MATLAB in first-year engineering mathematics

The paper details the integration of the mathematical software MATLAB into the teaching of core mathematics to first-year university engineering students. The engineering faculty requested that the mathematics staff allocate one hour per week for students to learn and use MATLAB in a computer pool and that group projects involving the use of MATLAB and other similar tasks be included in the overall assessment of the subject. The central concept in achieving this integration was the production of a guide A Focused Introduction to MATLAB which provides a bridge between the software and the core subject material covered in the textbooks and lectures. Recent extensions involve the construction of a Web site dedicated to this subject where students can obtain all information concerning the subject, including a copy of all MATLAB code and corresponding graphical demonstrations used in the lectures. Feedback has been very positive.     

More than multiplication in a 12 × 12 multiplication table

This note presents demonstrations of mathematics that emerges when problems are posed with respect to a combined 12?×?12 multiplication table?showing multiplier and multiplicand. Through processes such as recognizing and extending patterns, specializing and generalizing particular functional relationships between the diagonal and row sequences are compressed. Insights obtained from the various methods can be used to deepen teachers’ and students’ understandings of possible ways to bridge arithmetic and algebra.     

Some Thoughts on Computers and Introductory Statistics

Examination of results of an extensive project in the use of computers in undergraduate statistics reveals that the use is frequently unsuccessful because of insufficient maturity on the part of students. Nevertheless, even when students are not sufficiently mature to warrant hands‐on use, computers may still be used effectively. Indeed, when measured against the possible capabilities, instructor controlled computer usage can be just as effective as student use. An added bonus is economy of operation. Several suggestions for instructor controlled usage include graphical and numerical demonstrations, computer generated problem books with solutions, computer generated visual aids and loose‐leaf files of illustrations, examples and solutions.     

A Monte Carlo method for high dimensional integration

Summary A new method for the numerical integration of very high dimensional functions is introduced and implemented based on the Metropolis' Monte Carlo algorithm. The logarithm of the high dimensional integral is reduced to a 1-dimensional integration of a certain statistical function with respect to a scale parameter over the range of the unit interval. The improvement in accuracy is found to be substantial comparing to the conventional crude Monte Carlo integration. Several numerical demonstrations are made, and variability of the estimates are shown.     

Developing a statistical modeling framework to characterize Year 7 students’ reasoning

Some researchers advocate a statistical modeling approach to inference that draws on students’ intuitions about factors influencing phenomena and that requires students to build models. Such a modeling approach to inference became possible with the creation of TinkerPlots Sampler technology. However, little is known about what statistical modeling reasoning students need to acquire. Drawing and building on previous research, this study aims to uncover the statistical modeling reasoning students need to develop. A design-based research methodology employing Model Eliciting Activities was used. The focus of this paper is on two 11-year-old students as they engaged with a bag weight task using TinkerPlots. Findings indicate that these students seem to be developing the ability to build models, investigate and posit factors, consider variation and make decisions based on simulated data. From the analysis an initial statistical modeling framework is proposed. Implications of the findings are discussed.     

Calculus demonstrations using MATLAB

The note discusses ways in which technology can be used in the calculus learning process. In particular, five MATLAB programs are detailed for use by instructors or students that demonstrate important concepts in introductory calculus: Newton's method, differentiation and integration. Two of the programs are animated. The programs and the graphical user interface have been specifically designed to help the student understand the processes behind these important introductory concepts. Each program has a series of demonstrations that show unusual, difficult or important cases.     

A conceptual pathway to confidence intervals

Finding ways for the majority of students to better understand conventional normal theory-based statistical inference seems to be an intractable problem area for researchers. In this paper we propose a conceptual pathway for developing confidence interval ideas for the one-sample situation only from an intuitive sense to bootstrapping for students from about age 14 to first-year university. We make the case that conceptual development should start early; that probability and statistical instruction should change so that both orientate students towards interconnected stochastic conceptions; and that the use of visual imagery has the potential to stimulate students towards such a perspective. We analyse our conceptual pathway based on a theoretical framework for a stochastic conception of statistical inference based on imagery and some research evidence. Our analysis suggests that the pathway has the potential for students to become conversant with the concepts underpinning inference, to view statistics probabilistically and to integrate concepts into a coherent comprehension of inference.     

数学优生与差生的素质差异的比较研究

本文通过对初中学生基础素质指标与学习成绩的多元统计分析,探讨了数学差生与优生的素质差异,从不同群体的基础素质对数学学习成绩的影响,研究培养优生与转化差生的不同策略。 本项研究得到世界银行贷款的资助,获96年山东省普通高等学校优秀教学成果二等奖;98年1月通过了国家教委世行贷款办主持的鉴定。99年3月获“1998年度全国师范院校基础教育改革实验优秀成果”二等奖及世界银行贷款“师范教育发展”改革课题实验研究优秀项目二等奖。     

Statistical modeling to promote students’ aggregate reasoning with sample and sampling

While aggregate reasoning is a core aspect of statistical reasoning, its development is a key challenge in statistics education. In this study we examine how students’ aggregate reasoning with samples and sampling (ARWSS) can emerge in the context of statistical modeling activities of real phenomena. We present a case study on the emergent ARWSS of two pairs of sixth graders (age 11–12) involved in statistical data analysis and informal inference utilizing TinkerPlots. The students’ growing understandings of various statistical concepts is described and five perceptions the students expressed are identified. We discuss the contribution of modeling to these progressions followed by conclusions and limitations of these results. While idiosyncratic, the insights contribute to the understanding of students’ aggregate reasoning with data and models, with regards to samples and sampling.     

A quaternionic proof of the representation formula of a quaternary quadratic form

The celebrated Four Squares Theorem of Lagrange states that every positive integer is the sum of four squares of integers. Interest in this Theorem has motivated a number of different demonstrations. While some of these demonstrations prove the existence of representations of an integer as a sum of four squares, others also produce the number of such representations. In one of these demonstrations, Hurwitz was able to use a quaternion order to obtain the formula for the number of representations. Recently the author has been able to use certain quaternion orders to demonstrate the universality of other quaternary quadratic forms besides the sum of four squares. In this paper, we develop results analogous to Hurwitz's above mentioned work by delving into the number theory of one of these quaternion orders, and discover an alternate proof of the representation formula for the corresponding quadratic form.     

Interpreting latent variables in factor models via convex optimization

Latent or unobserved phenomena pose a significant difficulty in data analysis as they induce complicated and confounding dependencies among a collection of observed variables. Factor analysis is a prominent multivariate statistical modeling approach that addresses this challenge by identifying the effects of (a small number of) latent variables on a set of observed variables. However, the latent variables in a factor model are purely mathematical objects that are derived from the observed phenomena, and they do not have any interpretation associated to them. A natural approach for attributing semantic information to the latent variables in a factor model is to obtain measurements of some additional plausibly useful covariates that may be related to the original set of observed variables, and to associate these auxiliary covariates to the latent variables. In this paper, we describe a systematic approach for identifying such associations. Our method is based on solving computationally tractable convex optimization problems, and it can be viewed as a generalization of the minimum-trace factor analysis procedure for fitting factor models via convex optimization. We analyze the theoretical consistency of our approach in a high-dimensional setting as well as its utility in practice via experimental demonstrations with real data.     

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.     

Students’ emergent articulations of uncertainty while making informal statistical inferences

Research on informal statistical inference has so far paid little attention to the development of students?? expressions of uncertainty in reasoning from samples. This paper studies students?? articulations of uncertainty when engaged in informal inferential reasoning. Using data from a design experiment in Israeli Grade 5 (aged 10?C11) inquiry-based classrooms, we focus on two groups of students working with TinkerPlots on investigations with growing sample size. From our analysis, it appears that this design, especially prediction tasks, helped in promoting the students?? probabilistic language. Initially, the students oscillated between certainty-only (deterministic) and uncertainty-only (relativistic) statements. As they engaged further in their inquiries, they came to talk in more sophisticated ways with increasing awareness of what is at stake, using what can be seen as buds of probabilistic language. Attending to students?? emerging articulations of uncertainty in making judgments about patterns and trends in data may provide an opportunity to develop more sophisticated understandings of statistical inference.     

Dot plots and hat plots: supporting young students emerging understandings of distribution,center and variability through modeling

An important use of statistical models and modeling in education stems from the potential to involve students more deeply with conceptions of distribution, variation and center. As models are key to statistical thinking, introducing students to modeling early in their schooling will likely support the statistical thinking that underpins later, more advanced work with increasingly sophisticated statistical models. In this case study, a class of 10–11 year-old students are engaged in an authentic task designed to elicit modeling. Multiple data sources were used to develop insights into student learning: lesson videotape, work samples and field notes. Through the use of dot plots and hat plots as data models, students made comparisons of the data sets, articulated the sources of variability in the data, sought to minimize the variability, and then used their models to both address the initial problem and to justify the effectiveness of their attempts to reduce induced variation. This research has implications for statistics curriculum in the early formal years of schooling.     

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.     

Successful Students’ Perceptions of Secondary School Science

The attitudinal perceptions of successful college science students regarding their junior high and high school science experiences were compared with those of successful nonscience students. Particular attention was paid to recollections of teacher personality attributes and instructional methodology. Results indicate that science students were especially motivated by knowledgeable, enthusiastic, communicative, committed, friendly, competent, and creative science teachers, whereas the nonscience group preferred patient, knowledgeable, congenial, friendly, supportive, and enthusiastic instructors. Both groups agreed that, although traditional methods (textbooks, lectures, quizzes/tests) dominated their science experiences, their preferred instructional strategies included more dynamic methods, including laboratory activities, teacher demonstrations, and discussions. Both groups also agreed that high school science courses provided a closer match than did junior high/middle school courses in providing their preferred teacher attributes and instructional methods. Results supported the observation that, even for these academically gifted students, interest in science is relatively depressed during the junior high/middle school years. It was concluded that, although endogenous variables act in concert, the quality of the student-teacher interaction exerts the greatest influence on student attitudes, particularly if those students are not already “science-friendly.” Instructional implications are discussed.     

Statistical modelling and repeatable structures: purpose,process and prediction

Children have limited exposure to statistical concepts and processes, yet researchers have highlighted multiple benefits of experiences in which they design and/or engage informally with statistical modelling. A study was conducted with a classroom in which students developed and utilised data-based models to respond to the inquiry question, Which origami animal jumps the furthest? The students used hat plots and box plots in Tinkerplots to make sense of variability in comparing distributions of their data and to support them to write justified conclusions of their findings. The study relied on classroom video and student artefacts to analyse aspects of the students’ modelling experiences which exposed them to powerful statistical ideas, such as key repeatable structures and dispositions in statistics. Three principles—purpose, process and prediction—are highlighted as ways in which the problem context, statistical structures and inquiry dispositions and cycle extended students’ opportunities to reason in sophisticated ways appropriate for their age. The research question under investigation was, How can an emphasis on purpose, process and prediction be implemented to support children’s statistical modelling? The principles illustrated in the study may provide a simple framework for teachers and researchers to develop statistical modelling practices and norms at the school level.     

The effect of project-based learning on students’ statistical literacy levels for data representation

The point of this study is to define the effect of project-based learning approach on 8th Grade secondary-school students’ statistical literacy levels for data representation. To achieve this goal, a test which consists of 12 open-ended questions in accordance with the views of experts was developed. Seventy 8th grade secondary-school students, 35 in the experimental group and 35 in the control group, took this test twice, one before the application and one after the application. All the raw scores were turned into linear points by using the Winsteps 3.72 modelling program that makes the Rasch analysis and t-tests, and an ANCOVA analysis was carried out with the linear points. Depending on the findings, it was concluded that the project-based learning approach increases students’ level of statistical literacy for data representation. Students’ levels of statistical literacy before and after the application were shown through the obtained person-item maps.     

Understanding Distributions by Modeling Them

In current curriculum materials for middle school students in the US, data and chance are considered as separate topics. They are then ideally brought together in the minds of high school or university students when they learn about statistical inference. In recent studies we have been attempting to build connections between data and chance in the middle school by using a modeling approach made possible by new software capabilities that will be part of TinkerPlots 2.0 (TinkerPlots is published by Key Curriculum Press and has been developed with grants from the National Science Foundation (ESI-9818946, REC-0337675, ESI-0454754). Opinions expressed here are our own and not necessarily those of the Foundation.). Using a new Sampler object, students build “factories” to model not only prototypical chance events, but also distributions of measurement errors and of heights of people. We provide the rationale for having students model a wide range of phenomena using a single software tool and describe how we are using this capability to help young students develop a robust, statistical perspective.