Students’ interpretations of categorical data using dynamic graphical representations

Statistical association is an important concept in statistics. An exploratory study examined how students reason about statistical association utilizing graphical representations constructed with CODAP, a dynamic statistical graphing software. Task-based interviews were conducted with three 6th grade students prior to formal instruction. Students’ conceptions of a statistical relationship, proportional reasoning skill level, ability to interpret bivariate categorical graphs (particularly segmented bar graphs and two-way binned plots), and ability to identify association of two categorical variables were all investigated through interview tasks and responses to inquiry. Students were found to have developing proportional reasoning skills and struggled to correctly define and identify association. These results were compared to a previous study which asked students to analyze pre-constructed graphs. Students were more successful interpreting graphs that they constructed than pre-constructed graphs. These results have curricular and future research implications

Why is the World Short of Democracy? A Cross-Country Ananlysis of Barriers to Representative Government

This study first uses a set of graphs and tables to present the pattern of democracy in the world, using the Gastil Index. Then a statistical analysis is conducted by two techniques: Regression techniques are used to analyze the effect on democracy of a handful of variables. It shows that poverty, Communism and the Muslim culture are the main barriers to democracy. It then uses Bayesian probability methods to make explicit the concept of the “risk” of countries being ndemocratic. The analysis focuses on the dynamics of the income effect and of the democratic deficit of the Muslim countries to see if it is stationary or transitory. It is unstable, so it may be transitory, but it has been rising.Democracy, Lipset’s law, Western vs Muslim Culture

Vines and MAT-labeled graphs

The present paper explores a connection between two concepts arising from different fields of mathematics. The first concept, called vine, is a graphical model for dependent random variables. This concept first appeared in a work of Joe (1994), and the formal definition was given later by Cooke (1997). Vines have nowadays become an active research area whose applications can be found in probability theory and uncertainty analysis. The second concept, called MAT-freeness, is a combinatorial property in the theory of freeness of logarithmic derivation modules of hyperplane arrangements. This concept was first studied by Abe-Barakat-Cuntz-Hoge-Terao (2016), and soon afterwards investigated further by Cuntz-M{\"u}cksch (2020). In the particular case of graphic arrangements, the last two authors (2023) recently proved that the MAT-freeness is completely characterized by the existence of certain edge-labeled graphs, called MAT-labeled graphs. In this paper, we first introduce a poset characterization of a vine, the so-called vine. Then we show that, interestingly, there exists an explicit equivalence between the categories of locally regular vines and MAT-labeled graphs. In particular, we obtain an equivalence between the categories of regular vines and MAT-labeled complete graphs. Several applications will be mentioned to illustrate the interaction between the two concepts. Notably, we give an affirmative answer to a question of Cuntz-M{\"u}cksch that MAT-freeness can be characterized by a generalization of the root poset in the case of graphic arrangements.Comment: 32 pages; refined the definitions of the categories MG and LRV (Def. 6.2 & 6.3), hence improved the main result (Thm. 6.10); the term "vineposet" is no longer used, instead we distinguish the graphical and poset definitions of a vin

How Textbooks Influence Students’ Algebra Learning: A Comparative Study on the initial treatment of the concept of function

To give insights into cross national differences in schooling this study analyzed the initial treatment of the concept of function in three curricula: a US standards-based text--Connected Mathematic 2: Variables and Patterns, a US conventional text--Glencoe: Mathematics Applications and Concepts: Course 2, and a Chinese reform text--Shu Xue: Grade 8, first volume. This study examined content organization and problem features in the three textbooks. For content analysis, this study explored how the concept of function was introduced, defined, and developed. The results indicated both of the US textbooks introduce this concept at grade 7 whereas the Chinese text does so at grade 8. Connected Mathematics devotes more lessons than the Chinese text and Glencoe in the initial treatment of the concept of function. Connected Mathematics defines function as rule while Glencoe addresses it as relationship; the Chinese text introduces the concept of function as correspondence. Connected Mathematics pays equal an amount of attention to the four representations including tables, graphs, verbal descriptions, and equations examined in this study. In contrast, Glencoe employs the representations of tables, graphs, and equations and it focuses on the representation of graphs; the Chinese text also employs the representations of tables, graphs, and equations but it focuses on the representation of equation. The Chinese text provides many explanations and illuminations in worked-out examples to tell how the solutions are derived. Problems were then analyzed extensively with respect to three criteria: (1) contextual feature, (2) response type, and (3) cognitive expectation. Analysis results showed that all the three texts emphasize the cognitive expectation of representation. Connected Mathematics provides more real-world problems than other texts; and the problems aim at cultivating students’ mathematical reasoning. Most of the problems in Glencoe are embedded in pure math contexts to help students do procedure practice. The problems in the Chinese text emphasize problem solving. Implications for curriculum developers, teachers, and researchers have been discussed in accordance with the findings