Counterintuitive probability tasks in preservice mathematics teacher education: reasoning, misconceptions, and responses to student error

Abstract

The current study examines preservice mathematics teachers' answers, explanations, and evaluations of a hypothetical student's errors on six counterintuitive probability tasks. Three types of performance were assessed: selecting an answer, mathematically explaining the chosen answer, and analyzing a hypothetical student's error. Participants were 172 mathematics preservice teachers recruited through convenience sampling from two teacher training institutions in Kazakhstan – a primary sample of 145 from a private university in the Almaty region and a second-site sample of 27 third-year students from another private institution in the Zhetysu region. The research design involved selected-response analysis as well as coding of 1,032 written explanations and 1,032 evaluations with the optional reflections of participants' justifications for their answers and/or critiques of the hypothetical student's solutions. Quantitative analyses addressed task difficulty, block-level comparisons, comparison of course-exposure subgroups in the primary sample, as well as estimation of population-averaged logistic regression models with responses nested within participants. Qualitative analyses included task-based coding of explanations and evaluations, followed by expert audit of the coding scheme. Three main findings are reported. First, the split between Tasks 1–3 and Tasks 4–6 was particularly clear in the primary sample. Second, regardless of location, answer accuracy exceeded the quality of written explanations and pedagogical evaluations because many participants picked the right answer but could not explain why or properly evaluate a hypothetical student's error on the hardest tasks. Third, the second-site sample showed a relatively low task difficulty gradient resulting in a significant site by task block interaction effect.