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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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Mathematics Presentation Matters: How Superfluous Brackets and Higher‐order Operator Position in Mathematics Can Impact Arithmetic Performance
Abstract Perceptual learning theory suggests that perceptual grouping in mathematical expressions can direct students' attention toward specific parts of problems, thus impacting their mathematical reasoning. Using in‐lab eye tracking and a sample of 85 undergraduates from a STEM‐focused university, we investigated how higher‐order operator position (HOO; i.e., multiplication/division operators and the presence of superfluous brackets impacted students' time to first fixation to the HOO, response time, and percent of correct responses). Students solved order‐of‐operations problems presented in six ways (3 HOO positions × presence of brackets). We found that HOO position and presence of superfluous brackets had separate and combined impacts on calculating arithmetic expressions. Superfluous brackets most influenced undergraduates' performance when higher‐order operators were located in the center of mathematical expressions. Implications for learning and future directions are discussed about observing eye movements and gaining insights into students' processes when solving arithmetic expressions.
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- Award ID(s):
- 2320053
- PAR ID:
- 10521578
- Publisher / Repository:
- Mind, Brain, and Education
- Date Published:
- Journal Name:
- Mind, Brain, and Education
- ISSN:
- 1751-2271
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1111/mbe.12421
