Technology Tips: April 2001

Hand calculators have come into common use for performing calculations in high school and undergraduate courses in elementary statistics. Computers have also been used to do standard statistical computations and tests. However, computers can be used in elementary statistics courses not only to do standard statistical tests, but to perform simulations which clarify concepts and theorems of statistics. These simulations allow the nonmathematically oriented student in elementary statistics to have inductive experiences with statistical concepts in a time-efficient manner. A basic concept in statistics is that of a sample distribution. The sampling distribution of a sample statistic is defined to be the theoretical probability distribution of values for that sample statistic obtained from all possible samples of a certain fixed size from the population. Typically elementary texts illustrate this concept by using a very small finite population which has a small number of possible samples of a given size. Although large or infinite populations are of interest since they have a large or infinite number of samples of a certain size it is difficult to develop a sample distribution. Computer simulation, however, makes this easy to explore. The computer can simulate taking many samples of a certain size from a population. By taking many samples the law of large numbers guarantees that the observed frequency distribution of the sample statistic is close to the theoretical sample distribution. The sample distribution of the mean is the most important sample distribution. The computer simulations in Figure 1 explore a sample distribution of the mean. These simulations employ an elementary statistical package called MINITAB developed at Pennsylvania State University. The MINITAB command URANDOM simulates random observations from a population where numbers between 0 and 1 all have an equal probability of occurring. The first simulation consists of drawing 200 samples of size 2 and computing the means of the samples. A histogram of the results is given. Continuing with samples of size 2, drawing 750 samples is simulated and then 1500 samples. As the number of samples increases the student can see the isosceles triangle-shaped theoretical sample distribution emerging. Note that, because of the small sample size and the nonnormal population, the sample distribution should not be expected to be normal.

A "Reverse Order" Elementary Statistics Course

A collaborative, statistics education research project (Lovett, 2001) is discussed. Some results of the project were applied in the computer lab sessions of my elementary statistics course. I detail the process of applying these research results, as well as the use of knowledge surveys. Furthermore, I give general suggestions to teachers who want to put educational research results into effective use in their own classes.

Students in elementary statistics traditionally see experiments and data as words and numbers in a text. They receive little exposure to the important statistical activities of sample selection, data collection, experimental design, development of statistical models, the need for randomization, selection of factors, etc. They often leave the first course without a firm understanding of the role of applied statistics or of the statistician in scientific investigations. In an attempt to improve elementary statistics education, we have developed a statistics laboratory similar to those of other elementary science courses. We will discuss our experiences in teaching a laboratory component with the traditional elementary statistics course. In each lab session, students, working in teams, discuss the design of an experiment, carry out the experiment, and analyze their data using Minitab on a Macintosh or MS-DOS based computer. The students then individually either answer a series of short answer questions or write a formal scientific report. The labs are designed to be relatively inexpensive and portable. They do not require a prior background in science, statistics or computing.

<p style="text-align: justify;">Preservice mathematics teachers' beliefs about actions related to the use of the technological tools in teaching mathematics may affect how they are going to use them in their classroom activities. However, there is a limited evidence of what beliefs they hold on their intended actions of using technological tools in teaching mathematics. This study presents two preservice high school mathematics teachers' actional beliefs related to their intended actions in teaching geometric transformations (GTs) using Geometer's Sketchpad (GSP). The study comprised of a series of five task-based qualitative interviews with each of two senior undergraduate preservice teachers at a medium-sized public university in the Rocky Mountain Region of the United States. This study used a radical constructivist grounded theory (RCGT) with five assumptions—symbiosis, voice, cognition, adaptation, and praxis as a theoretical framework to guide the study process. The thematic findings of the study included four in vivo categories of their beliefs associated with actions of teaching GTs with GSP – assessment of student learning, engaging students in a group activity in exploring GTs with GSP, engaging students in individual activity in exploring GTs with GSP, and exploring GTs with GSP as 'suck it up and do it.' Pedagogical implications of these categories have been discussed.</p>

One of the main themes of statistics courses is to teach about variability, as well as location. This is especially important for non-statistics students, who often overlook variability. We consider particularly the problem of comparing variability among k samples (k > 2) that are not necessarily drawn from Gaussian populations. This can also be viewed as testing for homoskedasticity of samples. We examine tools for this problem from the perspective of their suitability for inclusion in elementary statistics courses for students of non-mathematical subjects. The ideas are illustrated by an example that arose in a student project.Supplemental data for this article can be accessed on the publisher's website.

The Statistics Anxiety Rating Scale has 51 items, each scored on a 5-point rating scale to measure statistics anxiety with six subscales, Worth of Statistics, Interpretation Anxiety, Test and Class Anxiety, Computational Self-concept, Fear of Asking for Help, and Fear of Statistics Teachers. Psychometric properties included analyses of construct and concurrent validities an internal consistency and test-retest reliability. 221 college students (74% women; M age=28 yr.) in elementary statistics courses at several southwestern state universities participated. The findings are consistent with previous reports and indicate adequate concurrent validity, internal consistency, and split-half reliability, but for construct validity confirmatory factor analysis yielded marginal support.

This article illustrates ways that dynamic software using some sophisticated techniques in Excel can be used to demonstrate fundamental ideas related to regression and correlation analysis to increase student understanding of the concepts and methods in elementary statistics courses and in courses at the college algebra/precalculus level that stress ideas on curve fitting techniques.

This study qualitatively investigates the nature of the challenges that a student who is highly successful in mathematics faces in learning college-level elementary statistics. The study draws on the constructs of eagerness, flexibility and willingness to characterize the necessary disposition for critical thinking that is essential in learning statistics. The case study is based on data collected through a survey assessment and a follow-up interview with a mathematics major enrolled in an elementary college statistics course at the time of the study. The qualitative analysis relies on the student's verbal descriptions of the challenges he was experiencing in the course. The findings suggest that while his strong inclination towards inquisitive learning and strong understanding of mathematical concepts supported this student's mathematics learning, the same characteristics might have been causing him difficulties in learning college-level elementary statistics.

Summary statistics used to describe a data set are some of the most commonly taught statistical concepts in the secondary curriculum. Mean, median, mode, range, and standard deviation are topics that can be found in nearly every program. Technology empowers us to access these concepts and easily to create visual displays that interpret and describe the data in ways that enhance students' understanding. Many graphing calculators allow students to display nonparametric statistical information using a box-and-whiskers plot or a modified box plot showing a visual representation of the median, upper and lower quartiles, and the range of the data. But how can students visually display the mean of the data or show what it means to be within one standard deviation of the mean? One way to create this type of visual display is with a bar graph and constant functions. Unfortunately, graphing calculators, and some computer programs, only display histograms and not bar graphs. The tips in this issue focus on using graphing calculators to draw bar graphs that can help students visualize and interpret the mean and standard deviation of a data set.

The power of The Geometer's Sketchpad® (GSP) for teaching and learning lies in its potential for enabling students to explore geometric concepts and recognize relationships among these concepts as illustrated through the use of multiple representations. This article explores two problems and examines how the use of tables along with GSP sketches may assist students in understanding these concepts. The first activity investigates the relationship of the length of a trapezoid's median (or midsegment) to that of its bases. The second involves showing how the area of a regular polygon approaches that of a circle when the number of sides increases. In both activities, we will start with the geometric object, look at the measurements, tabulate these, and make a graph to illuminate the situation.

Summary PCLogo and Geometer's Sketchpad are powerful tools that may be used by mathematics teachers who want to integrate technology with geometry instruction in the elementary classroom. The purpose of this study was to examine the usefulness of PCLogo and Geometer's Sketchpad to stimulate thinking about geometric concepts in elementary age children. We used a collective case study design that included four cases, two girls (ages 8 and 10) and two boys (ages 10 and 11). All participants were trained to use PCLogo and Geometer's Sketchpad to construct geometric shapes and measure the attributes of the shapes. After the training, participants used these technologies as tools to stimulate thinking about geometric concepts. As a result of our observations of children's performances as they reasoned about geometric concepts, we developed a learning model for teaching children about geometry.

In this month's tip, Jay Walgren, a physics teacher from Vernon Hills High School in Vernon Hills, Illinois, introduces his method of using a programmable calculator to convert grades that are based on learning objectives and mastery of content to a standard grading scale. Although he could use a spreadsheet for this practice, Walgren prefers the convenience and portability of a programmable calculator.

In this study, we focused on the graphing calculator to support the activity-oriented mathematics instruction with considering the accessibility of technology. The purpose of this study was to investigate the direction of the education of mathematics teachers. For this, we gave the teacher training for mathematics using graphing calculators for secondary mathematics teachers, and then examined the recognition for that of teachers. Teacher training of the graphing calculator was carried out three times in two years, we conducted a survey immediately at the time that has passed and after the 8 months or more after the training. As a result, we have obtained the suggestions of the advantages of using a graphing calculator in the learning mathematics, the difficulties of use of the graphing calculator in the classroom and the form of teacher training they want.

CIRDIS is an Italian Interuniversity Research Centre for Statistical Education located at the Universities of Padova, Palermo, Perugia and “Roma la Sapienza”. The Centre, constituted on 1991, has the general objective of promoting the development of research on statistical education in Italy, especially at the undergraduate level. For this purpose the Centre: carries out research on statistical education; develops teaching materials; carries out statistical training courses for school teachers. The Centre has recently renewed its internet site (http://cirdis.stat.unipg.it) to meet teachers’ increasing demand for an effective teaching of statistics in Italian schools. The site, which has been visited since April 2003 about 8.000 times, is an environment from which teachers and students can find information on Centre’s activities and can get a valuable support to motivate and improve the learning of statistics, by accessing to a wide variety of resources and links. The site contains materials to help teaching statistics in an active way, with a multidisciplinary approach, so showing the importance for every citizen of being statistically literate. The section “Teaching/learning resources” aims at providing real datasets and teaching materials from a variety of subjects, in order to give teachers interesting examples for their statistics course. Many of the teaching resources available on the site have been developed by CIRDIS researchers.