Mathematical Modeling of Free Recall in the Mental Attention Memory Task

1. Mathematical Modeling of Free Recall in the Mental Attention Memory Task J. Pascual-Leonea, E. M. Romero Escobara, J. Johnsona,& S. Morrab aYork University; bUniversità degli Studi di Genova Mental-attentional (M-) capacity is a content-free resource that serves to boost activation of task-relevant schemes. It is a causal component of working memory (Pascual-Leone & Johnson, 2005). M-capacity increases with age in normal childhood, growing by one symbolic scheme every other year from 3 to 15 years of age. M-tasks (i.e. measures of M-capacity are constructed using metasubjective task analysis (MTA) to estimate the mental (i.e., M-) demand of different levels of performance. We contrast three theory-driven models of performance on one M-task. RESULTS: Goodness-of-Fit Root-Mean-Scaled-Squared-Deviations (RMSSD) and Linear Regression Coefficients were used as GoF measurements of location. For GoF in terms of trend, a Pseudo R2 (wR2) was used to capture the differences in variance and number of participants at each level. As shown in Figure 1, Model 1 captures the trend of the observed data in terms of the difficulty levels of the subtasks and the developmental increase in recall, but neither the location nor the slope of the model fits the data well. For Model 2, parameter a (0.969) was estimated from SIMPLE data. It shows a considerable improvement in fit of both location and slope. Model 2 has a particularly close fit to STROOP subtask data. We tested a 3rd Model that excluded the M-capacity assumption but allowed activation levels to begin to decay as a function of equation 1 from the start. This Model 3 provided a poor fit to the data with no developmental increase in recall. TASK The mental attention memory task (MAM) isa verbal span measure. Its three subtasks vary in degree of interference with the basic recall task. On each trial participants see a supraspan list of consonants arranged in a circular pattern. List length is 4 above the predicted M-capacity for the person’s age. Participants read the consonants aloud at the rate of 1 per second. The SIMPLE subtask requires free recall of the consonants. In the TELEPHONE subtask, as the participant recalls each letter, s/he must also find and dial it on a rotary phone. In the STROOP subtask, s/he responds to a different Stroop card before recalling each letter (e.g., says “red” when shown the word GREEN printed in red ink). Score is mean recall across trials in each condition. MODEL 2 A sequential MTA was adapted from Morra’s (2000) model of short-term memory. It includes developmentally increasing M-capacity, which serves to activate fully a limited number of task-relevant schemes. Once capacity is exceeded, activation of mentally encoded stimuli that are not boosted by M begins to decrease. Morra proposed that decrease in activation would be a function of the number of intervening mental events (or M-steps, Burtis, 1982) and interference from other decaying schemes. Steps 1-9 in Table 2 model the encoding phase for a 9-10-year-old presented with 8 consonants. At each step, relevant operative and figurative schemes are denoted with capital and starred lower-case letters, respectively. Executive schemes (which do not need M-boosting) are coded with the letter E. Schemes boosted by M are enclosed in square brackets “[ ]”; those boosted by other operators appear between curly brackets “{}”. Letters are read one at a time and are encoded by the operative scheme CODE. Encoding results in a mental figurative scheme for the consonant, subindexed by list position (e.g., *c3). Schemes boosted by M have an activation weight of 1. Once the limit of M-capacity is reached, however, figurative schemes are dropped from M-space and their activation weight starts to decrease. Rate of decay is represented by a parameter a, and the activation weight (Wx) of a consonant scheme outside of M is: Wx  =  (Wx-1)*(a#DkScx) (Eq. 1) Here, x is the current step, x-1 is the previous step, and #DkSc is the number of decay schemes at a given step. Decay schemes are those in the field of activation, but outside M space. Steps 10-25 model the recall phase for the TELEPHONE task. Recall of each consonant requires two steps: One to select/retrieve and utter (e.g., *Uc1) the consonant and one to scan for and dial (*Dc1) it on the phone.During these steps consonant-schemes outside M continue to decay as a function of time/steps and interference; they are recalled in order of their activation levels (high to low) and with a probability equal to their final degree of activation. Table 1 Distribution of Participants by M-level • CONCLUSIONS • Our theory's complete model (Model 2), incorporating a developmentally increasing, limited attentional-capacity (M) and decay (I) and interference of schemes outside the focus attention provides a good fit to recall data in the MAM. • This is generally consistent with other findings re limited capacity (e.g., Barrouillet & Camos, 2001; Cowan et al., 2006), and supports in particular the capacity growth function proposed by Pascual-Leone (1970; Pascual-Leone & Johnson, 2005). MODEL 1 A dimensional MTA of the M-demand of the three subtasks predicted that average recall would be one beyond the person’s M-capacity in the SIMPLE, equal to capacity in the TELEPHONE, and one less than capacity in the STROOP task. For example, predicted mean recall for 9-10-year-olds (M-capacity= e+4) was 5 in SIMPLE, 4 in TELEPHONE, and 3 in STROOP.