The incoming statistical knowledge of undergraduate majors in a department of mathematics and statistics

Studies have shown that at the end of an introductory statistics course, students struggle with building block concepts, such as mean and standard deviation, and rely on procedural understandings of the concepts. This study aims to investigate the understandings entering freshman of a department of mathematics and statistics (including mathematics education), students who are presumably better prepared in terms of mathematics and statistics than the average university student, have of introductory statistics. This case study found that these students enter college with common statistical misunderstandings, lack of knowledge, and idiosyncratic collections of correct statistical knowledge. Moreover, they also have a wide range of beliefs about their knowledge with some of the students who believe that they have the strongest knowledge also having significant misconceptions. More attention to these statistical building blocks may be required in a university introduction statistics course.     

Introducing the Emerging Discipline of Statistics Education

Increasing attention has been given over the last decade by the statistics, mathematics and science education communities to the development of statistical literacy and numeracy skills of all citizens and the enhancement of statistics education at all levels. This paper introduces the emerging discipline of statistics education and considers its role in the development of these important skills. The paper begins with information on the growing importance of statistics in today's society, schools and colleges, summarizes unique challenges students face as they learn statistics, and makes a case for the importance of collaboration between mathematicians and statisticians in preparing teachers to teach students how to understand and reason about data. We discuss the differences and interrelations between statistics and mathematics, recognizing that mathematics is the discipline that has traditionally included instruction in statistics. We conclude with an argument that statistics should be viewed as a bridge between mathematics and science and should be taught in both disciplines.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

Describing critical statistical literacy habits of mind

Given the vast number of data representations that people encounter daily, it is imperative that people become critical consumers of the representations they will face. This requires the development of particular habits of mind. The purpose of this paper is to share the critical statistical literacy habits of mind framework. We articulate how we drew on the literature related to statistical literacy, critical mathematics, and critical statistical literacy to identify the habits of mind needed to enact critical statistical literacy in the context of consuming data representations. We describe the refinement process using qualitative interview data. To illustrate what each of the critical statistical literacy habits of mind looks like when enacted, we share examples from statistics teachers making sense of a data representation.     

Updating beliefs about variables given new information on how those variables relate

Bayesian techniques specify how to update beliefs about a variable given information on that variable or related variables. In many cases, statistical analyses also provide information about the relationship between variables, but the Borel Paradox prohibits many natural ways of updating beliefs conditioned on information about a relationship. This paper presents a method by which beliefs can be updated without violating the Borel Paradox under certain circumstances. We apply our approach to relationships specified by a statistical model (i.e., regression), and relationships described by statistical simulation.     

Mathematical literacy in undergraduates: role of gender,emotional intelligence and emotional self-efficacy

This empirical study explores the roles that Emotional Intelligence (EI) and Emotional Self-Efficacy (ESE) play in undergraduates’ mathematical literacy, and the influence of EI and ESE on students’ attitudes towards and beliefs about mathematics. A convenience sample of 93 female and 82 male first-year undergraduates completed a test of mathematical literacy, followed by an online survey designed to measure the students’ EI, ESE and factors associated with mathematical literacy. Analysis of the data revealed significant gender differences. Males attained a higher mean test score than females and out-performed the females on most of the individual questions and the associated mathematical tasks. Overall, males expressed greater confidence in their mathematical skills, although both males’ and females’ confidence outweighed their actual mathematical proficiency. Correlation analyses revealed that males and females attaining higher mathematical literacy test scores were more confident and persistent, exhibited lower levels of mathematics anxiety and possessed higher mathematics qualifications. Correlation analyses also revealed that in male students, aspects of ESE were associated with beliefs concerning the learning of mathematics (i.e. that intelligence is malleable and that persistence can facilitate success), but not with confidence or actual performance. Both EI and ESE play a greater role with regard to test performance and attitudes/beliefs regarding mathematics amongst female undergraduates; higher EI and ESE scores were associated with higher test scores, while females exhibiting higher levels of ESE were also more confident and less anxious about mathematics, believed intelligence to be malleable, were more persistent and were learning goal oriented. Moderated regression analyses confirmed mathematics anxiety as a negative predictor of test performance in males and females, but also revealed that in females EI and ESE moderate the effects of anxiety on test performance, with the relationship between anxiety and test performance linked more to emotional management (EI) than to ESE.     

Computer Literacy: An Evolving Concept

The term “computer literacy” became very popular in the 1980s as a catch phrase describing a new type of understanding. Prior to the phrase's popularity, “literacy” was reserved for the knowledge of basic skills in reading and writing and familiarity with the classical works and great books of ancient and modern cultures. To be literate means to be educated regarding the fundamental or basic ideas, beliefs and methods of communication in society. By applying the term “literacy” to the knowledge of computers, society is signifying that this sort of knowledge is as important to a person's education in contemporary society as knowledge of reading and writing has been in the past (Ringle, 1981). While many people seem to agree that the proliferation of computers and their application in extensive areas of human endeavor require us to take the notion of literacy seriously, there is still no consensus as to how this educational goal should be achieved. The difficulty stems from the fact that there is no universally accepted definition of computer literacy. Until educators are clear about the goal, effective ways to attain the objective of computer literacy will be clouded by confusion. The purpose of this research is to examine the historical evolution of the phrase “computer literacy,” to develop a chronological continuum of computer literacy, and to identify several competencies that will likely characterize the computer literate teacher in the 1990s.     

Multiple Priors and Asset Pricing

The asset pricing implications of a statistical model consistent with multiple priors, or beliefs about return distributions, are developed. It is shown that quite generally equilibrium differences in mean returns across priors are to be explained in terms of perceived risk differences between these priors. Advances in filtering theory are employed on time series data to filter all the multiple state conditional components of risks and rewards. It is then observed that excess return differentials across priors are broadly consistent with required risk compensations under these priors, though the sharp hypothesis of zero intercept and unit slope is rejected. The filtered results also deliver numerous other interesting statistics. Here we focus on the construction of long horizon return distributions from data on daily returns using a Markov chain approach to incorporate stochasticity in elementary risk characterizations like volatility, skewness and kurtosis.      

Technological Tools in the Introductory Statistics Classroom: Effects on Student Understanding of Inferential Statistics

While technology has become an integral part of introductory statistics courses, the programs typically employed are professional packages designed primarily for data analysis rather than for learning. Findings from several studies suggest that use of such software in the introductory statistics classroom may not be very effective in helping students to build intuitions about the fundamental statistical ideas of sampling distribution and inferential statistics. The paper describes an instructional experiment which explored the capabilities of Fathom, one of several recently-developed packages explicitly designed to enhance learning. Findings from the study indicate that use of Fathom led students to the construction of a fairly coherent mental model of sampling distributions and other key concepts related to statistical inference. The insights gained point to a number of critical ingredients that statistics educators should consider when choosing statistical software. They also provide suggestions about how to approach the particularly challenging topic of statistical inference. This revised version was published online in July 2006 with corrections to the Cover Date.     

The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting”.     

Every rose has its thorn: secondary teachers’ reasoning about statistical models

Statistical modeling is a core component of statistical thinking and has been identified by several countries as a curricular goal for secondary education. However, many secondary teachers have minimal preparation for teaching this topic. The goal of this research study is to learn about teachers’ perceptions of the role statistical models play in statistical inference and how these perceived purposes affect their reasoning about statistical models and inference. Problem-solving interviews were conducted with four in-service teachers who had recently taught a modeling and simulation-based introductory statistics course. Teachers’ responses suggest they may not see modeling variation as the primary purpose of statistical modeling and instead substitute two other purposes: making a decision and replicating the data collection process. Suggestions for how to build on teachers’ transitional conceptions and refocus attention on modeling variation are discussed.     

The Relationship Between Teacher and Student Beliefs About Mathematics

In this study the relationship between teacher (n = 7) beliefs about mathematics, the learning and teaching of mathematics and their respective students' beliefs about mathematics (n = 158) are examined. The data were collected by means of two instruments specifically designed to measure belief systems about mathematics. Teacher scores were adjusted so that a higher score reflected beliefs in aligntnent with the National Council of Teachers of Mathematics (NCTM) Standards. Results indicated that the students of teachers whose beliefs were in alignment with the NCTM Standards had significantly different beliefs about factors that lead to success in mathematics than did other students. Specifically these students felt that working hard to solve problems and striving for understanding would lead to success. No student differences were found for subscales of ego orientation, competitiveness, interest and extrinsic factors such as neatness and cooperation. These findings suggest that this group of teachers practiced what they believed and that these practices affected what their students believed about mathematics. We suggest that using these two assessments in tandem give a clearer picture of the mathematical environment within a classroom and can be used in professional development workshops to initiate teacher reflection about classroom practices.     

智能的统计软件──统计专家系统

计算统计是数理统计的一个重要分支，而统计专家系统是计算统计中一极热门的话题。本文简介了统计专家系统的概念及发展过程，并根据本人所作的工作介绍了统计专家系统的基本结构和构造过程等。最后对未来的统计专家系统提出了一点设想。     

Empowerment of Elementary School Teachers to Implement Science Curriculum Reforms

The authors evaluate the Science-Technology-Society (STS) views, attitudes and literacy of Israeli elementary science school teachers in three in-depth, longitudinal inservice programs. The programs were aimed at promoting teacher empowerment for the implementation, on a national scale, of a new STS-oriented elementary science curriculum reform. By using a blend of qualitative and quantitative research methods, the effectiveness of these similar but not identical programs has been determined. It was found that appropriately designed inservice science teacher training courses did induce a shift in the STS views/positions, beliefs/attitudes and literacy of elementary science teachers, and that this shift can be increased by using research-based data, obtained within an ongoing formative education process.     

Inductive and Deductive Justification of Knowledge: Flexible Judgments Underneath Stable Beliefs in Teacher Education

Personal epistemological beliefs are considered to play an important role for processes of learning and teaching. However, research on personal epistemology is confronted with theoretical issues as there is conflicting evidence regarding the structure, stability, and context-dependence of epistemological beliefs. We give evidence how theoretical and methodological issues can partly be resolved by distinguishing between relatively stable “epistemological beliefs” and situation-specific “epistemological judgments.” A qualitative content analysis of a series of semistructured interviews (study 1) with pre-service teachers, teachers, and teacher educators as well as a statistical analysis of pre-service teachers’ extensive answers in questionnaires (study 2), both on the topic of “mathematical discovery,” reveal not only beliefs of the participants but also different qualities of judgments. Therefore, in further research both aspects of beliefs should be considered in a more differentiated manner when categorizing belief structures.     

浅谈蒙特卡罗法在概率统计教学中的应用

在概率统计教学中利用蒙特卡罗法设计适当的模拟实验,结合多媒体在课堂教学中实时演示,能很好地加深学生对概率统计概念、思想的理解,进而培养学生的统计思维能力.     

Hybrid methodology for technical analysis

Hybrid methodology involving differential equations modeling and statistical regression is developed in order to test basic ideas in asset price dynamics. In particular, the method provides a mechanism for testing the relative importance of price trend compared with valuation. The significance of yearly highs in prices can also be understood through this procedure. A large data set of 52 closed-end funds comprising about 61,500 data points is used with the mixed effects model in SPlus. The model suggests that the role of the trend is as significant as the valuation. Upon determination of the coefficients, one has a model that can be used for short term forecasts of asset prices. The model incorporates the finiteness of assets and the importance of “liquidity”, or “excess cash”. The statistics utilize data on the number of shares and the national money supply. The methodology can easily be extended to other behavioral effects.     

Mathematics Learning and Participation as Racialized Forms of Experience: African American Parents Speak on the Struggle for Mathematics Literacy

This article draws on 3 ethnographic and participant observation studies of African American parents and adults from 3 northern California communities. Although studies have shown that African American parents hold the same folk theories about mathematics as other parents, stressing it as an important school subject, few studies have sought to directly examine their beliefs about constraints and opportunities associated with mathematics learning for both themselves and their children. I argue that, as they situate the struggle for mathematical literacy within the larger contexts of African American, political, socioeconomic, and educational struggle, these parents help reveal that mathematics learning and participation can be conceptualized as racialized forms of experience. As they attempt to become doers of mathematics and advocates for their children's mathematics learning, discriminatory experiences have continued to subjugate some of these parents, whereas others—as demonstrated in their oppositional voices and behaviors—resisted their continued subjugation based on a belief that mathematics knowledge, beyond its role in schools, can be used to change the conditions of their lives. The characterization of mathematics learning as racialized experience put forth in this article contrasts with culture-free and situated perspectives of mathematics learning often found in the literature. As a result of their experiences with oppression in this society, the concept of race has historically played a major role in the lives of African Americans. Although race has dubious value as a scientific classification system, it has had real consequences for the life experiences and life opportunities of African Americans in the United States. Race is a socially constructed concept which is [a] defining characteristic for African American group membership. (Sellers, Smith, Shelton, Rowley, & Chavous, 1998, p. 18)     

An applied statistics course for systematics and ecology PhD students

Statistics education is under review at all educational levels. Statistical concepts, as well as the use of statistical methods and techniques, can be taught in at least two contrasting ways. Specifically, (1) teaching can be theoretically and mathematically oriented, or (2) it can be less mathematically oriented being focused, instead, on application and the use of data to solve real-world problems. The second approach is growing in practice and new goals have recently emerged. At present, statistics courses stress probability concepts, data analysis, and the interpretation and communication of results. Understanding the process of statistical investigation is established as a way of improving mastery of statistical reasoning. In this context, a project-based approach allows the design and implementation of participating learning scenarios in order to understand the statistical methodology and, as a consequence, improve research. This approach points out that statistics is a rational methodology used to solve practical problems. The purpose of this paper is to present the design and results of an applied statistics course for PhD students in ecology and systematics using a project-based approach. Examples involving character coding, species classification, and the interpretation of geographical variation, which are the principal systematic analyses requiring statistical techniques, are presented using the results from student projects. In addition, an example from conservation ecology is presented. Results indicate that the students understood the concepts and applied the systematic and statistical techniques accurately using a data oriented approach.     

Innovations in statistical modeling to connect data,chance and context

Statistical modeling is emerging as a fertile research environment in which to promote and learn about student statistical reasoning processes. We outline a paradigm shift toward a modeling perspective that is occurring in statistics education research and how statistical modeling processes involve connecting data, chance and context. The innovative task and software designs and theoretical frameworks that are under development for explicating student reasoning and pedagogy are discussed. In conclusion, we reflect on statistics education research.