The Role of Mathematics and Metacognition in Gambling Judgments: 
A Test of Dual Process Theory

Eric Amsel & Fall Psy 3610 Class

        Dual process theories abound in developmental research these days.  There is a dual process theory from Steinberg (2008) which proposes dual neurological systems (executive control vs. reward system) and one from Reyna (2008) based on dual representational systems (verbatim vs. gist representations).  Each holds that development involves growth in the functioning of one system or the other, although both acknowledge context effects which can elicit one processing system or the other. 

          A cognitive-based Dual Process theory offers a different perspective on the nature and process of change (Amsel et al., 2008, 2009; Klaczynski, 2004, 2005, 2009). The theory proposes that information is processed by independent but interacting analytic (effortful, algorithmic, systematic) and experiential (automatic, heuristic-based, and intuitive) cognitive systems. The analytic system may be based on normative rules (mathematics and logic) which are often learned in a cultural context (e.g., school). The experiential cognitive system is the default, meaning that experientially-based responses on a task will be quickly produced and immediately applied. However, experientially-based responses may be inhibited in favor of analytically-based ones, if the latter are judged as optimal in the context and selected.  The process of inhibiting, comparing, and selecting optimal responses in a context is called metacognitive intercession. In addition to developmental change in each cognitive system, the ability for metacognitive intercession is thought to develop with the growth of conscious regulatory control (Klaczynski, 2004, 2005, 2009).

          Key in the growth of conscious regulatory control is the metacognitive skill to distinguish between experiential- and analytically-based responses. Metacognitive competence is central in order to identify analytical responses as normative. Metacognitive competence has been shown to a) develop during adolescence and young adulthood, b) be associated with the prevalence of optimal and absence of suboptimal responses ratio-bias judgment task, c) be stable over trials and across tasks, and d) be related to the avoidance of gambling behavior (Amsel et al., 2008, 2009). However, metacognitive competence was related to higher Mathematics ACT scores suggesting a possible interaction between metacognitive skill and mathematics ability.

          The present study explores whether college students performance on the ratio-bias judgment (R-BJ) task is related to mathematics ability and/or metacognitive skill.  The ratio-bias task presents a pair of equal gambles (1/10 vs. 10/100) to participants who are asked to choose between them or express no preference.  Participants are also asked whether they would be willing to pay to secure a particular gamble to avoid one being randomly selected for them.  Optimal performance on the R-BJ task is to express no preference, which conforms to the normative meaning of mathematical equivalence as the substitution of equivalent terms. Suboptimal performance is to express a willingness to pay for one of two equivalent gambles, which reflects a violation of normative probability theory.  

Participants and Methods

          One hundred and six (60% female and 60% freshmen) Introductory Psychology students were participants.  They completed a questionnaire in their classrooms which randomly varied whether the mathematics assessment was presented prior to or after the ratio bias tasks. The mathematics assessments involved 3 sets of 6 problems which require the conversion of fractions into ratios (1/10 = ___%), the conversions of ratios into fractions (10% = ___ /100), and the comparison of ratios (which is a high ratio, 2/10 vs. 20/100).  These mathematical abilities were assessed as they are required for optimal performance on the ratio-bias task.  

          The questionnaire also included 4 trials of the ratio bias judgment (R-BJ) and evaluation (R-BE) tasks. On each trial, participants imaged a carnival game (e.g., spinning wheel) in which they have an option to choose between two gambles, one which has a 1 in 10 chance of winning (10 slots on the wheel with one winner) or one which has a 10 in 100 chance of winning (100 slots on the wheel with 10 winners). Participants were asked to choose one gamble or the other, or express no preference without being told whether or not the gambles were mathematically equivalent. Participants were also asked whether they would pay for the opportunity to choose between gambles if one was going to be assigned to them.  Participants’ metacognitive knowledge was assessed on a R-BE task, which immediately followed each R-BJ task.  Participants rated their certain they each R-BJ response (1/10, 10/100, no preference) was a logical, reflective, and mathematically sound one on a 4 point scale (from not at all to very certain). 

Results

          As in previous studies (Amsel et al., 2008, 2009) most participants (92%) could be reliably categorized into a Metacognitive Status.  In the present study they were categorized into two metacognitive groups, Competent (certainty that only "no preference" judgments are analytic, N = 58) or Non-competent (any other reliable response pattern, N = 40).  Participants were further categorized as Mathematically Competent (a score at or above the median split score of 94%) or Mathematically Non-competent (a score less than the median split).

          A 2 (Order of Math Task) by 2 (Metacognitive Status) by 2 (Mathematics Ability) ANCOVA (age, sex, year in school as covariates) revealed that Metacognitive Status (M Competent = 3.69 vs. M Non-competent = 1.74) and Math Order (M Math Task First = 2.94 vs. M Math Task Second = 2.12) predicted optimal "no preference" responses. A similar 2 by 2 by 2 ANCOVA revealed that only Metacognitive Status (M Competent = .22 vs. M Non-competent = .81) predicted suboptimal "payment" responses.

Discussion

          The results demonstrate the role of unconscious (mathematics task priming) and conscious (metacognitive competence) regulatory processes on optimal responding. A priming effect was shown by participants making more no preference R-BJ responses when receiving the mathematics task first than last.  Although priming was sufficient to promote optimal (no preference) responses among some, it was insufficient to inhibit suboptimal (payment) responses. Metacognitive competence was related to making more optimal and fewer suboptimal R-BJ responses and was independent of mathematics ability and order effects.  This highlights the importance of conscious regulatory processes in judgment and decision-making. The findings support the cognitive Dual Process theory which emphasizes the development of conscious regulatory abilities during adolescence and young adulthood.