The obligatory activation of practiced complex multiplication facts and what it tells us about models of arithmetic processing

Loel Nicholas Tronsky

doi:10.5964/jnc.v2i2.22

Abstract

Three experiments were conducted in which adults practiced complex multiplication problems (e.g., 4 x 17). In Experiments 1 and 2, after practice participants completed a number-matching task in which two digits (cues) were followed by a single digit (probe) and had to determine whether the probe matched either of the cues. In simple arithmetic (e.g., 4 x 3), when the probe is the product of the cues (12), participants are slower/more error prone when determining whether there is a match. Results of Experiment 1 extended this effect to complex multiplication. In Experiment 2, participants practiced problems with the larger operand first (e.g., 17 x 4) or with the smaller operand first (e.g., 4 x 17). The number-matching interference effect from Experiment 1 was replicated, and was equal across the two groups whether cues were presented in their practiced or non-practiced order. Experiment 3 was conducted to determine if two additional simple multiplication effects, consistency and relatedness, could be documented for complex multiplication. After practice, in a verification task (4 x 13 = 56?) it was found that when presented answers shared a digit with the decade digit of the correct answer (consistency) or were a correct answer to another practiced problem (relatedness), participants rejected answers more slowly and/or less accurately. Together, findings from the three experiments support arithmetic models that posit that commuted pairs are not represented in long-term memory independently and that posit representations of two-digit multiplication answers are decomposed into decades and units during arithmetic processing.

Highlights

It should be noted that conventional null-hypothesis significance testing (NHST) was conducted throughout the manuscript; since many researchers have noted the shortcomings of NHST, Bayesian analyses were conducted and the resulting Bayes factors (BFs) reported
A lengthy exposition of the benefits of using Bayesian analyses is beyond the scope of this article, but a core benefit is that it enables one to determine the relative likelihood that the data fit the alternative hypothesis or the null hypothesis using the observed data, rather than rejecting or accepting the null hypothesis using hypothetical data as is done in NHST
The second goal of the set of experiments was to show that complex multiplication could be used to evaluate existing models of arithmetic fact representations

Summary

Introduction

The validity and promise of the operand recognition paradigm for determining arithmetic solution procedures has been extended to additional operations and samples, including children (Fanget, Thevenot, Castel, & Fayol, 2015; Thevenot, Castel, Fanget, & Fayol, 2010) Another technique to determine whether participants are solving problems via retrieval from long-term memory uses tasks for which fact retrieval is irrelevant. If answers were being automatically retrieved, participants’ ability to reject the probe as a match to one of the cues should have been interfered with, leading to slower response times and increased errors This is exactly what LeFevre et al (1988) found, and it occurred whether or not an addition sign was included between numbers in the cue. Since this seminal investigation that supported the obligatory activation of addition facts, subsequent studies have examined the obligatory activation of answers to other operations (e.g., De Brauwer, 2007; Thibodeau, LeFevre, & Bisanz, 1996), the role of working memory in automatic retrieval of arithmetic facts (Rusconi, Galfano, Speriani, & Umiltà, 2004), the extension of the obligatory activation to related problems’ answers (7 x 6 activates 42 and 48; Galfano, Rusconi, & Umiltà, 2003), and have examined the electrophysiological markers of task-irrelevant retrieval in simple multiplication (Galfano, Mazza, Angrilli, & Umiltà, 2004; Galfano, Penolazzi, Vervaeck, Angrilli, & Umiltà, 2009)

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Results

Conclusion