This study investigated the effects of 1) proximal grouping of numbers, 2) problemsolving goals to make 100, and 3) prior knowledge on students’ initial solution strategies in an interactive online mathematics game. In this game, students transformed an initial expression into a perceptually different but mathematically equivalent goal state. We recorded students’ solution strategies and focused on the productivity of their first steps—whether their initial action led them closer to the goal. We analyzed log data within the game from 227 middleschool students solving four addition problems and four multiplication problems consisting of a total of 1,816 problemlevel data points. Logistic regression modeling showed that students were more likely to use productive initial solution strategies to solve addition and multiplication problems when 1) proximity supported number grouping, 2) 100 was the problemsolving goal, and 3) students had higher prior knowledge in mathematics. Furthermore, when problemsolving goals were non100s, students with lower prior knowledge were less likely to use productive initial solution strategies than students with higher prior knowledge. The findings of the study demonstrated that perceptual and number features influenced students’ initial solution strategies, and the effect of number features on initial solution strategies varied by students’ prior knowledge. Results yield important implications for designing instructional activities that support mathematics learning and problemsolving.
Mathematics problems can often be solved by using several different strategies. As an example, students can solve the equation 2(3 +
Substantial empirical work suggests that mathematical reasoning is grounded in perceptual and embodied processes (
Here, we test a similar concept but operationalize proximal grouping as the location of numbers within the problem rather than the physical spacing of numbers and operators within the problem. For instance, in 3 + 5 + 4 = 8 + 4, 3 and 5 are adjacent to each other, whereas 3 and 5 are not adjacent to each other in the equation 3 + 4 + 5 = 8 + 4. Drawing on the Gestalt laws of perceptual organization (
The mathematical system in the U.S. has an underlying base10 structure. Base10 structure knowledge is a key predictor of students’ mathematics performance (
State mathematics standards tend to focus on students’ facility in making 10 or 100. In the U.S., one Kindergarten (age 5 or 6 years) standard is “For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation” (
Given this focus on 10s and multiples of 10s, we anticipate that 100 is a “friendly number” for which students can create combinations more easily as compared to other numbers. Thus, we hypothesize that students will be more likely to make productive transformations if the problem goal is to make 100 compared to problems in which the goal is to make numbers that are not 100 (e.g., 98; 102).
To tailor strategies to a mathematical equation flexibly and adaptively, students need to have mathematical content knowledge—understanding the concepts and procedures for potential solutions (
In addition to the main effects of problem structures and prior knowledge, students’ prior knowledge may moderate the effects of problem structures (proximal grouping of numbers, problemsolving goals to make 100) on students’ solution strategies. One study (
Here, we examine the influence of students’ prior knowledge on the association between problem structures and initial solution strategies. We hypothesize that there may be a significant interaction effect between students’ prior knowledge and the problem structures. Specifically, students with lower prior knowledge may benefit from the support in problem structures; they may use more productive initial solution strategies when the numbers to be added are adjacent to each other, and the goal is to make 100. Students with higher prior knowledge may use productive solution strategies regardless of whether the problem structures support these strategies.
The goal of this study is to understand the ways in which specific factors—the proximal grouping of numbers, problemsolving goals of making 100, and prior knowledge—affect students’ initial solution strategies in an interactive online mathematics game. In this game, students were presented with a series of problems consisting of a starting expression and a mathematically equivalent but perceptually different goal state; the objective was to transform the expression into the goal state. We focus on three main research questions:
Do proximal grouping, problemsolving goals of making 100, and prior knowledge uniquely influence students’ initial solution strategies?
Do these factors interact to affect students’ initial solution strategies?
Interaction between proximal grouping and problemsolving goals of making 100
Interactions between proximal grouping and prior knowledge as well as problemsolving goals of making 100 and prior knowledge
Threeway interaction among proximal grouping, problemsolving goals of making 100, and prior knowledge
Are these effects consistent across addition and multiplication problems?
First, we hypothesize that students may use a productive initial solution strategy when (a) the numbers to be combined are adjacent to each other, (b) the problemsolving goal is to make 100, and (c) students have higher prior knowledge. Second, students may be more likely to use a productive initial solution strategy when the numbers to be combined are adjacent to each other
This study used data from a larger randomized controlled trial conducted in six middle schools located in the southeastern U.S. In the larger study, we examined the efficacy of an interactive online mathematics game compared to an online problem set (see details in
In the final sample of 227 students (56% male, 44% female; age 11 to 13 years), most students (96%) were in sixth grade, and the rest (4%) were in seventh grade. In terms of instruction level, 85% of students were in advanced mathematics classes (i.e., accelerated mathematics programs) that were designed for students who excelled at mathematics and implemented more challenging course materials, 6% were in regular onlevel classes, and 9% were in support classes that were designed for students who needed additional help in learning mathematics. Although only 4% of the sample were seventhgrade students, these students were included in analyses because they were enrolled in seventhgrade support mathematics classes that were comparable to sixthgrade onlevel mathematics classes. The race/ethnicity of the final sample was 53% Asian, 36% White, 4% Hispanic, and 7% other races/ethnicities. This sample consisted of a larger percentage of Asian students in comparison with the districtwide population (52% White, 25% Asian, 15% Hispanic, 8% other races/ethnicities).
In the present study, we analyzed log data collected in an interactive online mathematics game in which students explore algebraic notations by performing mouse or touchbased gestures. In the game, mathematical symbols are turned into virtual objects that students can touch and move according to mathematical principles.
For each problem in the game, students are presented with two mathematical expressions—a start state, which is a transformable expression, and a goal state, which is perceptually different but mathematically equivalent to the start state (see
Within the game, we designed and embedded two quartets of problems—four addition problems (P10, P7, P13, P14) and four multiplication problems (P24, P32, P30, P26), that each varied on the proximal grouping of numbers in the start state and making 100 in the goal state (see
Making 100  Proximal Grouping 


Yes (1)  No (0)  
Addition  
Yes (1)  P10: (S) 44+56+ 
P7: (S) 11+55+ 
(G) 100+ 
(G) 100+ 

No (0)  P13: (S) 15+87+ 
P14: (S) 47+33+ 
(G) 102+ 
(G) 99+ 

Multiplication  
Yes (1)  P24: (S) 25*4* 
P32: (S) 10*20* 
(G) 100* 
(G) 

No (0)  P30: (S) 8*12* 
P26: (S) 4*6* 
(G) 96* 
(G) 96*96* 
For instance, for Problem 10 (Start state [S]: 44 + 56 +
Students’ prior algebra knowledge was measured with 11 items selected from two previously validated measures (
Initial solution strategies were measured by whether or not students made a productive mathematical transformation (i.e., change of expression) towards the goal state in their first transformation. All mathematical transformations made by the students were automatically logged in the database. We extracted and used students’ first mathematical transformation (i.e., “first step”) to measure the productivity of their initial solution strategies. We focus on students’ first step because it provides insights into students’ initial reaction to the problem, has impacts on their subsequent steps in reaching the goal state, and consequently influences their overall strategy efficiency on the problem.
For example, in Problem 14 (S: 47 + 33 +
Start state  Goal state  Productive first steps  Nonproductive first steps 

47+33+ 
99+ 
99+33+ 
80+ 
^{a}Transformations that involved commuting (i.e., moving numbers to be added adjacent to each other) were considered productive in bringing the student closer to the goal state.
We performed hierarchical binary logistic regression modeling because 1) the outcome variable (productivity of initial solution strategies) was binary, and 2) problemlevel data were nested within studentlevel data. First, unconditional models were estimated to examine the proportion of the variance explained in the outcome variable (i.e., the productivity of students’ initial solution strategies) between the schools, classrooms, and students. These intraclass correlation coefficients (ICCs) ranged between 0.00 and 0.03. Although the ICC values were low (less than 0.05), we conducted hierarchical binary logistic regression modeling because both Level1 predictors (proximal grouping, making 100) and a Level2 predictor (prior knowledge) were included in the models, and each student completed the same eight problems in the study (
The number of problemlevel data points included in the analyses was 1,816 problems, 908 for addition and 908 for multiplication. For students who tried the problems more than once, we used the data from their first attempts. We used R studio with the lme4 package for further analyses (
Before performing data analyses, we computed frequencies of students’ solution strategies for each problem (see
Problem  Problem structure  Number (%) of students with a productive first step  Number (%) of students with a nonproductive first step 

Addition  
Problem 10 
Proximal Grouping, 
218 (96.0%)  9 (4.0%) 
Problem 7 
NonProximal Grouping, 
210 (92.5%)  17 (7.5%) 
Problem 13 
Proximal Grouping, 
216 (95.2%)  11 (4.8%) 
Problem 14 
NonProximal Grouping, 
139 (61.2%)  88 (38.8%) 
Multiplication  
Problem 24 
Proximal Grouping, 
209 (92.1%)  18 (7.9%) 
Problem 32 
NonProximal Grouping, 
191 (84.1%)  36 (15.9%) 
Problem 30 
Proximal Grouping, 
213 (93.8%)  14 (6.2%) 
Problem 26 
NonProximal Grouping, 
135 (59.5%)  92 (40.5%) 
First, we tested whether students were more likely to make productive first steps when the numbers to be added were adjacent to each other, when the problemsolving goal was to make 100, and when students had higher prior knowledge. Specifically, we predicted the probability of making a productive first step as a function of two problemlevel predictors (Level1; proximal grouping and making 100) and one studentlevel predictor (Level2; prior knowledge) for addition problems (see
Variable  

Fixed effects  
Intercept  2.45*** (0.19)  11.59 
Proximal grouping  2.03*** (0.27)  7.61 
Making 100  1.69*** (0.25)  5.41 
Prior knowledge  0.12** (0.04)  1.12 
Random effects  
0.12 (0.35)  –  
Loglikelihood  295.5  – 
**
As shown in
We tested an intralevel interaction of proximal grouping by making 100 (Model 1.1), as well as two crosslevel interactions, proximal grouping by prior knowledge (Model 1.2) and making 100 by prior knowledge (Model 1.3;
Variable  2way Interactions 
3way Interaction 


Model 1.1 
Model 1.2 
Model 1.3 
Model 1.4 

Fixed effects  
Intercept  2.42*** (0.19)  11.22  2.45*** (0.19)  11.58  2.47*** (0.20)  11.87  2.45*** (0.20)  11.57 
Proximal Grouping  1.67*** (0.28)  5.29  1.99*** (0.27)  7.31  2.07*** (0.28)  7.91  1.69*** (0.29)  5.43 
Making 100  1.19*** (0.28)  3.30  1.70*** (0.26)  5.50  1.65*** (0.25)  5.21  1.19*** (0.29)  3.28 
Prior Knowledge  0.12** (0.05)  1.13  0.08 (0.05)  1.08  0.07 (0.05)  1.07  0.05 (0.06)  1.05 
Proximal Grouping × Making 100  1.96*** (0.56)  0.14  –  –  –  –  1.93*** (0.57)  0.15 
Proximal Grouping × Prior Knowledge  –  –  0.14 (0.10)  0.87  –  –  0.11 (0.11)  0.89 
Making 100 × Prior Knowledge  –  –  –  –  0.22* (0.10)  0.80  0.23* (0.11)  0.80 
Proximal Grouping × Making 100 × Prior Knowledge  –  –  –  –  –  –  0.02 (0.22)  0.98 
Random effects  
0.28 (0.53)  –  0.15 (0.38)  –  0.17 (0.41)  –  0.39 (0.63)  –  
Log likelihood  289.6  –  294.5  –  292.8  –  286.0  – 
*
Next, we added two crosslevel interaction terms with prior knowledge to the model, proximal grouping by prior knowledge (Model 1.2) and making 100 by prior knowledge (Model 1.3). Except for prior knowledge, all main effects were statistically significant, whereas the interaction between proximal grouping and prior knowledge was not (
In contrast, the interaction between making 100 and prior knowledge was statistically significant (
Next, we repeated the analyses for multiplication problems to test whether these effects were consistent across operations. Specifically, we tested the hypotheses that students were more likely to make productive first steps when the numbers to be combined were adjacent to each other, the problemsolving goal was to make 100, and when students had higher prior knowledge (see
Variable  

Fixed effects  
Intercept  1.97*** (0.15)  7.19 
Proximal grouping  1.79*** (0.23)  6.00 
Making 100  0.95*** (0.20)  2.59 
Prior knowledge  0.17*** (0.04)  1.19 
Random effects  
0.29 (0.54)  –  
Loglikelihood  363.6 
***
As shown in
For Model 2.1, the intralevel interaction term (proximal grouping by making 100) was added to the model (see
Variable  2way Interactions 
3way Interaction 


Model 2.1 
Model 2.2 
Model 2.3 
Model 2.4 

Fixed effects  
Intercept  2.00*** (0.15)  7.36  1.99*** (0.13)  7.29  1.98*** (0.15)  7.25  2.03*** (0.16)  7.63 
Proximal Grouping  1.69*** (0.23)  5.41  1.83*** (0.24)  6.26  1.82*** (0.23)  6.19  1.76*** (0.25)  5.82 
Making 100  0.58* (0.23)  1.78  0.95*** (0.20)  2.58  0.87*** (0.20)  2.40  0.54* (0.24)  1.71 
Prior Knowledge  0.18*** (0.04)  1.20  0.19*** (0.05)  1.21  0.16*** (0.04)  1.17  0.18*** (0.05)  1.20 
Proximal Grouping × Making 100  1.71*** (0.45)  0.18  –  –  –  –  1.68*** (0.49)  0.19 
Proximal Grouping × Prior Knowledge  –  –  0.05 (0.09)  1.06  –  –  0.07 (0.09)  1.07 
Making 100 × Prior Knowledge  –  –  –  –  0.17* (0.03)  0.84  0.16 (0.09)  0.86 
Proximal Grouping × Making 100 × Prior Knowledge  –  –  –  –  –  –  0.28 (0.18)  1.32 
Random effects  
0.40 (0.63)  –  0.28 (0.53)  –  0.32 (0.57)  –  0.50 (0.71)  –  
Loglikelihood  356.3  –  363.4  –  361.3  –  351.6  – 
*
Next, we added two crosslevel interaction terms with prior knowledge to the model (Model 2.2 and 2.3 in
In contrast, the interaction between making 100 and prior knowledge was statistically significant (
This study examined the effects of problem structures (proximal grouping of numbers and problemsolving goals to make 100) as well as students’ prior knowledge on middleschool students’ initial solution strategies within an online interactive mathematics game. The results indicated that proximal grouping of numbers, problemsolving goals to make 100, and students’ prior knowledge were significantly associated with students’ initial solution strategies. The findings of the current study were consistent with those of other studies which have shown that problem structures (
First, for both addition and multiplication problems, as we hypothesized, students were more likely to make a productive mathematical transformation when the numbers to be combined were proximal, the problemsolving goal was to make 100 (e.g., transforming 44 + 56 +
Second, we tested the hypothesis that there might be significant interaction effects among the proximal grouping of numbers, problemsolving goals to make 100, and prior knowledge on students’ initial strategies for solving addition problems. The results indicated significant interactions between “proximal grouping and making 100” and “making 100 and prior knowledge.” Specifically, students were less likely to use productive initial solution strategies when proximal grouping and problemsolving goals of making 100 were not present in the problem (e.g., transforming 47 + 33 +
Further, students with lower prior knowledge were less likely to use productive solution strategies when the problemsolving goals were nonfriendly numbers (e.g., 98, 101) compared to their peers with higher prior knowledge. Whereas students with higher prior knowledge may be fluent in solving problems involving friendly and nonfriendly numbers, students with lower prior knowledge may be less fluent in solving problems with nonfriendly, less practiced numbers. Considering that curricular activities often focus on addition combinations to reach friendly numbers like 100 (
Although the interaction between proximal grouping and prior knowledge was not statistically significant, the results showed that the problems that involved nonproximal grouping (e.g., transforming 47 + 33 +
Lastly, we examined if these effects extended to a more advanced topic by testing them in multiplication problems. Similar to addition problems, students were more likely to make a productive mathematical transformation for multiplication problems when the numbers to be combined were proximal, the problemsolving goal was to make 100, and when the students had higher levels of prior knowledge. Moreover, there were significant interaction effects between “proximal grouping and making 100” and “prior knowledge and making 100” in multiplication problems. Specifically, students with lower prior knowledge were less likely to make a productive first step when the number goals did not support efficient solution strategies (e.g., transforming 4 × 6 ×
Together, these results provide some support for the notion that novice students are less likely to notice and leverage problem structures in mathematical problem solving (
There were at least three limitations in this study. First, our measure of initial productive problem solving was only based on students’ first mathematical transformation. Future studies could investigate these effects with other measures of solution strategies (e.g., number of steps made, the sequence of students’ transformations).
Second, we analyzed only students’ decisions in the mathematical transformations they made, without considering the time they took to implement those transformations or other measures of mathematical skills (e.g., arithmetic skills). Another avenue for future research would be an examination of the relations between problem structure and other student behaviors (e.g., pause time before solving) or other cognitive skills that are related to mathematics performance (e.g., executive function skills, spatial reasoning skills) and their tradeoff with strategy productivity.
Third, our sample included a majority of Asian and White students as well as students in advanced mathematics classes, and thus was not representative of the U.S. population as a whole. However, even with this fairly homogenous, highperforming sample, we observed the effects of problem structures and prior knowledge. Future studies should examine the ways in which perceptual and conceptual features of problems may influence students’ solution strategies across a broader sample.
In sum, the findings of the current study demonstrate that students’ initial solution strategies vary by perceptual and number features of the problems as well as students’ prior knowledge. These results suggest that it may be helpful to teach students to notice important patterns in problem structures and build upon their familiarity with 100 to use it as an anchor for decomposition in multidigit problem solving, which may promote the use of more efficient solution strategies. Overall, these results extend past work demonstrating the effects of perceptual and number features on students’ problemsolving strategies from experimental settings into a digital, authentic learning context.
The research reported here was supported by the Institute of Education Sciences, U.S. Department of Education, through Grant R305A180401 to Worcester Polytechnic Institute. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education.
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 work, members of the Learning and Development Lab for their assistance in coding strategies, Erik Weitnauer and members of Graspable Math Inc. for programming and data support, and Neil Heffernan and the ASSISTments Team for their support.
The first three authors do not have competing interests. Erin Ottmar was a designer and a developer of From Here to There! and owns a 10% equity stake in Graspable Inc. This has been disclosed to WPI’s Conflict Management Committee, and a conflict management plan has been implemented.