Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

Filling an area discretely and continuously

The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under the curve and the limit of Riemann sums; these three conceptualizations of the definite integral are useful in different contexts and provide students with what it takes to interpret the definite integrals. Research shows that students rarely invoke the multiplicatively-based summation conception of the definite integral although it is essential for evaluating line integrals, surface integrals and volumes. This paper describes a teaching module that promotes understanding as well as activating all three conceptualizations of the definite integral through motivating the accumulation area function and the results in the Fundamental Theorems of Calculus.     

What Is STEM? A Discussion About Conceptions of STEM in Education and Partnerships

Educational reformation has proceeded slowly despite the many calls to improve science and mathematics for our students. The acronym STEM (science, technology, engineering, and mathematics) has been adopted by numerous programs as an important focus for renewed global competitiveness for the United States, but conceptions of what STEM entails often vary among stakeholders. This paper examines the conceptions of STEM held by faculty members from a public Research I institution in the middle of a regional “STEM movement.” Faculty members responded to two open‐ended questions: (1) What is STEM? and (2) How does STEM influence and/or impact your life? Although 72% of these faculty members possessed a relevant conception of STEM, the results suggest that they do not share a common conceptualization of STEM. Their conception is most likely based on their academic discipline or how STEM impacts their daily lives. STEM faculty members were likely to have a neutral or positive conception where non‐STEM faculty members often had negative feelings about STEM.     

Embodying mathematics and science: Microworlds as representations

This paper examines the category of open-ended exploratory computer environments that have been labeled “microworlds.” The paper reviews the ways in which the term “microworld” has been used in the mathematics and science education communities, and analyzes examples of specific computer microworlds. Two definitions of microworld are proposed: a structural definition that focuses on design elements shared by the environments, and a functional definition that highlights commonalties in how students learn with microworlds. In the final section of the paper, the notion that computer microworlds can be said to “embody” mathematical or scientific ideas is addressed, within the context of a re-evaluation of the general concept of representation.     

Science Attitudes of Selected Middle School Students in China: A Preliminary Investigation of Similarities and Differences as a Function of Gender

Evaluating the attitudes of science students is important for teachers, curriculum developers, and those working with preservice teachers. Although in the United States a great deal of attitudinal research has been conducted with regard to science education, in the People's Republic of China very little work concerning science attitudes has been completed. This study will report on an evaluation of Chinese boys' and girls' attitudes toward selected science topics. Students attended a middle school in the city of Shanghai. Analysis indicated that when the male and female Chinese students differ in their response patterns, females select more intense responses (“strongly agree” as opposed to “agree,”“strongly disagree” as opposed to “disagree”). Furthermore, the surveyed females often selected responses suggesting that they were more interested in the science topics and issues presented in the survey.     

Eichler integrals, period relations and Jacobi forms

This paper contains three main results: the first one is to derive two “period relations” and the second one is a complete characterization of period functions of Jacobi forms in terms of period relations. These are done by introducing a concept of “Jacobi integrals” on the full Jacobi group. The last one is to show, for the given holomorphic function P(τ, z) having two period relations, there exists a unique Jacobi integral, up to Jacobi forms, with a given function P(τ, z) as its period function. This is done by constructing a generalized Jacobi Poincaré series explicitly. This is to say that every holomorphic function with “period relations” is coming from a Jacobi integral. It is an analogy of Eichler cohomology theory studied in Knopp (Bull Am Math Soc 80:607–632, 1974) for the functions with elliptic and modular variables. It explains the functional equations satisfied by the “Mordell integrals” associated with the Lerch sums (Zwegers in Mock theta functions, PhD thesis, Universiteit Utrecht, 2002) or, more generally, with the higher Appell functions (Semikhatov et?al. in Commun Math Phys 255(2):469–512, 2005). Developing theories of Jacobi integrals with elliptic and modular variables in this paper is a natural extension of the Eichler integral with modular variable. Period functions can be explained in terms of the parabolic cohomology group as well.     

Student Perceptions of a Summer Ventures in Science and Mathematics Camp Experience

“As the world becomes increasingly technological, the value of (the ideas and skills of its population) will be determined in no small measure by the effectiveness of science, technology, engineering, and mathematics (STEM) education in the United States” and “STEM education will determine whether the United States will remain a leader among nations and whether we will be able to solve immense challenges in such areas as energy, health, environmental protection, and national security” (President's Council of Advisors on Science and Technology, 2010, p. vii). Research on the effectiveness of STEM‐focused school and other learning experiences (e.g., short‐term camps) on student attitudes and performance outcomes is sparse. In this study, we documented the influence of an intensive STEM summer program on high school students’ attitudes toward STEM concepts and interests in STEM careers. Attending the summer program was associated with gains on students’ attitudes toward some aspects of STEM as well as specific career interests. Notably, students reported statistically significant views of important aspects of STEM and their attitudes toward science and mathematics were more positive than their attitudes about engineering and technology.     

An exploratory study on student understandings of derivatives in real-world,non-kinematics contexts

Much research on calculus students’ understanding of applied derivatives has been done in kinematics-based contexts (i.e. position, velocity, acceleration). However, given the wide range of applications in science and engineering that are not based on kinematics, nor even explicitly on time, it is important to know how students understand applied derivatives in non-kinematics contexts. In this study, interviews with six students and surveys with 38 students were used to explore students’ “ways of understanding” and “ways of thinking” regarding applied, non-kinematics derivatives. In particular, six categories of ways of understanding emerged from the data as having been shared by a substantial portion of the students in this study: (1) covariation, (2) invoking time, (3) other symbols as constants, (4) other symbols as implicit functions, (5) implicit differentiation, and (6) output values as amounts instead of rates of change.     

Logic as a Science and Logic as a Theory: Remarks on Frege, Russell and the Logocentric Predicament

Since its publication in 1967, van Heijenoort??s paper, ??Logic as Calculus and Logic as Language?? has become a classic in the historiography of modern logic. According to van Heijenoort, the contrast between the two conceptions of logic provides the key to many philosophical issues underlying the entire classical period of modern logic, the period from Frege??s Begriffsschrift (1879) to the work of Herbrand, G?del and Tarski in the late 1920s and early 1930s. The present paper is a critical reflection on some aspects of van Heijenoort??s thesis. I concentrate on the case of Frege and Russell and the claim that their philosophies of logic are marked through and through by acceptance of the universalist conception of logic, which is an integral part of the view of logic as language. Using the so-called ??Logocentric Predicament?? (Henry M. Sheffer) as an illustration, I shall argue that the universalist conception does not have the consequences drawn from it by the van Heijenoort tradition. The crucial element here is that we draw a distinction between logic as a universal science and logic as a theory. According to both Frege and Russell, logic is first and foremost a universal science, which is concerned with the principles governing inferential transitions between propositions; but this in no way excludes the possibility of studying logic also as a theory, i.e., as an explicit formulation of (some) of these principles. Some aspects of this distinction will be discussed.     

Information Geometry and Interior-Point Algorithms in Semidefinite Programs and Symmetric Cone Programs

We develop an information geometric approach to conic programming. Information geometry is a differential geometric framework specifically tailored to deal with convexity, naturally arising in information science including statistics, machine learning and signal processing etc. First we introduce an information geometric framework of conic programming. Then we focus on semidefinite and symmetric cone programs. Recently, we demonstrated that the number of iterations of Mizuno–Todd–Ye predictor–corrector primal–dual interior-point methods is (asymptotically) expressed with an integral over the central trajectory called “the curvature integral”. The number of iterations of the algorithm is approximated surprisingly well with the integral even for fairly large linear/semidefinite programs with thousands of variables. Here we prove that “the curvature integral” admits a rigorous differential geometric expression based on information geometry. We also obtain an interesting information geometric global theorem on the central trajectory for linear programs. Together with the numerical evidence in the aforementioned work, we claim that “the number of iterations of the interior-point algorithm is expressed as a differential geometric quantity.”     

A simple model of imperfect competition,where 4 are few and 6 are many

The theory presented in this paper investigates the connection between the number of competitors and the tendency to cooperate within the context of a symmetric Cournot model with linear cost and demand, supplemented by specific institutional assumptions about the possibilities of cooperation. Cooperative forms of behavior are modelled as moves in a non-cooperative game. The proposition that few suppliers will maximize their joint profits whereas many suppliers are likely to behave non-cooperatively does not appear as an assumption but as a conclusion of the theory. For the simple model analyzed in this paper a definite answer can be given to the question where a “small group” of competitors ends and a “large group” begins: 5 is the dividing line between “few” and “many”.     

Trends in the evolution of models & modeling perspectives on mathematical learning and problem solving

In this paper we briefly outline the models and modelling (M&M) perspective of mathematical thinking and learning relevant for the 21st century. Models and modeling (M&M) research often investigates the nature of understandings and abilities that are needed in order for students to be able to use what they have (presumably) learned in the classroom in “real life” situations beyond school Nonetheless, M&M perspectives evolved out of research on concept development more than research on problem solving; and, rather than being preoccupied with the kind of word problems emphasized in textbooks and standardized tests, we focus on (simulations of) problem solving “in the wild.” Also, we give special attention to the fact that, in a technology-basedage of information, significant changes are occurring in the kinds of “mathematical thinking” that is coming to be needed in the everyday lives of ordinary people in the 21^st century—as well as in the lives of productive people in future-oriented fields that are heavy users of mathematics, science, and technology.     

Examination-Type Preferences of College Science Students and Their Faculty in Israel and USA: A Comparative Study

The science-examination preferences of college science students and their science faculty were surveyed, using the TOPE questionnaire at a teacher training and a community college in Israel and the U.S., respectively. The results obtained in the two countries were “intrally” and “interly” compared, in total and by gender, in terms of significant/no significant differences in the preferences made and the reasons provided by the students and faculty for their ranking. The findings suggest that: (a) college science students prefer mostly, the Israelis more so than the Americans, the nonconventional, written exams in which time is unlimited and any materials are allowed; (b) American college science students prefer the traditional class science examination (G) significantly more than their Israeli counterparts; (c) the preference of higher order cognitive skills (HOCS)-oriented exams (B. I and H) is significantly higher for female science students in Israel compared with no gender difference concerning the preferred examinations in the US, and rejection of oral examinations by all in both countries, significantly more by female students; and (d) there exists a significant gap between the preferred type of examinations of science students and their faculty in both countries. In view of the HOCS-orientation and the goal of conceptual understanding in current reforms of science education worldwide, the consonance between these curriculum objectives and examination practices is advocated. This, in turn, requires that provisions be made to lessen the gap between science teachers and their students' examination type preferences for better science learning to occur.     

Enriching Science and Math Through Engineering

This case study reviewed the collaborative efforts of university engineers, teacher educators, and middle school teachers to advance sixth‐ and seventh‐grade students' learning through a series of project‐based engineering activities. This two‐year project enriched regular school curricula by introducing real‐world applications of science and mathematics concepts that expanded opportunities for creativity and problem‐solving, introduced problem‐based learning, and provided after‐school programming (for girls only) led by engineering students from the local university. This engineering education initiative showed significant impact on students' (1) confidence in science and mathematics; (2) effort toward science and mathematics; (3) awareness of engineering; and (4) interest in engineering as a potential career. With regard to gender, there were no significant differences between boys' and girls' responses. The girls' confidence in their own skills and potential, however, was significantly more positive than the boys' confidence in the girls. These results gave rise to new questions regarding mentor/mentee relationships and the overall effect of “girls only” mentoring.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

Numerical solutions of diffusive logistic equation

In this paper we investigate numerically positive solutions of a superlinear Elliptic equation on bounded domains.The study of Diffusive logistic equation continues to be an active field of research. The subject has important applications to population migration as well as many other branches of science and engineering. In this paper the “finite difference scheme” will be developed and compared for solving the one- and three-dimensional Diffusive logistic equation. The basis of the analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from many authors these years.     

Stirling numbers and a geometric,structure from voting theory

In the spatial theory of voting, m candidates are each represented by a point in a p-dimensional Euclidean “attribute” space. The hyperplanes bisecting the line segments joining pairs of these points divide the space into regions, and each region corresponds to a definite ranking of the distances to the candidates. This paper discusses the combinatorics of such a structure and shows that the Stirling numbers have a geometrical significance.     

Statistical Hyperbolicity in Teichmüller Space

In this paper we explore the idea that Teichmüller space is hyperbolic “on average.” Our approach focuses on studying the geometry of geodesics which spend a definite proportion of time in some thick part of Teichmüller space. We consider several different measures on Teichmüller space and find that this behavior for geodesics is indeed typical. With respect to each of these measures, we show that the average distance between points in a ball of radius r is asymptotic to 2r, which is as large as possible. Our techniques also lead to a statement quantifying the expected thinness of random triangles in Teichmüller space, showing that “most triangles are mostly thin.”