THE EFFECT OF PURPOSEFUL MATHEMATICS DISCOURSE IN THE CLASSROOM ON STUDENTS’ MATHEMATICS LANGUAGE IN THE CONTEXT OF PROBLEM SOLVING

This study examines how one secondary school teacher’s use of purposeful oral mathematics language impacted her students’ language use and overall communication in written solutions while working with word problems in a grade nine academic mathematics class. Mathematics is often described as a distinct language. As with all languages, students must develop a sense for oral language before developing social practices such as listening, respecting others ideas, and writing. Effective writing is often seen by students that have strong oral language skills. Classroom observations, teacher and student interviews, and collected student work served as evidence to demonstrate the nature of both the teacher’s and the students’ use of oral mathematical language in the classroom, as well as the effect the discourse and language use had on students’ individual written solutions while working on word problems. Inductive coding for themes revealed that the teacher’s purposeful use of oral mathematical language had a positive impact on students’ written solutions. The teacher’s development of a mathematical discourse community created a space for the students to explore mathematical language and concepts that facilitated a deeper level of conceptual understanding of the learned material. The teacher’s oral language appeared to transfer into students written work albeit not with the same complexity of use of the teacher’s oral expression of the mathematical register. Students that learn mathematical language and concepts better appear to have a growth mindset, feel they have ownership over their learning, use reorganizational strategies, and help develop a discourse community.

From physics to metaphysics: philosophy and style in the critical writings of T. S. Eliot (1913-1935)

This thesis considers Eliot's critical writing from the late 1910s till the mid-1930s, in the light of his PhD thesis - Knowledge and Experience in the Philosophy of F. H. Bradley - and a range of unpublished material: T S. Eliot's Philosophical Essays and Notes (1913- 4) in the Hayward Bequest (King's College, Cambridge University); T. S. Eliot's Family Papers in the T. S. Eliot Collection at the Houghton Library (Harvard University); and items from the Harvard University Archives at the Pusey Library. 'Me thesis offers a comprehensive view of Eliot's critical development throughout this important period. It starts by considering The Sacred Wood's ambivalence towards the metaphysical philosophy of F. H. Bradley and Eliot's apparent adoption of a scientific method, under the influence of Bertrand Russell. It will be argued that Eliot uses rhetorical strategies which simultaneously subvert the method he is propounding, and which set the tone for an assessment of his criticism throughout the 1920s. His indecision, in this period, about the label 'Metaphysical' for some poets of the seventeenth century, reveals the persistence of the philosophical thought he apparently rejects in 1916, when he chooses not to pursue a career in philosophy in Harvard. This rhetorical tactic achieves its fulfilment in Dante (1929), where Eliot finds a model in the medieval allegorical method and 'philosophical' poetry. Allegory is also examined in connection with the evaluation of Eliot's critical writings themselves to determine, for instance, the figurative dimension of his early scientific vocabulary and uncover metaphysical residues he had explicitly disowned but would later embrace. Finally, it is suggested that, the hermeneutics of allegory are historical and it is used here to test the relationship between Eliot's early and later critical writings, that is the early physics and the later metaphysics.

SOME PROPERTIES OF THE MULTIPLICITY SEQUENCE FOR ARBITRARY IDEALS

In this work we prove that the Achilles-Manaresi multiplicity sequence, like the classical Hilbert-Samuel multiplicity, is additive with respect to the exact sequence of modules. We also prove the associativity formula for his mulitplicity sequence. As a consequence, we give new proofs for two results already known. First, the Achilles-Manaresi multiplicity sequence is an invariant up to reduction, a result first proved by Ciuperca. Second, I subset of J is a reduction of (J,M) if and only if c(0)(I(p), M(p)) = c(0)(J(p), M(p)) for all p is an element of Spec(A), a result first proved by Flenner and Manaresi.

Singular Perturbation for the Discounted Continuous Control of Piecewise Deterministic Markov Processes

This paper deals with the expected discounted continuous control of piecewise deterministic Markov processes (PDMP`s) using a singular perturbation approach for dealing with rapidly oscillating parameters. The state space of the PDMP is written as the product of a finite set and a subset of the Euclidean space a""e (n) . The discrete part of the state, called the regime, characterizes the mode of operation of the physical system under consideration, and is supposed to have a fast (associated to a small parameter epsilon > 0) and a slow behavior. By using a similar approach as developed in Yin and Zhang (Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach, Applications of Mathematics, vol. 37, Springer, New York, 1998, Chaps. 1 and 3) the idea in this paper is to reduce the number of regimes by considering an averaged model in which the regimes within the same class are aggregated through the quasi-stationary distribution so that the different states in this class are replaced by a single one. The main goal is to show that the value function of the control problem for the system driven by the perturbed Markov chain converges to the value function of this limit control problem as epsilon goes to zero. This convergence is obtained by, roughly speaking, showing that the infimum and supremum limits of the value functions satisfy two optimality inequalities as epsilon goes to zero. This enables us to show the result by invoking a uniqueness argument, without needing any kind of Lipschitz continuity condition.

What is the Problem of Ad Hoc Hypotheses?

The received view of an ad hoc hypothesis is that it accounts for only the observation(s) it was designed to account for, and so non-adhocness is generally held to be necessary or important for an introduced hypothesis or modification to a theory. Attempts by Popper and several others to convincingly explicate this view, however, prove to be unsuccessful or of doubtful value, and familiar and firmer criteria for evaluating the hypotheses or modified theories so classified are characteristically available. These points are obscured largely because the received view fails to adequately separate psychology from methodology or to recognise ambiguities in the use of 'ad hoc'.

The evolutionary structure of scientific theories

David Hull's (1988c) model of science as a selection process suffers from a two-fold inability: (a) to ascertain when a lineage of theories has been established; i.e., when theories are descendants of older theories or are novelties, and what counts as a distinct lineage; and (b) to specify what the scientific analogue is of genotype and phenotype. This paper seeks to clarify these issues and to propose an abstract model of theories analogous to particulate genetic structure, in order to reconstruct relationships of descent and identity.

The conserved quantity theory of causation and chance raising

In this paper I offer an 'integrating account' of singular causation, where the term 'integrating' refers to the following program for analysing causation. There are two intuitions about causation, both of which face serious counterexamples when used as the basis for an analysis of causation. The 'process' intuition, which says that causes and effects are linked by concrete processes, runs into trouble with cases of misconnections', where an event which serves to prevent another fails to do so on a particular occasion and yet the two events are linked by causal processes. The chance raising intuition, according to which causes raise the chance of their effects, easily accounts for misconnections but faces the problem of chance lowering causes, a problem easily accounted for by the process approach. The integrating program attempts to provide an analysis of singular causation by synthesising the two insights, so as to solve both problems. In this paper I show that extant versions of the integrating program due to Eells, Lewis, and Menzies fail to account for the chance-lowering counterexample. I offer a new diagnosis of the chance lowering case, and use that as a basis for an integrating account of causation which does solve both cases. In doing so, I accept various assumptions of the integrating program, in particular that there are no other problems with these two approaches. As an example of the process account, I focus on the recent CQ theory of Wesley Salmon (1997).

Causal loops and the independence of causal facts

According to Hugh Mellor in Real Time II (1998, Ch. 12), assuming the logical independence of causal facts and the 'law of large numbers', causal loops are impossible because if they were possible they would produce inconsistent sets of frequencies. I clarify the argument, and argue that it would be preferable to abandon the relevant independence assumption in the case of causal loops.

The advantages of theft over toil: The design inference and arguing from ignorance

Intelligent design theorist William Dembski has proposed an explanatory filter for distinguishing between events due to chance, lawful regularity or design. We show that if Dembski's filter were adopted as a scientific heuristic, some classical developments in science would not be rational, and that Dembski's assertion that the filter reliably identifies rarefied design requires ignoring the state of background knowledge. If background information changes even slightly, the filter's conclusion will vary wildly. Dembski fails to overcome Hume's objections to arguments from design.

Bringing the state to England: Andrew Tooke's translation of Samuel Pufendorf's 'De officio hominis et civis'

Andrew Tooke's 1691 English translation of Samuel Pufendorf's De officio hominis et civis, published as The Whole Duty of Man According to the Law of Nature, brought Pufendorf's manual fo statist natural law into English politics at a moment of temporary equilibrium in the unfinished contest between Crown and Parliament for the rights and powers of sovereignty. Drawing on the authors' re-edition of The Whole Duty of Man, this article describes and analyses a telling instance of how--by translation--the core political terms and concepts of the German natural jurist's 'absolutist' formulary were reshaped for reception in the different political culture of late seventeenth-century England.

How to be a chaste species pluralist-realist: the origins of species modes and the synapomorphic species concept

The biological species (biospecies) concept applies only to sexually reproducing species, which means that until sexual reproduction evolved, there were no biospecies. On the universal tree of life, biospecies concepts therefore apply only to a relatively small number of clades, notably plants and animals. I argue that it is useful to treat the various ways of being a species (species modes) as traits of clades. By extension from biospecies to the other concepts intended to capture the natural realities of what keeps taxa distinct, we can treat other modes as traits also, and so come to understand that the plurality of species concepts reflects the biological realities of monophyletic groups. We should expect that specialists in different organisms will tend to favour those concepts that best represent the intrinsic mechanisms that keep taxa distinct in their clades. I will address the question whether modes of reproduction such as asexual and sexual reproduction are natural classes, given that they are paraphyletic in most clades.

Listening to the Whispers of Matter Through Arabic Hermeticism: New Studies on the Book of the Treasure of Alexander

The Jabirian Corpus refers to the K. Thahirat Al-`Iskandar, ""The Book of the Treasure of Alexander"" (hereafter BTA), as one of several forgeries suggesting that alchemical secrets were hidden in inscriptions in various places. The book was neglected until 1926, when Julius Ruska discussed it in his work on the Emerald Tablet, placing the BTA within the literature related to the development of Arabic alchemy. His preliminary study became an essential reference and encouraged many scholars to work on the BTA in the following decades. Some years ago, we completed the first translation of the BTA into a Western language. The work was based on the acephalous Escorial manuscript, which we identified as a fourteenth-century copy of the BTA. This manuscript is peculiar, as part of it is encoded. After finishing our translation, we started to establish the text of the BTA. At present, the text is in process of fixation-to be followed by textual criticism-and has been the main focus of a thorough study of ours on medieval hermeticism and alchemy. A sample of the work currently in progress is presented in this paper: an analysis of the variations between different manuscripts along with a study and English translation of its alchemical chapter.