概率统计教学的几点建议

概率统计是大学数学的基础课程之一,我们通过具体的例子分析了概率统计课程教学应该重视的几个问题,并提出了解决问题的建议．这对概率统计课程的教学有一定的借鉴意义．     

Two-Way Tables: Issues at the Heart of Statistics and Probability for Students and Teachers

Some problems exist at the intersection of statistics and probability, creating a dilemma in relation to the best approach to assist student understanding. Such is the case with problems presented in two-way tables representing conditional information. The difficulty can be confounded if the context within which the problem is set is one where students have preconceived opinions on the direction of the potential association present. This article considers school students’ responses to two problems of association, with data presented in 2 × 2 tables. A hierarchical rubric is presented to document students’ understandings. Teachers’ pedagogical content knowledge is also considered in relation to the same two problems. Findings include a surprising relationship of outcomes for students across the problem contexts and some concern about teachers’ pedagogical content knowledge in this area of the curriculum.     

How students use the ‘see similar example’ feature in online mathematics homework

Homework is one of students’ opportunities to learn mathematics, but we know little about what students learn from homework. This study employs the instructional triangle and didactic contract to explore how students used the ‘see similar example’ feature in an online homework platform and how that use reflected their learning goals. Findings indicate students used similar examples to troubleshoot, to check if they were on the right track, and to see the form of the answer. Students also sought to unpack the reasoning in solution steps, used solutions as templates for solving their own problems, and sometimes copied answers. One student did a ‘see similar example’ problem for more practice. Students’ goals included completing the homework, maximizing their score, and understanding the content. This research lays groundwork for future work characterizing what students learn from homework and how features that provide students with similar examples help or hinder their learning.     

概率统计课程中拓广引申式教学法探索

概率论和数理统计研究的是随机现象,有许多本学科独有的思维方式和解题模式.学生初学时会面临全新的困难.如何让学生较快地适应本课程的学习?如何引导学生主动地发现问题和探索问题?着重介绍了作者从自己的教学实践中总结出的一套行之有效的方法——拓广引申式教学法.     

School mathematics and creativity at the elementary and middle-grade levels: how are they related?

In this study, we evaluated students’ creativity, as expressed in the solution methods of three problems for groups of students in different grades. Posing the same problems to students of similar (advanced) mathematical abilities in different grades allowed us to look for possible connections between creativity and mathematical knowledge. The findings indicate that at the elementary school level, the number of solution methods and creativity scores increased with age. The collective methods space of the eighth graders seemed to narrow almost exclusively to algebraic methods, but the increase in the number of solutions was renewed in the ninth grade.     

Effective collaboration in the productive failure process

Productive failure is a learning design that encompasses problem solving prior to instruction and the learning that occurs during and after this process. In the mathematics education literature, there is a need for analyses of students’ interactions that occur as they collaborate during the productive failure process. In this paper, we contribute to this area by taking a closer look at students’ interactions that characterize an effective productive failure process. In analyzing video footage of two different groups of students working on invention tasks in a flipped mathematics classroom, we observed that the productive failure process seemed to work best in groups of students among whom the instructional design evoked students’ intellectual need and curiosity. These students also developed a set routine for solving problems whose solutions are difficult to find without prior direct instruction on the topic, which proved valuable on follow-up in-class and posttest problems.     

Problem solving activities of two middle school students with distinct levels of units coordination

The aim of this study is to investigate relationships between students’ arithmetical knowledge and their proportional reasoning. Two of seven students for whom we conducted clinical interviews were selected as participants in the study. An analysis of their solutions to four different types of multiplicative problems (equal sharing, reversible multiplicative relationship, fraction composition, and proportional relationship) was conducted. Based on the analysis, we found that the student who coordinated two three-level units structures prior to activity in the first three problem types could also solve the proportion problem using the units coordination. In contrast, the student who coordinated two three-level units structures only in activity in the first three problem types could not solve the proportion problem. Given the importance of units coordinating operations in solving diverse problems, implications for further research on students’ construction of proportional reasoning are discussed.     

Beyond statistical methods: teaching critical thinking to first-year university students

We discuss a major change in the way we teach our first-year statistics course. We have redesigned this course with emphasis on teaching critical thinking. We recognized that most of the students take the course for general knowledge and support of other majors, and very few are planning to major in statistics. We identified the essential aspects of a first-year statistics course, given this student mix, focusing on a simple question, ‘Given this is the last chance you have to teach statistics, what are the essential skills students need?’ We have moved from thinking about statistics skills needed for a statistician to skills needed to participate in today's society. We have changed the way we deliver the course with less emphasis on lectures and more on alternative resources including on-line tutorials, Excel, computer-based skills testing, web-based learning materials and smaller group activities such as study groups and example classes. Feedback from students shows that they are very receptive and enthusiastic.     

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.     

Behind the scenes of pseudo-proportionality

In this study, we attempt to put in question students’ spontaneous and uncritical application of the simple and neat mathematical formula of linearity. This is impelled with the help of a written test involving geometrical problems, which is based on an original experimental setting. In this setting, grade 9 and 10 students were instructed first to solve all the geometrical problems and then to select only one problem as the appropriate for a given numerical answer. The difficulty of this choice lied in fact that a superficial handling of each problem would resolve to the same numerical answer as the one given. The results show that students’ choices are systematic and based on the solutions given to the tasks. However, the experimental setting has managed to help students question in some degree the applicability of the linear model in situations that appear to be linear but are not.     

Middle school students’ construction of quantitative unknowns

Two iterative, after school design experiments with small groups of middle school students were conducted to investigate how students constructed quantitative unknowns, conceived of as values of fixed quantities that are not known but can be determined. Students solved problems about an unknown height or length measured in two different units. Of 13 students who participated, 6 structured quantities into three levels of units. These students constructed an unknown as a height consisting of an indeterminate number of length units, each of which consisted of smaller length units, and they symbolized these relationships in their equations. The other 7 students structured quantities into two levels of units. Five of these students symbolized only the relationships between the measurement units, with two students demonstrating more basic and advanced solutions. The study shows that grappling with unknowns as measured and indeterminate is beneficial for students’ construction of variable.     

用数形结合法求解连续型随机变量的函数分布问题

针对概率统计课程教学中的关于"连续型随机变量函数的分布"这类难点,利用数形结合的思想,巧妙的通过一副图形完全涵盖了这一类问题所需的概率知识点,且帮助了学生后续积分的计算.表明数形结合法是一个值得在概率统计教学中推广的好方法.     

Computer-Assisted Statistics Education at Delft University of Technology

Abstract

At Delft University of Technology many students experience difficulties in mastering basic concepts of probability and statistics. In the past few years the lectures have undergone a radical change—the lecture notes now contain modern data analysis techniques, like kernel density estimation, simulation, and bootstrapping. In the TWI-Stat project, a computer-aided instruction course was developed to help students become more familiar with modern statistical analysis. The course presents itself as a dynamic, interactive, personal book. Highly interactive analysis tools are available. The software will be available for MS-Windows.     

A computational approach for understanding immune response to multiple epitopes based on optimal control formulation

The immune system does not response in equal probability to every epitope of an invader. We investigate the immune system’s decision making process using optimal control principles. Mathematically, this formulation requires the solution of a two-point boundary-value problem, which is a challenging task especially when the control variables are bounded. In this work, we develop a computational approach based on the shooting technique for bounded optimal control problems. We then utilize the computational approach to carry out extensive numerical studies on a simple immune response model of two competing controls. Numerical solutions demonstrate that the results of optimal control depend on the objective function, the limitations on control inputs, as well as the amounts of peptides. Moreover, the state space of peptides can be divided into different regions according the properties of the solutions. The developed algorithm not only provides a useful tool for understanding decision making strategies of the immune system but can also be utilized to solve other complex optimal control problems.     

高等院校概率论数理统计课程分级教学的实践与思考

系统介绍了我校概率论数理统计课程分级教学的改革实践与经验,深入分析了目前存在的主要问题,并初步探讨了构建合理的分级教学模式、全面提高教学质量的关键因素及解决途径.     

Do students really understand what an ordinary differential equation is?

Differential equations (DEs) are important in mathematics as well as in science and the social sciences. Thus, the study of DEs has been included in various courses in different departments in higher education. The importance of DEs has attracted the attention of many researchers who have generally focussed on the content and instruction of DEs. However, DEs are complex issues that students may find difficulty to understand. The limited research in this literature points to the need for more studies on students’ conceptions, and understanding of DEs and their basic concepts. The objective of this study is to fill this need by revealing the understanding, difficulties and weaknesses of the students who are successful in algebraic solutions, in relation to the concepts of DEs and their solutions. For this purpose, 77 students were asked 13 DE questions (6 of them about algebraic solution, and the rest about interpreting DEs and their solutions). From an analysis of the students’ answers, it was concluded that the students who were quite successful in algebraic solutions, indeed did not fully understand the related concepts, and they had serious difficulties in relation to these concepts.     

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

Quantitative stability in stochastic programming

In this paper we study stability of optimal solutions of stochastic programming problems with fixed recourse. An upper bound for the rate of convergence is given in terms of the objective functions of the associated deterministic problems. As an example it is shown how it can be applied to derivation of the Law of Iterated Logarithm for the optimal solutions. It is also shown that in the case of simple recourse this upper bound implies upper Lipschitz continuity of the optimal solutions with respect to the Kolmogorov—Smirnov distance between the corresponding cumulative probability distribution functions.     

An historical example of mathematical modelling: the trajectory of a cannonball

Classroom considerations of the concept and processes of mathematical modelling can do much to strengthen students’ problem solving skills. A systematic exposure to the techniques of mathematical modelling helps students formulate problems, re‐think those problems in mathematical terms, appreciate possible solution constraints and seek solutions that are realistic within the scope and conditions of the problem. While many mathematical modelling situations can be found in today's world, there are special pedagogical values in examining existing mathematical models that have an historical basis. Such an examination should reveal the mechanics of a modelling situation and how a model evolves or is refined to meet ever increasing human demands for accuracy or practicality. The trajectory of a cannonball provides such a modelling example. This topic captures the imagination of students and supplies the basis for a variety of classroom discussions and problem solving encounters.