically yielding language-related anterior negativities. To test this hypothesis, we explored three mathematical aspects: Order-relevant information, a parenthesis indicating the onset of an embedded calculation, and violations of the type of symbol displayed. Results yielded parieto-occipital instead of frontal negativities. Late posterior positivities were also found, largely comparable to linguistic P600 in topography, but dissociable in functional terms. Our data suggest that language-related anterior negativities may indeed reflect language-specific resources of the human brain and support recent claims that language and mathematical domains are more independent than previously thought.

Adult↗

The predictive relationship between beliefs, attitudes, intentions and secondary school mathematics learning: a theory of reasoned action approach.

This study investigated how attitudes and intentions about learning mathematics might be related to subsequent mathematics learning and achievement using the Ajzen and Fishbein theory of reasoned action. The sample consisted of 142 boys and girls between 12 and 14 years old in a large inner city comprehensive school who were assessed in a follow-up design over a nine-month period. Beliefs about the outcomes of learning, attitudes to learning, perceptions of significant others' prescriptions about learning, intentions to engage in learning behaviours, self and teacher reported learning behaviour and mathematics achievement were assessed at both stages. Regression analysis suggested that while the expectancy-value components of attitude did relate to learning behaviour intentions, perceived prescriptions did not relate to intentions. There was a weak relationship between the two measures of learning behaviour, but with neither measure did intention independently predict future behaviour once prior behaviour was taken into account. The best predictor of subsequent mathematics achievement was prior achievement, though teacher-reported learning behaviour did have an independent relationship with subsequent achievement. The findings are discussed in terms of the assessment of learning behaviours, the relevance of the behaviour intention construct for repeated multiple behaviours and future work on how affective variables might be related to cognitive achievements.

Achievement↗

Statistics and mathematics anxiety in social science students: some interesting parallels.

This study illuminates some interesting parallels between statistics anxiety and mathematics anxiety in social science students. Parallel to what is confirmed for mathematics anxiety, two factors were observed to underly statistics anxiety scores, namely, statistics test anxiety and content anxiety. The study revealed modest though significant correlations between student attributes and the two confirmed dimensions of statistics anxiety. Furthermore, parallel to the inverse correlation reported for mathematics anxiety and maths course performance, statistics anxiety correlated negatively with high school matriculation scores in maths as well as self perceptions of maths abilities. These data lend support to the hypothesis that aversive prior experiences with mathematics, prior poor achievement in maths, and a low sense of maths self-efficacy are meaningful antecedent correlates of statistics anxiety and thus lend some credence to the "deficit" interpretation of statistics anxiety.

Achievement↗

Motivational styles in English and mathematics among children identified as having special educational needs.

This study is concerned with the identification, development and prevalence of three different motivational styles: learned helplessness, self-worth motivation and mastery orientation in two National Curriculum core subjects: English and mathematics. These three motivational styles are concerned with the ways in which children respond in the face of difficult and challenging educational tasks. Using Craske's (1988) procedures, a total of 437 children in their first year of secondary school (aged 11-12) were categorised into one of the three motivational styles. Children with special educational needs were identified using a battery of cognitive ability tests (Thorndike, Hagen & France, 1986). This study compares the prevalence of each motivational style in English and mathematics in children of different abilities and presents empirical support for the view that learned helplessness and self-worth motivation are more prevalent among children identified as having special educational needs. Further analyses indicate that: overall, learned helplessness and self-worth motivation are more prevalent in English than mathematics; learned helplessness is more prevalent among girls and self-worth motivation among boys in mathematics. Implications for teachers in developing teaching strategies which foster mastery orientation in their children are discussed.

Adolescent↗

Experimental and mathematical models of Escherichia coli plasmid transfer in vitro and in vivo.

Little is known about the factors that govern plasmid transfers in natural ecosystems such as the gut. The consistent finding by earlier workers that plasmid transfer in the normal gut can be detected only at very low rates, if at all, has given rise to numerous speculations concerning the presence in vivo of various inhibitors of plasmid transfer. Plasmids R1, R1drd-19, and pBR322 were studied in Escherichia coli K-12 and wild-type E. coli hosts in two experimental systems: (i) gnotobiotic mice carrying a synthetic indigenous microflora (F-strains) which resemble in their function the normal indigenous microflora of the mouse large intestine, and (ii) anaerobic continuous-flow cultures of indigenous large intestinal microflora of the mouse, which can simulate bacterial interactions observed in the mouse gut. Mathematical models were developed to estimate plasmid transfer rates as a measure of the "fertility," i.e., of the intrinsic ability to transfer the plasmid under the environmental conditions of the gut. The models also evaluate the effects of plasmid segregation, reduction of the growth rates of plasmid-bearing bacterial hosts, repression of transfer functions, competition for nutrients, and bacterial attachment to the wall of the gut or culture vessel. Some confidence in the validity of these mathematical models was gained because they were able to reproduce a number of known phenomena such as the repression of fertility of the R1 plasmid, as well as known differences in the transmission and mobilization of the plasmids studied. Interpretation of the data obtained permitted a number of conclusions, some of which were rather unexpected. (i) Fertility of plasmid-bearing E. coli in the normal intestine was not impaired. The observed low rates of plasmid transfer in the normal gut can be explained on quantitative grounds alone and do not require hypothetical inhibitory mechanisms. (ii) Conditions for long-term spread and maintenance throughout human or animal populations of a diversity of conjugative and nonconjugative plasmids may be optimal among E. coli strains of low fertility, as are found among wild-type strains. (iii) E. coli strains carrying plasmid pBR322 plus R1drd-19 were impaired in their ability to transfer R1drd-19, but strains carrying pBR322 were significantly better recipients of R1drd-19 than a plasmid-free recipient E. coli. (iv) Long-term coexistence of plasmid-bearing and plasmid-free E. coli, in spite of undiminished fertility, appeared to be due to a detrimental effect of the plasmid on the growth rate of its host bacterium, rather than due to high rates of plasmid segregation. (v) Mathematical analysis of experimental data published by earlier investigators is consistent with the conclusion that plasmid transfer occurs consistently in the human gut, but that the resulting transconjugant E. coli populations are too small to be detected regularly with the culture methods used by earlier investigators. It is concluded that the long-term interactions observed were often the consequences of minor differences in parameters such as growth rates, fertility, rates of segregation, etc., which were too small to be detected except by precise mathematical analysis of long-term experiments, but which were nevertheless decisive determinants of the ultimate fates of the plasmids and their hosts.

Animals↗

Evidence that methylphenidate enhances the saliency of a mathematical task by increasing dopamine in the human brain.

OBJECTIVE: Methylphenidate is the most commonly prescribed drug for attention deficit hyperactivity disorder (ADHD), yet its therapeutic mechanisms are poorly understood. The objective of this study was to assess if methylphenidate, by increasing dopamine (neurotransmitter involved in motivation) in brain, would enhance the saliency of an academic task, making it more interesting. METHOD: Healthy subjects (N=16) underwent positron emission tomography with [(11)C]raclopride (dopamine D(2) receptor radioligand that competes with endogenous dopamine for binding) to assess the effects of oral methylphenidate (20 mg) on extracellular dopamine in the striatum. The authors compared the effects of methylphenidate during an academic task (solving mathematical problems with monetary reinforcement) and a neutral task (passively viewing cards with no remuneration). In parallel, the effects of methylphenidate on the interest that the academic task elicited were also evaluated. RESULTS: Methylphenidate, when coupled with the mathematical task, significantly increased extracellular dopamine, but this did not occur when coupled with the neutral task. The mathematical task did not increase dopamine when coupled with placebo. Subjective reports about interest and motivation in the mathematical task were greater with methylphenidate than with placebo and were associated with dopamine increases. CONCLUSIONS: The significant association between methylphenidate-induced dopamine increases and the interest and motivation for the task confirms the prediction that methylphenidate enhances the saliency of an event by increasing dopamine. The enhanced interest for the task could increase attention and improve performance and could be one of the mechanisms underlying methylphenidate's therapeutic effects. These findings support educational strategies that make schoolwork more interesting as nonpharmacological interventions to treat ADHD.

Adult↗

Metacognition and mathematical problem solving in grade 3.

This article presents an overview of two studies that examined the relationship between metacognition and mathematical problem solving in 165 children with average intelligence in Grade 3 in order to help teachers and therapists gain a better understanding of contributors to successful mathematical performance. Principal components analysis on metacognition revealed that three metacognitive components (global metacognition, off-line metacognition, and attribution to effort) explained 66% to 67% of the common variance. The findings from these studies support the use of the assessment of off-line metacognition (essentially prediction and evaluation) to differentiate between average and above-average mathematical problem solvers and between students with a severe or moderate specific mathematics learning disability.

Awareness↗

Making sense of number sense: implications for children with mathematical disabilities.

Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of "number sense." I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable.

Child↗

Comorbidity of reading and mathematics disabilities: genetic and environmental etiologies.

Although children with learning disabilities frequently manifest comorbid reading and mathematics deficits, the cause of this comorbidity is unknown. To assess the extent to which comorbidity between reading and mathematics deficits is due to genetic and environmental influences, we conducted a twin study of reading and mathematics performance. Data from 148 identical and 111 fraternal twin pairs in which at least one member of the pair had a reading disability were subjected to a cross-concordance analysis and also fitted to a bivariate extension of the basic multiple regression model for the analysis of selected twin data. Results of these analyses suggest that genetic and shared-environmental influences both contribute to the observed covariance between reading and mathematics deficits.

Adolescent↗

A life skills approach to mathematics instruction: preparing students with learning disabilities for the real-life math demands of adulthood.

Current mathematics instruction does not address the day-to-day needs of many students with learning disabilities. Although the vast majority of students with learning disabilities are not college bound, much of mathematics instruction provides college preparation. Too often, classes in mathematics ignore the skills needed in home and community and on the job. The present article examines the ways in which general mathematics instruction, focused on daily living skills, can easily be integrated into the classrooms of students with learning disabilities.

Activities of Daily Living↗

A twin study of mathematics disability.

Although results obtained from recent twin and adoption studies suggest that individual differences in mathematics performance are due in part to heritable influences, no genetic analysis of mathematics disability (MD) has been previously reported. In this article we present data from the first twin sample ascertained for mathematics deficits (40 identical and 23 same-sex fraternal twin pairs in which at least one member had MD). When mathematics performance data from these twin pairs were subjected to a multiple regression analysis, evidence for a significant genetic etiology was obtained. However, tests for the differential etiology of MD as a function of reading performance level were nonsignificant. Results of this first twin study of MD indicate that the condition is significantly heritable, but data from additional twin pairs will be required to test hypotheses of differential etiology more rigorously.

Achievement↗

Modeling stability of growth between mathematics and science achievement during middle and high school.

In this study, the authors introduced a multivariate multilevel model to estimate the consistency among students and schools in the rates of growth between mathematics and science achievement during the entire middle and high school years with data from the Longitudinal Study of American Youth (LSAY). There was no evident consistency in the rates of growth between mathematics and science achievement among students, and this inconsistency was not much influenced by student characteristics and school characteristics. However, there was evident consistency in the average rates of growth between mathematics and science achievement among schools, and this consistency was influenced by student characteristics and school characteristics. Major school-level variables associated with parental involvement did not show any significant impacts on consistency among either students or schools. Results call for educational policies that promote collaboration between mathematics and science departments or teachers.

Adolescent↗

The effects of computer-assisted instruction on the mathematics performance and classroom behavior of children with ADHD.

The present study examines the effects of computer-assisted instruction (CAI) on the mathematics performance and classroom behavior of three second-through fourth-grade students with ADHD. A controlled case study is used to evaluate the effects of the computer software on participants' mathematics performance and on-task behavior. Participants' mathematics achievement improve and their on-task behavior increase during the CAI sessions relative to independent seatwork conditions. In addition, students and teachers consider CAI to be an acceptable intervention for some students with ADHD who are having difficulty with mathematics. Implications of these results for practice and research are discussed.

Achievement↗

Executive functioning as a predictor of children's mathematics ability: inhibition, switching, and working memory.

Children's mathematical skills were considered in relation to executive functions. Using multiple measures--including the Wisconsin Card Sorting Task (WCST), dual-task performance, Stroop task, and counting span-it was found that mathematical ability was significantly correlated with all measures of executive functioning, with the exception of dual-task performance. Furthermore, regression analyses revealed that each executive function measure predicted unique variance in mathematics ability. These results are discussed in terms of a central executive with diverse functions (Shallice & Burgess, 1996) and with recent evidence from Miyake, et al. (2000) showing the unity and diversity among executive functions. It is proposed that the particular difficulties for children of lower mathematical ability are lack of inhibition and poor working memory, which result in problems with switching and evaluation of new strategies for dealing with a particular task. The practical and theoretical implications of these results are discussed, along with suggestions for task changes and longitudinal studies that would clarify theoretical and developmental issues related to executive functioning.

Child↗

The contribution of executive functions to emergent mathematic skills in preschool children.

Mathematical ability is related to both activation of the prefrontal cortex in neuroimaging studies of adults and to executive functions in school-age children. The purpose of this study was to determine whether executive functions were related to emergent mathematical proficiency in preschool children. Preschool children (N = 96) were administered an executive function battery that was reduced empirically to working memory (WM), inhibitory control (IC), and shifting abilities by calculating composite scores derived from principal component analysis. Both WM and IC predicted early arithmetic competency, with the observed relations robust after controlling statistically for child age, maternal education, and child vocabulary. Only IC accounted for unique variance in mathematical skills, after the contribution of other executive functions were controlled statistically as well. Specific executive functions are related to emergent mathematical proficiency in this age range. Longitudinal studies using structural equation modeling are necessary to better characterize these ontogenetic relations.

Aptitude↗

Role of mathematics in cancer research: attitudes and training of Japanese mathematicians.

An extensive survey of attitude towards scientific information of scientists in Japan was conducted in Japan. It was published in a technical report, and this survey is reviewed in this paper, with the hope that this will furnish findings important in working out the plan for promoting exploitation of mathematical talent in biomedical research. Findings are concordant with the impression of foreign visitors: (1) pure mathematicians tend to concentrate on mathematics only; (2) applied mathematics and statistics are heavily oriented toward industry; (3) mathematicians and pharmacologists are very different in their attitudes to scientific information. Based on the personal experience of the author, difficulties to be circumvented in utilizing aptitudes for mathematics and/or statistics in biomedical research are discussed.

Attitude↗

Changing levels of numeracy and other core mathematical skills among psychology undergraduates between 1992 and 2002.

Teaching statistics and research methods to psychology undergraduates is a major pedagogic challenge. Knowledge of students' conceptual problems in mathematics is important in the current climate of widening access, a burgeoning interest in psychology, and fears about declining standards of numeracy and other quantitative skills. This study compared the mathematical knowledge of two cohorts of undergraduates who entered psychology a decade apart--one in 1992, the other in 2002. Six broadly defined components of mathematical thinking relevant to the teaching of statistics in psychology were examined--calculation, algebraic reasoning, graphical interpretation, proportionality and ratio, probability and sampling, and estimation. Both cohorts were also compared with a 1984 cohort on a subset of items reported in a study by Greer and Semrau (1984). Results revealed highly significant differences between the two cohorts on all six components, with 1992 students outperforming their 2002 counterparts. Males were also found to perform significantly better than females on a majority of components. Level of qualification in mathematics was found to predict overall performance. Comparison with Greer and Semrau's (1984) sample revealed an alarming decline in performance across the two decades on a selection of test items.

Adolescent↗