The relationship between problem size and fixation patterns during addition, subtraction, multiplication, and division

Abstract

Eye-tracking methods have only rarely been used to examine the online cognitive processing that occurs during mental arithmetic on simple arithmetic problems, that is, addition and multiplication problems with single-digit operands (e.g., operands 2 through 9; 2 + 3, 6 x 8) and the inverse subtraction and division problems (e.g., 5 -3; 48 ÷ 6). Participants (N = 109) solved arithmetic problems from one of the four operations while their eye movements were recorded. We found three unique fixation patterns. During addition and multiplication, participants allocated half of their fixations to the operator and one-quarter to each operand, independent of problem size. The pattern was similar on small subtraction and division problems. However, on large subtraction problems, fixations were distributed approximately evenly across the three stimulus components. On large division problems, over half of the fixations occurred on the left operand, with the rest distributed between the operation sign and the right operand. We discuss the relations between these eye tracking patterns and other research on the differences in processing across arithmetic operations.Keywords: eye tracking, mental arithmetic, problem-size effect, addition, subtraction, multiplication, division Basic arithmetic is important for simple computational tasks as well as a scaffold for more complex mathematical abilities. Furthermore, mental arithmetic performance is of interest to cognitive psychologists because it provides a means to examine memory (a) in an applied context, (b) in conjunction with other cognitive processes (e.g., algorithms, strategy selection), and (c) in the context of a well-defined set of stimuli. We focused on the so-called simple arithmetic facts, defined as the set of addition and multiplication problems with single-digit operands from 2 through 9 (e.g., 2 + 3; 4 x 7) and the inverse subtraction and division problems (e.g., 5 -2; 28 ÷ 4). This 'standard set' of arithmetic facts has been the focus of research on simple mental arithmetic for approximately 30 years (Ashcraft & Guillaume, 2009;Campbell, 2005). People show considerable variability in arithmetic performance across operations (i.e., addition, subtraction, multiplication, and division; Campbell & Xue, 2001). Performanceas measured by response time and error rates -also varies based on the magnitude of the operands and solutions, a finding known as the problem-size effect (Ashcraft & Guillaume, 2009; Zbrodoff & Logan, 2005). The goal of the present research was to examine how attention is allocated to problems in different operations and of different size by measuring participants' eye movements.

Numerical cognition speaks up: Reflections and acknowledgements on the Journal of Numerical Cognition

“…The current issue completes the first full year of publication and accomplishes its target of three issues for the year, accompanying the initial issue that appeared near the end of 2015. I am particularly pleased that the Journal has been able to publish a range of articles that start to represent at least some of the diversity of work in numerical cognition, including for example developmental questions tested at age 4 and beyond (Knudsen, Fischer, Henning, & Aschersleben, 2015;Starr & Brannon, 2015), cross cultural issues (Morrissey, Liu, Kang, Hallett, & Wang, 2016), analysis of the trajectory of real world mathematics skills (Sullivan, Frank, & Barner, 2016) and the cognition that supports different arithmetic processes (Curtis, Huebner, & LeFevre, 2016). In addition, the Journal has addressed and discussed conceptual issues in the research priorities for the field (see Alcock et al, 2016, and accompanying commentaries).…”

Selection of Procedures in Mental Division: Relations Between Self-Reports and Eye-Movement Patterns

“…Research on procedure selection has consistently demonstrated that participants report greater use of non-retrieval procedures on division and subtraction than on addition and multiplication, especially for large problems (Campbell & Xue, 2001;Robinson et al, 2002). In accord with this finding, gaze duration and fixation count patterns appear to deviate substantially as problem size increases for division and subtraction problems (Curtis et al, 2014). Thus, the goal of this thesis was to assess how eye movement patterns are affected when adults of various skill levels (i.e., participants who primarily use memory retrieval vs. participants who readily use non-retrieval procedures) solve simple division problems.…”

“…Experiment 1A (i.e., the Self-Report condition) was designed to address two primary goals. First, I wanted to replicate patterns of performance for division that were reported by Curtis et al (2014). Replicating these findings would support the claim that overall performance is not drastically affected when participants provide self-reports.…”

“…To date, this is the first study to examine how the problem-size effect is distributed across problem components. Curtis et al (2014) postulated that the robust problem size by interest area interaction observed for subtraction and division may reflect a tendency for participants to switch from retrieval to non-retrieval procedures as problem size increases. This approach makes sense considering the available literature on the relationship between operations (LeFevre & Morris, 1999;Mauro et al, 2003;Campbell, 2008).…”

“…Evaluation of the raw data revealed 458 (7%) trials that were spoiled due to equipment failure (i.e., failing to detect an utterance or firing prematurely) and a further 517 (8%) trials in which participants responded incorrectly. The remaining 5,530 valid trials were used to examine whether overall fixation performance matched the results reported by Curtis et al (2014). Bonferroni post hoc tests were used to compare group performance and 95% confidence intervals (CIs)…”