Mathematical thought

This study investigates the structure and development of quantitative thought in Greece and India. A total of 297 Indian subjects and 269 Greek subjects, aged from 10 to 16 years, were examined by a battery addressing their ability to execute arithmetic operations, a battery addressing their proportional reasoning, and a battery addressing algebraic reasoning. The items in each battery addressed four developmental levels. Confirmatory factor analysis showed, as predicted, that the same model is able to account for the performance of both cultures. Some differences were observed in the relative strength of the various abilities. However, the developmental inter-patterning of abilities was generally the same in the two cultures. There were no differences in capacity-dependent sequences but there were some differences in strategy-dependent sequences. These findings are discussed in the context of our theory of cognitive development.

The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects' gender and SES. (a) Structural equations modelling revealed three specific ability factors (execution of arithmetic operations, proportional, and algebraic reasoning) and a higher-order general factor that loaded all specific factors. The higher-order factor may be abolished once the relations between the first order factors are specified. (b) Recently developed techniques (discrimination and rating scale analysis) basically confirmed the developmental levels initially assigned to most of the tasks. More importantly, the tasks pertaining to the different partial abilities could be ordered along a single developmental dimension with items from different abilities intertwined. This intertwining suggested that each ability's development precipitates and is precipitated by the development of the other abilities. (c) The structure of abilities was found not to be affected by individual differences factors. (d) Using the unified developmental scale as a frame of reference, gender and SES differences were found at age points associated with major developmental changes. Implications of these findings for theories of cognitive development and education are discussed within the framework of experiential structuralism.

This study investigates the structure and development of quantitative thought in Greece and India. A total of 297 Indian subjects and 269 Greek subjects, aged from 10 to 16 years, were examined by a battery addressing their ability to execute arithmetic operations, a battery addressing their proportional reasoning, and a battery addressing algebraic reasoning. The items in each battery addressed four developmental levels. Confirmatory factor analysis showed, as predicted, that the same model is able to account for the performance of both cultures. Some differences were observed in the relative strength of the various abilities. However, the developmental inter-patterning of abilities was generally the same in the two cultures. There were no differences in capacity-dependent sequences but there were some differences in strategy-dependent sequences. These findings are discussed in the context of our theory of cognitive development.

En este artículo se presentan los resultados de la investigación “La influencia de la tecnología en el aprendizaje de las asignaturas Cálculo Diferencial y Estadística Descriptiva en estudiantes del ciclo básico de Ingeniería”. Para ello se utilizó un diseño cuasiexperimental con grupos de control y experimental con aplicación sólo de un postest. Se utilizaron los programas Derive y Statgraphics en Cálculo Diferencial y Estadística Descriptiva, respectivamente. Se encontró que en los grupos experimentales en los cuales se enseñó a los estudiantes con el Derive y el Statgraphics (N = 25 y N =25) tuvieron un ligero mejor desempeño que los estudiantes de los grupos de control (N = 22 y N = 16) que realizaron el curso sin el uso de estos programas. Estadísticamente no se halló una diferencia significativa.

Mathematics researchers generally agree that algebra is a tool for problem solving, a method of expressing relationship, analyzing and representing patterns, and exploring mathematical properties in a variety of problem situations. Thus, several mathematics researchers and educators have focused on investigating the introduction and the development of algebraic solving abilities. However research works on assessing students' algebraic solving ability is sparse in literature. The purpose of this study was to use the SOLO model as a theoretical framework for assessing Form Four students' algebraic solving abilities in using linear equation. The content domains incorporated in this framework were linear pattern (pictorial), direct variations, concepts of function and arithmetic sequence. This study was divided into two phases. In the first phase, students were given a pencil-and-paper test. The test comprised of eight superitems of four items each. Results were analyzed using a Partial Credit model. In the second phase, clinical interviews were conducted to seek the clarification of the students' algebraic solving processes. Results of the study indicated that 62% of the students have less than 50% probability of success at relational level. The majority of the students in this study could be classified into unistructural and multistructural. Generally, most of the students encountered difficulties in generalizing their arithmetic thinking through the use of algebraic symbols. The qualitative data analysis found that the high ability students seemed to be more able to seek the recurring linear pattern and identify the linear relationship between variables. They were able to coordinate all the information given in the question to form the algebraic expression and linear equations. Whereas, the low ability students showed an ability more on drawing and counting method. They lacked understanding of algebraic concepts to express the relationship between the variables. The results of this study provided evidence on the significance of SOLO model in assessing algebraic solving ability in the upper secondary school level.

The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects' gender and SES. (a) Structural equations modelling revealed three specific ability factors (execution of arithmetic operations, proportional, and algebraic reasoning) and a higher-order general factor that loaded all specific factors. The higher-order factor may be abolished once the relations between the first order factors are specified. (b) Recently developed techniques (discrimination and rating scale analysis) basically confirmed the developmental levels initially assigned to most of the tasks. More importantly, the tasks pertaining to the different partial abilities could be ordered along a single developmental dimension with items from different abilities intertwined. This intertwining suggested that each ability's development precipitates and is precipitated by the development of the other abilities. (c) The structure of abilities was found not to be affected by individual differences factors. (d) Using the unified developmental scale as a frame of reference, gender and SES differences were found at age points associated with major developmental changes. Implications of these findings for theories of cognitive development and education are discussed within the framework of experiential structuralism.

A way to argue that something (e.g. mathematics, idealizations, moral properties, etc.) plays an explanatory role in science is by linking explanatory relevance with importance in the context of an explanation. The idea is deceptively simple: a part of an explanation is an explanatorily relevant part of that explanation if removing it affects the explanation either by destroying it or by diminishing its explanatory power, i.e. an important part (one that if removed affects the explanation) is an explanatorily relevant part. This can be very useful in many ontological debates. My aim in this paper is twofold. First of all, I will try to assess how this view on explanatory relevance can affect the recent ontological debate in the philosophy of mathematics—as I will argue, contrary to how it may appear at first glance, it does not help very much the mathematical realists. Second of all, I will show that there are big problems with it.

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics can do to make an explanation mathematical; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism (in this journal). I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist.

One way of thinking about explanations that are mind-independent in
science is that it answers “what-if-things-had-been-different” questions, supplying information about what changes in some phenomena if some change is made—these sort of statements are counterfactuals. Often in science there is physical phenomena that is not able to be explained in terms other than those of the mathematics, as opposed to explanations where mathematics seems to play the mere role of representing causal or physical facts. This raises a question: could the sort of dependence of physical facts on mathematical facts be counterfactual? In this dissertation I will argue that a counterfactual theory of extramathematical explanation can appeal to a metaphysical necessitation constraint to distinguish between cases where mathematics plays a merely representational role in explaining some physical phenomena, and cases where there is genuine mathematical explanations for the physical phenomena, a challenge raised by Baron (2016b) that I will call the explanatory genuineness problem. Using a new example of the usage of coprimeness facts in computer encryption algorithms, which is analogous to the usage of coprimeness facts in the explanation of cicada life-cycles, I argue that the fact that mathematical explanations can occur in multiple physically distinct systems is evidence in favour of the view that mathematical facts that play a genuine explanatory role represent truths about metaphysical
necessity, thus accounting for metaphysical necessitation. However, appealing to the metaphysical necessitation constraint introduces a further challenge: as evaluating a counterfactual under the metaphysical necessitation constraint will involve evaluating how a physical system is affected by the hypothetical changing of a mathematical fact at an impossible world, I argue that there is competing ways to model the effects of the same mathematical
intervention. This shows us that it is not exactly clear what aspects
of mathematical facts are representing the underlying metaphysical constraints. I conclude that even if a demarcation can be drawn between mathematical facts that play a representational and genuine explanatory role using metaphysical necessitation, a truly counterfactual theory of eme must give some epistemic position for why some evaluation of a mathematical counterfactual is agent independent.

I present an informationist alternative to Platonist, in re realist, and Nominalist appraisals of the nature of mathematical entities and structures and their explanatory power. Informational conceptions of the nature of mathematical entities are considered conceptually taxing due to problems with conceptions of information. I argue that, even given this difficulty, the causal informationist about mathematical entities can offer a story that is just as coherent as that given by the Aristotelian in re realist about mathematical entities, is more plausible than soft Platonism, and that debunks the Platonist petitio-principii-cum-strawman charge that intractable nominalism is the only viable alternative.

This presentation is based upon a paper originally produced in 2011 and presented in Otago, Dunedin at the Australasian Postgraduate Philosophy Conference in a different guise https://www.aap.org.au/Resources/Documents/conference/2011-2020/aap2011-aapnz2011-program-and-abstracts.pdf It is also derived from an original chapter in my PhD thesis originating 2010-11.

This paper reconstructs the relationship between the now, motion, and number in Aristotle to clarify the nature of the now, and, thereby, the relationship between motion and time. Although it is clear that for Aristotle motion, and, more generally, change, are prior to time, the nature of this priority is not clear. But if time is the number of motion, then the priority of motion can be grasped by examining his theory of number. This paper aims to show that, just as numbers are generated by the soul, time is not presupposed by motion, but emerges through the soul’s articulation of motion. Time is co-constituted by the soul and motion. The now is the key to understanding both the contribution of motion and soul to the being of time. The now is part of the soul’s articulation of motion, and sets the stage for an act that distinguishes a unit from its underlying motion. The now, then, sets up an abstraction by which the soul generates the temporal number from motion. Reconstructing this account of abstraction allows us to formulate more strongly Aristotle’s claim to the ontological dependence of time on motion. The paper then gives a systematic overview of the relationships between the now and number in order to address the question of whether the now might be extended. It closes with an examination of the possibility that motion depends on time, and how universal time is possible.

The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects' gender and SES. (a) Structural equations modelling revealed three specific ability factors (execution of arithmetic operations, proportional, and algebraic reasoning) and a higher-order general factor that loaded all specific factors. The higher-order factor may be abolished once the relations between the first order factors are specified. (b) Recently developed techniques (discrimination and rating scale analysis) basically confirmed the developmental levels initially assigned to most of the tasks. More importantly, the tasks pertaining to the different partial abilities could be ordered along a single developmental dimension with items from different abilities intertwined. This intertwining suggested that each ability's development precipitates and is precipitated by the development of the other abilities. (c) The structure of abilities was found not to be affected by individual differences factors. (d) Using the unified developmental scale as a frame of reference, gender and SES differences were found at age points associated with major developmental changes. Implications of these findings for theories of cognitive development and education are discussed within the framework of experiential structuralism.

The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects' gender and SES. (a) Structural equations modelling revealed three specific ability factors (execution of arithmetic operations, proportional, and algebraic reasoning) and a higher-order general factor that loaded all specific factors. The higher-order factor may be abolished once the relations between the first order factors are specified. (b) Recently developed techniques (discrimination and rating scale analysis) basically confirmed the developmental levels initially assigned to most of the tasks. More importantly, the tasks pertaining to the different partial abilities could be ordered along a single developmental dimension with items from different abilities intertwined. This intertwining suggested that each ability's development precipitates and is precipitated by the development of the other abilities. (c) The structure of abilities was found not to be affected by individual differences factors. (d) Using the unified developmental scale as a frame of reference, gender and SES differences were found at age points associated with major developmental changes. Implications of these findings for theories of cognitive development and education are discussed within the framework of experiential structuralism.

DEMETRIOU, ANDREAS, and EFKLIDES, ANASTASIA. Structure and Sequence of Formal and Postformal Thought: General Patterns and Individual Differences. CHILD DEVELOPMENT, 1985, 56, 1062-1091. According to Demetriou and Efklides's model, formal thought develops at 2 levels, those of strategy and tactics. This model postulates that tactics are structured in distinct capacity spheresthe relational, the experimental, and the correlational-probabilistic-which develop in a number of steps. The present study attempted to test most of these assumptions. For this purpose, 400 subjects, sampled from different age, sex, and SES groups, were examined by a battery of appropriate tasks. These tasks tested strategic, relational, experimental, and also postformal abilities. The application of factor analysis validated the existence of the relational and the experimental sphere. The strategic level was not clearly identified. However, both a late formal capacity and a postformal capacity for conceiving of possibilities and reflecting upon them were revealed. Joint application of discrimination levels analysis, and analysis of covariance, showed the various capacities developed in the steps described by the model. Sex and SES differences were also found. These results showed that the Demetriou and ElkHdes model could be integrated with the recent theories claiming that thought is structured in different inquiring systems, and that it develops postformally as systematic, metasystematic, and epistemic cognition. Flavell's notions of modification, inclusion, and mediation sequences were employed in order to interpret intra-and intersphere developmental relations. Individual differences were also discussed.

CITATIONS 28 READS 72 5 authors, including: Some of the authors of this publication are also working on these related projects: Specifying the role of cognizance (self-awareness about mental processes) in the development of intelligence from 4 to 16 years of age. View project please change my email address to [email protected] View project Abstract

This study investigates the structure and development of quantitative thought in Greece and India. A total of 297 Indian subjects and 269 Greek subjects, aged from 10 to 16 years, were examined by a battery addressing their ability to execute arithmetic operations, a battery addressing their proportional
reasoning, and a battery addressing algebraic reasoning. The items in each battery addressed four developmental levels. ConŽfirmatory factor analysis showed, as predicted, that the same model is able to account for the performance of both cultures. Some differences were observed in the relative strength of the various abilities. However, the developmental inter-patterning
of abilities was generally the same in the two cultures. There were no differences in capacity-dependent sequences but there were some differences in strategy dependent sequences. These findings are discussed in the context of our theory of cognitive development.

The present article discusses the nature of the quantitative-relational system, which involves the basic arithmetic operations, proportional reasoning, algebraic ability, and the integration
of these three abilities. Representative sets of tasks that assess each of these abilities were administered to 372 Greek subjects aged 9 to 16 to test hypotheses about (a) the structure
and (b) the development of this cognitive system, and (c) about the role of individual differences variables, such as subjects’ gender and SES. (a) Structural equations modelling revealed three specific ability factors (execution of arithmetic operations, proportional, and algebraic reasoning) and a higher-order general factor that loaded all specific factors. The higher-order factor may be abolished once the relations between the first order factors
are specified. (b) Recently developed techniques (discrimination and rating scale analysis) basically confirmed the developmental levels initially assigned to most of the tasks. More importantly, the tasks pertaining to the different partial abilities could be ordered along a single developmental dimension with items from different abilities intertwined. This intertwining suggested that each ability’s development precipitates and is precipitated by the development of the other abilities. (c) The structure of abilities was found not
to be affected by individual differences factors. (d) Using the unified developmental scale as a frame of reference, gender and SES differences were found at age points associated with major developmental changes. Implications of these findings for theories
of cognitive development and education are discussed within the framework of experiential structuralism

"A method of scaling called discrimination analysis was applied to data collected independently by the three authors on the performance of large samples of children aged 6 to 12 on a variety of concrete operational tasks in England, Greece, Australia, and Pakistan. The same task was found to scale at the same level, regardless of the country or the age of the child. Decalage between different children's performances on the same tasks was found to be confined within the limits of the three nodes of development described by Longeot (1978). A factor-analytical
study showed that the d£calage could be explained in terms of a simple mentalabilities model, with the tasks representing three capacities: quantitative-relational, qualitative-analytic, and imaginal-spatial. The scaling method yielded a measure of
the overall cognitive development of the child precise enough to be used in an intervention study, and a meta-analytical procedure was shown by which previously published studies may be scaled retrospectively. The results were discussed in relation to recent neo-Piagetian theories."

Dans cet article je propose d'exprimer et de défendre une conception des pratiques et du domaine de discours mathématiques qui soit sensible, d'une part, au pluralisme des relations entre pratiques inférentielles et intérêts, et d'autre part, à la structure objective et déterminante des concepts mathématiques. J'ébauche tout d'abord une caractérisation générale des pratiques, pour ensuite préciser certains phénomènes propres aux pratiques mathématiques. Suit un recensement des idées qui se dégagent des arguments pluralistes, et de celles qui sont à retenir. Mais je défends par la suite la nécessité d'une forme de réalisme mathématique, qui toutefois ne peut être le réalisme d'objets que prônent les partisans de l'argument d'indispensabilité. Je défends plutôt un réalisme des concepts, soutenu par ce que je baptise l'« argument de la contrainte ».