Constructivism and the Role of Skills in Mathematics Instruction for Academically At-Risk Secondary Students

Summary

Instructional uses of technology in special education have evolved considerably over the last two decades. Many researchers have moved away from stand alone uses (e.g., computer assisted instruction) toward an array of different technologies that serve as tools in complex learning environments. A change in thinking about teaching and learning has also occurred, as researchers have moved away from didactic instructional methods to constructivist approaches. Yet constructivism remains problematic for many in the field, in part because of the traditional emphasis on skills in day-to-day instruction. This article describes how skills instruction can be an important feature of constructivism for teaching special education students.     

Memory abilities in generally gifted and excelling-in-mathematics adolescents

This paper presents part of a multidimensional examination of mathematical giftedness. The present study examined the memory mechanisms associated with general giftedness (G) and excellence in mathematics (E) in four groups of 10th–12th grade students (16–18 years old) varying in levels of G and E. The participants first underwent the Raven test for general ability evaluation and SAT-M — the mathematical excellence tests in order to design the study groups. Afterwards, the students were tested on a battery of three memory tests including tests for short-term (STM) and working memory (WM). The results reveal that the G factor is related to high STM for both phonological loop and phonological central executive mechanisms. It was also found that the E factor is associated with high visual–spatial memory (VSM), in particular with the visual central executive mechanism. An interaction effect was found between G and E factors regarding WM. The central executive mechanism appeared to be related to both G and E factors. In addition, gender differences were shown within the groups. Male participants performed better than their female counterparts on a phonological storage task and a phonological central executive mechanism task. The results can contribute to the theoretical knowledge regarding similarities and differences in memory mechanisms in G and E groups.     

Derrida and Cavaillès: Mathematics and the Limits of Phenomenology

Abstract

This paper examines Derrida’s interpretation of Jean Cavaillès’s critique of phenomenology in On Logic and the Theory of Science. Derrida’s main claim is that Cavaillès’s arguments, especially the argument based on Gödel’s incompleteness theorems, need not lead to a total rejection of Husserl’s phenomenology, but only its static version. Genetic phenomenology, on the other hand, not only is not undermined by Cavaillès’s critique, but can even serve as a philosophical framework for Cavaillès’s own position. I will argue that Derrida’s approach to Cavaillès is fruitful, facilitating the exposition of some central Cavaillèsian ideas, including the notion of dialectics. Nevertheless, it is important to evaluate Derrida’s own arguments against static phenomenology. I undertake such an assessment in the last section of the paper, showing that Gödel’s theorems do not in themselves warrant rejection of static phenomenology. I base this conclusion in part on Gödel’s own understanding of phenomenology as a philosophical basis for mathematics.     

The Role of Mathematics in Deleuze’s Critical Engagement with Hegel

Abstract

The role of mathematics in the development of Gilles Deleuze’s (1925–95) philosophy of difference as an alternative to the dialectical philosophy determined by the Hegelian dialectic logic is demonstrated in this paper by differentiating Deleuze’s interpretation of the problem of the infinitesimal in Difference and Repetition from that which G. W. F Hegel (1770–1831) presents in the Science of Logic. Each deploys the operation of integration as conceived at different stages in the development of the infinitesimal calculus in his treatment of the problem of the infinitesimal. Against the role that Hegel assigns to integration as the inverse transformation of differentiation in the development of his dialectical logic, Deleuze strategically redeploys Leibniz’s account of integration as a method of summation in the form of a series in the development of his philosophy of difference. By demonstrating the relation between the differential point of view of the Leibnizian infinitesimal calculus and the differential calculus of contemporary mathematics, I argue that Deleuze effectively bypasses the methods of the differential calculus which Hegel uses to support the development of the dialectical logic, and by doing so, sets up the critical perspective from which to construct an alternative logic of relations characteristic of a philosophy of difference. The mode of operation of this logic is then demonstrated by drawing upon the mathematical philosophy of Albert Lautman (1908–44), which plays a significant role in Deleuze’s project of constructing a philosophy of difference. Indeed, the logic of relations that Deleuze constructs is dialectical in the Lautmanian sense.     

Intrinsic Explanation and Field’s Dispensabilist Strategy

Abstract

Hartry Field defended the importance of his nominalist reformulation of Newtonian Gravitational Theory, as a response to the indispensability argument, on the basis of a general principle of intrinsic explanation. In this paper, I argue that this principle is not sufficiently defensible, and can not do the work for which Field uses it. I argue first that the model for Field’s reformulation, Hilbert’s axiomatization of Euclidean geometry, can be understood without appealing to the principle. Second, I argue that our desires to unify our theories and explanations undermines Field’s principle. Third, the claim that extrinsic theories seem like magic is, in this case, really just a demand for an account of the applications of mathematics in science. Finally, even if we were to accept the principle, it would not favor the fictionalism that motivates Field’s argument, since the indispensabilist’s mathematical objects are actually intrinsic to scientific theory.     

Mathematics and Theology: A Stroll through the Garden of Mathaphors

Mathematics has long been considered the language of science and has even been acknowledged as the universal language of the future. Yet mathematics also plays a less-recognized religious role in that metaphors drawn from mathematics (mathaphors) can influence our spiritual perspectives by helping us to entertain new religious ideas and to challenge old ones. Mathaphors are therefore useful tools in matheology (the study of mathematics and theology). This essay explores ten ways in which contemporary mathaphors affect our spiritual lives. Specifically, mathaphors are: changing our metaphors for God; challenging our human role in the universe; helping us accept ambiguity; revamping our understanding of the one and the many; revising our thoughts about free will and determinism; moving us toward pluralistic, multi-world views; pushing the envelope on what consciousness is; altering our expectations for afterlife; offering the hope of a more compassionate future; encouraging faith perspectives that are always incomplete and in process.     

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.     

An example of formalizing recent mathematical results in Mizar

This paper describes an example of the successful formalization of quite advanced and new mathematics using the Mizar system. It shows that although much effort is required to formalize nontrivial facts in a formal computer deduction system, still it is possible to obtain the level of full logical correctness of all inference steps. We also discuss some problems encountered during the formalization, and try to point out some of the features of the Mizar system responsible for that. The formalization described in this paper allows also for contrasting the linguistic capability of the Mizar language and some of the phrases commonly used in “informal” mathematical papers that the Mizar system lacks, and consequently presents the methods of how to cope with it during the formalization. Yet, apart from the problems, this paper shows some definite benefits from using a formal computer system in the work of a mathematician.     

TPS: A hybrid automatic-interactive system for developing proofs

The theorem proving system Tps provides support for constructing proofs using a mix of automation and user interaction, and for manipulating and inspecting proofs. Its library facilities allow the user to store and organize work. Mathematical theorems can be expressed very naturally in Tps using higher-order logic. A number of proof representations are available in Tps, so proofs can be inspected from various perspectives.