This article contains two cautionary tales based on my experience working with students, adults, and teachers on research and professional development projects involving data and chance. The first arose from observing two grade 9 boys who ignored instructions and then tried to supplements samples of size two with other samples of size two in order to make samples of size four. The second was related to several observations of students and adults expressing beliefs about dice tossing that were contrary to my expectations: either expecting peaks in distributions that should be uniform or expecting uniformity in distributions that should be peaked. The solution to the dilemmas presented in these two tales would appear to be the creation of cognitive conflict to illustrate forcefully the importance of sample size and the difference between equally likely and non-equally likely outcomes. To handle the situations however, teachers need to be aware that these beliefs may be abroad, to experience the activities that can lead to conflict resolutions, and then plan their own strategies.
- Prof Dev