Opportunity to learn in the preparation of mathematics teachers: its structure and how it varies across six countries

Cross-national research studies such as the Program for International Student Assessment and the Third International Mathematics and Science Study (TIMSS) have contributed much to our understandings regarding country differences in student achievement in mathematics, especially at the primary (elementary) and lower secondary (middle school) levels. TIMSS, especially, has demonstrated the central role that the concept of opportunity to learn plays in understanding cross-national differences in achievement Schmidt et al., (Why schools matter: A cross-national comparison of curriculum and learning  2001). The curricular expectations of a nation and the actual content exposure that is delivered to students by teachers were found to be among the most salient features of schooling related to academic performance. The other feature that emerges in these studies is the importance of the teacher. The professional competence of the teacher which includes substantive knowledge regarding formal mathematics, mathematics pedagogy and general pedagogy is suggested as being significant—not just in understanding cross-national differences but also in other studies as well (Hill et al. in Am Educ Res J 42(2):371–406, 2005). Mathematics Teaching in the 21st Century (MT21) is a small, six-country study that collected data on future lower secondary teachers in their last year of preparation. One of the findings noted in the first report of that study was that the opportunities future teachers experienced as part of their formal education varied across the six countries (Schmidt et al. in The preparation gap: Teacher education for middle school mathematics in six countries, 2007). This variation in opportunity to learn (OTL) existed in course work related to formal mathematics, mathematics pedagogy and general pedagogy. It appears from these initial results that OTL not only is important in understanding K-12 student learning but it is also likely important in understanding the knowledge base of the teachers who teach them which then has the potential to influence student learning as well. This study using the same MT21 data examines in greater detail the configuration of the educational opportunities future teachers had during their teacher education in some 34 institutions across the six countries.     

Simple pictorial mathematics problems for children: locating sources of cognitive load and how to reduce it

ZDM – Mathematics Education - Pictorial representations are often used to help children understand the situation described in a given number-sentence scheme. These static pictorial problems...     

Local search for multiprocessor scheduling: how many moves does it take to a local optimum?

We analyze two local search algorithms for multiprocessor scheduling. The first algorithm is a job interchange algorithm for identical parallel machines due to Finn and Horowitz (Bit 19 (1979) 312). We construct instances for which this algorithm takes a quadratic number of iterations. This contradicts the original analysis of Finn and Horowitz who claimed a linear number of iterations.The second algorithm adds an additional rule to the Finn and Horowitz algorithm. Even for n jobs on m uniformly related machines, this modified algorithm takes only O(nm) iterations.     

Study on the aluminium phosphate bonded SiC refractory performance and how it is influenced by oxidation and chemical attack

Aluminium reduction cells service life and cell efficiency are strongly influenced by the side wall refractory lining. The aluminum phosphate bonded SiC refractories developed by us, instead of the carbon ones, have increased the cell capacity and productivity by the larger anode surfaces. In order to rank performances of various SiC products were carried out specific laboratory tests. The results of our research concerning materials made from aluminum phosphate bonded SiC showed that they are refractories with a high corrosion resistance and performances close to the bricks made of silicon nitride bonded silicon carbide. By extrapolation of the present study's results we can anticipate an increased service life of electrolysis cells that have such side walls. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The density function of the number of moves to complete the Towers of Hanoi puzzle

Any sequence of legal moves leads the Towers of Hanoi puzzle to an arrangement from which the final configuration must be built up. A recursive algorithm which finishes off the puzzle is considered and, assuming a uniform distribution on the possible unfinished situations, the density function of the number of moves it takes is derived.     

The mathematics teacher: Between solving equations and the meaning of it all

An attempt is made to suggest some ultimate goals for mathematics education beyond the narrow framework of the mathematics curriculum. On the other hand, there is an attempt to tie mathematics education to these goals by pointing at some principles which direct mathematical behavior as well as educated behavior. The main claim is that both mathematical behavior and educated behavior are supposed to be directed by rational thinking.     

Quorum Sensing: Complexity in the bacterial world

We discuss the mechanisms of bacterial Quorum Sensing, the biophysical phenomenon pointing out a social behavior in bacteria, highlighting thus the very complex structure of these systems. Actually, bacterial bioluminescence is an example of a quorum sensing mediated property. We show that the distribution of the activation times of the bioluminescent emission follows the universal behavior described by the Gumbel distribution of extreme value statistics. We provide further evidence on the system size scaling of bioluminescence, showing that the relation between cell density and total number of photons radiated by bacteria is highly non-linear.     

The monadic theory and the “next world”

Suppose thatV is a model of ZFC andU ∈ V is a topological space or a richer structure for which it makes sense to speak about the monadic theory. LetB be the Boolean algebra of regular open subsets ofU. If the monadic theory ofU allows one to speak in some sense about a family ofκ everywhere dense and almost disjoint sets, then the second-orderV ^B-theory of ϰ is interpretable in the monadicV-theory ofU; this is our Interpretation Theorem. Applying the Interpretation Theorem we strengthen some previous results on complexity of the monadic theories of the real line and some other topological spaces and linear orders. Here are our results about the real line. Letr be a Cohen real overV. The second-orderV[r]-theory of ℵ₀ is interpretable in the monadicV-theory of the real line. If CH holds inV then the second-orderV[r]-theory of the real line is interpretable in the monadicV-theory of the real line. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author thanks the United States-Israel Binational Science Foundation for supporting the research.     

The unknown in decision making: What to do about it

The unknown or “other” that affects our lives is what we usually very much want to know about to cope with uncertainty. We often suspect that it affects us with partial and indefinite evidence that it exists but we only have uncertain feelings about it. Even when we do not know what it is we would like to allow for its influence in our explaining the outcome of a decision. One way to deal with the many factors of a decision is to include the unknown as one of them and then determine its priority of influence on the outcome by comparing it with other factors. We are able to do that to the extent that we are sure of what we know and of the residual that remains outside our understanding that may also have some effect on what we do. Confidence from good understanding and past success are what we need in order to judge the potential significance of what we do not know on the outcome. We can then perform sensitivity analysis to see how much effect unknown factors can have on the stability of the choice we make. Pairwise comparisons make it possible to tackle this idea explicitly and rather simply. This note illustrates how to prioritize and test the effect of the unknown alongside the known.     

How to select and how to rank projects: The Promethee method

In this paper, we present the Promethee methods, a new class of outranking methods in multicriteria analysis. Their main features are simplicity, clearness and stability. The notion of generalized criterion is used to construct a valued outranking relation. All the parameters to be defined have an economic signification, so that the decision maker can easily fix them. Two ways of treatment are proposed: It is possible to obtain either a partial preorder (Promethee I) or a complete one (Promethee II), both on a finite set of feasible actions. A comparison is made with the Electre III method. The stability of the results given by the two methods is analysed. Numerical applications are given in order to illustrate the properties of the new methods and some further problems are discussed.