A Classification of Mathematical Modeling Problems of Prospective Mathematics Teachers

The purpose of this research is to classify the mathematical modelling problems produced by pre-service mathematics teachers in terms of the number of variables and to determine the mathematical modelling skills and mathematical skills used in solving the problems in each class. The current study is a qualitative research and the data was analyzed using descriptive analysis. The data of the study was obtained from the mathematical modelling problem written by 59 senior mathematics teachers. They were given a 1-week period to write the problems and solutions. The participants took mathematical modelling course for one semester period prior to the research. The problems are the original problems that the participants themselves produced. The mathematical modelling problems produced are categorically as follows: “Which option is more economical” problems, “Profit-making” problems, “Future prediction” problems and “Relationship between two quantities” problems. The mathematical modelling skills used are as follows: to be able to collect appropriate data, organize the data, write dependent and independent variables, write fixed values, visualize the real situation mathematically or geometrically, use mathematical concepts. The mathematical skills used are generally; to be able to do four operations with rational numbers, draw distribution and column graph, write algebraic expression, do arithmetic operation in algebraic rational expressions, write/solve equation and inequality in 1 or 2 variables, write an appropriate mathematical function explaining the data related to the data, solve 1st degree equations in 1 variable, establish proportion, use trigonometric ratios in right triangle, use basic geometry information, draw and interpret a 1st degree inequality in 2 variables.

Binary Covering Arrays on Tournaments

We introduce graph-dependent covering arrays which generalize covering arrays on graphs, introduced by Meagher and Stevens (2005), and graph-dependent partition systems, studied by Gargano, Körner, and Vaccaro (1994). A covering array $\hbox{CA}(n; 2, G, H)$ (of strength 2) on column graph $G$ and alphabet graph $H$ is an $n\times |V(G)|$ array with symbols $V(H)$ such that for every arc $ij \in E(G)$ and for every arc $ab\in E(H)$, there exists a row $\vec{r} = (r_{1},\dots, r_{|V(G)|})$ such that $(r_{i}, r_{j}) = (a,b)$. We prove bounds on $n$ when $G$ is a tournament graph and $E(H)$ consists of the edge $(0,1)$, which corresponds to a directed version of Sperner's 1928 theorem. For two infinite families of column graphs, transitive and so-called circular tournaments, we give constructions of covering arrays which are optimal infinitely often.

THE LAYERED NET SURFACE PROBLEMS IN DISCRETE GEOMETRY AND MEDICAL IMAGE SEGMENTATION

Efficient detection of multiple inter-related surfaces representing the boundaries of objects of interest in d-D images (d ≥ 3) is important and remains challenging in many medical image analysis applications. In this paper, we study several layered net surface (LNS) problems captured by an interesting type of geometric graphs called ordered multi-column graphs in the d-D discrete space (d ≥ 3 is any constant integer). The LNS problems model the simultaneous detection of multiple mutually related surfaces in three or higher dimensional medical images. Although we prove that the d-D LNS problem (d ≥ 3) on a general ordered multi-column graph is NP-hard, the (special) ordered multi-column graphs that model medical image segmentation have the self-closure structures and thus admit polynomial time exact algorithms for solving the LNS problems. Our techniques also solve the related net surface volume (NSV) problems of computing well-shaped geometric regions of an optimal total volume in a d-D weighted voxel grid. The NSV problems find applications in medical image segmentation and data mining. Our techniques yield the first polynomial time exact algorithms for several high dimensional medical image segmentation problems. Experiments and comparisons based on real medical data showed that our LNS algorithms and software are computationally efficient and produce highly accurate and consistent segmentation results.