A versatile approach to calculus and numerical methods

Traditionally the calculus is the study of the symbolic algorithms for differentiation and integration, the relationship between them, and their use in solving problems. Only at the end of the course, when all else fails, are numerical methods introduced, such as the Newton-Raphson method of solving equations, or Simpson’s rule for calculating areas. The problem with such an approach is that it often produces students who are very well versed in the algorithms and can solve the most fiendish of symbolic problems, yet have no understanding of the meaning of what they are doing. Given the arrival of computer software which can carry out these algorithms mechanically, the question arises as to what parts of calculus need to be studied in the curriculum of the future. It is my contention that such a study can use the computer technology to produce a far more versatile approach to the subject, in which the numerical and graphical representations may be used from the outset to produce insights into the fundamental meanings, in which a wider understanding of the processes of change and growth will be possible than the narrow band of problems that can be solved by traditional symbolic methods of the calculus

Exploring the episodic structure of algebra story problem solving

This paper analyzes the quantitative and situational structure of algebra story problems, uses these materials to propose an interpretive framework for written problem-solving protocols, and then presents an exploratory study of the episodic structure of algebra story problem solving in a sizable group of mathematically competent subjects. Analyses of written protocols compare the strategic, tactical, and conceptual content of solution attampts, looking within these attempts at the interplay between problem comprehension and solution. Comprehension and solution of algebra story problems are complimentary activities, giving rise to a succession of problem solving episodes. While direct algebraic problem solving is sometimes effective, results suggest that the algebraic formalism may be of little help in comprehending the quantitative constraints posed in a problem text. Instead, competent problem solvers often reason within the situational context presented by a story problem, using various forms of "model-based reasoning" to identify, pursue, and verify quantitative constraints required for solution. The paper concludes by discussing the implications of these findings for acquiring mathematical concepts (e.g., related linear functions) and for supporting their acquisition through instruction

Degrees of Propositionality in Construals of Time Quantities1

The paper investigates the possible conceptual bases of differences between seemingly synonymous and easily definable temporal expressions. Looking at the usage patterns of nominal temporal phrases in reference corpora of English and Polish we attempt to relate these subtleties to the different granularity of the cognitive scales on which construals of time quantities in general are based. More specifically, we focus on a subset of nominal temporal expressions which adhere to the “number + time unit” pattern, matching what Haspelmath (1997: 26) describes as “culture-bound artificial time units”. Using the British National Corpus (BNC) and the National Corpus of Polish (NCP), we first analyse both the variation and the regularity found in naturally-occurring samples of Polish and English. Finally, we compare the patterns of use emerging from the two corpora and arrive at cross-linguistic generalisations about the conceptualisation of time quantities

The use of colloquial words in advanced French interlanguage

This article addresses the issue of underrepresentation or avoidance of colloquial words in a cross-sectional corpus of advanced French interlanguage (IL) of 29 Dutch L1 speakers and in a longitudinal corpus of 6 Hiberno-Irish English L1 speakers compared with a control of 6 native speakers of French. The main independent variable analysed in the latter corpus is the effect of spending a year in a francophone environment. This analysis is supplemented by a separate study of sociobiographical and psychological factors that affect the use of colloquial vocabulary in the cross-sectional corpus. Colloquial words are not exceptionally complex morphologically and present no specific grammatical difficulties, yet they are very rare in our data. Multivariate regression analyses suggest that only active authentic communication in the target language (TL) predicts the use of colloquial lexemes in the cross-sectional corpus. This result was confirmed in the longitudinal corpus where a t-test showed that the proportion of colloquial lexemes increased significantly after a year abroad

Abstraction as a basis for the computational interpretation of creative cross-modal metaphor

Various approaches to computational metaphor interpretation are based on pre-existing similarities between source and target domains and/or are based on metaphors already observed to be prevalent in the language. This paper addresses similarity-creating cross-modal metaphoric expressions. It is shown how the “abstract concept as object” (or reification) metaphor plays a central role in a large class of metaphoric extensions. The described approach depends on the imposition of abstract ontological components, which represent source concepts, onto target concepts. The challenge of such a system is to represent both denotative and connotative components which are extensible, together with a framework of general domains between which such extensions can conceivably occur. An existing ontology of this kind, consistent with some mathematic concepts and widely held linguistic notions, is outlined. It is suggested that the use of such an abstract representation system is well adapted to the interpretation of both conventional and unconventional metaphor that is similarity-creating

Conceptual spatial representations for indoor mobile robots

We present an approach for creating conceptual representations of human-made indoor environments using mobile robots. The concepts refer to spatial and functional properties of typical indoor environments. Following findings in cognitive psychology, our model is composed of layers representing maps at different levels of abstraction. The complete system is integrated in a mobile robot endowed with laser and vision sensors for place and object recognition. The system also incorporates a linguistic framework that actively supports the map acquisition process, and which is used for situated dialogue. Finally, we discuss the capabilities of the integrated system

Learning by Seeing by Doing: Arithmetic Word Problems

Learning by doing in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children\u27s ability to see significant abstract structures in mathematical situations. The reflection necessary for such seeing is motivated by activities and contexts that emphasize affective and social aspects. Natural language, as a representation already familiar to children, is key in these activities, both as a means of mathematical expression and as a link between situations and various abstract representations. These tools support children\u27s ownership of a mathematical problem and its expression; remote sharing of problems and data; software interpretation of children\u27s own word problems; play with dynamically linked representations with attention to children\u27s prior connections; and systematic problem variation based on empirically determined level of difficulty

Loop quantum cosmology and inhomogeneities

Inhomogeneities are introduced in loop quantum cosmology using regular lattice states, with a kinematical arena similar to that in homogeneous models considered earlier. The framework is intended to encapsulate crucial features of background independent quantizations in a setting accessible to explicit calculations of perturbations on a cosmological background. It is used here only for qualitative insights but can be extended with further more detailed input. One can thus see how several parameters occuring in homogeneous models appear from an inhomogeneous point of view. Their physical roles in several cases then become much clearer, often making previously unnatural choices of values look more natural by providing alternative physical roles. This also illustrates general properties of symmetry reduction at the quantum level and the roles played by inhomogeneities. Moreover, the constructions suggest a picture for gravitons and other metric modes as collective excitations in a discrete theory, and lead to the possibility of quantum gravity corrections in large universes.Comment: 25 pages, 1 figur

Complexity over Uncertainty in Generalized Representational\ud Information Theory (GRIT): A Structure-Sensitive General\ud Theory of Information

What is information? Although researchers have used the construct of information liberally to refer to pertinent forms of domain-specific knowledge, relatively few have attempted to generalize and standardize the construct. Shannon and Weaver(1949)offered the best known attempt at a quantitative generalization in terms of the number of discriminable symbols required to communicate the state of an uncertain event. This idea, although useful, does not capture the role that structural context and complexity play in the process of understanding an event as being informative. In what follows, we discuss the limitations and futility of any generalization (and particularly, Shannon’s) that is not based on the way that agents extract patterns from their environment. More specifically, we shall argue that agent concept acquisition, and not the communication of\ud states of uncertainty, lie at the heart of generalized information, and that the best way of characterizing information is via the relative gain or loss in concept complexity that is experienced when a set of known entities (regardless of their nature or domain of origin) changes. We show that Representational Information Theory perfectly captures this crucial aspect of information and conclude with the first generalization of Representational Information Theory (RIT) to continuous domains