The Nctm Standards and the Philosophy of Mathematics

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy.     

Why psychology needs conceptual analysts: Wachtel's “discontents” revisited

Commenting on Wachtel's “Investigation and its Discontents” [Wachtel, P. L. (1980). Investigation and its discontents: Some constraints on progress in psychological research. American Psychologist, 35, 399–408], I agree that lack of support for a career path of conceptual analysis and critique is a serious problem. Psychology requires a strong conceptual component because it is subject to unusual distortions and self-deceptions in theory formation and evidential evaluation due to issues of power, self-esteem, and social ideology. I agree with Wachtel that pressures for quantity of publication are detrimental to scientific quality, but dispute his suggestion that excessive focus on quantity in assessing productivity can be addressed by having tenure and promotion candidates submit only their three best papers. Such reviews must be based on the entire record, so improvements must involve journals’ acceptance standards. Regarding Wachtel's concern about the influence of grants on reviews, I argue that grants should be relevant only to the extent they bear on the candidate's scholarly goals.     

On radical conceptual revolutions in social science

The paper considers arguments for and against correction and elimination of the basic conceptual categories as well as theories of social science. It is argued that some correction of at least some basic social notions is called for. A great part of the paper consists in a conceptual investigation of such notion of correction in terms of different notions of corrective explanation.     

Insight in frontotemporal dementia: conceptual analysis and empirical evaluation of the consensus criterion "loss of insight" in frontotemporal dementia

The objective of this study was to suggest a new formulation of the core research diagnostic consensus criterion "loss of insight" in frontotemporal dementia (FTD). Eight patients with FTD (diagnoses made by interviews, medical and neuropsychological examination, CT scan, and regional cerebral glucose metabolism measured by positron emission tomography (PET) participated in the study). The results indicated that insight was present in three out of eight patients, and that insight appears to be a heterogeneous concept. Two types of insight emerged: Emotional insight associated with frontotemporal functions, and cognitive insight, related to posterior cognitive functions. These results suggest that loss of insight should not serve as a core criterion on FTD, but serves well as a supportive criterion of the disease.     

The role of diverse instruction in conceptual change

This study examined how a fundamental principle of induction and scientific reasoning, information diversity, could be used to promote change in children's mental models of the earth's shape. Six-year-old children (N=132) were randomly allocated to a control or to one of two training conditions. Some training groups received instruction that simultaneously challenged children's beliefs concerning (a) why the earth appears flat to a surface observer and (b) the role of gravity. Others received instruction that repeatedly challenged only one of these beliefs. An adaptation of the Vosniadou and Brewer (1992, Cognitive Psychology 24, 535-585) protocol for identifying mental models of the earth was administered before and after instruction. Both instruction methods produced increases in factual knowledge. Only children receiving instruction about two core beliefs, however, showed an increased rate of acceptance of a spherical earth model at posttest. The findings show that instruction that challenges diverse aspects of children's na?ve scientific beliefs is more likely to produce conceptual change.     

Metamorphosed characters in dreams: constraints of conceptual structure and amount of theory of mind

Dream reports from 21 dreamers in which a metamorphosis of a person-like entity or animal occurred were coded for characters and animals and for inner states attributed to them (Theory of Mind). In myths and fairy tales, Kelly and Keil (1985) found that conscious beings (people, gods) tend to be transformed into entities nearby in the conceptual structure of Keil (1979). This also occurred in dream reports, but perceptual nearness seemed more important than conceptual nearness. In dream reports, most inanimate objects involved in metamorphoses with person-like entities were objects such as statues that ordinarily resemble people physically, and moreover represent people. A metamorphosis of a person-like entity or animal did not lead to an increase in the amount of Theory of Mind attribution. We propose that a character-line starts when a character enters a dream; properties and Theory of Mind attributions tend to be preserved along the line, regardless of whether, metamorphoses occur on it.     

Fictionalism and the attitudes

This paper distinguishes revolutionary fictionalism from other forms of fictionalism and also from other philosophical views. The paper takes fictionalism about mathematical objects and fictionalism about scientific unobservables as illustrations. The paper evaluates arguments that purport to show that this form of fictionalism is incoherent on the grounds that there is no tenable distinction between believing a sentence and taking the fictionalist's distinctive attitude to that sentence. The argument that fictionalism about mathematics is ‘comically immodest’ is also evaluated. In place of those arguments, an argument against fictionalism about abstract objects of any kind is presented in the last section. This argument takes the form of a trilemma against the fictionalist.

Revisiting preschoolers' living things concept: a microgenetic analysis of conceptual change in basic biology

Many preschoolers know that plants and animals share basic biological properties, but this knowledge does not usually lead them to conclude that plants, like animals, are living things. To resolve this seeming paradox, we hypothesized that preschoolers largely base their judgments of life status on a biological property, capacity for teleological action, but that few preschoolers realize that plants possess this capacity. To test the hypothesis, we taught 5-year-olds one of four biological facts and examined the children's subsequent categorization of life status for numerous animals, plants, and artifacts. As predicted, a large majority of 5-year-olds who learned that both plants and animals, but not artifacts, move in goal-directed ways inferred that both plants and animals, but not artifacts, are alive. These children were considerably more likely to draw this inference than peers who learned that the same plants and animals grow or need water and almost as likely to do so as children who were explicitly told that animals and plants are living things and that artifacts are not. Results also indicated that not all biological properties are extended from familiar animals to plants; some biological properties are first attributed to plants and then extended to animals.     

中小学生数学知识观的调查研究

本研究采用12道假设性情境题目和包含数学涉及运算、思考、实用性三个维度的数学认识问卷,选取90名六年级小学生和106名初二学生作为被试,探讨中小学生的数学知识观。结果表明:1)中小学生数学知识观形成及发展与学校数学课程内容紧密相关。2)中小学生数学知识观存在差异,主要表现在初二学生对假设性情境的认同程度显著地高于六年级学生;在数学认识问卷的数学实用性维度上,六年级学生的肯定程度显著地高于初二学生。3)从总体上看,学生的学业水平与数学知识观的关系不大。     

The development of conceptual interests in young children

We conducted a longitudinal analysis of the relative intensity and duration of interests associated with conceptual domains between the ages of 4 and 6 years, respectively. Results indicated a significant portion of preschool children do sustain an interest in conceptual domains during some portion of their childhood. Expected gender differences were found, with boys more likely to express an interest in a conceptual domain than girls. A latent growth curve analysis revealed that the probability of exhibiting a conceptual interest declined as school began, though the rate of that decline was similar for both boys and girls. Potential explanations for the decrease in conceptual interests as school begins are considered.     

How Gödel Relates Platonism to Mathematics

It is well known that Gödel takes his realistic world view as closely related to mathematics, especially to his own work in the foundations of mathematics. He reports, publicly as well as privately, that Platonism is fundamental to his major work in logic and set theory, and suggests that this philosophical position can be seen as a product of reflections on mathematics. These views of Gödel, however, are often regarded as being insufficiently formulated or argued for. In this article, the author tries to consider some points which are related to the understanding of the Gödelian mode of the interaction between mathematics and philosophy.     

The effects of ageing and visual noise on conceptual integration during sentence reading

The effortfulness hypothesis implies that difficulty in decoding the surface form, as in the case of age-related sensory limitations or background noise, consumes the attentional resources that are then unavailable for semantic integration in language comprehension. Because ageing is associated with sensory declines, degrading of the surface form by a noisy background can pose an extra challenge for older adults. In two experiments, this hypothesis was tested in a self-paced moving window paradigm in which younger and older readers’ online allocation of attentional resources to surface decoding and semantic integration was measured as they read sentences embedded in varying levels of visual noise. When visual noise was moderate (Experiment 1), resource allocation among young adults was unaffected but older adults allocated more resources to decode the surface form at the cost of resources that would otherwise be available for semantic processing; when visual noise was relatively intense (Experiment 2), both younger and older participants allocated more attention to the surface form and less attention to semantic processing. The decrease in attentional allocation to semantic integration resulted in reduced recall of core ideas in both experiments, suggesting that a less organized semantic representation was constructed in noise. The greater vulnerability of older adults at relatively low levels of noise is consistent with the effortfulness hypothesis.     

The effect of prosody on conceptual combination

Research into people's comprehension of novel noun-noun phrases has long neglected the possible influences of prosody during meaning construction. At the same time, work in conceptual combination has disagreed about whether different classes of interpretation emerge from single or multiple processes; for example, whether people use distinct mechanisms when they interpret octopus apartment as property-based (e.g., an apartment with eight rooms) or relation-based (e.g., an apartment where an octopus lives). In two studies, we manipulate the prosodic emphasis patterns of novel noun-noun combinations (placing stress on the modifier noun, the head noun, or dual stress on both nouns) and ask participants to generate an interpretation for the novel phrase. Results show that people are faster to generate property-based interpretations when dual emphasis stresses both nouns equally, with prosody having little effect on the speed of relation-based interpretations. These findings highlight a role for prosody during meaning construction and underline important differences between relation- and property-based interpretations that are difficult to reconcile with unitary process views of conceptual combination.     

The development of interpretations for novel noun-noun conceptual combinations during the early school years

This research investigates issues surrounding early school children's use of the similarity between head and modifier terms in deriving interpretations for novel noun-noun conceptual combinations. In these experiments, 6- and 9-year-olds and adults were asked to formulate interpretations of similar and dissimilar conceptual combinations. Both children and adults were sensitive to the similarity aspect of conceptual combinations, although the children had some difficulty with the property interpretations that high-similarity combinations require. Next, we examined 40 popular children's books for the presence of noun-noun conceptual combinations. Adult participants provided interpretations for these combinations and rated the similarity of the head and modifier nouns. Results indicated that there were few high-similarity combinations and few combinations requiring property interpretations, suggesting that children have limited exposure to highly similar combinations and property interpretations. Further analysis of children's interpretations indicates that they may have difficulty in selecting and integrating properties of the modifier onto the head, a process required by property interpretations.     

Knowledge, expectations, and inductive reasoning within conceptual hierarchies

Previous research (e.g. Cognition 64 (1997) 73) suggests that the privileged level for inductive inference in a folk biological conceptual hierarchy does not correspond to the “basic” level (i.e. the level at which concepts are both informative and distinct). To further explore inductive inference within conceptual hierarchies, we examine relations between knowledge of concepts at different hierarchical levels, expectations about conceptual coherence, and inductive inference. In Experiments 1 and 2, 5- and 8-year-olds and adults listed features of living kind (Experiments 1 and 2) and artifact (Experiment 2) concepts at different hierarchical levels (e.g. plant, tree, oak, desert oak), and also rated the strength of generalizations to the same concepts. For living kinds, the level that showed a relative advantage on these two tasks differed; the greatest increase in features listed tended to occur at the life-form level (e.g. tree), whereas the greatest increase in inductive strength tended to occur at the folk-generic level (e.g. oak). Knowledge and induction also showed different developmental trajectories. For artifact concepts, the levels at which the greatest gains in knowledge and induction occurred were more varied, and corresponded more closely across tasks. In Experiment 3, adults reported beliefs about within-category similarity for concepts at different levels of animal, plant and artifact hierarchies, and rated inductive strength as before. For living kind concepts, expectations about category coherence predicted patterns of inductions; knowledge did not. For artifact concepts, both knowledge and expectations predicted patterns of induction. Results suggest that beliefs about conceptual coherence play an important role in guiding inductive inference, that this role may be largely independent of specific knowledge of concepts, and that such beliefs are especially important in reasoning about living kinds.     

A conceptual metaphor framework for the teaching of mathematics

Word problems in mathematics seem to constantly pose learning difficulties for all kinds of students. Recent work in math education (for example, [Lakoff, G. &; Nuñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books]) suggests that the difficulties stem from an inability on the part of students to decipher the metaphorical properties of the language in which such problems are cast. A 2003 pilot study [Danesi, M. (2003a). Semiotica, 145, 71–83] confirmed this hypothesis in an anecdotal way. This paper reviews the implications of that study and of a follow-up one that is described here as well, in the light of how the metaphorical analysis of word problems allows learners to overcome typical difficulties in word problem-solving by teaching them how to flesh out the underlying concepts and convert them into appropriate representations.     

Attentional factors in conceptual congruency

Conceptual congruency effects are biases induced by an irrelevant conceptual dimension of a task (e.g., location in vertical space) on the processing of another, relevant dimension (e.g., judging words' emotional evaluation). Such effects are a central empirical pillar for recent views about how the mind/brain represents concepts. In the present paper, we show how attentional cueing (both exogenous and endogenous) to each conceptual dimension succeeds in modifying both the manifestation and the symmetry of the effect. The theoretical implications of this finding are discussed.     

Examining the Relationship Between Mathematics Anxiety and Mathematics Performance: An Instructional Hierarchy Perspective

This study investigated the relationship between mathematics anxiety, fluency, and error rates in basic mathematical operations among college students. College students were assigned to one of two groups (high anxiety or low anxiety) based on results from the Fennema-Sherman Mathematics Anxiety Scale (FSMAS). Both groups were then presented with timed tests in basic mathematical operations (addition, subtraction, multiplication, division, and linear equations). Results suggested that the higher mathematics anxiety group had significantly lower fluency levels across all mathematical operations tests. However, there were no significant differences in error rates between the two groups across any of the probes suggesting that mathematics anxiety is more related to higher levels of learning than to the initial acquisition stage of learning. Discussion focuses on a) stages of learning and their potential relationship to mathematics anxiety, b) the relationship between mathematics anxiety and mathematics performance, and c) directions for future research.