Cognition and Instruction, 27(3), 175-224.

ABSTRACT: This article reports on a case study from a design-based research project that investigated how students make sense of the disciplinary tools they are taught to use, and speciﬁcally, what personal, interpersonal, and material resources support this process. The probability topic of binomial distribution was selected due to robust documentation of widespread student error in comparing likelihoods of possible events generated in random compound-event experiments, such as ﬂipping a coin four times, for example, students erroneously evaluate HHHT as more likely than HHHH, whereas in fact these are 2 of 16 equiprobable elemental events in the sample space of this experiment. The study’s conjecture was that students’ intuitive reasoning underlying these canonical errors is nevertheless in accordance with mathematical theory: student intuition is couched in terms of an unexpanded sample space—that is, ﬁve heteroprobable aggregate events (no-H, 1H, 2H, 3H, 4H), and therefore students’ judgments should be understood accordingly as correct, for example, the combination “3H, 1T” is indeed more likely than “4H,” because “3H, 1T” can occur in four different orders (HHHT, HHTH, HTHH, THHH) but “4H” has only a single permutation (HHHH). The design problem was how to help students reconcile their mathematically correct 5 aggregate-event intuition with the expanded 16 elemental-event sample space. A sequence of activities was designed involving estimation of the outcome distribution in an urn-type quasi-binomial sampling experiment, followed by the construction and interpretation of its expanded sample space. Li, whose experiences were typical of a total of twenty-eight Grade 4–6 participants in individual semi-structured clinical interviews, successfully built on his population-to-sample expectation of likelihood in developing the notion of the expanded sample space. Drawing on cognitive-science, sociocultural, and cultural-semiotics theories of mathematical learning, I develop the construct semiotic leap to account for how Li appropriated as a warrant for his intuitive inference an artifact that had initially made no sense to him. More broadly, I conclude that students can ground mathematical procedures they are taught to operate even when they initially do not understand the rationale or objective of these cultural artifacts (i.e., students who are taught a procedure can still be guided to re-invent the procedure-as-instrument).