Contributed Papers

C101: A graphical approach to examine inferential reasoning development

Sharon Lane-Getaz   St. Olaf College, United States

This observational study examines inferential reasoning development in students taking a randomization-based introductory college course (n = 38). The proportions of students answering each item correctly are compared from Pretest to Posttest using a simple descriptive scatterplot. This “canoe plot” includes a superimposed 95% confidence band which differentiates items with no statistical difference between the proportion answering correctly on the Pretest and Posttest from those that do differ. A brief discussion follows highlighting the content for those items where inferential reasoning differs from Pretest to Posttest. The Reasoning about P-values and Statistical Significance (RPASS) scale is used to measure students’ inferential reasoning outcomes and gains. Directions for future research are discussed.

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C102: How to organize effective and efficient group work in tutorials using indicator strips

Toni C. Stocker   University of Mannheim, Germany

Many lectures in Statistics are accompanied by weekly tutorials for which exercises are to be solved and possibly submitted. One way to get students actively involved in tutorials is to organize them into small groups of about 3 or 4 students, in which they are to compare and discuss their individual solutions. I describe a simple but efficient method for how such groups can rationally be formed at the beginning of each tutorial and individually be supported by the tutor. This concept is based mainly upon the idea of homogenization. It turns out that this way the learning outcomes are much more sustainable than in traditional “lecture style tutorials” provided that some important frame conditions are met. Longtime experiences in large class Statistics courses with small class tutorials directed by teaching assistants are very positive.

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C104: The potential of a grounded theory approach to study teaching probability

Andreas Eckert   Linnaeus University, Sweden

An important part of teaching probability is teachers interacting with students about probability. Most of these interactions do not occur anywhere else but inside the classroom so that is where we should research teacher knowledge for future development of teacher training and professional development. To accomplish this I propose a research methodology founded on the theoretical assumptions of symbolic interactionism combined with a grounded theory approach. The purpose of this paper is to outline such a research methodology that focuses on teaching as classroom interaction between teachers and students. The discussion aims to emphasize the possibilities by this way of studying teachers’ knowledge for teaching probability and refine the methodological construct. Examples used are from lessons where two teachers work with unknown sample spaces and interact with students regarding chance, variation and the importance of sampling.

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C105: Distance education of teachers of statistics

Jeffrey Hovermill   Northern Arizona University, United States
Brian Beaudrie   Northern Arizona University, United States

A statistically literate citizenry is essential in any society. Teachers play a vital role in developing this expertise so it is very important to continually develop their capacity to teach statistics to the next generation. Teachers, however, are also often place- and time-bound and cannot commute to university cities between 9-5 to take face-to-face coursework. Connections: Statistics and Probability for Teachers, is an on-line course focused on preparing secondary and community college teachers to effectively teach statistics. This research-based course blends emphases on statistics content and pedagogy, and is part of a program focused on strengthening the depth of content knowledge, instructional strategies, and reflective practice of middle, high school, and community college teachers. This paper describes the structure and format of the course and how it fits into the overall program. Examples from the course and reflections from students are also shared.

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C107: Challenges in teaching medical statistics in developing country

Ashwani Kumar Mishra   All India Institute of Medical Science, India

“Biostatisticians” work in diversified medical settings. The process of teaching biostatistics is extremely challenging due to the absence of a formal procedure of statistical education at most places. Irrespective of the nature of statistical education, some of the general challenges encountered are understanding the perspectives of statisticians and clinicians, recognizing the importance of statistical ingredients in the formulation of the research protocol, knowing the methodologies for setting up a research database, acquainting the clinicians with an accurate statistical test to apply, assessing the clinical and statistical significance of the proposed research findings and finally the dissemination of results. The present deliberation specifically aims to address challenges faced by the mid-level biostatistician from India, a developing country. The younger statistics and biostatistics professionals at various levels constitute the target audience.

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C108: The wonderful lamp of Aladdin? Project work in engineering courses

Maria Manuel da Silva Nascimento   University of Tras-os-Montes and Alto Douro, Portugal
José Alexandre dos Santos Vaz Martins   Polytechnic Institute of Guarda, Portugal
Assumpta Estrada   University of Lleida, Spain

In the first semester of the 2012/2013 academic year we approached teaching and learning statistics using project work. Here we will present and discuss the proposal made to the students in the Probability and Statistics course for different engineering degrees – Civil, Energy and Mechanics Engineering from a Portuguese university. In the first cycle a survey from a Master’s thesis in civil engineering was the basis of a hands-on project work developed by second year students in their courses. We examined the results of the project work written reports based on the approach of Pimenta (2006) and Nascimento and Martins (2008) and related it to the Dublin Descriptors of the Bologna Process for the Higher Education in Europe.

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C109: After statistics reform: should we still teach significance testing?

Tony Hak   Rotterdam School of Management, The Netherlands

In the longer term null hypothesis significance testing (NHST) will disappear because p-values are not informative and not replicable. As with any reform, the question can be asked whether we should continue to teach the procedures of abolished routines (i.e., NHST), not as a commendable practice but as a means of understanding what our (more or less statistically ignorant) predecessors did. Three arguments are discussed for not teaching NHST in (introductory) undergraduate courses in inferential statistics: experience shows that NHST is too difficult for introductory courses; dichotomous thinking inherent to NHST is a cognitive obstacle for interpretation; and students can find relevant information in research reports without knowing NHST.

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C111: Measuring the basics of probabilistic reasoning: the IRT-based construction of the probabilistic reasoning questionnaire

Caterina Primi   University of Florence, Italy
Kinga Morsanyi   School of Psychology Queen's University Belfast, United Kingdom
Francesca Chiesi   University of Florence, Italy

Some of the difficulties students have in learning basic data analysis stem from a lack of basic understanding of probabilities. The aim of the present study was to develop a scale to measure basic probabilistic reasoning skills, which are deemed necessary to successfully complete introductory statistics courses. Specifically, our aim was to accurately measure low levels of ability in order to identify students with difficulties. Item Response Theory was applied to construct a scale. The validity of the scale was studied with reference to math skills and self-efficacy, attitude towards statistics, and numeracy. Finally, the equivalence of the scale across genders was assessed by analyzing Differential Item Functioning. The scale has advantages in introductory statistics teaching. Indeed, once students who are more likely to encounter difficulties are identified, ad-hoc training courses can be developed, focusing on probability knowledge that is required at the exam, in order to improve students’ performance.

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C116: A review of probability and statistics apps for mobile devices

Howard Edwards   Massey University, New Zealand

Mobile devices such as tablets and smartphones are rapidly replacing laptops and notebooks as the primary student e-learning device. This paper reviews some the mobile apps currently available which enable a user to either learn Statistics or to carry out the sorts of summaries and analyses encountered in an undergraduate Statistics course. Implications of these apps for both teaching and learning are discussed.

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C117: Games as a locus of self-empowered collaborative learning

Shonda Kuiper   Grinnell College, United States
Rod Sturdivant   Ohio State University, United States

This paper presents web-based games and corresponding investigative laboratory modules (labs) to effectively teach statistical thinking and the process of scientific research. We demonstrate game-based labs that follow the GAISE guidelines and bridge the gap between short, focused homework problems and the open-ended nature of a research project. Each game-based lab presents a research question in the context of a case study and encourages students to follow a complete process of statistical analysis. These materials consist of one- or two-day activities designed for introductory college courses as well as more complex projects geared toward upper level undergraduate courses. These game-based labs encourage students early in their studies to experience the role of a research scientist and to understand how the field of statistics helps advance scientific knowledge.

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C119: Fostering deep statistical learning using diversity content: the “statistics of sexual orientation” course

Michele DiPietro   Kennesaw State University, United States

This course is the first of its kind, and to this day it remains unique. It was developed in 2004 after the realization that all the questions people ask in the public debates surrounding LGBT rights are (Is homosexuality genetic or learned? Are LGBT couples good parents? How many people are LGBT? Can you change your sexual orientation? And so on) are in fact statistical questions—they involve gathering data from a sample, measuring relevant variables, drawing a conclusion from the sample, and generalizing to a larger population to the extent possible. Based on this realization, my hope was that the diversity content and the statistical tools would act synergistically and stimulate deeper leaning in both areas. Statistical knowledge would enable students to take informed positions on important societal debates, and real life applications would demonstrate the power of statistics.

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C120: Excel simulation as a tool in teaching sampling distributions in introductory statistics

Leslie Chandrakantha   John Jay College of Criminal Justice of CUNY, United States

Many instructors are adopting computer simulation to introduce key concepts which many students in introductory statistics classes struggle to understand. Research has shown that the use of computer simulation methods as an alternative to traditional methods of books and lecture enhance conceptual understanding. Computer simulation using spreadsheets such as Excel allows students to experiment with data and to visualize the results. The concepts of sampling distributions are essential for the understanding of later inferential statistics topics such as confidence intervals and hypothesis testing. In this paper, we will describe how to use simulation using the Excel Data Tables facility and standard functions to teach sampling distributions in introductory statistics classes. This approach gives a better understanding of the Central Limit Theorem which describes the sampling distribution. Our preliminary assessment shows that this approach would enhance the understanding of the concepts.

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C121: Perceptions of undergraduate and executive students about the use of an exercise-based statistics website

André Samartini   Business School of São Paulo, Brazil

The purpose of this paper is to analyze the perceptions of undergraduate and executive students of a business course regarding the experience with an exercise-based statistics website. This particular website has multiple-choice questions designed for students who are taking basic statistics courses. By providing other types of resources (videos, textbooks, lists of exercises, lecture notes) to the students, it was possible to verify, at the end of the course, preference, efficacy and use of the site compared to all available resources. Results of a survey with the students show that offline resources were slightly preferable than online resources but, whether online or printed, most of the students considered exercises more helpful than any other resource type. Regarding the experience with the site, students agreed that the site collaborated to their learning and most of them were more motivated to learn statistics.

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C122: Students’ understanding of confidence intervals

Robyn Reaburn   University of Tasmania, Australia

The aim of this study was to gain knowledge of students’ beliefs and difficulties in understanding confidence intervals and to use this knowledge to develop improved teaching programs. This study took place over four consecutive teaching semesters of a one-semester tertiary statistics unit. The study was cyclical, in that the results of each semester were used to inform the instructional design for the following semester. Over the semesters the following instructional techniques were introduced: simulation with and without a computer, encouraging students to write about their work, and the use of alternative representations. As the interventions progressed, a higher proportion of students successfully defined and used confidence intervals to estimate the value of the population mean. This study also identified sources of confusion for students that can be a basis for further research.

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C123: Learning statistics in the first grades

Susana Colaço   University of Lisbon, Portugal

Statistics Education has received some attention in recent years, with particular emphasis in the first grades. We present an experiment that follows an exploratory approach performed in a 2nd grade class. The purpose is to understand how students in first grades undertake the collection, organization and analysis of data. The work presented aims to identify the main difficulties of these students and to understand to what extent the real context may or may not be a facilitator in this process. Finally, it will also be discussed, though briefly, the teacher’s role during this process. The results show that students have difficulties in representing the data using charts but, as far as the analysis of the data is concerned, the students, in most cases, can read and interpret the data presented. The data that refer back to a real context may facilitate carrying out the task but on the other hand, provides the existence of misinterpretations, often derived from the students’ own experience.

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C126: Restructuring of disciplines in statistics based on students’ evaluation in online education programs

Suzi Samá Pinto   Federal University of Rio Grande, Brazil
Mauren Moreira Porciúncula da Silva   Federal University of Rio Grande, Brazil

This paper aims to investigate the contribution of a student evaluation of Statistics courses in the restructuring of an online degree. With the students’ evaluation as a base, it was possible to make adaptations and changes to the organization and pedagogy of the courses. Discursive textual analysis was utilized to analyze the students’ evaluations. Some modifications to the virtual platform were implemented to accommodate the symbolic and language characteristics of statistics classes.. These changes improved the virtual interaction between professors, students, and tutors. The discussions on virtual forums allowed the students to rethink their ideas, because of the contributions of peers, tutors, and professors. These reflections obtained through the virtual dialogues helped to build students’ statistical knowledge.

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C128: Introductory statistics students’ conceptual understanding of variation and measures of variation in a distribution

Rachel Chaphalkar   University of Montana, United States
Cindy Leary   University of Montana, United States

The Guidelines for Assessment and Instruction in Statistics Education College Report (Aliaga, et. al., 2010) encourage a focus on conceptual understanding and statistical thinking in introductory statistics courses. One of the main components of Wild and Pfannkuch’s (1999) model of statistical thinking is consideration of variation. The concept of variation is extremely important when students learn about sampling methods, probability, distributions, and sampling distributions (Shaughnessy, 2007). Previous studies have focused on either students’ conceptual understanding about variation in a distribution (Cooper & Shore, 2008) or measures of variation (delMas & Liu, 2005; Turegun, 2011). In this observational study, students responded to conceptual questions asking them to compare the variability in histograms in general or using numerical measures. In this paper, we will compare students’ ability to reason conceptually about variation and with measures of variation in a distribution.

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C129: Exploration of problem solving processes of students with learning disabilities

Megan Mocko   University of Florida, United States
Melissa Crow   University of Florida, United States

Students with learning disabilities (LD) comprise an increasing portion of the university population. These students often face a disproportionate host of educational and attitudinal challenges, including dysgraphia, ADHD, reading comprehension issues, problems with information retention, and learned helplessness. Current instructional practices may not address obstacles faced by these students. In this study, students with LD will verbalize their thought processes and strategies as they work through a confidence interval word problem. We first identify several primary obstacles to student success. We also explore students’ response patterns and their attempts to use aids to overcome obstacles. By gaining an improved understanding of how learning disabilities affect students’ efforts toward statistical problem solving, we hope to develop more effective methods to sustain statistical literacy for their long term success.

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C130: Normal inverse function in teaching inference about population mean and population proportion

Xiaoyi Ji   Utah Valley University, United States

If a simple random sample is drawn from a normal distribution or the sample size 𝑛 is large, the normal inverse function can be used to obtain a confidence interval of the population parameter 𝜇 or 𝑝. The traditional teaching method, finding a confidence interval indirectly through a standardized normal variable, is not necessary. The study shows that once the normal distributed random variables, 𝑋~ or 𝑃~, have been specified, the lower and upper quantiles of the distribution of the variable form an interval estimation of the population parameter 𝜇 or 𝑝, and the lower and upper quantiles can be directly computed by using normal inverse function in statistics or mathematics software packages. Two examples given in my teaching show, furthermore, the investigation of normal inverse function in teaching the inference about the population parameter provides a more quick, direct and natural manner to solve the problems.

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C131: Reduced major axis regression: teaching alternatives to least squares

William Harper   Otterbein University, United States

The theoretical underpinnings of standard least squares regression analysis are based on the assumption that the independent variable (often thought of as x) is measured without error as a design variable. The dependent variable (often labeled y) is modeled as having uncertainty or error. Both independent and dependent measurements may have multiple sources of error. Thus the underlying least squares regression assumptions can be violated. Reduced Major Axis (RMA) regression is specifically formulated to handle errors in both the x and y variables. It is an excellent topic to teach students the importance of understanding the assumptions underlying the statistical procedures commonly used in practice as well as showing them that alternatives may better satisfy the actual needs.

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C134: It is time to include data management in introductory statistics

Robert H Carver   Stonehill College & Brandeis University, United States
Mia Stephens   SAS Institute, United States

There has been widespread adoption of real data sets and computational software in the teaching of introductory statistics. To sustain these two developments and to maintain currency with the explosion in freely available data from public sources, it is important for students to learn methods for obtaining, cleaning, organizing and manipulating large datasets from multiple sources prior to analysis. Though experimental studies remain central, the practice of data analytics in many disciplines begins with observational data. This paper begins with a rationale for including foundational concepts of data management such as joining tables, selecting rows, and inserting rows, as well as practice with automated approaches to missing and dirty data. It then goes on to provide illustrative examples of how such topics can be taught in engaging and accessible ways, and to suggest course topics that can be suppressed to open room for these topics.

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C135: Environmental interfaces in teaching statistics

Celso Ribeiro Campos   PUC-São Paulo, Brazil
Otávio Roberto Jacobini   PUC-Campinas, Brazil
Maria Lúcia Lorenzetti Wodewotzki   UNESP-Rio Claro, Brazil
Denise Helena Lombardo Ferreira   PUC-Campinas, Brazil

The objective of this article is, based on the Critical Statistics Education assumptions, to value some environmental interfaces in teaching Statistics by modeling projects. Due to this, we present a practical case, one in which we address an environmental issue, placed in the context of the teaching of index numbers, within the Statistics discipline in an undergraduate course in Economic Sciences. In this project, we discuss the Human Development Index (HDI) and we propose the creation of an environmental index in order to evaluate the countries concern level in following some ecological and/or preservation practices.

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C137: Teaching statistics in a crisis zone

Irene David   University of Canterbury, New Zealand
Hilary Seddon   University of Canterbury, New Zealand
Jenny Harlow   University of Canterbury, New Zealand
Jennifer Brown   University of Canterbury, New Zealand

In February 2011 Christchurch, New Zealand, was hit by a large earthquake. Within weeks of this event we were challenged to provide learning and teaching in large tents or using on-line and flexible formats. With audio-visual facilities limited to whiteboards and microphones, moving to an on-line environment was a better choice for the introductory statistics course. For two years we have used the lessons we learned to implement a blended learning environment with opportunities for students with different learning styles to achieve success. The on-line environment allows for monitoring engagement and for timely interventions to increase the level of student achievement. On-going improvements provide novel learning opportunities as technology upgrades offer new possibilities. In our presentation we discuss the changes we have made to the introductory statistics course, using feedback from the students to monitor the effectiveness of the learning experience.

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C139: Formative assessment and learning analytics in statistics education

Dirk Tempelaar   Maastricht University, The Netherlands

Learning analytics seeks to enhance the learning process through systematic measurements of learning related data, and informing learners and teachers of the results of these measurements, so as to support the control of the learning process. Learning analytics has various sources of information, two main types being intentional and learner activity related metadata. This contribution discusses the potential to apply Buckingham and Deakin’s theoretical framework of a learning analytics infrastructure that combines learning dispositions data with data extracted from computer based, formative assessments. In a large introductory statistics course based on the principles of blended learning, combining face-to-face problem-based learning sessions with technology enhanced education, we demonstrate that students learning choices can profit from providing students with feedback based on learning analytics, as to optimize individual learning.

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C140: Quality engineering: an experience in teaching statistics for engineers

Elisa Henning   Santa Catarina State University, Brazil
Adelmo Anselmo Martins   Santa Catarina State University, Brazil
Marcelo Savio Ramos   Santa Catarina State University, Brazil

In quality engineering, statistics focuses on manufacturing and process control, analyzing variation and quality in products, seeking to track process stability. In this sense, the main applications deal with statistical quality control in the analysis of the measurement system, experiment design and the Six Sigma methodology. All these methods use standardized procedures for data collection and analysis in order to identify, treat and eliminate sources of error in pursuit of continuous improvement in the quality of products and processes. Industrial engineers have contact with quality engineering at the end of the undergraduate program, and this article aims to present and analyze some experiments in the classroom developed in an engineering program. They are lab classes, applied exercises, real experiments, seminars and reviews of scientific articles. Since the program’s classes are at night, many students are already working in the industries, giving a different, more collaborative character to the class.

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C141: Learning objects for teaching statistics in engineering courses

Elisa Henning   Santa Catarina State University, Brazil
Dátila Cristina Mondini   Santa Catarina State University, Brazil
Luciane Mulazani dos Santos   Santa Catarina State University, Brazil

In engineering, statistics has wide application in process planning and control, product development and production strategies. Thus, it is critical that engineering students have significant contact with statistics during their undergraduate studies. However, students often do not exercise in practice what would help them better understand the theory. In this sense, the creation, delivery and dissemination of learning objects can fill in some gaps that arise due to this situation. Learning objects are reusable digital resources that contain information for which meanings are produced in the process of knowledge construction and are important elements for the systems of teaching and learning that use the computer and that rely on information technology. They can be used both in distance learning and in the classroom. Thus, this paper is a survey of existing learning objects aimed at teaching engineering statistics, focusing mainly on statistical quality control, design of experiments and the Six Sigma methodology.

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C142: Students’ conceptions of average and standard deviation

Véronique Dubreil-Frémont   Catholic University of the West, France
Christelle Chevallier-Gaté   Catholic University of the West, France
Noëlle Zendrera   Catholic University of the West, France

As statistics teachers, we investigated undergraduate students’ conceptual understanding of two major measures of descriptive statistics, average and standard deviation. First, 352 students were asked to define average and standard deviation before their first statistics class. The answers confirmed that most of the students had an algorithmic conception of average and failed to explain what standard deviation is. We then focused on the way students’ understanding of average and standard deviation improved, which is still largely unexplored: so we repeated the same experiment after the last statistics lesson of the year. Students’ preconceptions seem to have disappeared after statistics teaching; the final results showed a better understanding of average and standard deviation with some lingering difficulties for standard deviation.

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C143: Meanings of probability in Spanish curriculum for primary school

Emilse Gómez-Torres   National University of Colombia, Colombia
J Miguel Contreras   University of Granada, Spain

Recent curricula in many countries suggest that children are introduced to chance and probability in early years of schooling. Probability has had various meanings throughout history, most of which are complementary; however their epistemological differences have been a source of philosophical debates and cognitive conflicts. In this paper we analyze the meanings suggested for probability in the primary school Spanish curricular guidelines. Using ideas from the onto-semiotic approach, we identify the mathematical objects (problems, concepts, propositions and procedures) suggested in these curricular documents for different meanings of probability. We finally establish a reference meaning that may be useful to understand and analyze teaching practices, as well as to predict and help overcome children’s possible learning conflicts.

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C144: Development of integrated quantity judgments: means of distributions appear more different than they are

Franca Agnoli   University of Padova, Italy
Gianmarco Altoè   University of Cagliari, Italy
Tatiana Marci   University of Cagliari, Italy

Inferential statistics are used to decide whether two or more samples are from the same or different distributions, a decision that is generally difficult to make by visual inspection of sample frequency distributions. We investigate how children (8, 10, and 12 year old) and adults compare two sets of five vertical bars, similar in appearance to histograms, to determine which set represents a greater quantity or whether they were equal quantities. Our bars are conceptually simpler than histograms because only bar lengths matter, not their position. Participants of all ages correctly identified about 75% of the sets with greater quantity. Judgment accuracy was, however, strongly affected by age when sets contained equal quantities, increasing from 13% correct for 8 year olds to 61% for adults. Recognizing equality is difficult when integrating across multiple bars, and difficulty increases with variability. Implications for comparing statistical distributions are discussed.

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C145: Observations of implementations of an active learning module in introductory statistics

Nancy McCormick   Middle Tennessee State University, United States
Jeremy Strayer   Middle Tennessee State University, United States
Lisa Green   Middle Tennessee State University, United States
Scott McDaniel   Middle Tennessee State University, United States
Ginger Holmes Rowell   Middle Tennessee State University, United States
Natasha Gerstenschlager   Middle Tennessee State University, United States

Middle Tennessee State University’s Modules for Teaching Statistics with Pedagogies using Active Learning (MTStatPAL) project team is developing teaching modules for introductory statistics courses that help faculty create active learning classroom environments. MTStatPAL instructor materials include: 1) a video of an experienced statistics instructor implementing the classroom activity, 2) an online pre- or post-class activity that helps students develop a conceptual understanding of the topic, 3) a teacher activity implementation guide, and 4) a set of pre- and post-activity testing instruments for the assessment of student learning. Observations of implementations of the regression MTStatPAL module were made by persons trained in using the Reformed Teaching Observation Protocol (RTOP) instrument. This paper discusses how the MTStatPAL regression module helped instructors implement instruction that encouraged students make conceptual sense of statistical content.

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C146: Training prospective teachers for teaching of probability at secondary school in Colombia

Emilse Gómez-Torres   National University of Colombia, Colombia

This paper presents a study of current guidelines for training mathematics teachers in Colombia. Its particular focus is the teaching of probability. The structure of the Colombian educational system and the rules for becoming a secondary school teacher are described. The mathematical and pedagogical training required to teach probability is analyzed into eleven bachelor programs for mathematics teachers. Because these programs spend only a few hours on training about probability and how to teach probability, and because there is a lack of specific mathematical training required for entering the teaching profession, the analysis concludes that there is a general lack of specific pedagogical training for teaching probability. One implication is that programs to educate teachers should extend class time to enhance the competence of prospective teachers in the design of educational activities and in recognition of students’ difficulties, mistakes and misconceptions about probability.

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C147: Examining Venezuelan secondary school mathematics teachers’ professional competencies to teach statistics: focusing on the instruction of descriptive statistics

Orlando González   Hiroshima University, Japan

This article reports on a study of mathematics teachers’ professional competencies for teaching statistics at secondary school, which are of critical importance to achieve the aims of the mathematics curriculum regarding statistics education. In this study, such competencies are defined using an eight-dimensional construct, comprised of the six aspects of professional knowledge identified by Ball, Thames and Phelps (2008), as well as conceptions of variability and beliefs about statistics teaching and learning. Based on this framework, a survey instrument was designed and administered to fifty-three secondary school mathematics teachers working at the metropolitan area of Caracas, Venezuela. The collected data not only revealed strengths, weaknesses and misconceptions in participants’ professional knowledge base, but also led to a deeper understanding about how the identified dimensions of professional competencies for teaching contents in the field of descriptive statistics might affect each other. Furthermore, some other interesting findings, trends and implications yielded from the data analysis are discussed.

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C148: Simulation illogic repaired

Patricia B. Humphrey   Georgia Southern University, United States

Simulations are becoming more important in introductory statistics courses because of new emphasis on resampling procedures (which are now included in the US Common Core for high school students). There are many applets (and apps) that will perform simple simulations and resampling. All too often students don’t understand what is being done. For example, they ignore the difference between resampling for a confidence interval and resampling for a hypothesis test. More complex simulations require more thinking and capable software. At an introductory level for a probability or statistics class, R (free) or SAS (expensive) are often too complex for students to use effectively. I'll show how simple three- or four-line execs and macros in Minitab can do the job effectively, and further serve to eliminate the “black box” of many applets. Students who can perform these understand the situation!

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C149: Preservice teachers’ awareness of variability

Edward Mooney   Illinois State University, United States
Dhimitraq Duni   Illinois State University, United States
Elisha VanMeenen   Illinois State University, United States
Cynthia Langrall   Illinois State University, United States

There have been changes in the recommended content for data analysis at the K–12 level in the U.S. with the publication of Common Core State Standards (2010) and Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K–12 Curriculum Framework (2005). In particular, students need to understand the nature and sources of variability, which raises the question about how well teachers understand or recognize variability. The purpose of this study was to (a) examine the awareness of variability exhibited by mathematics education majors at the elementary (Grades K–5), middle (Grades 6–8) and secondary (Grades 9–12) levels and (b) to determine what that might imply about their preparation to teach data analysis. Results indicate that the preservice teachers had a limited awareness of variability even after instruction.

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C150: A glimpse of two year college instructors’ understanding of variation in histograms

Monica Dabos   College of the Canyons, United States

Statistics education researchers are urging teachers of statistics to help students develop a more sophisticated understanding of variation, since variation is the core of statistics. However, little research has been done into the conceptions of variation held by instructors of statistics at the post-secondary level. This exploratory study was designed to map the conceptions of variation held by two-year college mathematics instructors. A total of 52 instructors from 33 different California community colleges responded to a survey designed to reveal instructors’ conception of variation. The results indicate that a tendency to focus only on the center of the distribution seemed to prevail and very few instructors gave explanations integrating different aspects of the distribution. The majority focused on the center or on the range, and instructors’ lack of consideration for context was also notorious. This study has opened the gate and laid the groundwork for understanding conceptions of variation held by two-year college instructors.

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C151: Middle and high school teachers’ transformative learning of center

Susan Peters   University of Louisville, United States
Jonathan D. Watkins   University of Louisville, United States
Victoria Miller Bennett   University of Louisville, United States

We describe a project to design and implement professional development and investigate how dilemma, critical reflection, and rational discourse affect middle and high school teachers’ reasoning about center. Framed by transformative learning theory, the study highlights how teachers’ engagement with activities focused on these elements can enhance understandings of traditionally problematic content. Results reveal that critically reflecting on activities, engaging in rational discourse while working on activities, and examining multiple perspectives through multiple representations and conceptualizations broadened teachers’ perspectives on the mean. This study contributes to teacher education by identifying circumstances conducive to deepening statistical understandings and supporting reasoning in increasingly sophisticated ways using a theoretical frame for adult learning to investigate teacher learning.

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C155: Investigation about curricular orientations in teaching statistics in Brazil and Mexico

Hugo Hernandez   National Autonomous University of Mexico, Mexico
Verônica Yumi Kataoka   State University of Santa Cruz, Brazil
Claudia Borim da Silva   University of São Judas Tadeu, Brazil
Irene Cazorla   State University of Santa Cruz, Brazil

The objective of this paper is to investigate similarities and differences between curricular orientations for teaching statistics at a Brazilian high school and the School of Sciences and Humanities (CCH) in Mexico. CCH is an important Mexican institution and one out of three high school systems offered by the National Autonomous University of Mexico. In Brazil, the curricular orientations are recommended in the National Curricular Parameters (PCN). Among the differences, the CCH’s curriculum includes Inferential Statistics, particularly Estimation and Test of Hypothesis. Curricular orientations are similar in recommending collecting, organization and presentation of data; construction, reading and interpretation of graphics, tables, measures of central tendency and dispersion; observation of the random character of phenomena, construction of sample space and probabilities calculations, aiming to develop the statistics literacy of the students. Comparison between both curricular orientations may contribute to thinking about teaching statistics in high school.

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C157: An early start on inference

Edith Seier   East Tennessee State University, United States

Starting to teach inference early in an introductory statistics course means that the students have more time to assimilate the new concepts involved in estimation and hypothesis testing, especially if they are exposed to them in a sustained way throughout the semester. I teach a special section of the algebra-based course in which we start writing statistical hypotheses during the first week. We tell the students the general idea about testing hypotheses and that the details on how to calculate or approximate the p-value will depend on the context and the tools available. Randomization methods (permutation tests and bootstrapping) are introduced first because they require less background. After covering the basics of probability, the binomial distribution is used to do inference about one proportion. The classical methods using the normal, t-student and Chi-square distributions are studied at the end of the semester after these distributions have been introduced.

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C158: Computer-aided graphics to teach eigenvalues and eigenvectors

Jaime Curts   University of Texas Pan American, United States

This paper presents a model for teaching vector orthogonal rotation with the use of a computer-aid statistical graphics software. The rotation transformations are defined as: x’ = x cosθ + y sinθ and y’ = -x sinθ + y cosθ. The purpose is to facilitate students’ understanding of a rotation matrix by allowing them to explore and plot different values of theta (θ) given a point P. Students will learn that the maximum variance is reached when θ = 45 degrees. They will also examine the graphical and statistical properties of these transformations, in particular that the variability along y’ is the largest and that x’ and y’ are uncorrelated (orthogonal). Finally students explore how to determine the angle theta so that the variability of a set of observations along the y’ axis is maximized and x’ is orthogonal to y’.

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C159: Sustaining progress in statistics education in the United States through an analysis of the past 30 years of advancement

Katherine Halvorsen   Smith College, United States
John McKenzie, Jr.   Babson College, United States

This comprehensive account of statistics education in the USA describes the changes in the American Statistical Association’s (ASA) statistics education efforts during the past 30 years. These changes include the creation of committees that support the work of statistics educators in mathematics departments in four-year colleges and universities, in two-year colleges, and in schools, along with ASA recommendations on course requirements for undergraduate statistics majors and guidelines for assessment and instruction in statistics education in kindergarten through secondary school and in the first-year of college (the GAISE reports). The paper concludes with continuing and new challenges, along with future opportunities, for statistics educators in the United States and other countries.

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C160: Exam question evaluation with Item Response Theory

Elisabeth Korendijk   Wageningen University, The Netherlands
Evert Jan Bakker   Wageningen University, The Netherlands

We describe our first results with the analysis of the answers to 21 exams of the same Statistics course using Item Response Theory. We used a model with one-parameter (difficulty) and a model with two parameters (difficulty and discriminatory power). The second one appeared to be significantly better in 20 out of 21 exams. Using the outcomes of the analysis with the two-parameter model, we discuss some of the elements that can be analyzed with this model: the discriminatory power of individual questions, the information value of an exam, and we draw some conclusions about the type of exam questions used in this course of which the students’ correct-wrong answering pattern aroused suspicion about how good the question was.

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C162: Students’ social representation of statistics in the humanities and social sciences

Jean-Marie Marion   Catholic University of the West, France
Alain Bihan-Poudec   Catholic University of the West, France

Our starting point is how undergraduates in the humanities and social sciences perceive statistics when they discover the topic through their university curriculum. We highlighted in previous papers that they associate statistics with mathematics, numbers, calculus… (Marion & Bihan-Poudec, 2012; Bihan-Poudec, 2013). Our theoretical framework is to consider learning as a social and individual construction. To that effect, we did a statistical analysis on a survey with 147 students majoring in Education either initial training or adult education. We first investigated the relationship between the type of course chosen by the students and their conception of statistics; second, we examined the students’ answers concerning their attitude towards statistics. Results reveal the link between attitude and cognitive factors, and between the students’ conceptions of statistics and their own experience of the discipline.

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C163: Communication and collaboration in support of a significant mathematical learning

Joyce Furlan   University of San Francisco, Brazil
Lia Marques Marocci   University of San Francisco, Brazil

Recurrently we observe that the practice of teaching probability prioritizes the use of rules and formulas. In other words, teachers develop the concept of probability by repetitive exercises, reproducibility, like a mechanical process. In this way the teacher is an enunciator while the students become listeners, reproducing the information. However, we believe in a more collaborative and communicative process to teach the topic. This article is a result of two research projects. Although they had different purposes, themes and subjects, they were developed concomitantly, using the same tools and with mutual cooperation between the two researchers. In this paper we discuss the process of developing problem solving ability in the students, where they act as the protagonists of their own process of conceptual development, and how cooperation between teacher and researcher can help in this process.

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C164: Teaching precursors to data science in introductory and second courses in statistics

Nicholas Horton   Amherst College, United States
Ben Baumer   Smith College, United States
Hadley Wickham   Rice University-RStudio, United States

Statistics students need to develop the capacity to make sense of the staggering amount of information collected in our increasingly data-centered world. Data science is an important part of modern statistics, but our introductory and second statistics courses often neglect this fact. This paper discusses ways to provide a practical foundation for students to learn to “compute with data” as defined by Nolan and Temple Lang (2010), as well as develop “data habits of mind” (Finzer, 2013). We describe how introductory and second courses can integrate two key precursors to data science: the use of reproducible analysis tools and access to large databases. By introducing students to commonplace tools for data management, visualization, and reproducible analysis in data science and applying these to real-world scenarios, we prepare them to think statistically in the era of big data.

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C167: Attending to students’ thinking on bivariate statistical data at secondary level: two teachers’ pedagogical content knowledge

Sandra Quintas   University of Lisbon, Portugal
Hélia Oliveira   University of Lisbon, Portugal
Rosa Tomás Ferreira   University of Porto & CMUP, Portugal

In this study we compare and contrast aspects of the pedagogical content knowledge of two experienced secondary mathematics teachers when they teach bivariate data under the topic of statistics. In particular, we aim to understand how their knowledge of students is expressed in their instructional process. The data collected for this study was obtained from observation of some lessons taught by each teacher, semi-structured interviews and resources used in the lessons. This study highlights the need for these teachers to help students in reasoning more deeply about bivariate relationships, namely regarding aspects of structure and strength, model fitting, and the role of the linear regression model in predicting events. It also shows the importance of fostering students’ comprehension of concepts related to correlation and regression when they work, in the classroom context, on tasks that incorporate bivariate data.

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C168: Teaching innovations in introductory statistics courses

Ramon Gomez   Florida International University, United States

The emphasis while teaching introductory statistics courses at university level has shifted lately from computations to a deeper discussion of statistical concepts and their association with real life problems. In this regard, the use of technology resources has been generally identified as a facilitator of students’ understanding. This paper describes an innovative approach to teach introductory Statistics courses, implemented by the present author at Florida International University. This approach consists of an interactive method involving the daily use of PowerPoint presentations for lectures and the SPSS software for computations of real life data. Passing and retention rates with this method showed significantly better results compared to a more traditional approach based on a teacher-centered style and low level use of technology.

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C169: High school teachers’ statistical reasoning about comparison of distributions of data in a computer environment

Santiago Inzunsa   Independent University of Sinaloa, Mexico
José A Juárez   Independent University of Sinaloa, Mexico

The article presents results of a study about statistical reasoning developed by seven high school Mexican teachers on how teachers compare distributions of data in a computer environment. The evaluation instruments adopted were four activities carefully chosen as well interviews with two of the teachers involved. The results indicate that the statistical reasoning of the teachers was predominantly oriented to inferior levels of the model SOLO (Structured Observed Learning Outcomes); in particular the process of informal inferences was the one that produced more difficulties for them. The teachers did better on working with tables and averages in comparison to the distributions. Perhaps the reason why is that they didn’t make the most of the software potential for the manipulation of several graphic representations that would allow a better and more complete analysis.

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C171: The professional knowledge of German secondary school teachers about descriptive statistics

Stefanie Schumacher   Bielefeld University, Germany

For many years, (descriptive) statistics had an inferior standing in schools worldwide. However, in recent years, a considerable shift from probability to statistics is apparent with regard to National Standards for Mathematics. Nevertheless, studies about teachers’ professional knowledge in this new domain are scarce, particularly in Germany, although it is verifiable that the teachers’ knowledge has an impact on their classroom practice and also on students’ learning. Therefore, a main focus of the research presented here is to generate a questionnaire to survey teachers’ professional knowledge in the domain of descriptive statistics. The theoretical basis is on the one hand Ball’s model of professional knowledge and the concept of ‘statistical literacy’ on the other hand. Thus, the theoretical background and its realization in the test instrument itself are presented and discussed.

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C172: Peer assessment as active learning method

Tatjana von Rosen   Stockholm University, Sweden

The importance of feedback in the student learning process is well understood among educational researchers (theorists) and teachers (practitioners). Its positive effects on the students learning and achievements have extensively been discussed in the theoretical research and illustrated by many empirical studies. It is also recognized that student assessment of other students’ work is a useful tool for activating student engagement in the learning process. This work presents the results of a pilot study investigating the attitudes toward peer-assessment of examination papers and exploring experiences of the peer-assessment process of students pursuing a bachelor degree in statistics.

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C173: The use of web resources in statistics education

Svetlana Tishkovskaya   University of Central Lancashire, United Kingdom
Gillian Lancaster   Lancaster University, United Kingdom

It is widely recognised that Internet technologies have opened new possibilities and directions for teaching, in general, and for the teaching of statistics in particular. Even though the use of online learning environments in education is expanding, there are very few studies where statistical issues and the Web have been discussed systematically. This article emphasises the choices available to teachers of applied statistics at the tertiary level for using Internet technology. In this paper, we outline the potential of the Internet for teaching and learning statistics, provide a systematic general overview of statistics on the Web and list some problems in incorporating Web technology into the classroom. The paper aims to contribute to the understanding of the current state of the art in Web-based learning environments in statistical education and helps teachers understand and implement Internet technology in a variety of innovative ways for the enhancement of the teaching of statistics.

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C176: Teaching statistics concepts through stock market contexts

Larry Weldon   Simon Fraser University, Canada

Most students of statistics will learn that variability is the key to separating transient differences from reproducible differences. But many students will not understand the role of variability in the financial markets. Concepts of “risk” and “diversification” are often misused in this context, and this misuse can be blamed on a faulty application of basic statistics. In this talk, I describe how to introduce students to portfolio management though virtual portfolios, and how to assess risk and diversification. The talk will also show how to introduce students to the idea of options without getting into the arcane details. An exposure to these ideas will help students to leverage their understanding of basic statistics in selling their expertise to future employers. Some programs in R can be made available to teachers who wish to use this approach.

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C177: Rethinking the intersection of statistics education and social justice

Lisa Poling   Appalachian State University, United States
Nirmala Naresh   Miami University, United States

Critical consumers of data are able to think and reason about statistics and use statistical tools to better explore and understand issues that are significant to their immediate community, the larger society, and the broader world. Promoting statistical knowledge by understanding and investigating social issues that create disparities provides an entry point for an individual to take the first step in making positive change and engaging in equitable practices (Gutstein, 2006a). Drawing from the research domains of statistics and social justice we designed a statistics learning module for prospective teachers to help them emerge as critical and caring consumers of data. The amalgamation of statistics and social justice issues created a sustainable learning environment where data were interpreted and analyzed in a meaningful way.

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C178: On-line homework in probability and statistics: WeBWorK incorporating R

Davor Cubranic   University of British Columbia, Canada
Bruce Dunham   University of British Columbia, Canada
Djun Kim   University of British Columbia, Canada

WeBWorK is an open source on-line homework application supported by the Mathematical Association of America. There are presently tens of thousands of problems freely available on mathematical topics in WeBWorK, but very few in the areas of probability and statistics. An on-going project has developed a wide range of homework questions for courses in the statistical sciences and has augmented WeBWorK to enable its communication with the statistical computing software R. This integration allows WeBWorK access to R’s rich facilities for statistical data manipulation, analysis, and visualization and hence permits the creation of probing and diverse problems in statistical science. The application is described in detail here, including examples of questions, technical issues, and student and faculty feedback.

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C182: How students relate residuals behavior with parameter significance tests in a linear model: A case of study in a higher education Mexican institution

José Luis Ángel Rodríguez Silva   National Institute of Statistics and Geography (INEGI), Mexico

In the implementation of a linear regression model, it is crucial to carry out significance tests to assure the proper inclusion of parameters. In addition, in order to perform such a hypothesis test, students should verify the properties of the residuals and the most important is undoubtedly the normality of the residuals. However, it is clear that there is, in general, an important gap between the hypothesis testing and residuals verification stages. In this paper we present an empirical study about the most probable confusions and misunderstandings students have in relating these two mental statistical constructs in a course of Administrative Statistics in a Mexican Education Higher Institute. One of the main results is that the order in which these two concepts are taught is essential to guarantee there is not a disconnection between the two stages, and perhaps it would be better to first analyze the properties of the residuals and then to engage in the hypothesis testing phase.

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C183: Stat321, an on-line statistics textbook based on intuitive inferential reasoning

Nicolas Greliche   INSA Lyon, France

Research in human cognition has shown that people possess intuitive notions of statistics, but may fail to generate correct judgments in a number of situations. These characteristics of the human mind can be used to introduce statistical concepts and warn against misconceptions by showing parallels and divergences between proper statistical thinking and intuition. To date, this approach has been little investigated resulting in a lack of resources for the interested teacher. Here, I present Stat321, a project of an atypical online statistics textbook which uses the naive statistical thinking of the reader to explain the what, why and how of Statistics. Stat321 is designed to evolve with comments, ideas and the advances in statistics education research. Initial feedback from undergraduate students is discussed.

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C184: A case study of Cypriot primary school students’ use of a dynamic statistics software package for analyzing and interpreting data

Irene Kleanthous   Cyprus Ministry of Education, Cyprus
Maria Meletiou-Mavrotheris   European University Cyprus, Cyprus

The use of dynamic learning environments, which are designed explicitly to facilitate the visualization of mathematical concepts, provides an enormous potential for making complicated mathematical ideas accessible to young learners. This paper explores the potential of dynamic statistics software for supporting the teaching and learning of probabilities. It shares the experiences from a case study that implemented a data-driven approach to mathematics instruction using the dynamic data-visualization software InspireData©, an educational package specifically designed to meet the learning needs of students in the middle and high school grades (Grades 4-12). We report on how a group of Grade 4 (about 9-year-old) students used the affordances provided by the dynamic learning environment to gather, analyze, and interpret data, and to draw data-based conclusions and inferences. The role of the technological tool in scaffolding and extending these young students’ stochastic and mathematical reasoning is discussed.

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C187: Fostering changes in confidence intervals interpretation

Luisa Andrade   National Pedagogical University, Colombia
Felipe Fernández   National Pedagogical University, Colombia
Ingrith Álvarez Alfonso   National Pedagogical University, Colombia

As literature has reported, it is usual that university students in statistics courses and even statistics teachers interpret the confidence level associated with a confidence interval as the probability that the parameter value will be between the lower and upper interval limits. To confront this misconception, a class activity has been designed with the aim to realize that this application of confidence level explicitly violates the basic laws of probability. We consider two non-overlapping confidence intervals, that could plausibly correspond to two random samples from the same population, where the probability of events within this interpretation contradicts the probability rule for disjoint events and the rule of monotonicity (P[E] ≤ P[F] if E ⊆ F). We use simulation to help students shift to a frequentist interpretation of confidence intervals.

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C188: Individualised project assessments for statistics courses – the best of both worlds?

Robert Grant   University of London & Kingston University, United Kingdom
Ahmed Younis   University of London & Kingston Universit, United Kingdom

Examinations in statistics have been criticized for failing to assess analytical thinking and practical problem-solving skills. Project-based assessment is a widely used alternative, but detection of plagiarism is a concern as students should arrive at the same results. Definition of plagiarism is also difficult; sharing ideas on methods is a positive learning experience, while sharing results and computer output is not. Creating a different dataset for each student can resolve these problems but requires automation to be feasible. We describe the experiences gained from programming a general algorithm for this in R and SPSS and piloting in two years of postgraduate healthcare research methods students. There is potential to introduce unfairness if the requirements of analysis, such as post-hoc testing, are not identical in all datasets. Our algorithm creates multiple datasets that are constrained to differ enough to be identifiable, while also sharing exactly the same analytical requirements.

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C189: Statistics content and pedagogy in a course for pre-service secondary math teachers

Kady Schneiter   Utah State University, United States

The statistics content in pre-college math courses in the US has risen considerably in recent years. In particular, the Common Core Mathematics Curriculum, which has been adopted by 45 of the 50 states, includes statistics objectives at all levels, 6-12. Nevertheless, many pre-service and in-service mathematics teachers are uncomfortable with statistical topics and apprehensive about teaching them. The prevalence of statistics in the math curriculum has created a need for teachers who are familiar with appropriate methods for teaching statistics and who have a solid understanding of statistical topics. This paper describes a teaching methods course that integrates learning about statistics content and pedagogy to prepare future secondary math teachers to teach statistics effectively.

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C190: Teaching statistics with lectures or activities: a comparative study

Jennifer L Loveland   Utah State University, United States
Kady Schneiter   Utah State University, United States

Many recommendations for teaching statistics with less lectures and more student-centered methods have been proposed. Nonetheless, there are only a few comparative studies in which an entire course was taught using lectures in one section and student-based methods in another section. In order to gather information about which method produces higher student understanding of statistical topics and ability to apply statistical procedures, a sample of 74 students in a university introductory statistics course was divided into two sections. One section was taught entirely with lectures. The other section was taught using minimal teacher-centered activities; methods included hands on activities, discovery based units, and group work. Students’ results on exams were analyzed. The lecture and activity sections were compared to determine which method produced higher results on conceptual questions, and which method facilitated higher results on procedural questions. Implications for using student-centered teaching methods to improve introductory statistics classes are discussed.

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C191: Technology tools and success in statistics: interpreting the research

Jason Schenker   Kent State University, United States
Shawn Fitzgerald   Kent State University, United States
Jian Li   Kent State University, United States
Emtiaz Rony   Kent State University, United States

The purpose of the present research study was to examine the effectiveness of using various technology tools to enhance statistics instruction using meta-analytic techniques. A total of 148 effect sizes were obtained from 62 individual articles. A mean study-weighted effect size of 0.280 was found across all studies and this value was statistically significant, t (62.991) = 4.467, p < 0.001. However, considerable variance in effect sizes remained unexplained suggesting that the mean study-weighted effect size found when considering all studies was moderated by one or more study characteristics. Further analyses examined differences in effect sizes across online vs. face-to-face courses, function of technology use (supplemented or substituted instruction), duration of instruction, instructor bias (same vs. different instructions), research design, and publication status (published vs unpublished studies). Only the difference in effect sizes between online and face-to-face courses was statistically significant. Implications for using various technology tools are discussed.

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C192: How is success in statistics moderated by instructional practices used at the university level: interpreting 30+ years of research on the topic

Hanna Wilson   Kent State University, United States
Shawn Fitzgerald   Kent State University, United States
Jason Schenker   Kent State University, United States
Jian Li   Kent State University, United States

Results from this study provide information regarding the effectiveness of various “innovative” instructional approaches employed in university level statistics courses. The average effect across all “innovative” instructional approaches, when compared to the traditional lecture approach, indicates that these strategies influence student achievement in a positive manner (d = 0.3389). However, this should be interpreted with caution as it was determined that several study characteristics moderated the relationship between instructional type and statistics achievement. Average effects of the innovative instructional practices on statistics achievement were moderated by type of publication (i.e., journal, presentation, dissertation), subject assignment to courses, (i.e., random, nonrandom) and length of implementation of innovative instructional practice. Implications for using various instructional approaches are discussed.

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C195: Teaching undergraduates through statistical consulting

Hyun-Joo Kim   Truman State University, United States
K Scott Alberts   Truman State University, United States
Scott Thatcher   Truman State University, United States

Statistical consulting is a challenging and rewarding part of statistics. A consultant uses the art and science of statistics to solve practical problems. However, most consultants at universities and in industry have graduate degrees in statistics or are working on graduate degrees leaving undergraduates little opportunity to engage in this rewarding part of statistics. CASE (Center for Applied Statistics and Evaluation) is an undergraduate driven statistical consulting center. Students working with clients ranging from grant recipients to community organizations experience transformative, active, and civic engagement learning, as well as an experience similar to undergraduate research and internships. CASE can help students build a strong technical background in statistics, but more importantly provides the opportunity to build communication skills, common sense and a good business sense with faculty and peer mentoring.

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C196: The status of reform in statistics education: A focus on the introductory course

Rossi Hassad   Mercy College, United States

For almost two decades, the statistics education community has been actively focused on reform, particularly regarding introductory statistics. Reform is intended to improve student learning outcomes, and in general, this approach is guided by the constructivist philosophy, and emphasizes the use of active learning and student-centered strategies, so as to foster statistical literacy. Nonetheless, formal assessment data on the effectiveness of reform efforts are lacking. As well, there is no recognized consensus on acceptable measures and indicators, in this regard. This paper presents the results of a qualitative systematic review of literature on statistics education, and concludes that the discipline must urgently heed a call for change toward a more organized and scientific approach to measurement, research, and setting of priorities.

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C197: A 21st century teaching approach in statistics

Nelia S. Ereno   University of the Philippines Cebu, Philippines
Allen F Vicente   University of the Philippines Cebu, Philippines

We surveyed 146 out of 217 students who took basic statistics course at the post secondary level. The objectives were to asses the current pedagogy in teaching statistics based on Bloom’s Digital Taxonomy; determine software tools that facilitated the learning of statistics and identify the extent of the use of these resources; and revise the course syllabus in basic statistics to incorporate content and approaches to better match 21st century teaching-learning. The survey revealed that teachers in statistics were using a mixed approach – the traditional and digital approaches; that students were provided with computers in the laboratory with installed licensed statistical software, Word processing, and internet connection, and that they have good knowledge in using them. After the assessment, the course syllabus which contains recommended teaching strategies on the topic(s) was developed.

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C198: Assessment of students’ knowledge and skills in learning statistics in the 21st Century

Allen F Vicente   University of the Philippines Cebu, Philippines
Nelia S. Ereno   University of the Philippines Cebu, Philippines
Liberty Grace Baay   University of the Philippines Cebu, Philippines

This paper investigated the ways on how students understood the various topic areas in statistics in order to have better learning experiences in gaining knowledge and skills in statistics. The relationship between the students’ attitudes to statistics; their abilities and learning outcome were also examined. Results showed that students whose biases and misconceptions were being corrected tend to have excellent grades. Also, students who agreed that they like statistics were likely to have very good and excellent grades. Several activities that improved awareness, developed talents and potential, enhanced quality of life and contributed to the realization of dreams and aspirations in the field of statistics in the 21st century were also presented. The students agreed that they were encouraged to ask questions, to experiment and to formulate their own plan of action. In addition, they were also taught to use statistical software in their respective classes.

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C199: Probability and statistics in access exams to Spanish universities

P. Díaz   University of Oviedo, Spain
V. Mier   University of Oviedo, Spain
P. Alonso   University of Oviedo, Spain
Luis J Rodríguez-Muñiz   University of Oviedo, Spain

Exams to get access to Spanish universities are performed region-wide every year. These exams are based on the curriculum established for the Spanish Baccalaureate. We perform a crossed analysis between curricular guidelines and items appearing in the exams, in four Spanish regions, so as to detect prevalent units, similarities and differences. Thus, we check what competencies and curricular units appear in the official curriculum related to Statistics and Probability, and how they are assessed in the exams. We show how these exams influence teaching methodologies; over-representing mechanical exercises instead of interpretive questions, especially applications of Statistics and Probability in students’ everyday life. This fact considerably lessens the importance of a teaching/learning methodology based on competencies.

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C200: Graduate teaching assistants’ beliefs, practices, and preparation for teaching introductory statistics

Nicola Parker   University of Minnesota, United States
Elizabeth Fry   University of Minnesota, United States
Joan Garfield   University of Minnesota, United States
Andrew Zieffler   University of Minnesota, United States

Graduate teaching assistants (GTAs) are responsible for the instruction of approximately 15% of introductory courses in U.S. statistics departments (Blair, Kirkman & Maxwell, 2013). Little is known about graduate teaching assistants’ beliefs and practices regarding the teaching of introductory statistics. As part of an NSF-funded project (eATLAS, DUE 1044812 & 1043141), an online survey was developed to explore the beliefs and practices of statistics GTAs, including course content priorities, teaching methods, use of technology and assessment strategies. The data collected from the survey were based on responses from 213 GTAs across over 35 Ph.D. granting institutions in the United States. The results can inform future programs for GTA development.

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C202: Introductory statistics instructors’ practices and beliefs regarding technology and pedagogy

Elizabeth Brondos Fry   University of Minnesota, United States

The development of technology tools has created many possibilities for the introductory statistics classroom, but student learning outcomes may be influenced by how instructors use technology in their teaching. The Statistics Teaching Inventory (STI) was developed to assess introductory statistics instructors’ practices and beliefs as part of an NSF-funded project (e-ATLAS, DUE 1044812 & 1043141), to evaluate the effect of reform efforts in statistics education. The STI was administered to a national sample of teachers of introductory statistics courses across various disciplines in U.S. colleges and universities. Results from a preliminary sample of 96 instructors will be presented, and the relationship between technology use and pedagogical practices will be explored.

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C203: A shiny new opportunity for big data in statistics education

Karsten Maurer   Iowa State University, United States

As the availability of truly massive data sets proliferates it is enticing to incorporate these data sources into the curriculum of an undergraduate statistics course. Major barriers exist for inclusion of big data due to the computationally intense nature of working with large databases. Difficulties include gaining access to the database, interacting with database management software and obtaining manageable subsamples from the database for student use. This paper describes a web based application, the Shiny Database Sampler, which allows instructors to bypass these barriers using a simple JavaScript based tool. The tool is constructed using R and the R packages Shiny and RMySQL to allow the instructor and/or students to sample observations from a number of different large databases, using selected sampling schemes, for use in the statistics classroom.

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C206: A graphical illustration of binomial distributions

Kang Sup Lee   Dankook University, Korea
Dong Joo Yang   Dankook University, Korea

Binomial distribution, along with normal distribution, plays an important role in school mathematics. However, in all reality, students rely simply on memorization even for its simple properties such as average or variance therefore most students struggle to study binomial distribution and its applications. Although there are difficulties in studying binomial distribution, since it plays a vital role in increasing problem solving skills and mathematical balance in real life, one cannot neglect studying this subject. In this paper we suggest a program that shows various graphical illustrations of binomial distributions. The developed program could be contributes to not only increasing the student’s interest but a simulator for combined binomial distributions.

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C208: The interactive submarine: using boxplots as a likelihood approach

Pedro Campos   University of Porto, Portugal

In this work, we explore the capacity of students and professionals with different education levels to cope with boxplots and to draw conclusions. An experiment has been conducted where specimens of a subspecies of a reptile are distinguished by weight, age and size. Participants compare sample data with population data in order to identify the subspecies that have been collected. Young students seem to be more prepared to analyze and interpret boxplots than older students and adults which reinforces the need for the training of statistical reasoning in adult ages. The submarine simulator (or Interactive Submarine) is included in Exploristica (www.exploristica.com), an itinerant exploratory exhibition consisting of various interactive modules with the aim of bringing the fundamentals of Statistics and Probability to basic and secondary schools, conveying the statistical concepts in a practical and experimental way.

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C209: Analysis of linear regression in Spanish baccalaureate textbooks

P. Díaz   University of Oviedo, Spain
Luis J Rodríguez-Muñiz   University of Oviedo, Spain

In this paper we study the presentation of linear regression in Spanish textbooks, from different publishers, written for the first year of Bachillerato (16 years old), Spanish post-compulsory Secondary Education (Baccalaureate). We perform a three-way analysis, based on conceptual, educational-cognitive, and phenomenological points of view. Furthermore, we perform a study of the type of problems and exercises proposed in the textbooks. Finally, we analyze correspondences between contents in textbooks and Spanish official curriculum for this year. Our goal is to check whether textbooks include innovative key points recently introduced in the official curriculum, because, in Spain, textbooks are still very important tools in the classroom.

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C210: Just sum the marks: spurious wisdom

Timothy Dunne   University of Cape Town, South Africa

Both formative and summative assessments may involve three-fold purposes: to rank performances of all participants as a norm-referencing task, to order test items by difficulty and make diagnostic inferences for intervention, and to adjudicate which respondents have attained performance of a specified criterion level (pass or distinction). The use of a weighted or unweighted total score (percentage) as reportable outcome is an unquestioned convention in statistics teaching and examinations, and elsewhere. Several issues arise: under what conditions is a sum convention adequate, and when is it deeply problematic? How can the test data disclose evidence against the use of the sum? What then can be done to offer paths to justifiable outcomes that can stand both ethical and legal scrutiny? The case is made for the importance of prior relative difficulty judgments recorded within the marking memo rubrics, and the use of appropriate statistical methods for ordinal data.

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C212: An evaluation of the statistical methods used by business researchers in South African publications

Gary D Sharp   Nelson Mandela Metropolitan University, South Africa
Warren J Brettenny   Nelson Mandela Metropolitan University, South Africa
Chantelle M Clohessy   Nelson Mandela Metropolitan University, South Africa
Catherine G Logie   Nelson Mandela Metropolitan University, South Africa

Statistical training to service disciplines is common practice at tertiary institutions. In South Africa, training is generally restricted to under-graduate, first-year level commonly found in introductory texts. This is typical for service courses offered to students enrolled in the commercial based disciplines of management and economics. This research investigates the manner in which statistical methods have changed business related publications in South Africa and compares this to the curriculum currently on offer to students. A suggestion for future training and content design is considered.

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C213: Statistical consultation as part of statistics education

Lizelle Fletcher   University of Pretoria, South Africa

The Department of Statistics at the University of Pretoria offers an internal consultation service to faculty and postgraduate students with the aim to support the research function of the University. This service offers unique training opportunities to our own students and from 2015 we are planning to incorporate training a statistical consultant as an integrated part of the course content of a capstone module. The goal is to develop students’ statistical thinking and intuition, and to equip them with experience in the practice of statistics. Students will be expected to actively participate in the internal consultation service as a form of internship under supervision. Furthermore, they will have to familiarize themselves with the literature on statistical consulting and demonstrate their understanding by presenting a case study based on their experience in the consultation service. This paper will discuss the development of this initiative.

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C214: The role of lecturers in students’ performance

Klára Kazár   University of Szeged, Hungary

Creating positive attitudes toward statistics has a key role in sustainable statistical knowledge. However, a lot of students have negative attitudes toward and fear of statistics. Lecturers have a key role in identifying these possible problems and taking the necessary steps. An important step in this process is making a course evaluation and analyzing the lecturer’s role and effect on students’ opinion and performance. This paper introduces the results of a course evaluation focusing on the role of lecturers. Applying PLS (partial least squares) path analysis we came to the conclusion that the evaluation of a lecturer has an effect on the satisfaction with the course, and the latter influences the course performance of the students. According to the results, the higher the evaluation of the lecturer is the higher the evaluation of the subject and the performance of the course prove to be.

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C215: Sustaining student engagement in a college statistics course through a reflective teaching model using youth statistics

Liza Lorena Jala   University of Cebu, Philippines
Enriqueta Reston   University of San Carlos, Philippines

This paper described how micro data on youth statistics generated by official statistical agencies in the Philippines and from various international surveys are used in teaching college students to sustain their engagement in learning statistics through contextualized data-based activities. Through a reflective teaching model, statistical concepts and methods are introduced using a creative integration of multiple data sets on youth statistics from various interdisciplinary perspectives of the social sciences. Teaching-learning activities are modular, which incorporate features of constructivist teaching and learning using inductive and active learning approaches. Student motivation and ownership of learning is also sustained as students make sense of youth statistics which reflect their own sector of the population. Portfolio assessment is used to provide students the opportunity to reflect on their learning with data and for the teacher to assess their understanding of statistical concepts and development of statistical literacy skills.

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C216: Teachers’ confidence in teaching statistical ideas

Rosemary Callingham   University of Tasmania, Australia
Jane M. Watson   University of Tasmania, Australia

Teachers lack confidence in their ability to teach statistical ideas. Although understanding of school students’ development of underpinning ideas in statistics has grown, this has not been matched by a deeper realisation of how best to develop teachers’ confidence and competence in teaching statistics. As part of a larger project, 42 teachers completed a profile instrument that included a 20-item confidence inventory and, a 5-item scale addressing beliefs about statistics in everyday life. A factor analysis of the teaching confidence items indicated four factors that could be interpreted as procedural statistics, probability, the application of statistical ideas, and underlying ideas of variation and inference. To explore teachers’ confidence further individual “KidMaps” provided a profile of items unexpectedly endorsed as high or low confidence. Dissimilar patterns of response to items that loaded onto the identified factors were identified among teachers who had different overall levels of confidence.

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C217: The impact of an inverted traditional teaching model on first level statistics students

Fransonet Reyneke   University of Pretoria, South Africa
Lizelle Fletcher   University of Pretoria, South Africa

In the past decade various intervention strategies have been adopted by the Department of Statistics at the University of Pretoria to address the problem of low success rates of its first level Statistics students. In January 2013 we embarked on the challenge of inverting traditional teaching methods. The new generation students come to class armed with smartphones and tablets. Technological innovation can hence be roped in to facilitate the teaching and learning of Statistics. In a flipped classroom environment, the transfer of information takes place online in the form of podcasts or YouTube videos, as well as pre-reading and using the Aplia online-homework system. This approach allows for more quality time to be spent interacting with students inside the lecture hall, thus enhancing the deep learning process which takes place when students participate in tutorials to re-enforce and consolidate knowledge. The impact of the inverted teaching model will be discussed.

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C218: Istat’s new tools in teaching statistics to digital natives

Barbara Ascari   Italian National Institute of Statistics, Italy
Marina Peci   Italian National Institute of Statistics, Italy

Istat (Italian National Institute of Statistics) has been actively engaging in strongly disseminating statistical literacy towards civil society and, to this aim, established in 2011 the Advanced School for Statistics and Socio-economic Analyses. According to this mission, one of the main targets is represented by the world of education, students and teachers. The use of the Web is vital in order to catch the attention of digital natives and to attract them to statistics. Istat’s current strategy is to find out innovative didactic solutions for the different school levels, through learning and collaborative environments and emerging technologies in teaching statistics. According to this idea, Istat has created an interactive virtual laboratory – ScuoladistatisticaLab (SchoolofStatisticsLab) for the secondary school. For the lower school levels, Istat has realized other didactic tools developed in collaboration with Italian institutions and Istat’s regional offices.

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C219: Teaching NHST vs Bayesian inference in post-secondary technology programs

John H. Mott   Purdue University, United States
Erin E. Bowen   Embry-Riddle Aeronautical University, United States

While literature demonstrating the weaknesses inherent in null hypothesis significance testing (NHST) and Frequentist statistical analysis is extant, NHST is still the predominant statistical methodology employed for research in the social sciences. Although Bayesian inference as a means of statistical analysis has made inroads in the scholarly literature in some social science disciplines, the use of Bayesian data analysis in the area of technology is limited. This article examines the advantages and disadvantages of the introduction of Bayesian methods in postsecondary technology programs, and concludes that there are significant advantages to the teaching of such methods. The authors recommend a blended approach, whereby both techniques are taught and applied to practical problems.

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C220: Integrating learning technologies for applied statistics in distance education at the graduate level: challenges and strategies

Erin E. Bowen   Embry-Riddle Aeronautical University, United States

Distance-based graduate education programs in technology and related disciplines (e.g., aviation) are rapidly expanding; a cornerstone of these professional programs includes proper education in statistical analysis. A unique consideration of such programs is their primary focus on professional students – those who intend to pursue industry careers rather than a role in academia. For these students, a thorough grasp of the practical application of statistical analysis and statistical software tools is essential; however, conventional graduate statistics teaching methods are ill-suited to this audience. The author, who has won university-level awards for teaching statistics via distance technologies, discusses the challenges and strategies facing instructors and students in a technology-mediated educational environment. The role of statistical software tools for applied learning is evaluated, and available learning technologies are explored.

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C223: What’s missing in teaching probability and statistics: building cognitive schema for understanding random phenomena

Sylvia Kuzmak   University of Pennsylvania, United States

Teaching Probability and Statistics is more than teaching the mathematics. Historically, the math was first developed through analyzing popular games of chance such as involving the rolling of dice. This paper makes the case that the development of the understanding of Probability and Statistics is dependent upon building a “mature” understanding of common random phenomena such as the rolling of dice or the blind drawing of colored balls from an urn. An analysis of verbal protocols of 24 college students, who interact with and describe random phenomena involving the mixture of colored marbles, is presented, using cognitive schema to represent the subjects’ expressed understanding. A cognitive schema representing a “mature” understanding of the random phenomenon is contrasted to a diversity of schema representing observed “immature” understanding. Teaching to explicitly build the mature cognitive schema is proposed.

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C226: How I learned to stop worrying about it and just teach MBA 5800 (with apologies to Dr. Strangelove)

Alan S Chesen   Wright State University, United States

This course is a requirement for the MBA students at Wright State University who have not studied statistics previously. It is taught in order to allow these students to attain the level of competence of those students who have taken two semesters of undergraduate statistics. By completing MBA 5800, students will be able to understand the more complex statistical concepts contained in their other graduate courses. What has been developed is a methodology that includes a set of procedures that utilize lecture, discussion and problem solving that uses software demonstration techniques to teach the concepts of the two undergraduate courses in one course. In the spirit of the theme of the 2009 USCOTS conference, “Letting Go to Grow,” and also in the spirit of the theme of the 2013 USCOTS conference, “Making Change Happen,” a paper will be presented concerning how the course is taught in order to best educate the students.

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C227: Business management students’ attitude and performance in statistics learning in Nigeria Metropolitan College of Technology

Adeyemi D Aromolaran   Yaba College of Technology, Nigeria
Abiodun Karim   Yaba College of Technology, Nigeria
Emmanuel Ikegwu   Yaba College of Technology, Nigeria
Uneke Okoroafor   Yaba College of Technology, Nigeria
Yemisi Ajiboye   Yaba College of Technology, Nigeria

Attitudinal apathy among undergraduate management students in Nigeria towards learning of statistics and its consequential poor performance is of great concern. 300 structured questionnaires were administered by stratified random sampling method across three different departments in the school of management. The categories of students involved in the study were the year-two students who had the course in their curriculum in the session. Chi-square test and log-linear statistical tools in IBM SPSS was used for the data processing and analysis. Findings showed that most student preferred statistics to be removed from their course list, attributed their poor performance to the teaching approach being adopted by the lecturers and existence of association between previous knowledge in mathematics and performance in statistics. It is therefore recommended that students’ interest in mathematics at secondary education level should be worked upon to enhance their interest in the learning of statistics at the tertiary education level.

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C228: Confirmatory and exploratory factor analyses of students’ developmental levels in learning statistics

Rini Oktavia   Syiah Kuala University, Indonesia
M. Alejandra Sorto   Texas State University, United States

This study provides an empirical based-explanation on how students develop their understanding of statistical concepts and investigations. We developed an instrument that measures students’ developmental levels in learning statistics and administered it to 797 high school and middle school students in central Texas. We conducted exploratory and confirmatory factor analyses by applying structural equation modeling (SEM) approaches. In this paper, we discuss the results of these analyses.

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C229: Using Calibrated Peer Review™ in introductary statistics courses

Melissa Q Pittard   University of Kentucky, United States
William S Rayens   University of Kentucky, United States

The two most common challenges when grading written assignments in large undergraduate statistics courses are finding the time, and getting students meaningfully involved in feedback provided by the effort. Simple, paper-exchange peer grading is one option, but it can be difficult to implement, and more difficult to ensure that students are taking the process seriously. Calibrated Peer Review™ (CPR), is a “web-based, instructional tool” that was designed to electronically facilitate peer grading, while also attempting to address the integrity of the students’ effort in the grading process and provide students with an extended learning experience. This article discusses students’ and instructors’ perceptions of and experiences with CPR in large undergraduate statistics classes at the University of Kentucky.

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C232: Metonymy as a lens into student understanding of sampling distributions

Stacey Hancock   University of California, Irvine, United States
Jennifer Noll   Portland State University, United States

We use metaphor and metonymy on a regular basis in everyday language, but what sorts of metaphors and metonymies materialize as part of students’ conceptions of sampling distribution and informal inference in an introductory statistics course? This research aims to answer this question. Student difficulties in understanding ideas of sampling distributions and informal inference are well documented in the literature. Although we know that these ideas are complex and difficult for students, little is known about the ways in which students’ language mediates their statistical problem solving activities. Based on semi-structured interviews with undergraduate students in an introductory statistics course, this research explores students’ use of metonymy when speaking about distributions, sampling ideas, and statistical inference, allowing us to gain deeper insight into these students’ statistical thinking.

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C234: IPUMS International: a data resource for statistics education

Patricia Kelly Hall   University of Minnesota, United States
Lara Cleveland   University of Minnesota, United States
Matthew Sobek   University of Minnesota, United States

IPUMS-International is the world’s largest collection of high-precision census data samples containing individual-level information on 544 million people in 74 countries spanning five decades. These data are available for download at no cost to educators, students and researchers for scholarly, educational, and policy-related analysis. The database, built in cooperation with national statistical offices, provides remarkable access to data for educators wishing to expose students to real-world governmental statistics. Variables with distinct census responses for each person are coded consistently across time and place; documentation is thorough, harmonized and easily accessible; and the web delivery system allows registered users to create and download customized data sets pooled across time and place. Individual level responses mean that data can be used in analyses that range from simple descriptive tables to advanced statistical modeling.

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C237: Teaching scientific integrity through statistics

Marijtje A.J. Van Duijn   University of Groningen, The Netherlands
Wendy J. Post   University of Groningen, The Netherlands

In the past years, Dutch academia was confronted with several cases of fraud. The Stapel investigation revealed that the prevailing research culture allowed questionable research practices (QRP). As a consequence, there is an ongoing debate on how to prevent academic misconduct. Teaching scientific integrity is an evident solution, although its implementation may be less obvious. In our workshops and classes we have used the principles of statistical reasoning and methodology, especially validity, to help students understand the importance of scientific integrity and the dangers and consequences of QRP. We feel that this approach is more effective than merely discussing principles of scientific integrity, such as verifiability and independence. The explanation may be that students are sufficiently aware of the ethical norms, but fail to see how they apply to or might challenge their own behavior. We will present an outline of our lectures.

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C238: Teaching hypothesis testing: a necessary challenge

Marijtje A.J. Van Duijn   University of Groningen, The Netherlands
Wendy J. Post   University of Groningen, The Netherlands

The last decades a debate has been going on about the use of hypothesis testing. This has led some teachers to think that confidence intervals and effect sizes need to be taught instead of formal hypothesis testing with p-values. Although we see shortcomings of the use of p-values in statistical inferences and the difficulties in really understanding hypothesis tests, we take a different view. We think that it is essential to understand what the fundamental principles are behind hypothesis testing in order to obtain correct statistical inference by interpreting confidence intervals (and, at the same time, p-values). In our course “Applied Statistics” for graduate students we designed course material in which we explain the three main approaches of hypothesis testing, Fisher, Neyman-Pearson and Bayesian, using a popular chance game as illustration. In this paper, we will shortly present the highlights of the course material, the results of the evaluation of our teaching, and suggestions for extensions.

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C240: Elements of variance analysis, evaluation of difficulties by questionnaire

Osmar D. Vera   University of Quilmes, Argentina
Carmen Díaz Batanero   University of Huelva, Spain

In this paper we present a study where we assess the difficulties and errors in understanding some elements of Variance Analysis (ANOVA) by a sample of 224 Psychology students, after finishing a Data Analysis course. We analyze the selection of an ANOVA model, the understanding of assumptions, and the associated linear model, the computation of ANOVA in different models, the interpretation of results and software outputs. These results provide information in an area where no much previous research is available.

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C242: Developing conceptual understanding: the role of interactive dynamic technology

Gail Burrill   Michigan State University, United States

The U. S. Common Core State Standards recommend that all students should analyze univariate and bivariate data, two-way contingency tables with categorical data, margin of error and whether there is a significant difference between two proportions. This paper describes a project to develop materials to support the teaching and learning of these “hard-to-teach/hard to learn” topics in introductory statistics. The materials, intended to complement core course materials, consist of interactive dynamic documents, student worksheets, and teacher notes that build foundations for the concepts and confront typical student misconceptions. They were designed to reflect the content and pedagogy advocated by GAISE where dynamic interactive technology provides the opportunity to rethink how students might come to better understand central statistical concepts.

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C244: Invariance and descriptive statistics

Guido del Pino   Catholic University of Chile, Chile

The main focus of teaching statistics at school is statistical literacy, which uses as little mathematics as possible. This fact, together with an unappealing training emphasizing scattered definitions and computational recipes, makes mathematics teachers to strongly dislike statistics. Although the main issue is statistical literacy, providing an underlying mathematical structure for the statistical concept should help making the subject more interesting to mathematically trained people. It is shown that the mathematical concepts of invariance and equivariance under a family of transformations, including some concrete and intuitive interpretations, provide an insight on frequency distributions, graphical displays, and summary measures. The latter can then be rigorously defined, which makes it possible to construct new measures.

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C248: Teaching box plots: An intervention using refutational text and multiple external representations

Stephanie Lem   KU Leuven, Belgium
Goya Kempen   KU Leuven, Belgium
Eva Ceulemans   KU Leuven, Belgium
Patrick Onghena   KU Leuven, Belgium
Lieven Verschaffel   KU Leuven, Belgium
Wim Van Dooren   KU Leuven, Belgium

Despite the omnipresence of box plots in education and research, many misinterpretations have been reported on this representation for data distributions. Previous studies did not succeed in remediating these misinterpretations, leading to the present study in which two teaching methods are tested: refutational text and multiple external representations (MERs). Refutational text explicitly refutes an incorrect conception and provides an alternative, correct conception. MERs combine various external representations in order to improve the interpretation or understanding. These teaching methods were not only used separately, but also in combination. A posttest showed that students in the control condition scored weakest, and students in the combination condition scored best. The students in the MERs and refutational text conditions scored in between. The implications of these results for theory and educational practice are discussed.

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C249: Is an active learning space better than traditional classroom for learning experience in a first year statistics tutorial class?

Ayse Bilgin   Macquarie University, Australia
Greg Robertson   Macquarie University, Australia
David Bulger   Macquarie University, Australia

Service teaching of first year statistics classes is common in many universities due to the importance of statistical knowledge for many disciplines. However, large classes and students’ varied demographic and discipline backgrounds create challenges for academics. How can we ensure a satisfactory learning experience for these students, who usually are not in the class by choice? Could active and collaborative learning spaces be the missing ingredient? In this research we investigated students’ perceived learning experiences in a large first year statistics unit at Macquarie University. Data from a quantitative survey, administered to students in a traditional classroom in 2012 (n=226) and in an active learning space in 2013 (n=619), were used to address the question, “Are students more satisfied with their learning in an active and collaborative learning space as compared to a more traditional classroom?”

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C251: Vergnaud's theory applied to basic school students’ statistical representations

Soledad Estrella   Pontificial Catholic University of Valparaíso, Chile
Arturo Mena   Pontificial Catholic University of Valparaíso, Chile

This study addresses the conceptualization of the table and its teaching possibilities at school level. The theoretical framework adopted is the theory of conceptual fields, which allows for a cognitive analysis of conceptualization in learning. Vergnaud (1990, 2013) affirms that mathematics is needed to characterize with minimum ambiguity the knowledge contained in ordinary mathematical competencies and highlights the fact that this knowledge, although intuitive and implicit, should not hide the need for mathematical concepts and theorems for analyzing it. The study explores primary-school children’s comprehension and how they make sense of statistical frequency tables by examining their productions when they are faced with a data analysis situation. From a qualitative research standpoint, the mathematical relations underlying the operations students use to resolve a situation were analyzed. The principal result of this study is the identification of different levels of conceptualization of the table for students in the same age group.

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C252: Examining graduate students’ prior mathematics/statistics experiences and their statistics self-perceptions: a third order structural model with latent variable analysis

Karen Larwin   University Plaza Youngstown, United States

The proposed presentation will demonstrate the causal link between prior mathematics/statistics coursework experiences (PME) and Statistics Self-Perception (SSP), using structural equation modeling (SEM) techniques. This was accomplished with data from n = 238 graduate students enrolled in their first graduate statistics class. Students were asked to complete inventories: Statistics-Related Self-Efficacy (Finney, 2003), Statistics-Related Attitudes (Schau et al, 1995), and Statistics-Related Anxiety (Cruise, Cash, & Bolton, 1985). SEM was used to demonstrate that these three constructs support a higher order construct of SSP. The resultant model demonstrates a strong link between PME and SSP. Additionally the relationship to participant demographic variables (gender, age, ethnicity, prior college mathematics/statistics classes, etc.) is examined.

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C254: Designing opportunities for students to reason about the relationship between sources and structures of data

Jennifer Nickell   North Carolina State University, United States
Hollylynne S. Lee   North Carolina State University, United States

This paper presents a rationale for designing opportunities for students to reason about the relationship between sources of data and their resulting structures. Students completed a survey about their personal information). Responses were entered into TinkerPlots and attribute names were disguised (i.e., A, B, C, etc.). The students’ task was to examine data from several attributes to analyze their structures and then propose which source (question) from the survey was the best match for a particular attribute. We describe how students reasoned and supported their arguments with a focus on how they used different perspectives in describing data as a pointer, case-value, classifier, and aggregate (Konold et al., 2004). Additionally, we propose a new perspective, data as measurement, used by students to make connections between the data and its source.

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C256: Graphical statistics as an option for the improvement of learning in Psychology

Mirian Agus   University of Cagliari, Italy
Maribel Peró-Cebollero   University of Barcelona, Spain
Joan Guàrdia-Olmos   University of Barcelona, Spain
Maria Pietronilla Penna   University of Cagliari, Italy

Many students have difficulties in the appreciation of concepts related to statistical problems. Various researches have determined how students’ aptitude to solve statistical proofs can be affected by the methods of displaying data. The application of distinct visual aids could improve statistical reasoning, sustaining the principle of graphical facilitation. Some authors did not agree with this point of view, highlighting the complications related to the use of illustrations; they upheld that visual aids could burden the cognitive system with unserviceable information. We confront the basic level of statistical reasoning on probabilities regarding two methods of problem arrangement: verbal-numerical and graphical. Students in Spain and Italy solved the homolog and paired problems in a verbal-numerical and graphical way, in different sequences. Analysis of the correctness of responses and the reasoning applied, managed to compare these ways of presentation and to clarify the cognitive process applied in the problem solving.

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C257: Education statistics for professional specialization: use of knowledge at the workplace of postgraduate in statistics

José Paúl Carrasco Escobar   National Polytechnic Institute, Mexico

The need to analyze the results of training and development programs in Statistics—by government agencies—around the world has led to a research in the following field “Training in Official Statistics”. This research pretends to show the professional development relationship at the workplace of agencies’ employees which are specialized with graduate level education (Techniques and Methodologies) for the generation, analysis and dissemination of official statistics. This research intends to show evidence of the relationship between their academic background and their work activities (statistical knowledge perspective) as well as its application in the workplace. It proposes a series of activities that demonstrate the continuity of the study, along with some observed discussions that must be carried out during the research process. All of this supported by means of a statistically educational framework at a workplace scenario.

Paper

C258: A comparison of outlier labeling criteria in univariate measurements

Menus Nkurunziza   University of Burundi, Burundi
Lea Vermeire   KU Leuven Kulak, Belgium

For the detection of potential outliers in univariate measurements, undergraduate statistics courses often refer to the boxplot. In the workfield, various other sector-linked criteria for outliers are also popular, e.g. Chauvenet’s criterion in engineering. We compare statistical properties of five current criteria – the 3-sigma rule, the Z-score, Chauvenet’s criterion, the M-score or median criterion, and the boxplot or Tukey’s criterion. In particular, in case of a normal population, a joint structure of the five criteria is detecte,d and large sample asymptotic properties of their non-outlier intervals are derived. Pointing at these results should help students to match the statistics course and the lab practice during their education or in their future professional environment. Next, for mathematical statistics students, proving these results may be an instructive activity.

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C259: Statistical and mathematical self-efficacy of incoming students at a large public university

Ulrike Genschel   Iowa State University, United States
Andee Kaplan   Iowa State University, United States
Alicia Carriquiry   Iowa State University, United States
Elgin Johnston   Iowa State University, United States
Wolfgang Kliemann   Iowa State University, United States
Kenneth Koehler   Iowa State University, United States
Ian Mouzon   Iowa State University, United States

All participants in the ongoing STEM education discussion agree that, in addressing national priorities, a key concern is the critical transition of students from high school (or community college) to a four-year college program in the mathematical sciences in particular. Failure in college-level mathematics and statistics courses may discourage students from pursuing STEM majors or perhaps lead to complete college dropout. In fact, even a mediocre performance in these courses often restricts student career choices to fields outside of STEM disciplines. This presentation is aimed at summarizing a statistical investigation of student self-efficacy and self-confidence in mathematics and statistics, particularly with regard to gender differences. Interestingly, preliminary results indicate much higher levels of self-efficacy/self-confidence in statistics compared to mathematics where differences occur. Additionally, for statistics the self-efficacy results are consistent across gender, which is seemingly not the case for mathematics.

Paper

C260: Clickers for engagement in the large undergraduate statistics classroom: do clickers improve final grades?

Krista Wilde   University of Saskatchewan, Canada

With the increase in the number of undergraduate students, the ability to actively engage all students in a large classroom is challenging. The use of student response systems or Clickers is a relatively new technology that allows the students in a large classroom to select individual responses to questions posed to them during class, thereby increasing classroom engagement. In this study, students in a large undergraduate statistics class were separated into two groups: those that used Clickers and those that did not use Clickers. Analysis revealed that there was no difference in the final grades between the students who used Clickers and the students who did not use them.

Paper

C261: Teaching statistics to non-statistics majors

Kolentino N. Mpeta   North West University, South Africa
Diteboho Xaba   North West University, South Africa
Ellen Materechera   North West University, South Africa
Mamolahluwa Mokoena   North West University, South Africa

It is not always easy teaching statistics to students from varied backgrounds who are pursuing a variety of careers and thus have different motivations. What makes it even more difficult is the attitude that some of the students might have towards mathematics and statistics in particular. The situation is further compounded when teaching students who previously failed the statistics course in question and hence have no hope in as far as passing statistics is concerned. One approach that has been effective in assisting the students at the North West University is the use of supplementary instructors. A sample of 310 first year students is used to show the effectiveness of the supplementary instruction. This paper seeks to highlight the challenges faced in teaching statistics to students from diverse faculties as well as approaches, including supplementary instruction, that are found to be effective in such situations.

Paper

C262: Overview of prospective mathematics teachers’ probabilistic thinking

Amable Moreno   National University of Cuyo, Argentina
José María Cardeñoso   University of Cádiz, Spain

This paper presents an overview of the models of probabilistic thinking constructed by 583 prospective mathematics teachers in the province of Mendoza, Argentina. The goal was to gain insight into the personal meanings that these future teachers attribute to random phenomena and to the estimation of their probability. This knowledge should enable one to design the probabilistic education of prospective teachers from more appropriate conceptual and didactic standpoints, and thus contribute to the statistical literacy of secondary school pupils (Cardeñoso et al. 2008; Meletiou-Mavrotheris et al., 2008; Carmichael, Callingham et al., 2009; Vega, et al., 2010, 2011). For this reason all the prospective mathematics teachers responded to a twenty-four item questionnaire which was based on the category system proposed by Cardeñoso (2001). The responses were analysed using multivariate statistical techniques. These revealed four levels or trends in probabilistic thinking which fitted the descriptive labels of deterministic, personalistic, uncertainty, and contingent.

Paper

C263: Real-time educational interpreting in statistics

Thomas L. Berning   University of Stellenbosch, South Africa

The issue of language in education has been a contentious issue in most developing countries. This is due to the critical role that language and communication play in the learning process. In order to offer second language learners the opportunity to learn in their language of choice, a project has been launched to promote multilingualism at a South African university. The institution offers real-time educational interpreting (in the form of simultaneous whisper interpreting between Afrikaans and English) to statistics students in various undergraduate years of study. This offers an additional advantage that lecturers are also able to communicate in their home language. The purpose of this paper is to evaluate educational interpretation from the student and the lecturer’s perspectives as well as provide a brief overview of the technology employed.

Paper

C264: The application of blended learning to large groups

Ivona E. Contardo-Berning   University of Stellenbosch, South Africa

Information and communication technologies have the potential to transform education in developing countries from the traditional methodologies to a more modern and flexible learning environment. The application of information and communication technologies can play a significant role in improving the learning experience of students. In our introductory statistics courses we encounter large, diverse, heterogeneous groups of students from various disciplines and backgrounds which create some specific challenges. We would like to challenge and stimulate strong students while providing students with a weaker background with additional learning opportunities and materials. Blended learning offers us a unique opportunity to create a blend of face-to-face and technology enhanced learning opportunities to cater for these diverse groups of students. We present results from a pilot study using blended learning for a large first year service course in statistics at a South African university.

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C266: The garden sprinkler: an interactive web-based application for teaching design of experiments

Bart De Ketelaere   KU Leuven, Belgium
Karl Siebertz   , Germany
David van Bebber   , Germany
Koen Rutten   KU Leuven, Belgium

Most exercises for teaching students Response Surface Methodology deal with setting up the experiment itself or analyzing the data after the fact. As such, the experience of “learning-by-doing” and drawing intermediate conclusions is lost. For this reason a web-based application was developed, based on a real-life problem dealing with the multi-objective optimization of a garden sprinkler having 8 design parameters and 3 responses. Participants interact with the tool to get the response data for their own designs. Depending on the choice of the considered response(s), different design parameters are active and should be analyzed. By inserting a design into the application, the corresponding responses are generated and can be used for analysis and follow-up experimentation. The tool has been successfully used both in company training and academic courses.

Paper

C270: Using the SETS instruments to investigate sources of variation in levels of pre-service teacher efficacy to teach statistics

Leigh Harrell-Williams   Georgia State University, United States
M. Alejandra Sorto   Texas State University, United States
Rebecca Pierce   Ball State University, United States
Teri Jo Murphy   Northern Kentucky University, United States
Larry Lesser   The University of Texas at El Paso, United States

The authors developed the middle grade and high school Self-Efficacy to Teach Statistics (SETS) instruments based on statistical concepts in United States national and state guidelines for student and teacher knowledge, such as the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Pre-K-12 Report and the Common Core State Standards for Mathematics (CCSSM). The items on both SETS instruments ask teachers to rate their self-efficacy to teach a particular concept using a Likert scale from 1 (“not confident at all”) to 6 (“completely confident”). To investigate the sources of variation in levels of self-efficacy, the authors designed open-ended questions asking participants to describe why they feel less/more confident about certain statistical concepts. This paper discusses results from multiple U.S. public institutions of higher education.

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C271: Graduate students’ learning experience

Josefina Almeda   University of the Philippines, Philippines

There is great demand from private and government institutions for statisticians and many individuals are interested in post graduate education to enhance their career prospects. However, in recent years, only a slight improvement in graduation rate from the graduate programs in the School of Statistics has been seen. This study evaluates post graduate students’ learning experiences in their masters’ degree programs to assist the administrators and teachers in taking proactive steps to address students’ concerns and to further enhance support for their post graduate students. A phenomenological case study was undertaken considering the different perspectives of post graduate students on their learning experiences in the School. The emergent themes were eagerness to learn; career advancement; time management; coping with obstacles; faculty competency; faculty advising; learning environment; and peer influence. Data triangulation was utilized by interviewing other stakeholders to gain insights into their perspectives on the graduate program and the experiences of their students.

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C273: Assessing dimensionality of the communication, language and statistics survey: a multi-group analysis with introductory statistics students near the US-Mexico border

Amy Wagler   The University of Texas at El Paso, United States
Lawrence M. Lesser   The University of Texas at El Paso, United States

Preliminary results of the theoretical and empirical characteristics of the third generation of the Communication, Language, And Statistics Survey (CLASS) are presented. Though validity is a multi-faceted concept, this manuscript focuses just on providing evidence of the dimensionality and internal consistency of the CLASS III. The information from this analysis will be used, in conjunction with more complete analysis, to demonstrate the valid use of this scale on multi-cultural student populations. This study indicates that the items asked of both ELL (English language learner) and non-ELL students have six identifiable dimensions and the ELL-only items have three. It is also concluded that revisions to the CLASS III are necessary in order to promote this survey as a research tool for statistics educators studying cross-cultural and language issues.

Paper

C274: Thinking out of the box in teaching statistics

Tim Low   University of Cape Town, South Africa

Students entering University in South Africa are generally diverse in cultural background, motivation, academic orientation, learning sophistication and stages of emotional maturity. They may have home or first languages other than the language of the tertiary classroom. The home language may possibly also be without a published text in statistics, or even without a statistical vocabulary of its own. Students’ interests and the school curricula to which they were exposed mean they will also vary in access to and experience of technology of any kind (smartphones, pc, internet, etc.). This presentation will describe ideas used to capture the imagination of students, using classroom examples and techniques to create the experience of small learning environments where every student matters even within large lectures.

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C275: Teaching statistics in a mathematics course in middle school: interdisciplinarity, really?

Annie Savard   McGill University, Canada
Dominic Manuel   McGill University, Canada

Statistics is a domain combined with probability that is taught in Mathematics in all school levels. We suggest a potential in using an interdisciplinary approach with this concept, which, according to Fourez & Larochelle (2003), implies that the development of one discipline contributes to the development of others. Thus the development of the understanding of a situation might mean to use both mathematical and statistical reasoning. In this paper, we present a case study where a middle school Mathematics teacher created a lesson in Statistics where the students had the task to study a pack of colored candy and create a pie graph representing its distribution. The teacher hoped that her students made links between mathematical concepts while doing this task. Results show us that her procedural vision of Statistics lead her to focus more on a graphical representation and thus led her to avoid all statistical reasoning development (Garfield, 2002).

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C276: A view of statistics education research in Brazilian extreme south coast

Mauren Moreira Porciúncula da Silva   Federal University of Rio Grande do Sul, Brazil
Suzi Samá Pinto   Federal University of Rio Grande do Sul, Brazil
Daiane Lemos de Sá   Federal University of Rio Grande do Sul, Brazil
Lidiane Santos de Freitas   Federal University of Rio Grande do Sul, Brazil

This paper presents the research performed by the Group of Statistic Education Research from the Federal University in Brazilian South extreme coast. Following Piaget's Genetic Epistemology and Biology of Cognition by Maturana, we still believe in the potential of digital technologies as an alternative to provoke the emotion and contribute to the brain's retention and recall of information. The group produce interactive learning objects and multimedia activities, to promote Statistical Literacy. These objects are designed to offer the opportunity to work with various statistical methods in classroom. The group develop a project called Multimedia Statistical Literacy (LeME), aiming to literate statistically, including virtually 300 citizens annually in social vulnerability. These University projects are promoting the development of statistical literacy in the region. Now the group is innovating, using the potential of quantitative research methods such as Item Response Theory and Principal Component Analysis to research learning and Statistical Literacy.

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C277: Twitter as a learning tool in higher education

Juan M. López-Zafra   University College of Financial Studies, Spain
Sonia de Paz-Cobo   King Juan Carlos University, Spain

As far as the second semester of 2010-2011 we started to communicate with the students in the course of Business Statistics in the Business Administration grade in a private school in Madrid, Spain. The goal was to surpass the email service and check if Twitter could be the weapon. At the end of the course a survey was presented and 95% of the enrolled students filled it out. The questions were divided into three different categories, teacher, technological environment and Twitter, and the student academic and personal background. We present the results on the second one, with 15 questions ranging from “It’s my first experience in Twitter within a subject” (82% answered “yes”) to “Globally considered, the experience has been positive” (63% fully agreed or agreed). As our experience shows, Twitter is very powerful in motivating students in a business statistics course.

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C279: Students' inferential reasoning about sample size

Ethan Brown   University of Minnesota, United States

Intuitions that guide judgments about sample size and sampling variability have been studied for over 40 years, but the implications for statistics instructors are not clear. The administration of the second version of Goals and Outcomes Associated with Learning Statistics (GOALS-2), with 1,165 US undergraduate statistics students participating in a pilot test in December 2013, provides opportunities for an exploration of students’ inferential reasoning about sample size across multiple contexts. The assessment contains items regarding the impact of sample size on sampling variability, confidence interval widths, and p-values. For each item, about half of participants answered the item correctly, and performance was positively correlated across items. However, correlations were small and patterns of responses revealed little consistency across formally analogous distractors. More fundamental research on the linkages between these concepts appears to be needed.

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C283: Against All Odds: Inside Statistics

Marsha Davis   Eastern Connecticut State University, United States

Given the importance of statistics and statistical reasoning in an increasingly complex and information rich-world, ways must be found to engage students with real-world contexts and activities that support learning the basic elements of statistical thinking and the important concepts that underlie statistical reasoning, particularly concepts that students find difficult. Online resources from the newly released Against All Odds (AAO): Inside Statistics (funded by Annenberg and the Consortium for Mathematics and Its Applications (COMAP)) are designed to do just that. The videos, activities, interactive applets, exercises, and written support in AAO provide a rich learning environment that makes statistics come alive and promotes deeper understanding. This presentation will focus on a video and an applet and related class activities. Results from class testing will be shared.

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C284: A suggested theoretical basis for teacher learning in statistics

Fayez Mourad Mina   Ain Shams University, Egypt

The suggested theoretical basis for teacher learning in statistics has many dimensions. Some of the most important of them are paradigm shifts in science, mathematics, education, mathematics education and curricula. All these paradigm shifts have been described and their implications on teacher learning in statistics have been identified. Some of these implications are: Rejecting linear statistical models, applying statistics in real life situations and employing advanced technology both in teaching and learning. The suggested strategy for learning is “professionalization” which is basically means that teacher education should be conducted in an atmosphere comparable to what ought to be at school in its ideal form and analyzing reality of schools with the intension to develop it. The paper was concluded with exploring the futuristic nature of its findings.

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C286: Teaching statistics in the Arab countries: the ambitions and the needs

Hanan Innabi   UAE University, Jordan

For last two decades, most of the Arab countries have been involved in Mathematics Education reform. Data Analysis and Probability is one of the standards upon which the new mathematics school curriculum is built. There are many changes related to statistics content and objectives. A new vision concerning learning statistics can be found in the mathematics frameworks and outlines such as: formulating questions, collecting, organizing, representing, analyzing, and interpreting data. In spite of this apparent change, careful examination reveals that in practical terms, teaching and learning statistics still reside within an old tradition of procedural knowledge. In order to achieve the new goals of teaching statistics, teachers need to embrace the view that statistics is a tool for a daily critical thinking and problem solving. They also need statistical content and pedagogy knowledge and skills to achieve the new vision of statistics education.

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C287: Enhancing statistical literacy and thinking through analysis of scientific journal articles

Mahtash Esfandiari   University of California, Los Angeles, United States
Kekona Sorenson   University of California, Los Angeles, United States
Dave Zes   University of California, Los Angeles, United States
Kevin Nichols   University of California, Los Angeles, United States

One of our major goals in teaching “applied statistics” is to train the “novice” to think and act like a statistical “expert”. We have shown that: 1) Teaching from scientific articles plays an invaluable role in demonstrating how “experts” use statistical methods to solve “real world” problems, and communicate scientific findings. 2) Having students analyze scientific articles in groups, along with virtual or actual office hours with the instructor, and writing the results of their critiques trains them to “think” and “act” like a statistical expert. We have developed an online “article bank” with articles from multiple disciplines. For each article, we have designed multiple-choice and open-ended questions and answers for testing knowledge of statistical methods plus issues related to design, sampling, results, and conclusions.

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C289: Bridging the gap of manpower training for statistics education in Nigerian colleges of education: an empirical evaluation of some selected colleges in south-western Nigeria

Oluokun Kasali Agunloye   University of Botswana, Botswana

Despite the monumental awareness campaign mounted on the importance of statistics education by various national professional statistical organizations and some international agencies committed to the advancement of statistics education, statistics education is not yet accorded due priority attention that it deserves as an independent discipline at both primary and post-primary levels of education in Nigeria. In Nigeria as in most African countries, statistics is still being taught as an appendage under mathematics at primary and post-primary levels of education and this precarious situation explains the prevalent under-development of statistics education as an independent discipline. This paper examines the state of statistics education in some selected colleges of education in south-western Nigeria with respect to manpower training for statistics education at colleges of education level.

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C290: Teaching critical thinking through the introductory statistics class

Mark Ferris   Saint Louis University, United States
Sherri Cheng   , United States
Frank Wang   , United States
Jessica Perolio   , United States

One of the main expectations that society has from a college education is that the student will graduate knowing how to “think”, and not just “think,” but think critically. Yet research reported in the book “Academically Adrift” (2011) by authors Richard Arm and Josipa Roksa, show strong evidence that critical thinking skills of undergraduate college students evidence a discouraging lack of improvement in these very same skills, over the entire college experience. In fact, there is strong evidence that a substantial number of students are leaving college without any appreciable gains in critical thinking, complex reasoning and writing skills. The authors explore using the introductory statistics class as a vehicle to address these shortcomings.

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C291: Disabilities people: Quotas Law which may exist behind the RAIS statistics and census of IBGE

Paulo Oliveira   University of São Paulo, Brazil

In this paper, we consider RAIS data and IBGE Census, both 2010, studied as different situations that are on the margins of the Quota Law for employment of people with disabilities in the labor market. For this study, we took into account variables such as gender, education level, income in minimum wage, the main type of employment, physical, intellectual, visual and auditory disabilities and multiple disabilities.

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