A significant number of students in introductory statistics courses may function at Piaget's concrete operational level of thought. These students may find it difficult to understand the complex correlations and interactions between variables that typify many statistical procedures. A technique for introducing analysis of variance (ANOVA) in a concrete fashion is presented. This technique leads students to an intuitive understanding and their relationship to each other.
Students in introductory statistics courses often commit similar errors in computation and interpretation. A handout that lists such common errors is described. Students appreciated the handout, thought it reduced errors, and recommended it for future classes.
Most statistics teacher strive to make the course material meaningful. This article presents specific examples for elaborating the statistical concepts of variability, null hypothesis testing, and confidence interval with common experience.
It is not unusual today - even among people who consider probability as a concern of pure mathematics - to start a probability course with an attempt to uncover the experimental roots of the probability concept. In fact, it is not a new feature. The story about tossing a coin with the happy result of a fair distribution of heads and tails in the long run has been the custom for quite a long time. What is new about it, is that the story is dramatized and acted out - I mean, by the author or the textbook. Maybe even the teachers or the students are expected to try out this experiment - following the highly encouraging examples given by the textbook authors. It is a pity that by showing one experiment without asking themselves whether it is typical, textbook authors lead the teachers up the wrong path and help to create wrong attitudes towards probabilistic problems.
Not long ago probability was investigated, taught, and published in the way sheep are counted, distances, time and money are accounted for, and physics is pursued, that is tacitly supposing that everybody - investigators, teachers, students, authors and readers - knew what dice, heads and tails, stakes, chances, events, dependency and independency are. There are didactical reasons to believe that at school level probability cannot be taught in any other way, but at any account it may be taken for granted that intuitional knowledge about such tools and concepts is indispensable whenever probability ought to be related to reality.
In the article "The Probabilistic Abacus", by A. Engel (Educational Studies in Mathematics 6, 1975, p. 1 - 22), we have introduced the probabilistic abacus and we have applied it to absorbing Markov chains. We have explained in detail how to compute absorption probabilities and expected times to absorption, but we gave no proofs. In this paper we give more applications of the abacus and we supply proofs for most of its properties.
All too often mathematics is considered to be the study of certainties: certain truth and certain falsity. We need to overcome this misleading impression and to show that mathematics can describe the uncertainties of everyday life. Much of our daily life is unpredictable and uncertain. A branch of mathematics, probability theory, can help us cope with this aspect of life. However, if the theory of probability is taught formally, then students may not make the connection between the theory and its usefulness in daily life. The following introduction helps ensure that this connection is made.
In this study, 48 subjects who had no previous exposure to probability or statistics read one of three texts that varied in the degree of explanation of basic concepts of elementary probability. All texts contained six formulas, each accompanied by an example as well as definitions and information logically required to solve all problems. The high-explanatory test differed from the low-explanatory and standard texts in that it emphasized the logical basis underlying the construction of the formulas, the relations among formulas, and the relations of variables to real-world objects and events. On both immediate and delayed performance tests, subjects in the low-explanatory and standard-text conditions performed better on formula than on story problems, whereas the subjects in the high-explanatory text condition did equally well on both types of problems. It was concluded that explanation did not improve the learning of formulas but rather facilitated the application of what was learned to story problems.
Students of Statistics, whether they plan to be statisticians, or only to use statistics as a tool in their professions, often fail to grasp the big ideas of statistics from their courses. "Service" courses concentrate on methods, while "mainstream" courses emphasize mathematical structure, and in both types of course, the powerful concepts most useful in practice are not given much emphasis. The textbooks that guide our teaching style do not seem to include a broad appreciation of statistical ideas among their objectives. Statistics courses that do provide some pedagogic emphasis to the big ideas, may still fail to convey these ideas if the examination does not require their comprehension. In this paper, I give some examples of "big ideas" and exam questions that would assess students' comprehension of them, and argue that even though they are the most important aspects of a course, that they will not be absorbed from courses following currently available textbooks. I suggest the use of a project-based teaching technique with which I have had some experience and success, and how to use traditional textbooks as support for such a project-based course.
Computers and software can be used not only to analyze data, but also to illustrate essential statistical topics. Methods are shown for using software, particularly with graphics, to teach fundamental topics in linear regression, including underlying model, random error, influence, outliers, interpretation of multiple regression coefficients, and problems with nearly collinear variables. Systat 5.2 for Macintosh, a popular package, is used as the primary vehicle, although the methods shown can be accomplished with many other packages.