Mathematics achievement and interest in mathematics from a differential perspective

In this article, we present results of an empirical study with 500 German students of grades 7 and 8. The study focussed on students' mathematics achievement and their interest in mathematics as well as on the relation between these two constructs. In particular, the results show that the development of an individual student's achievement between grade 7 and grade 8 depends on the achievement level of the specific classroom and therefore on the specific mathematics instruction Interest in mathematics could be regarded a predictor for mathematics achievement Moreover, our findings suggest that the students show hardly any fear of mathematics independent of their achievement level.     

The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Adsorption of dicarboxylates on nano-sized gibbsite particles: effects of ligand structure on bonding mechanisms

The adsorption of the dicarboxylates o-phthalate, maleate, fumarate, malonate, and oxalate (representing ligands with the general composition ⁻O₂C---C_n---CO₂⁻; n=0, 1, or 2) on gibbsite were studied by means of quantitative batch adsorption experiments and attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy. The interpretations of ATR-FTIR spectra were aided by comparison with IR spectra of solution species and by results from theoretical frequency calculations. The main objectives of the study were to identify the molecular level bonding mechanisms of the dicarboxylates to gibbsite, and to investigate how these were influenced by the composition and structure of the ligands. Carboxylates with n=2 formed predominantly outer sphere complexes, whereas the importance of inner sphere complexes progressively increased for n=1 and 0. The inner sphere structures were identified as mononuclear chelates with one oxygen from each carboxylate group bonded to Al(III) at the surface. This showed the importance of chelate ring structure for the formation of inner sphere surface complexes, with stabilities of the complexes increasing in the order seven-membered ring less than six-membered ring less than five-membered ring. For ligands with n=2 only small variations in surface speciation were observed as a function of steric factors; irrespective of the relative positions of the carboxylate groups and bulkiness of the ligands outer sphere bonding is the dominating adsorption mode. Adsorption experiments were also conducted with gibbsite particles exhibiting differences in shape and surface roughness. These experiments showed that inner sphere complexes were favored on the less well-developed and more irregular gibbsite particles.     

Correction to: On Webs of Maximum Rank (Geom. Dedicata 31 (1989), 19–35)

We correct an error in Example (3.4) in Geom. Dedicata 31 (1989), 19–35.     

Limits of Functions as They Developed Through Time and as Students Learn Them Today

This article compares first-year university students' development of the concept of limits to mathematicians' historical development of the concept. The aim was to find out if students perceive the notion as mathematicians of the past did, as understandings of the concept evolved. The results imply that there are some similarities—for example, the struggle with rigor and attainability. Knowledge of such critical areas can be used to improve students' opportunities of learning limits of functions. Some teaching aspects related to the study are also discussed.     

The mixed-aldehyde synthesis of difunctional tetraarylporphyrins

The synthesis is reported of nine unsymmetrical, meso-substituted porphyrins. Among the compounds prepared are the following 5-(R)-10,15,20-tri-p-tolylporphyrins; R = 2,6-dinitrophenyl, 4-hydroxy-3-ethoxy-phenyl, 4-hydroxy-3-methoxy-5-nitrophenyl, 5-hydroxy-2-nitrophenyl and 4-hydroxy-3-nitrophenyl. Other porphyrins reported include 5-(2-(1-butoxy)phenyl)-15-(2-nitrophenyl)-10-15-di-p-tolylporphyrin and the two 5-(R)-10-15,20-tripropylporphyrins in which R = 2-nitrophenyl and 2-hydroxyphenyl. The disubstituted porphyrins offer a rational route to the synthesis of difunctional “tailed-porphyrins”.