What teacher of logic has not experienced frustration in trying to convey the basics of propositional logic or Boolean algebra? The title of this paper certainly got my attention, and reading it certainly gave me some insights into where some of our students’ problems might be coming from. It also got me to think about how it might be possible to improve the way that I teach logic.

While the title of the paper refers to a very general problem, the focus of the text is inevitably more specific, namely, the translation of verbal statements into Boolean expressions. This area has long been seen as problematic, and it underlies the construction of Barwise and Etchemendy’s program, Tarski’s World, which was designed to teach the semantics of predicate logic with reference to an idealized blocks world inside a computer program rather than in the real world. Nevertheless, this paper moves from an anecdotal identification of a general problem to a specific and multifaceted understanding of how students behave when trying to complete this sort of task.

The paper begins with a comprehensive overview of previous work in the area, as well as an explanation of their approaches. It is not enough to identify the misconceptions that students have. It’s also necessary to know why they make the mistakes that they do. To do this, the authors use a force concept inventory--a technique first developed in the physics education community--to design multiple choice tests that are then solved by participants in the study under the observation of the researchers. Based on this, the researchers identified nine themes about how students deploy their knowledge by, for instance, reducing problems to Karnaugh map manipulation, showing confusion about the role of antecedents in logical statements, and applying proofs by the (potentially incomplete) enumeration of cases. On the basis of this, three research questions get answered.

(1) Are misconceptions consistent? Students’ behavior was identified as “chaotic,” and based on ambiguous conceptual frameworks, perhaps because of the myriad different conceptual cues presented by the different real-world scenarios.

(2) Was failure to completely enumerate all cases a particular source of errors? Participants didn’t like using these approaches, but when constrained to ensure that an enumeration was complete they were able to solve problems much more successfully.

(3) Does the difficulty lie in a poor understanding of the underlying logic or in the translation process? Here, too, it seemed to come down to understanding verbal statements in a variety of contexts, and not simply understanding the specifications or the Boolean logic itself.

The answers to these questions inform advice given to instructors, which constitutes the final section of the paper. Students should explicitly address the process of finding and using the correct cues in making a translation, as well as employ exhaustive enumerations to fully understand the import of informal specifications and to render them formally.

I would recommend this paper to anyone interested in improving their understanding of some of the limits of teaching logic, and seeking ways their practice might be improved.