Statistics and Probability in High School

As the title promises, this book provides an overview of the teaching of probability and statistics in the high schools. The title does not mention that it does this from an international perspective, including research and programs from around the world. The authors are from Spain and Austria. The review that follows will offer many criticisms, but make no mistake: this is a valuable resource for anyone teaching statistics at any level.

The book begins with an introductory overview chapter. The remainder of the book is made up of four chapters. These partition the content being taught into four groups and follow a common format with each. First, there is a brief summary of the history of how that content was developed outside of the schools. Then there is a somewhat less brief history of how the content came to be included in high school curricula. The strength of the latter is the international perspective already mentioned. The weakness is that it overlooks some events in the United States, and seriously overlooks the educational efforts of professional statisticians. Prior to World War II, statistics was taught in the United States primarily to graduate students and current researchers in the disciplines where statistics was used. The now common introductory statistics course often taken by college freshpeople probably had its roots in S. S. Wilks’ Elementary Statistical Analysis published in 1948.

Wilks was at Princeton. The chair of the statistics department at Harvard, Fred Mosteller, was a pioneer in bring statistics to the high schools. In 1961, he offered an introductory statistics course nationally in the United States on the NBC television network. Mosteller estimated that more than 75,000 students took that course and around 320 colleges (as well as many high schools) granted credit for it. We can think of this as the forerunner of the current AP Statistics course, which also gets little attention in the book at hand. As with the TV course, one could argue that AP Statistics is really a college course rather than a high school course, but AP is taught in high schools, and the many who teach it become key resources for any other teachers seeking to integrate statistics into the high school curriculum. And the TV course could have been followed by anyone, including high school teachers and students. With various coauthors, often including high school teachers, Mosteller wrote a textbook for the TV course, a textbook for use in a high school course, a college introductory statistics textbook, and a series of paperbacks with series title Statistics by Example (1973) that were well suited to high school use. Other than these omissions, the book at hand provides good coverage of the history of probability and statistics in the high schools.

The second chapter covers the topics generally referred to as “(exploratory) data analysis” (EDA). The authors do an outstanding job here. They make the key point that the goal is not to learn to make stem and leaf or box plots, but rather to learn to make sense of data using those plots as tools. (Those tools are chosen because they are both more concrete and easier to make by hand than more traditional plots such as histograms.) Research shows that both the students and their teachers are much more comfortable making the plots than interpreting the results, yet there is no point in making the plots if we cannot learn anything about the data from the plots. Hand in hand with the emphasis on interpreting rather than computing is the use of technology to do much of the computing. The authors stress this, but do not emphasize that appropriate software can be hard to come by. Professional statistical packages often are very expensive, and usually emphasize more traditional plots. A welcome, free exception is Stats Homework, which even does back-to-back stem and leaf plots which are hard to come by in software.

The section on common difficulties and misunderstandings of EDA is closely aligned with your reviewer’s experience with both college students and current teachers.

The next chapter is on probability and is more of a mixed bag. In the colleges, recommendations have for decades suggested less probability in the introductory statistics course. Much research suggests this is a difficult topic for students. Yet the authors address a wider range of topics than are included in AP Statistics or have ever been included in many introductory college courses. This is offset by extensive use of tables of counts. These are more concrete, and it is easier for most people to deal with specific counts and proportions than abstract probabilities. In addition, tables are how such information is commonly presented in the research literature, with trees and Venn diagrams relatively uncommon. The authors advocate turning abstract probabilities into hypothetical tables, a method long used by teachers, and advocated for widespread use in recent years by Gerd Gigerenzer.

On the other hand, the later parts of the probability chapter delve into material that is part of the AP Statistics course, but is rarely seen in American high schools otherwise, and in fact is unusual in college introductory statistics courses. The material is quite abstract and formula-laden compared to the previous chapter, or most recommendations for statistics in high schools. Most extreme is a discussion of subjective probabilities and Bayesian methods in more detail than found in most any first course in the US. That discussion is undermined by an unfortunate choice for its main example. That concerns a person who thinks their probability of an auto accident differs from that for people in general. That may well be true, but insurance companies know this, and adjust rates based on age, gender, driving record, and a variety of other factors. At least one company offers policy holders the option to install a computer in their car that monitors their driving with the hope that the results could affect rates on an individual basis. Of course, all these adjustments are made on a frequentist basis. Perhaps a simpler example of the issue is the fact that human height varies from individual to individual, but that does not make it “subjective” in the sense of Bayesian statistics. Your height is not just a matter of subjective opinion.

Despite these issues, this chapter can be highly recommended to teachers of statistics with the proviso that they focus on the material actually included in their text or program, and ignore the rest.

The final chapter deals with associations between variables. It wisely includes both categorical and quantitative data, and connects the two better than any textbook your reviewer has seen. Otherwise this chapter exhibits no outstandingly good or bad features, and continues to do well the things done well in earlier chapters.

The statistical topics treated in this book are well aligned with what the statistics profession believes people should know. The authors give many examples of activities to help students learn these topics at a useful conceptual level (though many teachers in the U.S. might prefer more details, such as ready-to-go student handouts). The discussion and references to the research literature on how students learn, and the topics in probability and statistics that give them trouble, will be useful to any teacher of statistics. Required reading for that audience.

After a few years in industry, Robert W. Hayden (bob@statland.org) taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He contributed the chapter on evaluating introductory statistics textbooks to the MAA’s Teaching Statistics.