Alfred B. Manaster

Mathematics Teaching in the United States Today (and Tomorrow): Results From the TIMSS 1999 Video Study:

The Third International Mathematics and Science Study (TIMSS) 1999 Video Study examined eighth-grade mathematics teaching in the United States and six higher-achieving countries. A range of teaching systems were found across higher-achieving countries that balanced attention to challenging content, procedural skill, and conceptual understanding in different ways. The United States displayed a unique system of teaching, not because of any particular feature but because of a constellation of features that reinforced attention to lower-level mathematics skills. The authors argue that these results are relevant for policy (mathematics) debates in the United States because they provide a current account of what actually is happening inside U.S. classrooms and because they demonstrate that current debates often pose overly simple choices. The authors suggest ways to learn from examining teaching systems that are not alien to U.S. teachers but that balance a skill emphasis with attention to challenging mathematics and conceptual development.

Understanding and Improving Mathematics Teaching: Highlights from the TIMSS 1999 Video Study

portion of the TIMSS 1999 Video Study included Australia, the Czech Republic, Hong Kong SAR, Japan, the Netherlands, Switzerland, and the United States. In this article, we focus on the mathematics lessons; the science results will be available at a later date. Stimulated by a summary article that appeared in the Kappan and by other reports, interest in the TIMSS 1995 Video Study focused on its novel methodology and the striking differences in teaching found in the participating countries. In particular, the sample of eighth-grade Understanding and Improving Mathematics Teaching: Highlights from the TIMSS 1999 Video Study

EFFECTIVE MATCHMAKING AND ¿-CHROMATIC GRAPHS

ALFRED B. MANASTER1 AND JOSEPH G. ROSENSTEIN2Abstract. In an earlier paper we showed that there is a recursivesociety, in which each person knows exactly two other people,whose marriage problem is solvable but not recursively solvable.We generalize this result, using a different construction, to the casewhere each person knows exactly k other people. From this wededuce that for each k^2 there is a recursive 2(Ar—l)-regular graph,whose chromatic number is k but which is not recursively k-chromatic.

Mathematical features of lessons in the TIMSS video study

The video component of the Third International Mathematics and Science Study marked the first time that national samples of teachers were videotaped at work in their classrooms. In this article we review some of the results of the study, with special attention to the nature of the mathematics evident in these eighth-grade lessons from Germany, Japan, and the United States. We conclude by proposing that many lessons within a country follow a cultural pattern of teaching, and that differences among countries on individual indicators of teaching must be understood in reference to these patterns.

Recursive categoricity and recursive stability

Versions effectives de la categoricite. Categoricite des ordres lineaires. Programme de mathematiques effectives

Small recursive ordinals, many-one degrees, and the arithmetical difference hierarchy

Let W be the Markwald-Spector [ 10] set of GiSdel numbers for recur. sive well-ordering relations; for each recursive ordinal e, let W(a) denote the set of Gbdel numbers for recursive well-orderings of order type /1) and deg m W(a) the m-degree of W(t~). Then

Two-dimensional partial orderings: Recursive model theory

In this paper and the companion paper [9] we describe a number of contrasts between the theory of linear orderings and the theory of two-dimensional partial orderings. The notion of dimensionality for partial orderings was introduced by Dushnik and Miller [3], who defined a partial ordering 〈 A, R 〉 to be n -dimensional if there are n linear orderings of A , 〈 A, L 1 〉, 〈 A, L 2 〉 …, 〈 A, L n 〉 such that R = L 1 ∩ L 2 ∩ … ∩ L n . Thus, for example, if Q is the linear ordering of the rationals, then the (rational) plane Q × Q with the product ordering (〈 x 1 , y 1 〉 ≤ Q × Q 〈 x 2 , y 2 , if and only if x 1 ≤ x 2 and y 1 ≤ y 2 ) is 2-dimensional, since ≤ Q × Q is the intersection of the two lexicographic orderings of Q × Q . In fact, as shown by Dushnik and Miller, a countable partial ordering is n -dimensional if and only if it can be embedded as a subordering of Q n . Two-dimensional partial orderings have attracted the attention of a number of combinatorialists in recent years. A basis result recently obtained, independently, by Kelly [7] and Trotter and Moore [10], describes explicitly a collection of finite partial orderings such that a partial ordering is a 2dpo if and only if it contains no element of as a subordering.

Partial orderings of fixed finite dimension: Model companions and density

The model companions of the theories of n -dimensional partial orderings and n -dimensional distributive lattices are found for each finite n . Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n -dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n -dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory. In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T * theories in a common language, T * is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T * are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T * is model complete, which means that if and are both models of T * and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T * admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ 0 -categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

A universal embedding property of the RETs

?0. Introduction. Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950s. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.

Planarity and minimal path algorithms

In 1981 two notions of effective presentation of countable connected graphs were formulated by J. C. E. Dekker—namely, edge recognition algorithm graphs and minimal path algorithm graphs. In this paper we show that every planar graph has a minimal path algorithm presentation but that some graphs have no minimal path algorithm presentations. We introduce the notion of a shortest distance algorithm graph, show that it lies strictly between the two notions of Dekker, and show that every graph has a shortest distance algorithm presentation. Finally, in contrast to Dekkers result about minimal path algorithm graphs, we produce a shortest distance algorithm graph which has no spanning tree which is an edge recognition algorithm graph.