Sixty (35 girls) ninth graders were assessed on measures of algebraic reasoning and usage of visual and symbolic representations (with a prompt for visual use) to solve equations and inequalities. The study grouped visual representations into two categories: arithmeticvisual, which entailed the use of realworld objects to represent specific values of variables, and algebraicvisual, which involved formal representations like the number line and the coordinate plane. Symbolic representations, on the other hand, encompassed the use of standard algorithms to solve equations, such as changing the place of terms in an equation. The results reveal that the use of algebraic visuals, as opposed to arithmetic visuals, was associated with enhanced algebraic reasoning. Further, although the students initially relied on standard algorithms to explain equations and inequalities, they could produce accurate algebraicvisual representations when prompted. These findings suggest that students have multiple representations of equations and inequalities but only express visual representations when asked to do so. In keeping with the general relationship between visuospatial abilities and mathematics, selfgenerated algebraicvisual representations partially mediated the relation between overall mathematics achievement and algebraic reasoning.
Competence with algebra is the gateway to advanced mathematics and provides the foundation for work in science, technology, engineering, and mathematics (STEM) disciplines and many other occupations (
One of the recurring themes in intervention studies is the use of manipulatives and object pictures to represent quantity and variables, followed by a gradual transition to standard algebraic form (e.g.,
That is a substantial limitation because the use of studentgenerated representations in mathematics is crucial for tracking students’ progress and supporting their mathematical development (
Consequently, we studied the relationship between students’ algebraic reasoning and their selfgenerated visual and symbolic representations of equations and inequalities following their initial problemsolving attempts. We focused on studentgenerated visual representations because they should reveal the mental models students used to understand algebraic relationships, which may be an integral part of their algebraic reasoning. Further, this approach would provide insight into how students utilize their visuospatial abilities to understand mathematics, which is particularly relevant given the welldocumented relationship between visuospatial abilities and various mathematics outcomes (
Although psychometric visuospatial measures are useful, they do not provide a comprehensive understanding of how students employ their visuospatial abilities to comprehend mathematics. Therefore, studying studentgenerated visual representations of algebraic relationships offers a unique perspective on the role of visuospatial abilities in mathematical development.
Mathematical ideas can be communicated and demonstrated in various ways using representations (
Current mathematical knowledge is associated with variations in the representation and problemsolving methods used across different achievement levels (
Visual representations refer to those involving the use of imagery to illustrate problems and find solutions (
Bloom’s Taxonomy outlines six stages of learning outcomes: remembering, understanding, applying, analyzing, evaluating, and creating (
In addition to aiding comprehension, selfgenerated drawings can also contribute to mathematical development by serving as both an analysis and a solution medium. They make problems more concrete, allow for the rearrangement of the given information, abstract problem features, direct attention to problem semantics, and facilitate the correct representation of quantitative relationships (
Third, the contents of selfgenerated visual representations used during problemsolving can provide valuable insights into the problemsolving process, including potential misunderstanding of mathematical content (
Despite these findings, the specific ways in which visuospatial abilities contribute to mathematics learning, particularly in algebra (
Mixed results may stem from individual differences in students’ understanding of visuospatial representations of mathematical concepts and relationships. However, prior studies did not assess students’ mental models of algebraic concepts before instruction, particularly through students’ selfgenerated drawings, making it difficult to infer how students’ models (including visual representations) changed as they gained competence in algebra. There is reason to believe that these representations do change with instruction. For instance,
Based on the types of representations used to teach and solve word and patterning problems (
It is crucial to examine the use of algebraicvisual representations as they might aid in bridging the gap between arithmetic and algebraic reasoning, which is difficult to close (
For instance, the visual representation used to teach the arithmetic series in
In any case, it should be noted that the difference between arithmetic and algebraicvisual representations might be analogous to students’ word problem drawings of pictorial versus schematic representations mentioned earlier. While the pictorial representations depict an image of the story problems, schematic visuals show patterns within the problem, and only the latter has a significant positive relation with word problem performance (
Mathematical concepts can also be represented symbolically (
Working with algebraic equations and inequalities depends partly on competence in using symbolic representations arithmetically, such as using the distributive law (
Further,
Here, we assume that students can represent and solve problems in multiple ways, including generating symbolic and visual representations. This assumption follows
In the current study, we aimed to investigate how students’ selfgenerated visual and symbolic representations were related to their algebraic reasoning, particularly following prompts to use visualizations after solving algebraic inequality and equation problems. While we examined the relationship between all types of representations and algebraic reasoning, our main focus was on the potential predictive power of selfgenerated visual representations. We hypothesized that visual representations, specifically those related to algebra, would be better predictors of algebraic reasoning than arithmetic visuals, as has been found in previous research on word problems (
Our focus on visual representations assumed that the ability to form sophisticated visuospatial representations of mathematical relations mediates mathematics gains (
The participants were a convenience sample of 60 ninthgrade students in Turkey (
Primary  Middle  High  College  Graduate^{a}  

Father  22% (13)  8% (5)  23% (14)  37% (22)  10% (6) 
Mother  25% (15)  12% (7)  33% (20)  23% (14)  7% (4) 
^{a}graduate = at least one graduate school.
We used the Pearson online version of Raven’s Progressive Matrices (Raven’s 2) as an index of general ability (
High School Entrance Exam (LGS, Turkish acronym), a rigorous and highstakes test in Turkey, was taken by 1,472,088 students in 2020, of which only 212,485 were granted admission to high school based on their scores. This comprehensive exam assesses students’ knowledge in mathematics, science, Turkish, revolution history and Kemalism, religious culture and moral knowledge, and foreign language. The Turkish Ministry of National Education (2020) published a report detailing the exam results (for descriptive statistics, see Table 5, p. 19), revealing that the mean score for all testtakers was
The mathematics subsection of the LGS included questions covering algebra, data and probability, and radical numbers. Three questions, which constituted fifteen percent of the exam, involved algebra. However, it is worth noting that the algebraic questions in the exam differed from those in the algebraic interview, as they were multiplechoice and involved measurement concepts such as determining the area, length, and circumference of shapes, whereas the interview focused purely on algebraic concepts.
During the interview, the students were asked eleven questions related to their comprehension of variables, and there was no time limit. Each question was designed to explore a different aspect of students’ understanding of variables. However, we paid extra attention to three questions for two reasons. Firstly, these questions involved variables that could have multiple values, such as x = 2 or x = 0 for the equation 5x < 30. Secondly, the questions could be solved using either visual aids like a number line or symbolic representations such as manipulating symbols in equations:
Q1. What values of
Q2. What happens to the value of
Q3. Ben said, “
Once the students had provided their initial response, typically involving standard symbolic algorithms, we inquired whether they could represent the equations or inequalities in an alternative way, explicitly prompting the use of visual representations. In cases where the students did not have a visual representation, we emphasized that such representations were not compulsory and that other forms of representation were acceptable. Subsequently, some students responded with additional symbolic representations, while others employed various forms of visual representation.
We utilized a threecategory rubric, comprising 0 points, 1 point, and 2 points, to evaluate the initial responses to the three questions. The categories are explained in detail in
We used two dimensions to evaluate the representations provided by students in response to the followup question. The first dimension focused on the type of representation used, which fell into three categories: algebraicvisual, arithmeticvisual, and symbolic. Algebraicvisual included solutions that used algebraic representations, including the number line, coordinate plane, and algebraic diagrams and tiles. Arithmeticvisual involved using reallife examples such as apples and balls to represent specific values for variables. Symbolic representation involved using and manipulating symbols in algebraic expressions.
The second dimension we considered was the accuracy of the response, which was scored as 0 (no targeted representation), 1 (partially accurate representation), or 2 (accurate and correct representation). The maximum score for each representation category was 6, but the scores were mutually exclusive. This means that if a student correctly used algebraicvisual representations for all three questions, their score would be 6 for algebraicvisual and 0 for arithmeticvisual and symbolic. Examples of student responses are presented in
Category  Question  Scoring 


2 Points  1 Point  0 Points  
Algebraic  What happens to the value of y as x increases in value give the equation y = 2 
NR  
Arithmetic  What values of x make the following statement true: 5 
NR  
Symbolic  Ben said, “ 
NR 
The study was conducted online via Zoom due to COVID19, with the legal guardians’ permission. The session began with an algebra interview, where the questions were presented on the screen, and the students solved them on blank papers. They showed their work on the screen and explained their answers. Once the interview was completed, the students took pictures of their work and sent them to the researchers via phone messages. Following the interview, the students completed Raven’s test digitally, with the researchers providing remote control. This study was approved by the local institution’s ethics committee (IRB# E84391427050.01.0418335/ 202131).
Three graduate students, who had at least a bachelor’s degree in mathematics education and experience teaching diverse students, coded the students’ responses. The first author, who was also one of the coders, developed the rubrics based on research findings. According to the literature, students use two solution methods when solving algebraic problems: algebraic and nonalgebraic (
The development of the algebraic representation codebook was based on the primary and middle school curriculum and standards (
The coding was also informed by previous research on the use of two types of representations in word problems: schematic and pictorial (
To ensure the reliability of the coding process, two researchers independently coded all responses using the rubric. Any discrepancies were discussed between the two coders, and if they could not agree on a code, a third researcher was consulted. The final score was determined by the third researcher if their code matched one of the initial coder’s codes. There were no instances where all three coders disagreed. The initial coders had a high level of agreement, with 92% to 95% agreement on the algebraic reasoning measure and 75% to 85% agreement on the algebraic representations scores. Any disagreements were resolved using the aforementioned method.
The study aimed to test the hypothesis that algebraic reasoning is related to selfgenerated visual representations of the associated concept, and this relation is not due to general cognitive ability or general mathematics achievement. To test this hypothesis, algebraic reasoning was regressed on general mathematics achievement, general ability scores, and the three representation types (algebraicvisual, arithmeticvisual, symbolic).
We performed post hoc power analysis to ensure that the sample size was sufficient using the GPower software package (
Two mediation analyses were used in the study. The first analysis aimed to determine if generating algebraicvisual representation mediated the link between general mathematics achievement and algebraic reasoning, which would stem from the relation between visuospatial abilities and mathematics. The second analysis aimed to test if algebraic reasoning mediated the association between overall mathematics achievement and generating algebraicvisual representations. The analyses were conducted using R (
Variable  1  2  3  4  5  

1. General ability  109.79  14.07  
2. Mathematics Achievement  10.35  5.52  0.73**  
3. Algebraic Reasoning  3.17  1.70  0.59**  .074**  
4. Algebraicvisual Representation  0.98  1.63  0.38**  0.56**  0.58**  
5. Arithmeticvisual Representation  0.73  1.42  0.23  0.07  0.04  0.21  
6. Symbolic Representation  0.85  1.27  0.21  0.21  0.09  .25*  0.10 
*
Predictors  β  

General ability  0.01  0.11  0.02  0.84  .41 
Arithmeticvisual  0.05  0.05  0.12  0.46  .65 
Symbolic  0.05  0.04  0.14  0.37  .71 
The results of the first mediation analysis, presented in
In the second mediation analysis, there was a significant direct relationship between mathematics achievement and generating algebraicvisual representations, as shown in
In this study, we explored the link between algebraic reasoning and different selfgenerated representations (i.e., algebraicvisual, arithmeticvisual, and symbolic) of algebraic equations and inequalities, following the solution of these problems and after receiving a prompt to construct a visual representation. While we examined the relationships between all kinds of representations and algebraic reasoning, we specifically focused on the relationship between students’ selfgenerated visual representations and their ability to reason algebraically. Our findings shed light on the visuospatial aspect of algebraic reasoning and help to improve our understanding of the wellestablished correlation between visuospatial abilities and mathematical competence (
Higher algebraic reasoning scores were associated with a greater number of accurate selfgenerated algebraicvisual representations of equations and inequalities, above and beyond general mathematics achievement and general cognitive ability. Performance on the latter likely reflects, at least in part, individual differences in crystallized intelligence (general knowledge) that, in turn, is influenced by fluid intelligence and working memory (
Recall, Bloom’s Taxonomy is a framework for categorizing learning outcomes into six stages: remembering, understanding, applying, analyzing, evaluating, and creating (
It is also worth mentioning that the algebraicvisual representations generated by the students were like the visualizations commonly used in algebra instruction, which suggests that students who received higherquality or more advanced algebra instruction may have been exposed to these visualizations through instructional methods. However, this is not likely to be a systematic bias because the same curriculum and similar instructional materials were used in all schools attended by the students in this study. The key finding is that even with similar instruction and materials, students still differed in their ability to integrate conventional algebraic visual representations into their understanding of equations and inequalities.
The contents of selfgenerated visual representations also provided valuable insights into unique aspects of algebraic processes. The link between algebraic reasoning scores and selfgenerated algebraic, rather than arithmetic, visual representations of equations and inequalities aligns with differences in arithmetic and algebra processing, referred to as the cognitive gap between the two problemsolving approaches (
Arithmetic visuals cannot represent more abstract numbers, such as irrational numbers that do not have a physical analog (
The algebraicvisual representations appear to become an integral part of students’ algebraic reasoning once they have a strong understanding of the generalized features of algebra, including the concept of variables. However, researchers suggest that there is no direct transition from arithmetic to algebraic reasoning, and instead, students have access to two types of representations (
Furthermore, accurate initial responses, indexed as solid algebraic reasoning skills, involved efficient symbolic usage. Therefore, the robust association between selfgenerated algebraic visuals and algebraic reasoning also suggests that students generating algebraicvisual representations are typically adept at using conventional symbolic algorithms flexibly. In line with that, a negative relationship was observed between algebraicvisual and symbolic representations. As students generated more algebraicvisual representations, they tended to prefer producing symbolic representations less. The pattern is consistent with prior studies of representational flexibility (
In summary, students with a solid understanding of algebra can flexibly produce symbolic representations to solve equations and inequalities but may inhibit their symbolic repertoire in favor of algebraicvisual representations, particularly when prompted to use visuals. However, some students who correctly use standard symbolic representations may not provide an algebraicvisual representation, even when prompted to do so. It is unclear why these students do not use algebraicvisual representations, but one possibility is that they do not have strong visuospatial abilities, which could make generating visual representations more challenging compared to using symbolic manipulations. This would be an interesting avenue for future research.
The results of the study suggest several educational implications for teaching algebraic reasoning. The findings reveal that use of algebraic visuals is more advanced than algebraic reasoning, which seems to emerge after students have some foundational algebraic knowledge. When such foundational knowledge is established, visualization appears to enhance algebraic reasoning. Studies show that providing training to enhance students’ visualization skills leads to an improvement in their mathematical performance (
Students may not spontaneously generate algebraic visualizations but may need to be prompted to do so. It is important for teachers to recognize that individual differences in students’ mental models of algebraic equations and inequalities may affect their ability to generate these visualizations. For students who can effectively use the provided algebraic visuals (in textbooks and during instruction), instruction and practice in generating their own visualizations may be beneficial. However, for those who struggle with using the provided visuals, instruction on interpreting them and generating their own may be beneficial. It might be the case, however, that some students have difficulties generating visual representations, even with instruction, and for them a focus on efficient use of standard symbolic problemsolving strategies might be preferrable, but this is unclear at this point.
Intervention studies have shown that using manipulatives and pictorial representations of quantities and variables, with a gradual transition to the conventional algebraic form, can be effective in teaching algebra (
Selfgenerated visuals might also provide insight if students fall back on using arithmetic visual representations. Prior research suggests arithmetic representation use can interfere with algebraic representation use; that is, competence with arithmeticbased representations will not automatically facilitate and may impede the learning and use of algebraic representations (
The study had a few limitations, such as the fact that students could not attend regular inperson classes due to COVID19, which may have affected their responses to the questions. Although we were unable to test if the differences in LGS scores between our sample and the entire population of test takers were statistically significant, this is likely not a significant issue given the strong correlations observed between their LGS mathematics scores and their performance on our tasks. Additionally, the data collection was online, which may produce different responses than an inperson interview. The study was also correlational, so future longitudinal and experimental research is necessary to understand the nature of the association between different visuospatial representations and algebraic reasoning and make causal inferences. Furthermore, the number of questions was limited, so the generalizability of the results to all areas of algebra needs to be further investigated. For instance, subtopics like quadric equations with a visuospatial component are worth examining.
Despite these limitations, this study extends our understanding of variation in how students think about and represent algebraic equations and inequalities as related to their ability to reason algebraically. The study also provides valuable insights into the link between visuospatial and mathematical abilities. Specifically, it suggests that competence with algebraic reasoning is dependent, in part, on the ease with which visuospatial systems can be used to represent algebraic relationships above and beyond the contributions of domaingeneral abilities and general mathematics achievement.
Questions  Scoring 


2 points 
1 point 
0 points 

Providing an accurate answer through an accurate general statement.  Providing a partially accurate answer or an answer through substitution. The answer might have missing values, such as irrational numbers.  Completely wrong statements; no answer.  
What values of x make the following statement true: 5x is less than 30?  … (the student’s written work: “x<6”) I wrote 5x is less than 30. So then, I divided both sides by 5. x was less than 6. It is valid for any value less than 6.  … (the student’s written work: “5,4,3,2,1,0,1,2,3,4,5…”) When we solved the equation, it asked xs less than 30. It asked about 5xs less than 30. It wanted the xs. It can be 1,2,3,4,5 for positives. It becomes equal; if it is 6, it becomes equal, it asks us for the smaller ones. In the continuation, it meets all the negative numbers, which can be small.  … (the student’s written work: “x = 5”) Miss, it says here 5x is less than 30, it says for which value of x, it is correct. It says the value of x here, I don't quite understand it. Since 5x is a small number, of course it is less than 30 
What happens to the value of y as x increases in value give the equation y = 2x + 3?  … (the student’s written work: “y2x=3”) When x increases, that is, when the subtracted number increases, this number also increases so that the result is the same, and the initial number also increases. Therefore, the value of y is also affected by increasing.  … (the student’s written work: “y=2x+3, x=1, y=5, x=2, y=7”) If x increases, y also increases. I wrote an equation. So, I wrote the equation by giving a value. When I give 1 to x, 2x, 2 times 1 is 2 plus 3, y becomes 5. Then, I increased x. I gave 2 to x. 2 times 2 is 4, plus 3 is 7, which is y. So if x increases, y also increases.  … (the student’s written work: “y=2x+3, x=2, y=3”) Here, miss, I didn't do any calculation, but I just wrote and thought. So, it says y equals 2x plus 3, miss, if x is 2, for example, y is 3, which one would increase uhm, if x increases continuously, I think y decreases, that is, I think it decreases. 
Ben said, “w + 3” is less than “5 + w.” 
… (the student’s written work: “2>0, w+2>w, w+5>w+3”) First, it's always true because I know that 2 is greater than 0. If so, if I add w to each side of the equation, w plus 2 is greater than w, so w plus 0 is. If I add 3 to both sides of this inequality again, w plus 2 plus 3 uhm is greater than w plus 3. That is w plus 5 is greater than w plus 3.  … (the student’s written work: “2+3=1, 52=3”) 
… Miss, I couldn’t understand this question. 
This project was supported by Josephine Mitchell Smith Fellowship to Zehra E. Ünal and DRL1659133 from the National Science Foundation to David C. Geary.
We thank Lara Nugent, Mary Hoard, Lütfiye Lütüncü, Sıdıka Uzunköprü, Sacide Öncel, Hafize Ünal, Serra Ulusoy, Büşra Ergün, Belgin Eriz, Beyzanur Yalvaç, and Emel Uçar for their help during the research process.
This study had ethical approval from Boğazici University (IRB# E84391427050.01.0418335/ 202131).
For this article, a data set is freely available (
The Supplementary Materials contain the data and R code for this study (for access see
The authors have declared that no competing interests exist.