Orchestrating Semiotic Leaps from Tacit to Cultural Quantitative Reasoning—The Case of Anticipating Experimental Outcomes of a Quasi-Binomial Random Generator

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).

ACKNOWLEDGMENTS

This article builds on several conference papers (Abrahamson, 2008a, 2008b; Abrahamson et al., 2008; Abrahamson & Cendak, 2006). The Seeing Chance project was supported by a National Academy of Education/Spencer Postdoctoral Fellowship. I am grateful to the students who volunteered to participate in this study and to the school principal and staff for opening their doors to this collaboration. A big thanks to Jeanne Bamberger, Allan Collins, Andy diSessa, Maria Droujkova, Alan Schoenfeld, Tobin White, and Betina Zolkower for commenting on earlier drafts of this article—shortcomings of this article reflect my failure to heed their advice. Thank you Cliff Konold for helping me be more precise and consistent with my mathematical terminology. Special thanks to members of the Center for Connected Learning and Computer-Based Modeling (Uri Wilensky, Director) at Northwestern University who assisted in implementing my design, and especially Paulo Blikstein, for his help in engineering and producing my design of the marbles scooper. At UC Berkeley, thank you to members of the Embodied Design Research Laboratory in the Graduate School of Education, including participants in the Undergraduate Research Apprenticeship Program, and to students of the graduate course Learning Chance. I wish to extend my gratitude to the prior and current editors in chief and executive editors of Cognition and Instruction as well as Susan Jurow and several anonymous reviewers for their highly effective comments. Finally, I thank Michael Monroe for “Simple Life” and “Wintersong,” pensive musical art that inspired much of the aforementioned search for coherence.

Notes

¹I have chosen to describe the sample space as a collection of elemental events (TTTT, TTTH, TTHT, THTT, …. HHHT, HHTH, HTHH, THHH, HHHH) parsed into a subsets of aggregate events (no-H, 1H, 2H, 3H, 4H) so as to eschew common terminology confusions around the term “outcome,” which is often used for both theoretical and empirical probability. Generally, there is much overlap and ambiguity, in mathematical and educational texts on probability, with respect to the precise meanings of the terms “event,” “outcome,” etc., and these terminology challenges are related to different epistemologies (i.e. classicist vs. frequentist probability). One concern is a confusion between potential (theoretical) and actual (empirical) outcomes, and another is with regards to what an “event” encompasses within the sample space. In fact, the ambiguity of outcomes is mathematical and not only psychological. Weisstein (2006) writes, citing Papoulis (1984, pp. 24–25), “Experimental outcomes are not uniquely determined from the description of an experiment, and must be agreed upon to avoid ambiguity.” My particular choice of the adjective “aggregate,” as opposed to the more conventional “class,” “set,” or “group,” is an intentional nod to educational research on the cognition and instruction of complex phenomena, wherein agent- and aggregate-based perceptual/conceptual frames play a key role in understanding emergence—these perspectives have been implicated as instrumental in understanding natural distributions, for example, the normal curve, as emerging from multiple instances of random agent-based interactions (e.g., see Jacobson & Wilensky, 2006; Wilensky, 1997). Also, by using the same noun—“event”—for both the naïve and the expanded sample spaces, I am preparing lexical grounds for a claim that the naïve sample space is as legitimate as the mathematical sample space—they are contingent on different ways of seeing the random generator, and these contingencies must be acknowledged, embraced, and leveraged in educational design.

²There is no broad agreement over the precise semantics of “intuition” (e.g., whether it is fixed or can develop). For example, Fischbein (1987) maintains that whereas “primary intuitions” are stable, “secondary intuitions” can develop with learning, such that initially fragmented or counterintuitive situations may become patterned schemes of expert practice (see also Dreyfus & Dreyfus, 1986, 1999). Interestingly, such learned skills are colloquially termed “second nature.”

³I am by no means claiming that students cannot in principle attend to order. (Indeed, Tversky and Kahneman also documented assertions of type “P(HTHT) > P(HHTT)” that treat two different permutations on “2H, 2T.” However, note that this comparison item makes the property of order salient because order is the only parameter distinguishing the two sequences—a pragmatic framing suggesting that the items are at least nominally distinct along the dimension of order and that this distinction is somehow meaningful to the task at hand. Also, note that the notion of “sequence” is not necessarily salient in a string of four symbols, such as “HHHT,” if one is not privy to the process that is captured in the inscription (e.g., a temporal series of four coin flips). In like vein, when four coins are flipped simultaneously, the notion of order is complicated—order is rarely a spatial feature of the coins themselves and is, instead, an analytic construct.) Rather, I am suggesting that students do not attend to order because they do not conceptualize that property of the symbol sequence as relevant to the task of determining relative likelihoods, just as, say, one might not know to rap on a watermelon to hear if it is ripe—attending to this acoustic property of watermelons is not part of what Goodwin (1994) calls one's professional vision of this domain of scrutiny. Indeed, what people see when they look at objects is greatly contingent on the context and goals of their perceptual activity. This idea is probably self-evident and is certainly a tenet of phenomenology philosophy (e.g., Heidegger, 1962), but see, for example, Shinoda, Hayoe, and Srivastava (2001) for an overview of half a century of cognitive-science empirical studies relevant to this idea of goal-based selective attention.

⁴“Top-down coherence: Symbolic and verbal propositions are prominent in instruction. It is possible to view these as being learned prior to the broader coordinations in intuitive knowledge that are eventually required. … The subtleties and reliability of top-down coherence generation as a developmental principle are important to understand. Most schooling seems to count heavily on explicitly and literally rememberable elements. My working assumption is that this only works well within subsystems that already involve a sufficiently rich and reliable network. … I would like the theory sketch developed here to be capable of expressing the difficulties in top-down development” (diSessa, 1993, pp. 115–116).

⁵Initial results appear in Abrahamson and Cendak (2006), implications for learning theory appear in Abrahamson (2008a, 2009b), and implications for design-based research methodology appear in Abrahamson and Wilensky (2007), Abrahamson and White (2008), and Abrahamson (2009a).

^{6 6}It would be outside the focus of this article to survey the wealth of prior constructivist design for probability learning. Here, the design is taken mainly as setting the context for data analysis. Also, the learning activities that are the focus of this study are only the first of several activities in the full sequence that also includes computer-based interactive modules, all part of the ProbLab experimental unit (Abrahamson, 2009a, Abrahamson, Janusz, & Wilensky, 2006; Abrahamson & Wilensky, 2002, 2004a, 2005).

⁷The difficulty that probability concepts present to learners, and therefore its aptness for a study of intuition and learning, is perhaps augured by the tumultuous history of this mathematical topic, which was fraught with resistances from unlikely quarters going far beyond lay people's naïve beliefs and including religious authorities (e.g., Hacking, 1975).

⁸In this study, there were equal numbers of marbles of each of the two colors. The rationale was to begin the instructional sequence with the case in which it is equally likely to draw a marble of each color (p = .5), so that all 16 compound events are equiprobable (see also Falk & Lann, 2008). This scaffold would subsequently be removed, at which point the equiprobability would need to be qualified as obtaining only within aggregate-event classes—not between them (Abrahamson, 2009a).

⁹The scooper's spatial template is an indigenous–structural format for inscribing the outcomes, similar to the left-to-right order of independent events in Vegas-style fruit machines. In the case of coin flipping, however, one must apply an exogenous–analytic format. Note that the “HHTH” left-to-right linear textual inscription commonly used to designate discernable compound-event outcomes does not directly translate spatial/physical features of the actual coin-flipping experiment. Rather, the spatial configuration of the actual four flipped coins that have landed on a desk is a necessary, if prosaic, mechanical feature of this random generator that is irrelevant for our current probability analyses. Finally, note that, strictly speaking, the marbles-box experiment is only a hypergeometric approximation of the binomial. That is, a single scoop is in fact the result of four dependent without-replacement trials and so is not precisely commensurate with the concurrent flipping of four independent coins or a sequence of four coin flips. Thus, the true expected outcome distribution from the experiment has lower variance than the binomial distribution (e.g., it is even more difficult to draw a 4H scoop than in a truly with-replacement experiment). That said, the ratio of the scooper sample size (4) to the population (hundreds of marbles) renders this issue practically negligible for the purposes of simulating the binomial. Also, our computer modules implement truly binomial experiments. Henceforth in this article, I shall treat the marbles-box experiment as though it were truly binomial.

¹⁰The two activities can be further contrasted. In the first activity, participants are asked to cast a judgment about a specified property of an available object, but they are not told how to go about making this judgment. In the second activity, participants are guided to build and assemble an initially unavailable object, but they are not told why they are engaged in this activity or what they are to do with its product.

¹¹The data analysis process has bootstrapped our research group into expanding our reading of cognitive science, sociocultural theory, cognitive- and cultural semiotics, pragmatics, philosophy, design-based research methodology, and critical theory. The caveat of making sense of the data impelled us to consider how these additional resources, which each introduced cogent perspectives, could possibly be integrated into the evolving model of guided tool-based learning. As one result of this collaborative process, we published on the potential efficacy of this theory integration (Abrahamson, 2008a; Abrahamson et al., 2008; Abrahamson & White, 2008). As a second result, I am developing a graduate seminar that uses the Seeing Chance data as a fulcrum for a survey of learning-sciences literature.

¹²Tversky and Kahneman's “representativeness heuristic” was originally articulated to explain how people judge an outcome presented in the absence of the random generator that produced it. In our design, conversely, the participants are asked to guess an outcome in the presence of the random generator. Nevertheless, I submit that these two decision-making processes both involve object-to-sample inferences, because I assume that some mental imagery and simulation are involved in each of them (cf. Xu & Vashti, 2008, in which both the population and the sample are displayed).

¹³It is this sense of an event class that underlies mathematical phrases such as “the chance of getting any of these” or “the chance of getting two green and two blue in any order”—phrases that may be challenging due to the ostensible assignment of a single chance value to a collection of objects (whose respective chances add up to the event's chance).

¹⁴Was Li's “6/16” mathematical proposition indeed grounded in his initial “2g2b” intuitive inference? That is, what role, if any, does the initial interaction with the marbles box play in participants' learning path? Could the initial interaction in fact be a superfluous activity such that we could cut to the chase and begin the design directly with the combinatorial analysis? To examine this question empirically, we ran an experiment in which 23 middle-school students were randomly assigned to three conditions that framed the nature of their initial interactions with the marbles box: (a) “leading question,” in which essentially the current protocol was enacted; (b) “no question,” in which participants were shown the marbles box but no additional problem or context were invoked; and (c) “distracter question,” in which participants were asked to estimate how many without-replacement scoops are required to empty the box. We found that leading-question participants were more likely to question the necessity of permutations in the sample space as well as eventually interpret the sample space as meaning that 2g2b would be the most likely event (Mauks-Koepke, 2008; Mauks-Koepke, Buchanan, Relaford-Doyle, Sushkova, & Abrahamson, 2009).

¹⁵Evolutionary sociocultural theorists might argue in return that, on a phylogenetic scale, innate capacities can nevertheless be attributed to the survival of gene carriers whose chance innate capacities were best adapted to socially emergent needs and hence were naturally selected and further honed over subsequent generations, recursively.

¹⁶I wish to emphasize the potential contributions of Sfard's magnum opus to my own research program. The Seeing Chance interview can be viewed as an asymmetrical negotiation, in which the interviewer creates an opportunity for the student to consider the relevance of the analytic procedure as complementary to, and enhancing of, his naive inference that was based on perceptual judgment. Pivotal to the completion of this negotiation was that the student be able to view the thematic mathematical object—the 2-by-2 matrix—as one of 16 (equiprobable) events and not only as one of five “things you can get,” and that he understand the implications of alternating between these views. Combinatorial analysis and the discussion around the products of this analysis—the sample space, first loosely grouped on the desk and then assembled into the orderly combinations tower structure—constituted the context in which the student was expected to expand his repertory of views toward the events so as to accord with the interviewer's. However, seeing an event as one of 16 rather than as one of 5 is complexly contingent on understanding why one might wish to adopt a new view of the object. That is, relinquishing or modifying a world view begs a willingness to participate in a discursive practice that entails an adoption of what initially appears to be an arbitrary construal of material substance. In this sense, combinatorial analysis, as opposed to direct perceptual judgment, can be viewed as a

new [and incommensurate] discourse ̈̈governed by meta-rules different from those according to which the student has been acting so far, ̈̈[thus entailing] a situation in which communication is hindered by the fact that different discussants are acting according to different meta-rules (and thus possibly using the same words in differing ways). … Only too often, [discursive] commognitive conflicts are mistaken for factual disagreements, that is, as a clash between two sentences only one of which can be correct. … Without other people's example, children may have no incentive for changing their discursive ways. From the children's point of view, the discourse in which they are fluent does not seem to have any particular weaknesses as a tool for making sense of the world around them. (Sfard, 2007, pp. 574–575)

Sfard concludes that a student's willingness to engage in other people's discourse is complexly related to the student's sense of identity. Sfard thus paves methodological avenues for investigating relations between identity and learning, relations that are becoming increasingly central to the study of diversity in mathematics education.

¹⁷The Vygotskiian legacy, and in particular the ethnographical work of Alexander Luria in the Ural, has demonstrated that “naïve” logical reasoners do not necessarily reason syllogistically as “scientific” reasoners do. Therefore, I never judge middle-school students' reasoning by the extent to which they subscribe to the ineluctable deductivity of the formal syllogistic format, because I view this format as a cultural tool in which the participant may well be unfluent. Indeed, whereas the content of the interlocutors' turn taking in the transcribed excerpts of this interview might be construed as constructing a syllogistic sequence of statements, these structures are at most suggested and never explicit.

¹⁸A Peircean analysis of Li's behavior as exemplifying abductive reasoning is further elaborated elsewhere (Abrahamson, 2009b). To read further on the history, roles, and mechanisms of C. S. Peirce's “generative abduction” and diagrammatic “hypostatic abstraction” in mathematical [re]-discovery, see Bakker and Hoffmann (2005), Heeffer (2006), and Radford (2008).

¹⁹See Abrahamson and Wilensky (2007) on a design-oriented formulation of principles for fostering content learning as the reconciliation of vying perceptual constructions of mathematical objects.

²⁰I have been repeatedly asked by leading scholars why the four-marbles scoop is structured in 2-by-2 form rather than 4-by-1, given that the normative mathematical listing of a sample space is as linear strings, for example, “HHTH,” and that we may wish to scaffold students toward that normative form. My prosaic reply is that the 2-by-2 configuration is a heritage of the square samples used in our computer-based statistics activity for networked classrooms (S.A.M.P.L.E.R., Statistics As Multi-Participant Learning-Environment Resource, Abrahamson & Wilensky, 2004b; Abrahamson & Wilensky, 2007), and we wished to create a uniform format across our curricular material, ProbLab (Abrahamson, Janusz et al., 2006; Abrahamson & Wilensky, 2002). Auspiciously, though, when square samples are rotated as a form of combinatorial expansion, they produce a set of four permutations as compared to only two that would be produced by rotating a card representing a linear scoop. Notwithstanding, a comparison of the learning affordances of square and linear scoops would make for an intriguing study. In any case, a combinations-tower format utilizing linearly configured compound events has been adopted by the Model Chance project and thus integrated into an experimental version of the TinkerPlots computer-based learning environment (Cliff Konold, personal communication, January 22, 2007).