For students to advance beyond arithmetic, they must learn how to attend to the structure of math notation. This process can be challenging due to students' left-to-right computing tendencies. Brackets are used in mathematics to indicate precedence but can also be used as superfluous cues and perceptual grouping mechanisms in instructional materials to direct students’ attention and facilitate accurate and efficient problem solving. This online study examines the impact of operator position and superfluous brackets on students’ performance solving arithmetic problems. A total of 528 students completed a baseline assessment of math knowledge, then were randomly assigned to one of six conditions that varied in the placement of higher-order operator and the presence or absence of superfluous brackets: [a] brackets-left (e.g., (5 * 4) + 2 + 3), [b] no brackets-left (e.g., 5 * 4 + 2 + 3), [c] brackets-center (e.g., 2 + (5 * 4) + 3), [d] no brackets-center (e.g., 2 + 5 * 4 + 3), [e] brackets-right (e.g., 2 + 3 + (5 * 4)), and [f] no brackets-right (e.g., 2 + 3 + 5 * 4). Participants simplified expressions in an online learning platform with the goal to “master” the content by answering three questions correctly in a row. Results showed that, on average, students were more accurate in problem solving when the higher-order operator was on the left side and less accurate when it was on the right compared to in the center. There was also a main effect of the presence of brackets on mastery speed. However, interaction effects showed that these main effects were driven by the center position: superfluous brackets only improved accuracy when students solved expressions with brackets with the operator in the center. This study advances research on perceptual learning in math by revealing how operator position and presence of superfluous brackets impact students’ performance. Additionally, this research provides implications for instructors who can use perceptual cues to support students during problem solving.
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The effects of operator position and superfluous brackets on student performance in simple arithmetic.
For students to advance beyond arithmetic, they must learn how to attend to the structure of
math notation. This process can be challenging due to students' left-to-right computing
tendencies. Brackets are used in mathematics to indicate precedence but can also be used
as superfluous cues and perceptual grouping mechanisms in instructional materials to direct
students’ attention and facilitate accurate and efficient problem solving. This online study
examines the impact of operator position and superfluous brackets on students’ performance
solving arithmetic problems. A total of 528 students completed a baseline assessment of
math knowledge, then were randomly assigned to one of six conditions that varied in the
placement of higher-order operator and the presence or absence of superfluous brackets: 1)
brackets-left (e.g., (5 * 4) + 2 + 3), 2) no brackets-left (e.g., 5 * 4 + 2 + 3), 3) brackets-center
(e.g., 2 + (5 * 4) + 3), 4) no brackets-center (e.g., 2 + 5 * 4 + 3), 5) brackets-right (e.g., 2 + 3
+ (5 * 4)), and 6) no brackets-right (e.g., 2 + 3 + 5 * 4). Participants simplified expressions in
an online learning platform with the goal to “master” the content by answering three
questions correctly in a row. Results showed that, on average, students were more accurate
in problem solving when the higher-order operator was on the left side and less accurate
when it was on the right compared to the center. There was also a main effect of the
presence of brackets on mastery speed. However, interaction effects showed that these
main effects were driven by the center position: superfluous brackets only improved
accuracy when students solved expressions with brackets with the operator in the center.
This study advances research on perceptual learning in math by revealing how operator
position and presence of superfluous brackets impact students’ performance. Additionally,
this research provides implications for instructors who can use perceptual cues to support
students during problem solving.
more »
« less
- Award ID(s):
- 1931523
- NSF-PAR ID:
- 10443583
- Date Published:
- Journal Name:
- Journal of Numerical Cognition.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Subtle visual manipulations to the presentation of mathematical notation influence the way that students perceive and solve problems. While there is a consistent impact of perceptual cues on students’ problem-solving, other cognitive skills such as inhibitory control may interact with perceptual cues to affect students’ arithmetic problem-solving performance. We present an online experiment in which college students completed a version of the Stroop task followed by arithmetic problems in which the spacing between numbers and operators was either congruent (e.g., 2 + 3×4) or incongruent (e.g., 2+3 × 4) to the order of precedence. We found that students were comparably accurate between problem types but might have spent longer responding to problems with congruent than incongruent spacing. There was no main effect of inhibitory control on students’ performance on these problems. However, an exploratory analysis on a combined performance measure of accuracy and response time revealed an interaction between problem type and inhibitory control. Students with higher inhibitory control performed better on congruent versus incongruent problems, whereas students with lower inhibitory control performed worse on congruent versus incongruent problems. Together, these results suggest that the relation between inhibitory control and arithmetic performance may not be straightforward. Furthermore, this work advances perceptual learning theory and contributes new findings on the contexts in which perceptual cues, such as spacing, influence arithmetic performance.
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