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' lefttoright 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 higherorder operator and the presence or absence of superfluous brackets: [a] bracketsleft (e.g., (5 * 4) + 2 + 3), [b] no bracketsleft (e.g., 5 * 4 + 2 + 3), [c] bracketscenter (e.g., 2 + (5 * 4) + 3), [d] no bracketscenter (e.g., 2 + 5 * 4 + 3), [e] bracketsright (e.g., 2 + 3 + (5 * 4)), and [f] no bracketsright (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 higherorder 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.
As students progress beyond arithmetic in middle school, they are challenged to learn to attend to the structure of math notation (
Aside from structure sense, the position of terms within a math expression may play a role in how students reason about math. Students may also struggle to resist the urge to solve math problems from left to right. Early middle school students have a strong tendency to adhere to the lefttoright principle when solving problems, which may lead them to overlook and violate the order of operations (
One potential way to support students’ structure sense and help them notice the order of operations when solving math problems is to increase the visual salience of important cues in an expression. Brackets are often used in mathematics to group numbers together, to emphasize, and/or to identify precedence. In some cases, brackets are necessary to indicate precedence (i.e., (a + b) * c + d) and in other times, brackets are used only for emphasis, or are
A number of previous studies have investigated how
Given these mixed results, it is important for researchers and educators to better understand how adding superfluous brackets to math notations could be used to help support student performance and learning. In addition to understanding superfluous brackets’ influences on outcomes, it is also important to better understand how brackets and HOO position within a math expression influence students’ performance to identify the situations where these brackets may help versus hinder students. The current study aims to advance our understanding of how operator position and superfluous brackets within a math expression can affect students’ problemsolving performance in an online learning environment.
The goals of our study are threefold. First, we aim to examine the isolated impact of HOO position on students’ performance on simple arithmetic problems in an online activity. Second, we aim to extend prior research by testing whether the presence of superfluous brackets can support student performance. Third, we assess if the presence of superfluous brackets moderates the impact of HOO position. Specifically, we examine how the presence of superfluous brackets and the position of HOOs (i.e., multiplication and division) within an expression independently and simultaneously influence student performance.
Learning math is naturally dependent on our perceptual processes; the way that we perceive incoming stimuli informs the way we think about it (
Although multiple visual features may have no bearing on the mathematical meaning of notation, they can direct individuals’ attention towards structures within the notation and act as perceptual grouping mechanisms, impacting learners’ performance on problem solving. For example, visual features that are proximal, as opposed to distal, to one another are more likely to be perceptually grouped together by our visual systems according to the Gestalt principles of perceptual grouping (
In addition to the spatial proximity of symbols, several other visual features can also serve as attentional cues in math notation. For example, strategic coloring can be used as a perceptual grouping mechanism to support students’ generation of problemsolving strategies (
In arithmetic, brackets can be used in several ways. First, within math expressions, brackets typically signify grouping in which the math content in the group takes precedence over its surroundings based on the order of operations (e.g., in “10 ÷ (2 + 3)”). Second, brackets can be used superfluously to visually highlight the structural elements of an expression without changing the mathematical meaning of the expression (
One explanation for why superfluous brackets could work as a visual grouping mechanism is that they create a common visual region within math expressions that draws students’ attention to a specific area of math notation (
Research on the use of superfluous brackets as a visual cue has been primarily conducted in classrooms (
The current study aims to provide additional empirical evidence about how superfluous brackets affect student performance with two important extensions of past work. First, this work explores the differential impacts of higherorder operator position on math performance. Second, much of the prior work has been conducted using paper and pencil tasks. Online learning and use of technology platforms in the math classroom have grown significantly and become more centric to education due to the COVID19 pandemic, motivating us to explore the impact of superfluous brackets in problem sets implemented in an authentic online educational technologybased learning environment. Thus, our study aims to provide insights on how superfluous brackets may act as visual and perceptual support for students while solving problems in an online learning environment.
In this study, we compare performance on simple arithmetic problems among fifth to seventhgrade students. Students completed math problems that were presented in one of six different ways varying the presence or absence of brackets and the position of the HOO. Based on previous findings on students’ weak structure sense, we hypothesize that when solving math problems related to the order of operations, students will be more likely to subscribe to the lefttoright tendency in computing. Thus, students will perform better when they see the HOO on the left, compared to when the operator is in the center or on the right side of the expression. In line with perceptual learning theory, we also hypothesize that superfluous brackets will act as a perceptual cue that primes students and draws their attention to the HOO; thus, students will perform better on problems
Does the position of a HOO (i.e., multiplication or division) impact student performance on simple arithmetic problems in an online homework assignment, as measured by student mastery speed and average response time?
Does the presence of superfluous brackets impact student performance, as measured by student mastery speed and average response time?
Is there an interaction between the effects of operator position and superfluous brackets on student performance, as measured by student mastery speed and average response time?
We received approval from our university’s ethics committee for this research project.
Additionally, we preregistered the study design and data analysis plan for this project on Open Science Framework (see
We recruited students by advertising this study to existing fifth to seventhgrade teacherusers of ASSISTments (
A total of 690 students from 24 middle school classrooms in the U.S. initially opened the assignment. Of those students, 19 students were immediately dropped from the sample because they did not complete the pretest and were therefore not assigned to a condition. An additional 71 students were dropped from the sample because they took an older and longer version of the pretest or had data not logged due to an error. A total of 600 students completed the threeitem baseline assessment, were randomly assigned to a condition, and were included in our preliminary analysis examining mastery. Of the 600 students, 46 students quit the assignment before completion, meaning they did not reach content “mastery”. These students were included in preliminary analyses then dropped from the sample for the primary analysis. We then checked the distribution of average response time and mastery speed to identify outliers. Of the 554 students who did reach “mastery”, 17 students had average response times well over five minutes per problem and nine additional students had mastery speeds that exceeded three standard deviations from the sample mean; these 26 students were dropped from the sample. These exclusions resulted in a final sample of 528 students for the primary analysis.
A post hoc power analysis in G*Power showed that a sample size of 528 students would provide 78.59% power to detect a smalltomedium effect size of
We created this randomized controlled trial as a problem set in ASSISTments, an online tutoring system with free K12 content that focuses on math (
Once students clicked the link to open the problem set, they completed a threeitem baseline assessment on simplifying orderofoperations expressions (
After completing the baseline assessment, students were randomly assigned to one of six conditions, described below. Within condition, students simplified orderofoperations expressions that were presented in a randomized order within an ASSISTments’ Skill Builder, where the goal was to “master” the content by answering three questions correctly in a row. In the Skill Builder, once students correctly answered three problems in a row, they were considered to have “mastered” the topic and received a message indicating that they completed the assignment.
We used a 3 (HOO position: left, center, or right)
Each condition varied in the placement of the HOO and the presence or absence of superfluous brackets in math expressions (
Condition Name  Structure  HOO Position  Presence of Brackets  Example 

BracketsLeft  *++  Left  Yes  (1 * 6) + 2 + 5 
No BracketsLeft  *++  Left  No  1 * 6 + 2 + 5 
BracketsCenter  +*+  Center  Yes  2 + (1 * 6) + 5 
No BracketsCenter  +*+  Center  No  2 + 1 * 6 + 5 
BracketsRight  ++*  Right  Yes  2 + 5 + (1 * 6) 
No BracketsRight  ++*  Right  No  2 + 5 + 1 * 6 
The problems used in this study were based on the Common Core Standards for fifth grade content on “Operations and Algebraic Thinking” (
ASSISTments recorded whether each student completed the pretest as a binary measure and calculated their performance on the pretest as the number of correct answers across the three items.
ASSISTments provided a binary measure of whether students reached “mastery” as defined by correctly answering three problems in a row. Students may have dropped out of the assignment before reaching mastery. This measure was used as the outcome in the preliminary analysis to check attrition rates by condition.
For each student that achieved mastery, the system recorded their assignment mastery speed, which was measured as the count of problems that a student saw (after the pretest) to successfully complete three problems in a row. For example,
For each experimental problem in the ASSISTments Skill Builder, the system recorded the time from which the problem window opened until the student submitted the correct answer to the problem. Students’ response time for each problem was summed and divided by the number of problems that they solved to calculate each student’s average response time per problem. Previous studies have explored response time as an outcome variable of student performance during math problem solving (
Prior to conducting primary analyses, we checked for differential mastery rates across conditions to see whether one condition may have been significantly more challenging for students to the point of not completing the assignment.
We then conducted a logistic regression to examine whether students were more likely to have mastered the assignment when assigned to a condition with superfluous brackets and/or a particular operator position (left, center, or right). The logistic regression model, controlling for pretest, was statistically significant, χ^{2}(5, 593) = 26.29,
Condition  Estimate  Wald Statistic  95% CI 


(Intercept)  2.31**  0.57  10.03  4.04  16.32  1.19  3.42 
Pretest correct  0.21  0.16  1.23  1.31  1.71  0.10  0.52 
Brackets  0.91  0.75  2.49  1.23  1.50  0.55  2.38 
Position Center  0.55  0.57  0.58  0.97  0.93  1.67  0.57 
Position Right  1.30**  0.53  0.27  2.45  5.99  2.33  0.26 
Brackets * Position Center  0.15  0.96  1.17  0.16  0.03  1.74  2.04 
Brackets * PositionRight  0.57  0.91  1.76  0.62  0.39  1.12  2.36 
*
To answer our first and second research questions, we investigated how the position of the HOO and the presence of superfluous brackets may have separately impacted students’ performance among those who completed the problem set. Using the analytic sample of 528 students, we compared differences across conditions in students’ mastery speed and average response time as two indicators of student performance. Specifically, we conducted a Poisson regression to predict mastery speed and a linear regression to predict average response time. We used students’ pretest scores as a covariate to control for prior knowledge. We chose to conduct a Poisson regression for mastery speed since the variable represents count data. We did not use a multilevel model accounting for the nesting of students in teachers (
For each analysis, we analyzed the main effect of operator position (left and right compared to the center), the main effect of superfluous brackets (superfluous brackets vs. no brackets), and two Operator Position * Presence of Bracket interactions. The main effect of operator position (left and right compared to center) revealed whether and how the position of the HOO (i.e., multiplication or division) in math expressions impacted student performance on simple arithmetic. The main effect for the presence of brackets (superfluous brackets vs. no brackets) informed us whether and how superfluous brackets impacted student performance. Lastly, the interactions indicated whether there was an interaction between the impact of operator position (left and right compared to the center) and the presence of superfluous brackets on student performance.
All students who were included in the primary analyses achieved “mastery” (i.e., answering three problems correctly in a row) at some point in the study assignment (
Condition  Average Pretest Performance ( 
Average Mastery Speed ( 
Average Response Time ( 


Overall  528  2.15 (0.93)  4.49 (2.66)  38.57 (34.31) 
No BracketsLeft  78  2.28 (0.91)  3.68 (1.55)  34.29 (26.23) 
BracketsLeft  109  2.15 (0.93)  3.92 (2.13)  36.15 (30.90) 
No BracketsCenter  83  2.08 (0.90)  5.17 (3.17)  35.04 (29.37) 
BracketsCenter  99  2.18 (0.90)  3.66 (1.53)  40.13 (37.95) 
No BracketsRight  77  2.20 (1.01)  5.60 (3.64)  46.84 (41.29) 
BracketsRight  82  2.02 (0.96)  5.29 (2.82)  39.77 (37.58) 
To first examine the effects of brackets and HOO position on students’ mastery speed, we conducted a Poisson regression controlling for students' pretest performance, with the center position as the reference group (
Predictor  Model 1 
Model 2 


β  β  
Intercept  1.74***  0.06  29.02  1.83***  0.06  28.83 
Pretest  0.10***  0.02  4.54  0.09***  0.02  4.28 
Brackets  0.12**  0.04  2.99  0.34***  0.07  4.77 
PositionLeft  0.11*  0.05  2.22  0.32***  0.08  4.29 
PositionRight  0.24***  0.05  4.90  0.11  0.07  1.66 
Left*Brackets  0.26**  0.10  2.70  
Right*Brackets  0.40***  0.10  3.87 
The results revealed a significant effect of superfluous brackets presence on students' mastery speed,
Second, the results revealed a significant effect of HOO positions on students' mastery speed. Specifically, the mastery speed of students who solved expressions with the HOO on the left was significantly lower (i.e., quicker) compared to those who solved problems with the operator in the center (
Next, to examine whether the effects varied by condition, we added two interaction terms with bracket and position to the model (
The second interaction comparing brackets and no brackets and the center and right HOO position on mastery speed was also statistically significant (
Additionally, after including the two interaction terms pretest performance remained a significant predictor of mastery speed (
Next, we conducted a regression predicting average response time and controlling for students' pretest performance (
First, there was no significant effect of HOO position on students' average response time, both
Predictor  Model 3 
Model 4 


β  β  
Intercept  238.80  789.70  0.30  601.50  860.40  0.70 
Pretest  69.60  276.20  0.25  46.20  277.10  0.17 
Brackets  379.50  518.30  0.73  387.80  879.40  0.44 
PositionLeft  327.80  616.80  0.53  116.00  932.90  0.12 
PositionRight  398.50  641.90  0.62  440.10  934.90  0.47 
Left*Brackets  818.00  1289.20  0.66  
Right*Brackets  1590.40  1289.20  1.23 
The goal of this study was to explore whether the position of higherorder operator and the presence of superfluous brackets within math expressions may separately and simultaneously impact student performance on orderofoperations problems in an online tutoring system. Three notable findings emerged from this study. First, students were more likely to not complete (i.e., “not master”) the assignment if they were in a condition that had the HOO on the right, suggesting that this presentation of arithmetic expressions may have posed more challenges to students during problem solving than the other position conditions. Second, main effects show that, on average, students who were assigned conditions where the HOO was in the center of the expressions had slower mastery speeds than when it was on the left, but quicker mastery speeds than students who solved expressions with the HOO on the right. Further, students who saw expressions with brackets tended to have quicker mastery speeds than those who did not see brackets. Third, interaction effects revealed that these main effects were largely driven by the presence of superfluous brackets on the center position which moderated the impacts of HOO position on mastery speed. Among students in the two conditions with the HOO in the center, students who solved expressions with superfluous brackets achieved mastery more quickly (comparable to students who were in the left position conditions) than students who solved expressions without brackets (comparable to the right position conditions).
Based on previous work on perceptual cues within ASSISTments (
In our preregistration, we predicted that the bracketsleft condition would be the easiest for students to solve arithmetic problems, and that the no bracketsright condition would be most difficult, as indicated by students’ higher (i.e., slower) mastery speeds. The logistic regression showing that students assigned to the right position condition were significantly less likely to complete the assignment provides support for this hypothesis. We interpret this finding to mean that the position of the HOO may be very influential in how students reason about math; in particular, solving expressions may seem more difficult when the position of the HOO is on the right side of the expression. One plausible explanation for students dropping out more often in this condition may be that they became frustrated by the difficulty of the assignment or getting more problems incorrect.
Since the primary analyses conducted only included students who did achieve mastery in the study activity, the findings on mastery speed and average response time need to be interpreted with the context that there was differential attrition between our six conditions. However, we contend that by dropping students who did not achieve mastery from the analytic sample, the findings may actually present a more conservative estimate of how the position and presence of perceptual cues within arithmetic expressions impact student performance. Future research should explore itemlevel data to better understand factors such as time on task, accuracy of initial responses, and students’ behaviors while problem solving to unpack the mechanisms behind why students might have dropped out of the assignment before reaching mastery.
We predicted that solving math expressions with the HOO on the left and expressions with superfluous brackets would lead to (a) quicker mastery speeds and (b) quicker response times than solving expressions with the operator in the center or on the right and expressions without superfluous brackets. The main effect result supports the first hypothesis: seeing math expressions with the HOO on the left was, on average, related to quicker mastery speeds (i.e., higher accuracy rates) than the center. The presence of superfluous brackets also independently impacted students’ mastery speed during simple arithmetic. However, these variables did not significantly predict response times. Students had comparable response times across conditions, suggesting that neither operator condition nor superfluous brackets impacted students’ problemsolving speed.
The main effect finding that HOO position impacted student performance aligned with previous research showing that students have a lefttoright tendency during math problem solving (
This study advances research on the roles of perceptual factors in math notation by isolating the effect of higheroperator position on students’ performance. While the effects of superfluous brackets and HOO position on student performance are much smaller than we anticipated based on prior work (
The most notable finding is that the presence of superfluous brackets moderated the effect of HOO position, specifically when students solved math expressions with the operator in the center. While students, on average, demonstrated the quickest mastery speeds in the HOOleft conditions, students in the center position with brackets condition performed comparably well to the two HOOleft conditions, suggesting that superfluous brackets may increase students' accuracy on problems when the HOO is in the center. Conversely, students in the no bracketscenter condition had comparable mastery speeds to students in the HOOright conditions, suggesting that the absence of brackets with the HOO in the center posed challenges to students.
One possible interpretation of these results is that position of HOO might be a type of visual and perceptual feature that works congruously with the lefttoright calculating tendency to impact students’ problem solving: leftsided HOOs facilitate higher accuracy, while center and rightsided HOO (without brackets) elicit more errors. Students were highest performing when they were able to apply a lefttoright solving strategy. In cases when they could not (i.e., center and right conditions), their performance (mastery speed) dropped. The exception to this trend was for students who saw brackets in the center position, suggesting that in cases when students could not apply a lefttoright solving strategy, the superfluous brackets may have naturally and visually grouped the numbers to prevent a left to right calculation, or shifted their attention and helped them identify the groupings to apply the first steps for problem solving. The brackets could have prevented students from compulsively performing lefttoright calculations by visually breaking up the structure of the math expressions. Specifically, brackets around the center terms may be the most impactful because, in that position, it breaks up the structures into three distinct parts, where the brackets naturally block the flow of left to right computations (i.e., 1+ (7 * 5) – 4).
When solving orderofoperations problems, students seem to rely on both HOO position and superfluous brackets presented in the expressions; however, this work suggests that operator position, particularly when placed in the center, may play a strong attentionguiding role. Aligned with our findings, HOO position seemed to be a salient factor that impacted students’ performance when calculating left to right. Superfluous brackets, while significant, specifically seemed to impact students’ performance when the operator position was in the center. Having brackets on the left or right did not seem to impact performance. Thus, if students relied on left to right calculations, they would use more inaccurate problem solving when the operators were not located on the left side of the expressions. However, when the operators were in the center of the expressions, the presence of the brackets could have helped students attend the HOO first, breaking up the expression into chunks, and helping to facilitate more accurate problem solving.
Another alternative explanation of this finding is that, for students who have not yet conceptually mastered the order of operations, they may have memorized a simpler procedural rule that parentheses must be computed before other operations (i.e., PEMDAS). Therefore, it is plausible that the brackets may not be serving solely as a visual perceptual cue, but rather students could be relying on the PEMDAS (or BEDMAS) rule that parenthesis must be calculated first as a foolproof way for students to perform simple procedures without understanding. However, this explanation is challenged by the finding that, in the case of HOOleft and HOOright expressions, the presence of brackets did not help students perform more accurately.
The findings from this study demonstrate how operator position and superfluous brackets may impact students’ participation in, and performance on, orderofoperations problems in an online homework environment. Importantly, this work provides considerations for designing standalone assignments: using problems with leftmost HOO may increase students’ performance while using rightmost HOOs in expressions may decrease students’ likelihood of finishing the assignment. We posit that variations in perceptual features like HOO position and superfluous brackets (particularly in the center position) may help students quickly infer which operations to address in a given expression, similar to creating semantic alignment in the structure of word problems (
Several decades of research on desirable difficulties has shown that learning conditions which are more difficult in the moment and decrease individuals’ performance improve longterm learning and retention (
Taken together, we consider the possibility that variations in expression structures and perceptual cues that lend themselves to higher problemsolving performance (i.e., leftmost HOO, superfluous brackets) may not be the same features that lead to longterm learning, retention, and flexibility or efficiency. Instead, more difficult mathematical structures or the presence or absence of perceptual cues that
This study had multiple limitations. First, the differential attrition (i.e., students who achieved mastery was different by condition) can be seen as problematic as the attrition was not random. However, although more students were dropped from the analytic sample in the no bracketsright condition, the findings of this study are notable, especially given that the negative effects of no bracketsright were still present, even when dropping those who did not achieve mastery.
Second, given the current data, it is difficult to specify who is impacted by superfluous brackets and HOO position and when. While we intentionally recruited teachers of incoming fifth to seventhgrade students, the online platform used to deploy the study does not permit collecting any individual participants’ demographic information (e.g., gender, race, age, grade, inperson vs. remote learning status) due to privacy concerns. While we acknowledge that the lack of demographic information is not ideal and limits our ability to understand individual differences, it is a tradeoff for using open educational platforms like ASSISTments for conducting educational research at scale. Further, teachers were aware of the content targeted in the study and may have assigned the content to students in other grades if the content was appropriate for their knowledge levels (e.g., in advanced lower grades or remedial higher grades). Additionally, conducting this study in an online platform provided ecological validity by testing how these experimental manipulations impacted students’ performance on an online class or homework assignment.
Third, the focal dependent variable in this study, mastery speed, is specific to the type of problem set built in ASSISTments. As a result, these findings are not directly generalizable to other online tutoring platforms or contexts. However, estimating treatment effects on students’ mastery speed provides suggestions for another measure for accuracy and a unique analysis of how perceptual cues may impact students’ performance at a granular level.
Looking ahead, future research may consider collecting more demographic information from participants to control for individual differences among students that may affect their susceptibility to perceptual cues. This approach would help uncover when, and for whom, perceptual scaffolding may be the most effective. Further, this approach may be most effective when paired with other theoretical perspectives that explain cognitive and developmental factors of math performance and learning. Additionally, future research may consider replicating this work with students in classrooms or using additional methodologies to tease out the mechanisms and practical significance of presenting and implementing perceptual supports to facilitate enhanced problem solving in everyday math learning. Current work utilizing eye tracking is underway by our team to identify whether students who are presented with brackets do in fact look at the brackets and HOO first or fixate longer within expressions in various positions. This approach can help confirm or challenge more procedural or perceptual/attentional explanations for why students demonstrate improved performance with brackets in the center position.
Broadly, this study adds to the growing body of literature on the importance of perceptual grouping, and variations in structures for developing structure sense and reasoning in mathematics (
These findings support that adding visual information in notation, such as manipulating the position of HOOs and adding superfluous brackets, may act as perceptual supports that can positively influence students’ reasoning and performance in math. Our findings also highlight the importance of helping students develop structure sense in math learning and identifying students who may be underperforming due to a lefttoright solving tendency. Regarding the implications of this work, instructors and educators may be able to intentionally make changes to the visual presentation of notation, such as adding superfluous brackets, to guide students' attention to relevant features of math problems, lead them to correct solution strategies, and enhance their performance on tasks. Since students tend to solve math problems from left to right, which might lead to inaccurate solutions, it may be useful for teachers to utilize superfluous brackets in the center position to improve students' weak structure sense in arithmetic by helping inhibit their compulsion to compute from left to right. A second possible way to support students' development of structure sense may be to introduce them to superfluous brackets early and slowly in instructional practice. As students first learn how to solve problems with HOOs in an expression (e.g., 2 + (1 * 6) + 3), guiding them to notice the higher order operator by using superfluous brackets or instructing them to put the brackets around the multiplication or division to break up structure and signs might help students gradually build an understanding that these brackets serve as a visual cue to highlight important elements of math problems. From frequent exposure to superfluous brackets as a perceptual cue, students might better understand the importance of finding significant elements within a math expression and comprehending the overall structure of a problem prior to solving. As students become more comfortable with simplifying orderofoperation expressions with different structures, the need for those supports may become less necessary.
The current study applies and integrates work from cognitive science, math education, and educational technology to explore the impact of superfluous brackets and HOO position on students’ math performance simplifying expressions. We found that, generally, students tend to demonstrate the highest performance on expressions with the HOO on the left but performance decreases as the HOO moves to the right. However, we found that adding superfluous brackets in the center position supports learning, while having brackets on the right or left does not provide additional impacts for students. Overall, this study highlights the importance of providing variation in subtle structural and perceptual variations in math notation, showing that the position and presence of perceptual grouping structures, such as superfluous brackets, impacts students’ completion and performance on online assignments.
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program to Avery Harrison Closser under Grant No. 1645629. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
We would like to thank the teachers and students for their participation; members of the Math, Abstraction, Play, Learning, and Embodiment (MAPLE) Lab for their help, and Neil Heffernan for his support. We thank the ASSISTments Foundation and Worcester Polytechnic Institute for hosting the
All procedures followed were in accordance with the ethical standards of the Institutional Review Board (IRB) at Worcester Polytechnic Institute to be minimal risk.
For this article, a data set is freely available (
The Supplementary Materials contain the data set, code, and preregistration information for this study (for access see
The authors have declared that no competing interests exist.
This research was conducted as a partial fulfillment of the Master’s thesis requirement for Vy Ngo at Worcester Polytechnic Institute in the Learning Sciences and Technologies program (