1. Introduction
Visual representations are commonly used in the teaching of mathematics (
Bolden et al., 2015;
Schraw et al., 2013) because they can help students assign meaning to the concepts being taught (
Rau, 2017b;
Schoenherr & Schukajlow, 2024). Various factors influence the impact of visual representations on learning, including the extent to which their perceptual features are relevant to the target concept (
Belenky & Schalk, 2014;
Glenberg et al., 2013;
Goldstone & Son, 2005;
Kaminski et al., 2009) and the amount of prior domain knowledge the learner possesses (
Cook, 2006;
Cooper et al., 2018). Our research aims to investigate the impact of visual representations of LEGO
® bricks on children’s learning of fraction division. LEGO bricks can be used to capture children’s interest in mathematics (
Lenzner et al., 2013;
Moyer, 2001), but they possess perceptual features, such as the studs (i.e., the cylindrical bumps that attach the bricks together), that may not be relevant to the target concepts.
Tellos and Osana (
2025) found that the studs on images of LEGO bricks prompted fifth and sixth graders to use inappropriate strategies on fraction division problems. In particular, some students inappropriately used the studs instead of the bricks as units to solve the problems, which, in some cases, negatively impacted their problem-solving performance.
Despite recent research on the effects of perceptual features in visual representations on learning (
Kaminski & Sloutsky, 2013,
2020;
Menendez et al., 2020), questions about how prior domain knowledge influences the effects of perceptual features on LEGO bricks, which have become increasingly popular for mathematics instruction within the educational community (
Kidspot, 2016;
TheDadLab, 2019,
2020;
We Are Teachers, 2018), remain unanswered. Therefore, the first objective of the present study was to extend the research conducted by
Tellos and Osana (
2025) on the extent to which the studs in visual representations of LEGO bricks prompt inappropriate strategies in the context of fraction division. The second objective was to contribute to current understandings of how prior domain knowledge impacts children’s response to materials that contain irrelevant perceptual features. Specifically, in this study, we aimed to investigate the role of prior knowledge in children’s strategies when solving fraction division problems with LEGO bricks.
1.1. Visual Representations in Fraction Division
A meaningful explanation of fraction division involves measurement division concepts, specifically, determining how many times the divisor fits into the dividend (
Barnett-Clarke et al., 2011;
Gregg & Gregg, 2007). In the context of measurement division, the divisor is conceptualized as a unit of measure to determine the size of the dividend (
Zembat, 2017). In elementary mathematics instruction, however, fraction division is frequently taught using the “flip and multiply” algorithm, often without presenting associated concepts that explain why the algorithm works (
Olanoff et al., 2014). To address this typical shortcoming in the teaching of fraction division, teacher educators have recommended the use of visual representations (e.g., bar diagrams) to enhance students’ understanding of measurement division concepts when learning fraction division (
Barnett-Clarke et al., 2011;
Reys et al., 2015). Both physical LEGO bricks and images of bricks can serve as useful visual tools in this context (e.g.,
Tellos & Osana, 2025;
TheDadLab, 2019). LEGO bricks can depict the whole, the dividend fraction, and the divisor fraction, facilitating the process of determining how many times a divisor brick can be stacked onto a dividend brick to reach a solution. In an exploratory study,
Tellos and Osana (
2025) provided instruction to fifth- and sixth-grade students on how to use images of LEGO bricks to solve fraction division problems (e.g., 1/4 ÷ 1/16 =). An error analysis suggested that 18% of the sample inappropriately counted the studs on the bricks rather than the bricks themselves, which may have been one of the reasons they had difficulty applying concepts of measurement division on a transfer task administered after the instruction.
1.2. Extraneous Details in Visual Representations
Visual representations used to teach mathematics often include perceptual details that are not directly related to the to-be-learned concept (
Belenky & Schalk, 2014;
Kaminski & Sloutsky, 2013). Such perceptual details have been referred to as “extraneous” (
Kaminski & Sloutsky, 2013). In teaching fractions to children, teachers have a strong preference for using images of real-world items, such as pizzas and cookies (
Kaminski & Sloutsky, 2020). Many such representations include irrelevant perceptual information, such as pepperonis and chocolate chips that have been shown to impede learning and transfer (
Kaminski & Sloutsky, 2013;
Macevicius & Osana, 2024). Consistent with the cognitive load theory, extraneous information can overwhelm learners, leaving fewer cognitive resources to attend to the task (
Sweller et al., 2011). Further, extraneous details in visual representations can prompt learners to activate inappropriate schemas that are then used to erroneously interpret the information in representations (
Harp & Mayer, 1998). Interpretations based on the extraneous information can, in turn, disrupt learning and transfer (
Mayer, 2019;
Rey, 2012;
Son & Goldstone, 2009).
Researchers have observed young children using inappropriate counting schemas when presented with instructional materials that contain extraneous details that are presented in the form of discrete elements.
Shipley and Shepperson (
1990) presented children aged 3 to 7 years with arrays of identical images of forks that were either intact or spatially detached (e.g., prongs and handle). When asked how many forks there were, children often inappropriately counted the separated parts as distinct, countable objects. In other words, children tended to treat spatially separated elements as individual units, suggesting that perceptual cues, such as spacing, may reinforce object-by-object enumeration as a default strategy. In another study,
Wege et al. (
2023) asked 4- and 5-year-old children to sort animal toys (i.e., discrete objects) by color (e.g., brown, pink) and kind (e.g., horses, pigs), and to count the number of sets they created. The results revealed that some children struggled to count the grouped sets of animals as units and instead counted the individual animals within each set.
With school-aged children,
Kaminski and Sloutsky (
2013) provided instruction to kindergartners, first graders, and second graders on how to use the y-axis to read quantities in bar graphs. They found that when the bars in the graphs contained extraneous images of shoes and flowers, children inappropriately counted the images in the bars instead of using the y-axis to read the quantities represented. In the context of fractions more specifically,
Kaminski and Sloutsky (
2020) asked first graders to write a fraction symbol that corresponded to shaded portions of circular area models, some of which were generic grey circles and others were images of pizzas with extraneous details (e.g., pepperonis, mushrooms). Children who used the representations of pizzas with extraneous details tended to focus only on the number of slices present (i.e., the numerator), suggesting that they emphasized counting the slices rather than thinking relationally about fractional quantities. Together, these findings suggest that children’s inclination to count individual elements may be a perceptually-driven response to the discrete nature of the items and that spatial separation encourages children to perceive elements—such as studs on LEGO bricks—as countable, even when appropriate group-based strategies may be more accurate or efficient.
Children’s early numeracy experiences in the home and in educational contexts frequently involve object-by-object counting (
Clements & Sarama, 2007). For this reason, the tendency for children to count discrete elements in representations may be traced back to their early educational experiences. Children’s initial encounters with numbers are shaped by mathematical tasks that center around individual object counting, which establishes a strong association between numerical value and physical items in their environment (
Alibali & DiRusso, 1999;
Graham, 1999). Early instruction typically emphasizes one-to-one correspondences between each individual item and a number word in the counting sequence (
Mix, 2008), which then creates a well-rehearsed routine on which children repeatedly rely when faced with counting tasks (
Mix, 2008;
Sophian, 2008). Thus, when teachers use concrete and visual representations to teach mathematical concepts, children can default to counting perceptual details, even when the details are extraneous to the learning objective.
The impulse to attend to discrete elements in instructional materials appears to persist in later elementary years when children are introduced to multiplicative concepts in the mathematics curriculum (
Fitzpatrick & Hallett, 2019). In one study,
Tellos and Osana (
2024) presented children with multiplication expressions (e.g., 2 × 5) and asked the children to rate the extent to which visual displays of LEGO bricks represented the expressions. The authors found that 8-year-olds attended to the studs in their ratings more so than 10-year-olds, who more often used entire bricks when rating the visual displays. In another study,
Jeong et al. (
2007) compared the influence of discrete and continuous visual representations of quantities on the proportional reasoning of children aged 6 to 10 years. Participants were shown circular spinners with red and blue regions representing different proportions. In one condition, both regions were presented as undivided areas, whereas in the other condition, the regions were divided into equal partitions that could be individually counted but would result in the incorrect solution. The results demonstrated that the children used more appropriate strategies when reasoning with the undivided regions than the divided ones. Similar findings were reported by
Boyer et al. (
2008), who found that children between kindergarten and the fourth grade counted discrete partitions in images of juice mixtures instead of correctly identifying and applying the underlying ratios in the mixtures. Collectively, these studies highlight that the tendency to count discrete elements in visual representations—while rooted in early mathematical experience—can persist into later childhood, and that extraneous details in the form of discrete quantities may continue to prompt inappropriate counting strategies, possibly hindering children’s ability to grasp more advanced mathematical concepts.
In sum, discrete extraneous details in visual representations can prompt children to fall back on familiar, yet inappropriate, counting strategies that can deflect attention away from the underlying mathematical structure. An example of an inappropriate counting strategy would be using the studs on the LEGO bricks as units to solve certain measurement division problems. The tendency to count objects highlights how easily children’s attention can be drawn to extraneous details, even in mathematical tasks where counting individual items is not relevant. In the context of fraction division with LEGO bricks, for instance, learners’ attention may be drawn to perceptual details on the bricks, such as the individual studs, that in some cases may be irrelevant to the solution.
1.3. Prior Domain Knowledge
Factors beyond the perceptual characteristics of visual representations themselves can explain children’s learning from visual displays (e.g.,
Cooper et al., 2018), such as age and prior domain knowledge. We use
Simonsmeier et al.’s (
2022) definition of domain knowledge as “specialized knowledge of a topic” (p. 31). We define prior domain knowledge, therefore, as information related to a topic that is stored in long-term memory. Theoretical accounts highlight the critical role of prior knowledge in shaping how learners interpret and process incoming information (e.g.,
Kalyuga, 2007). When individuals receive visual and verbal inputs, as is often the case in mathematics, they must select information relevant to the task and integrate it with what they already know (
Bransford & Johnson, 1972;
Brod et al., 2013;
Mayer, 2008). With respect to visual representations specifically, prior domain knowledge can help learners direct their attention toward the most relevant features in visual representations (
Chi et al., 1981;
Cook, 2006;
Kalyuga, 2013), leading to the accurate encoding and interpretation of new information (
Brod et al., 2013;
van Kesteren et al., 2014). Prior knowledge not only guides attention but also supports the organization of new material into meaningful chunks (i.e., bundles of information), which facilitates retrieval and application during problem solving (
Gobet, 2005;
Gobet et al., 2001).
Glaser (
1990) further argued that these knowledge-driven chunks enable learners to perceive the underlying structure of a task, which is essential for successful learning and performance (
Richland et al., 2012;
Verschaffel et al., 2020). Therefore, prior knowledge not only aids in filtering out extraneous information, but also enables learners to accurately apply appropriate strategies, leading to more effective problem solving and a deeper understanding of associated concepts.
The role of prior knowledge as directing learners’ attention to the relevant information in representations has received empirical support. For example,
Magner et al. (
2014) investigated how secondary students’ understanding of angles influenced learning after a geometry lesson. Some students received instructional materials containing extraneous details, such as an illustration of a person riding a bicycle, while others did not. In the extraneous details condition, students with high prior knowledge outperformed those with low prior knowledge on a near transfer task. Similarly,
Cooper et al. (
2018) examined the effect of students’ mathematics ability, as measured by standardized test scores, on their accuracy on trigonometry problems. Half of the problems contained diagrams that presented problem-relevant information, and the other half included no diagrams. Half of the diagrams were accompanied by an illustration that highlighted the real-world context described in the problem text but was not required for the solution. The remaining diagrams were not accompanied by an illustration. The results indicated that students with low prior knowledge had more difficulty when illustrations contained additional, yet extraneous, information than students with high prior knowledge. These studies demonstrated that prior knowledge not only helps learners filter out extraneous details but also enables them to focus on the essential mathematics concepts.
The previous research has examined how prior knowledge influences learners’ responses to the presence of visual representations that are irrelevant to the target concepts (e.g., an image of the Statue of Liberty;
Cooper et al., 2018). Less is known, however, about how learners respond to individual, discrete details within such representations. Nonetheless, studies reporting developmental trends in children’s counting strategies highlight the potential role of prior knowledge in attention to such individual details.
Kaminski and Sloutsky (
2013), for example, found that younger children were more likely to count individual elements, such as images of flowers and shoes, in their graph reading task, whereas older children more frequently attended to the relational structure of the task, suggesting a developmental shift possibly influenced by prior knowledge. Similarly, both
Jeong et al. (
2007) and
Boyer et al. (
2008) found that younger children inappropriately counted the discrete segments in visual representations more frequently than older children. The younger children in
Tellos and Osana (
2024) also relied on the individual studs on LEGO bricks to a greater extent than the older children when rating multiplicative structures. Finally,
Barmby et al. (
2009) found that the 8-to-9-year-olds in their sample attended more to the lower-level units in visual representations of dot arrays than the 10-to-11-year-olds, who used strategies that demonstrated greater attention to the multiplicative structures in the arrays. Thus, to the extent that age effects can be explained by an accumulation of domain-specific knowledge as children progress through school, we hypothesized that prior knowledge influences the way children respond to discrete perceptual details that are irrelevant to the target concept. As children gain more conceptual knowledge, they become better equipped to recognize which aspects of a visual representation are structurally meaningful and which are not (
Cook, 2006). Building on this literature, we hypothesized that prior knowledge plays a critical role in children’s ability to focus on the visual elements in LEGO bricks that are conceptually aligned with measurement division when solving fraction division problems.
1.4. Present Study
Little is known about the role of prior domain knowledge when learners interact with visual representations that contain extraneous details. The aim of the present study, therefore, was to investigate the extent to which children’s prior domain knowledge would play a role in their attention to extraneous details in visual representations of LEGO bricks when solving fraction division problems. Specifically, we hypothesized that various types of prior knowledge would influence children’s application of the inappropriate strategy of counting the individual studs on LEGO bricks, namely, knowledge of the following: (a) fractions concepts, (b) fraction magnitude, (c) fraction representation with LEGO bricks, and (d) measurement division concepts. This was assessed experimentally by assigning participants to two different instructional conditions: demonstrations only and demonstrations plus explanations. These categories of domain knowledge differ in terms of the extent to which they are specific to the task of solving fraction division with LEGO bricks (
Brod, 2021). More specifically, general knowledge of fractions concepts and fraction magnitude are ostensibly broader constructs than those more directly related to fraction division with LEGO bricks. In contrast, knowing how to represent fractions with images of LEGO bricks may be more directly related to the task at hand. Finally, instruction in the meaning of measurement division may have a direct impact on counting the correct representations of the divisors in fraction division problems.
We used a fractions test and a number line estimation task to assess participants’ knowledge of fractions concepts and fraction magnitude, respectively. Further, we designed a task to measure knowledge of fraction representation with LEGO bricks, and we manipulated the delivery of explanations of measurement division using an instructional intervention. All participants received instruction on how to use visual representations of LEGO bricks to solve fraction division problems. In one condition, the instruction also included explanations of measurement division concepts, whereas in the other condition, no explanations were provided.
To what extent do children use inappropriate counting strategies (i.e., counting studs) on images of LEGO bricks when solving fraction division problems (RQ1)?
Does knowledge of fractions concepts, fraction magnitude, fraction representation with LEGO bricks, along with the children’s instructional condition, predict the extent to which the children use inappropriate counting strategies when solving fraction division problems with LEGO bricks (RQ2)?
In terms of RQ1, we expected that some of the participants would count the studs instead of the bricks when solving fraction division problems in a measurement division context. With respect to RQ2, we predicted that each type of fractions knowledge and the children’s instructional condition would contribute uniquely to the children’s counting strategies when solving fraction division problems with LEGO bricks. Higher prior knowledge of fractions concepts would facilitate learners’ attention to the relevant features in the visual representations, namely the bricks as the unit of measure, which would lead to correctly encoding them as the divisor quantity in a measurement division context. The abstraction of the measurement division context would in turn support the grouping of the studs on the bricks as units and then counting how many times the divisor brick fits onto the dividend brick. In contrast, those with lower prior knowledge would be more likely to attend to irrelevant features in the visual representations, which would then mask the underlying measurement division context. Without an accurate problem structure, children would then fall back on well-practiced strategies of counting discrete elements in visual representations.
2. Materials and Methods
2.1. Participants
The participants were 39 fourth- (n = 23) and fifth- (n = 16) grade students (14 girls and 9 boys in Grade 4; age: M = 10.04, SD = 0.33, and 5 girls and 11 boys in Grade 5; age: M = 11.14 years, SD = 0.28) from two fourth-grade classrooms and two fifth-grade classrooms in one French-speaking, private suburban school in eastern Canada. A majority of the sample (81.5% of 27 participants who responded) reported a family income of over 100,000 CAD. No participant was excluded from the study.
2.2. Design
In sum, the design of the study included both correlational and experimental elements. We assessed the influence of three continuous predictor variables, namely knowledge of fractions concepts, fraction magnitude, and fraction representation with LEGO bricks, on the outcome measure, which was the extent to which the participants counted the studs instead of entire divisor bricks to solve fraction division problems. We also tested the effect of an experimentally manipulated variable—the presence or absence of explanations providing the meaning of measurement division (MD) in a fraction division context—on the participants’ counting strategies. As has already been noted, participants were randomly assigned to two instructional conditions, one in which the procedure for solving fraction division problems with LEGO bricks was demonstrated, accompanied by explanations of the meaning of MD; and the other in which no explanations of MD were provided beyond demonstrations of the fraction division procedure.
2.3. Instructional Intervention
The instructional intervention consisted of three phases, namely Phase 1 (introduction to fractions lesson), Phase 2 (fraction division lesson), and Phase 3 (procedure consolidation). All instruction was delivered in French in online individual sessions with participants. The lessons in Phases 1 and 2 were delivered using a pre-recorded video with narration and animation. In Phase 1, the video lesson demonstrated how to represent nine target fractions (1/2, 1/4, 1/8, 2/4, 3/4, 1/2, 1/3, 1/12, 2/3) using bricks of different colors to represent the target fraction and the whole. In Phase 2, the video lesson consisted of demonstrations of how to use LEGO bricks to solve three fraction division problems using measurement division (1/4 ÷ 1/16 =, 5/6 ÷ 1/12 =, and 3/4 ÷ 1/8 =).
Figure 1 presents screenshots from the lesson for the problem 1/4 ÷ 1/16 =. Different colored bricks were used to represent the whole, the dividend, and the divisor (see
Figure 1a). The divisors were always unit fractions represented by a LEGO brick with one stud in all three problems.
The procedure was shown through an animation that showed the yellow divisor brick sliding onto the blue dividend brick until no other divisor bricks could fit. Once the dividend brick was fully covered by divisor bricks, an image of a hand appeared that pointed to each divisor brick one at a time (see
Figure 1b). Then, the solution (e.g., “4”) appeared to the right of the equal sign.
The objective of Phase 3 was to consolidate the procedure of solving fraction division problems as demonstrated in Phase 2. The researcher presented three problems (2/3 ÷ 1/12 =, 1/2 ÷ 1/16 = and 3/4 ÷ 1/16 =
1) to the participants, all of which again required the use of a one-stud divisor brick for the solution. Each problem was presented with the symbolic representation of the fraction division problem placed at the top of the screen (e.g., 1/2 ÷ 1/16 =), the correct bricks placed underneath the dividend (e.g., 1/2) and divisor (e.g., 1/16), and a selection of bricks of various sizes at the bottom of the screen (see sample item in
Figure 2). Because the intervention was delivered online, the participants directed the researcher to select specific bricks from among those at the bottom of the screen and asked her to move them around in ways that would lead to a solution. The researcher provided corrective feedback by repeating the condition-specific instructions in the fraction division lesson, if required. Phase 3 was terminated when the last problem was solved correctly regardless of performance on the other two problems, a criterion reached by all participants. To avoid a potential confound of color, the colors of the bricks were identical to the ones used in the instructional intervention and were the same across all items used in the study (e.g., the whole bricks were always red).
Experimental Conditions
Within each grade, the participants were randomly assigned to one of two conditions: (a) MD explanation (n = 19; 11 in Grade 4 and 8 in Grade 5) and (b) no-MD explanation (n = 20; 12 in Grade 4 and 8 in Grade 5). Assignment to conditions resulted in approximately even distributions in each classroom: in classroom 4A, there were 12 participants, with 7 in the MD explanation condition. In classroom 4B, there were 11 participants, with 4 in the MD explanation condition. In classroom 5A, there were 7 participants with 3 in the MD explanation condition, and in classroom 5B, there were 9 participants with 5 in the MD explanation condition.
Participants in both conditions received the same instructional intervention described earlier, except that in the MD explanation condition, additional verbal explanations about the meaning of measurement division were given. The additional explanations were delivered during the demonstrations in the fraction division lesson in Phase 2. In two of the explanations, the narrator said, “I have to find how many [divisors] are in [dividend]” (e.g., “I have to find how many one-sixteenths are in one-fourth.”). In the animation, when the hand began pointing at the divisor bricks on the dividend brick, the narrator then said, “So I am going to count the [divisors]” (e.g., “So I am going to count the one-sixteenths.”). The same explanations were provided for all problems at the same points in the lesson.
In the no-MD explanation condition, participants viewed the identical video lesson used in the MD explanation condition, with one key difference: The verbal explanations about the meaning of measurement division were omitted from the narration. Specifically, the explanations of measurement division during each demonstration problem (e.g., “I have to find how many one-sixteenths are in one-fourth”) were removed. As a result, participants in the no-MD explanation condition did not hear any verbal explanation about the meaning of measurement division during those segments, but the visual content remained unchanged.
2.4. Measures
All to-be-described measures were administered in French.
2.4.1. Fractions Concepts
The Fractions test is a paper-and-pencil test based on
Saxe et al.’s (
2001) work that assesses participants’ conceptual and procedural knowledge of fractions. Only the conceptual subscale was used in the present study. The test consists of nine items: (a) three items that require writing a fraction that corresponds to a shaded amount in an area model; (b) two estimation items that require circling a fraction from a choice of three that best represents the unpartitioned shaded region in a circle; (c) three fair-sharing items, each of which requires partitioning area models evenly among a given number of people; and (d) one item that requires the participants to use drawings to compute 2/5 + 3/5. The fractions score is the proportion of correct responses. The internal reliability of the fractions concepts measure was 0.68.
2.4.2. Fraction Magnitude
The Number Line Estimation (NLE) task (
Hansen et al., 2017) assesses participants’ fraction magnitude. We administered the pencil-and-paper version of the task. The participants placed a hash mark on a number line where they thought a target fraction was located. Participants completed nine items on a 0–1 number line (i.e., 1/5, 13/14, 2/13, 3/7, 5/8, 1/3, 1/2, 1/19, 5/6) and 10 items on a 0–2 number line (i.e., 1/3, 7/4, 5/6, 7/6, 1, 3/8, 1 5/8, 1 1/5, 7/9, 4/3). The Percent Absolute Error (PAE) was computed for each item using the formula PAE = (|child answer − correct answer|)/numerical range. Two separate NLE scores were computed by averaging the PAE across relevant trials: (a) magnitude 0–1 score for the 0–1 number line items, and (b) magnitude 0–2 score for the 0–2 number line items. Each score ranged from 0 to 1, with lower scores indicating greater accuracy. The internal reliability of the number line estimation measure was 0.87 for the magnitude 0–1 score and 0.78 for the magnitude 0–2 score.
2.4.3. Representation Task
The Representation task assessed participants’ knowledge of how to represent fractions with LEGO bricks immediately after the introduction to fractions lesson. Knowledge of how to represent fractions with LEGO bricks was operationalized by the rate at which the studs were used in their responses on six multiple-choice items. Participants chose the fraction among three that correctly represented a quantity shown with LEGO bricks. On each item, one of the distractors was 1 over the number of studs found on the whole (i.e., 1/16 in the sample item in
Figure 3) and the other was the number of studs on the target brick (i.e., 8 in the sample item in
Figure 3). The representation score was the proportion of correct responses. The internal reliability of the representation measure was 0.75.
2.4.4. Extraneous Details Task
The Extraneous Details task was used to assess the participants’ understanding of measurement division concepts, operationalized by the extent to which participants counted the studs instead of the bricks when solving fraction division problems. The task consisted of 10 problems: 1/2 ÷ 1/6 =, 1/6 ÷ 1/12 =, 2/3 ÷ 1/6 =, 1/2 ÷ 1/8 =, 4/6 ÷ 1/3 =, 1/4 ÷ 1/12 =, 1/4 ÷ 1/8 =, 1/2 ÷ 1/4 =, 1/3 ÷ 1/6 =, and 1/3 ÷ 1/12 =
2. On each item, the participants directed the researcher to select specific bricks from among those at the bottom of the screen and to move them around in ways that would lead to a solution. No reminders or corrective feedback was provided.
The items on the Extraneous Details task were presented in the same way as those in the procedure consolidation phase of the intervention with two exceptions. First, the brick used to represent the divisor contained more than one stud (see sample item in
Figure 4). Because the fraction division lesson in Phase 2 of the intervention always involved one-stud divisors, using divisors with more than one stud on the Extraneous Details task served to evaluate whether the participants counted studs, which would suggest attention to individual discrete elements on the images, or the bricks. That in turn would indicate a conceptual misunderstanding of the unit required to solve the problem. Of the 10 items, seven items involved a divisor brick with two studs, two items used a four-stud divisor brick, and one item used an eight-stud divisor brick. Second, the bricks at the bottom of the screen did not offer the choice of a one-stud brick to solve the problem. The counting strategy score was the proportion of items on which the studs were counted.
2.5. Procedure
The Fractions test and the NLE task, in that order, were administered in person at the participants’ school. Five participants at a time were removed from the classroom for individual testing in the school cafeteria. After the in-person testing, the researcher contacted the participating families to schedule individual online sessions for the instruction and remaining testing. The online session was delivered on average 17 days after the in-person testing. There were no instructional condition differences in the number of days between the in-person testing and the day on which the online session took place, t(37) = 0.63, p = 0.53.
The researcher’s camera and microphone remained on during the online sessions and all sessions were recorded using the software’s recording feature. Participants shared their video and audio. The researcher shared a slide deck to deliver the instructional intervention and to administer the Representation task and the Extraneous Details task. The researcher scored the latter two tasks using a scoring sheet during administration and all scoring was confirmed by reviewing the recordings.
2.6. Analytic Plan
To address the first research question, we first computed the proportion of all items on the Extraneous Details task that were solved by inappropriately counting the studs. To address the second research question, we conducted a multiple-regression analysis with the counting strategy score as the criterion measure and the fractions knowledge measures and instructional condition as predictors in the model.
4. Discussion
Visual representations are widely used in mathematics instruction, yet relatively little is known about how individual differences, such as prior knowledge, influence children’s attention to the perceptual details in the representations. The objective of the present study was to examine the impact of perceptual features in visual representations that contain extraneous information about the target concepts. We examined children’s mathematical problem solving and the role of prior domain knowledge in reducing any negative effects of extraneous features. We provided instruction to fourth- and fifth-grade students on how to use images of LEGO bricks to solve fraction division problems. After instruction, the participants were assessed on the extent to which they counted the individual studs on the divisor bricks in a task where such a strategy would lead to the incorrect solution.
Extraneous details in visual representations can have a negative impact on mathematics performance, particularly when those details are in the form of discrete elements within representations (e.g.,
Kaminski & Sloutsky, 2013,
2020;
Tellos & Osana, 2025). We hypothesized that images of LEGO bricks would hamper children’s fraction division performance by their considering the studs rather than the bricks themselves as the units of measure. The results revealed that almost one-third of our sampled children inappropriately counted the studs on at least some fraction division problems. Researchers have documented children’s tendency to count discrete elements in visual representations of multiplicative structures (e.g.,
Barmby et al., 2009;
Bolden et al., 2015;
Boyer et al., 2008), and that such counting can negatively impact performance (
Jeong et al., 2007;
Kaminski & Sloutsky, 2013). One-to-one counting, including that which we observed in our study, may be a result of the participants’ early experiences with numbers and extensive practice counting individual objects in school (
Mix, 2008;
Sophian, 2008).
Next, we explored the extent to which prior domain knowledge accounted for the variance in the participants’ strategies. In line with theoretical accounts (e.g.,
Bransford & Johnson, 1972;
Mayer, 2008;
Simonsmeier et al., 2022), we expected that students with more fractions knowledge would be better equipped to interpret visual representations meaningfully, attend to structurally relevant details (i.e., the bricks as units), and filter out extraneous or misleading features (i.e., the studs). The data partially supported our predictions. First, knowledge of fractions concepts, such as estimating fractional quantities and fair sharing as assessed on the Fractions test, accounted for a significant proportion of the variance in participants’ counting strategies. One interpretation of this finding is that participants with more fractions knowledge may have been better able to identify which features in the LEGO representations were relevant to the new information about measurement division concepts delivered in the instruction. Greater attention to the correct fractional units (i.e., the bricks) with which to measure the dividend during instruction (
Brod, 2021;
Sidney & Thompson, 2019) were then successfully applied to solve the fraction division problems. Participants with less fractions knowledge, on the other hand, may not have known which aspects of the visual representation were relevant. As such, their proclivity to count the discrete elements in the images may have drawn their attention to the studs and masked the underlying measurement division structure used in the instruction to demonstrate fraction division (
Sidney & Alibali, 2015).
In addition, because of its operationalization, the Representation task served as an additional indicator of the extent to which the participants used either bricks or individual studs in their responses. A secondary regression analysis allowed us to draw similar conclusions about their strategies, namely that prior knowledge of fractions concepts accounts for a substantial proportion of the variance in their attention to the studs when representing fractions with LEGO bricks. More specifically, students with greater fraction knowledge were more likely to ignore extraneous features, thereby demonstrating the filtering function of prior knowledge, as explained by
Kalyuga (
2007).
The data also supported our hypothesis that students’ representational knowledge would predict their counting strategies. The participants’ knowledge of how to represent fractional quantities may have helped them recognize which bricks in the visual representations correctly corresponded to the divisors that were presented symbolically (e.g., 1/16) in the instruction and in the problems they solved after instruction. Accurate encoding of the divisor bricks was critical to counting the number of bricks, rather than studs, that fit on the dividend.
Contrary to our predictions, knowledge of fraction magnitude as assessed by number line estimation was unrelated to the participants’ strategies. This finding was unexpected, particularly because performance on the number line task was correlated with the participants’ knowledge of fractions concepts and also because of existing evidence that number line estimation predicts fractions arithmetic performance (
Bailey et al., 2017;
Schneider et al., 2018).
Brod’s (
2021) framework outlining the effects of prior knowledge on learning emphasized that domain knowledge must be relevant to the specific task at hand. Although
Schneider et al. (
2018) found in their meta-analysis that number line estimation correlates with mathematical tasks that typically involve symbolic representations, such as counting, mental and written arithmetic, as well as standardized tests of mathematical achievement, we propose that number line estimation may not be useful for learning about the conceptual underpinnings of fraction division. The instruction and the fraction division problems administered in our study were presented using visual representations, which were not within the scope of
Schneider et al.’s (
2018) meta-analysis. It is thus possible that even if knowledge of fraction magnitude was activated during instruction, it might not be congruent with the conceptual demands of fraction division using visual representations (
Brod, 2021). Thus, our findings are in line with
Brod’s (
2021) account that not all types of prior domain knowledge are equally accessible or functionally relevant in a given learning context.
Finally, we predicted that instruction that included explanations of measurement division concepts would be a stronger predictor of children’s counting strategies than instruction on the fraction division procedure alone. The conceptual emphasis on measurement division was expected to facilitate the children’s problem-solving performance. This hypothesis was not supported by the data, however. The instructional manipulation did not yield the anticipated benefits on the dependent measure, nor did it result in differences on any of the other measures used in the study. There are several possible explanations for this finding. First, the differences between the instructional conditions may have been too subtle to produce measurable effects. Second, we did not administer a direct measure of participants’ understanding of measurement division, nor did we include a pretest in the study design. As a result, we cannot determine whether the instructional condition that included explanations of measurement division in fact enhanced participants’ conceptual understanding relative to the condition that did not include explanations. Consequently, our data do not allow for conclusions about the extent to which participants’ solutions to the fraction division problems with LEGO bricks were facilitated by instruction in measurement division.
Despite our results, it remains unclear whether prior knowledge has a direct positive influence on how children manage extraneous perceptual features, such as the studs on LEGO bricks. Research on executive functioning has shown that inhibition—the ability to suppress irrelevant or misleading information—is a key predictor of early numeracy and mathematics achievement (e.g.,
Fuhs & McNeil, 2013;
Gilmore et al., 2015;
Merkley et al., 2016). Moreover, some evidence suggests that children with greater mathematics knowledge may demonstrate more inhibitory control (
Ashkenazi & Blum-Cahana, 2023;
Gilmore & Cragg, 2018). Thus, it is possible that prior domain knowledge not only supports problem-solving but also enhances children’s ability to inhibit perceptual features that are not directly related to the to-be-learned concept. Future studies should further examine the relationship between prior knowledge of mathematics and inhibitory control to better understand how children learn to extract relevant information from visual representations that possess perceptual details.
4.1. Contributions to the Literature
The present study is one of the first to investigate how images of LEGO bricks used as visual representations of mathematical concepts are related to children’s problem-solving performance. Prior research has examined other educational uses of LEGO bricks, such as the relation between construction in a play context and spatial reasoning (e.g.,
McDougal et al., 2023;
McDougal et al., 2024), but none to our knowledge has tested the use of bricks as mathematical tools for teaching concepts in the elementary curriculum. In particular, we explored how children respond to perceptual features in visual representations of LEGO bricks when learning about fraction division. An important contribution is that children attended to different aspects of the perceptual features in the LEGO images, which was associated with varying levels of performance. In some cases, children attended to the bricks as the units and appropriately counted them to solve fraction division problems. In other cases, the children were drawn to extraneous details on the images of the LEGO bricks, namely the studs, when solving the problems. The results contribute to a small but growing body of research (e.g.,
Boyer et al., 2008;
Jeong et al., 2007;
Kaminski & Sloutsky, 2013,
2020;
Tellos & Osana, 2024,
2025;
Wege et al., 2023) demonstrating that under certain conditions, the presence of discrete perceptual elements in visual representations can prompt children as old as 12 years of age to count them.
These findings are noteworthy first because children’s attention to discrete elements in visual representations can derail learning in mathematical domains other than multiplication and proportional reasoning (e.g.,
Boyer et al., 2008), namely, fraction division. Second, we showed that undue attention to discrete features can also occur when representations of LEGO bricks are used in the teaching and learning of mathematics. Investigating what other perceptual features of LEGO bricks, such as color, shape, and spatial configurations (
Tellos & Osana, 2023), capture students’ attention in instructional contexts is a consideration for future research.
Prior research on the effects of extraneous details in visual representations have predominantly used stimuli that have arguably stronger real-world referents than studs on LEGO bricks, such as pepperonis and mushrooms on pizzas (
Kaminski & Sloutsky, 2020) and images of beetles and ladybugs (
Menendez et al., 2022). Researchers have explained that children’s attention can shift away from the targeted mathematical concept when familiar and interesting details are inserted into visual representations (e.g.,
Cooper et al., 2018;
Kaminski & Sloutsky, 2020), particularly when they are irrelevant to the task (
Kaminski & Sloutsky, 2013;
Magner et al., 2014;
Menendez et al., 2022). In contrast, one would be hard-pressed to consider a stud on a LEGO brick, a relatively generic feature, as possessing inherently “seductive” (see
Harp & Mayer, 1998;
Mayer, 2019;
Rey, 2012) or visually distracting qualities. Nonetheless, we found that studs prompted the children to use inappropriate strategies when solving fraction division problems. This finding is important because there are few studies that have examined how children respond to details in instructional materials in mathematics that are not perceptually rich or “realistic” (e.g.,
Carbonneau & Marley, 2015;
Petersen & McNeil, 2013). In sum, our findings show that even perceptual features that do not have real-world referents can also interfere with learning and performance in mathematics.
Finally, our data suggest that different types of prior knowledge play an important role in how children interpret and apply visual representations. This finding is noteworthy, first because it suggests an explanation based on individual differences in children’s tendency to count discrete extraneous details beyond a purely developmental one (e.g.,
Barmby et al., 2009;
Jeong et al., 2007;
Wege et al., 2023). Nevertheless, more research is needed to identify more precisely the cognitive and task-related factors that impact learners’ interpretations of visual information. Second, the finding that more than one dimension of prior knowledge contributes to children’s responses to extraneous details extends previous research, which has focused primarily on a single type of prior knowledge (e.g., mathematics achievement,
Bolden et al., 2015;
Cooper et al., 2018; principles related to angles,
Magner et al., 2014). Our results suggest that at least two domain-specific knowledge structures–knowledge of general fractions concepts and representation knowledge–form key elements of the conceptual landscape of fraction division. We speculate that these forms of prior knowledge provided the children with efficient searches of the problem space that guided their attention to the relevant information in the representations, thereby facilitating appropriate counting strategies (
Kalyuga, 2007).
4.2. Strengths and Weaknesses
A strength of the present study is the way in which participants’ counting of the studs on LEGO bricks was operationalized. We designed a task that provided more direct evidence of inappropriate counting strategies than measures of performance accuracy used in previous studies (e.g.,
Kaminski & Sloutsky, 2020;
Magner et al., 2014). More specifically, our measure involved solving fraction division problems using divisor bricks with more than one stud, which provided a clearer indication of whether studs or bricks were counted. Confirmation with observational data (e.g., verbal counting, pointing, or tagging behaviors), however, would provide a direct verification of the participants’ counting strategies. Another strength of the study is the use of a video-based intervention, which ensured that all participants received identical instruction. This type of delivery strengthened the implementation fidelity by eliminating potential differences in how the content was presented to the participants.
We note that although our outcome measure produced a more valid indicator of the extent to which the participants used inappropriate counting strategies than what has been used in prior research, it did not directly indicate a conceptual understanding of fraction division. The act of counting the bricks only indicated that the participants knew to count something other than the individual studs, but not the reasons for doing so. On the one hand, they may have counted the bricks because of their grasp of measurement division, which directed them to use the appropriate units to solve the problems. On the other hand, they may have simply used the brick as a strategy or “trick” to best represent the divisor fraction without relying on an understanding of measurement division. A task that directly assesses measurement division concepts likely would provide insight into these speculations. Along the same lines, the participants who counted the individual studs may have done so for reasons other than an impulse to count discrete elements in the images of LEGO bricks. A plausible alternate explanation may be that they learned to count the studs because it was demonstrated in the instruction. Thus, measures that could better reveal the cognitive mechanisms underlying participants’ behaviors, such as additional qualitative data based on think-aloud protocols (
Ericsson & Simon, 1980), could provide a deeper understanding of how children engage with perceptual features of instructional materials.
Another potential methodological weakness pertains to the design of the study, which could be argued did not yield sufficient statistical power to detect the expected effects. As the argument goes, conducting the multiple-regression analyses with a larger sample would have increased our ability to detect relations between prior knowledge and children’s responses to extraneous details in visual representations if such effects were actually to exist in the population. Given the small sample, the likelihood of Type II errors was increased, reducing the credibility of our conclusions. That said, we remind the reader that the present design and analyses proved capable of detecting a number of omnibus test and statistically controlled pairwise relationships that addressed our primary research questions.
An additional weakness of the present study was that it was conducted in a private school, which restricts the study’s external validity. Our findings were based on a sample from a relatively high socioeconomic level with mathematical competencies that may not have reflected those of children in the same grades in public schools. Further, a sample drawn from a population with wider variability in prior mathematical knowledge might have produced differences between the experimental conditions. Replicating the current study in more diverse settings would allow for more generalizable findings.
We also note several limitations because of the online procedures that were required because of COVID restrictions. First, because the delivery of the instruction and the administration of the measures were carried out online in students’ home environments, any data on the extent to which the students paid attention to the video lessons—such as documenting their gaze during the instruction—might not have adequately captured their engagement levels. Furthermore, we could not ensure that the students completed the prior knowledge assessments and the fraction division tasks independently. Second, the online format limited the conclusions to visual representations of LEGO bricks. Given that learning opportunities provided by mathematical tools depend heavily on the types of actions they support (
Donovan & Alibali, 2022;
Manches et al., 2010), the extent to which our findings can be generalized to physical LEGO bricks is limited. Physical objects enable sensorimotor engagement: students can touch, move, or arrange bricks, which can help ground abstract concepts in concrete experiences (
Abrahamson et al., 2020;
Martin, 2009;
Pouw et al., 2014). In contrast, when learners use digital manipulatives, like those used in the current study, they tend to rely more on visual and pattern-based reasoning (
Pouw et al., 2014). The difficulty in transferring our conclusions to physical LEGO bricks also constrains the practical implications of the research because teachers might be more likely to use physical bricks in the mathematics classroom than their digital images. Third, and as an aside, even a mathematically sophisticated adult who was unfamiliar with the present study experienced difficulty in examining the visually depicted LEGO bricks and performing the requested division task.
4.3. Educational Implications
The results of the present study highlight the importance of identifying not just whether learners possess prior knowledge, but whether they have access to the specific knowledge that is most relevant and readily activated to support problem solving. Simply having prior knowledge is not always sufficient insofar as educators must consider how instructional contexts and representations can cue the use of that knowledge in productive ways. Further, educators interested in using LEGO bricks in fractions instruction may wish to establish students’ knowledge of related concepts and how to represent fractional quantities with the bricks. Without this foundation, students might not possess the conceptual tools necessary to accurately identify the visual elements that will assist in abstracting the relevant conceptual structure.
Beyond these points, our findings raise additional considerations for instructional practice. First, teachers may need to explicitly draw students’ attention to the elements of visual representations that are mathematically meaningful and those that are not. In the context of fraction division, this could involve guiding students to focus on the conceptual units (e.g., bricks) and helping them filter out perceptual details that may be irrelevant or misleading for the task at hand (e.g., studs). Second, the knowledge that perceptual information in visual representations can impact the ways in which children interpret them is an important component of teachers’ mathematical knowledge for teaching (
Ball et al., 2008). Therefore, teachers are advised to select or adapt instructional materials with the foresight that reducing extraneous perceptual details—particularly for children with less prior knowledge—can facilitate more appropriate teaching and learning strategies. When teachers use images of LEGO bricks in particular, they might benefit from observing which perceptual features are capturing their students’ attention and if necessary, modify the representations accordingly.
Furthermore, we recommend that teachers be cautious when generalizing our findings, which were based on visual representations, to classroom practice with concrete LEGO materials. The ways in which students engage with visual materials might be different from the ways they interact with concrete manipulatives, such as LEGO bricks.
Pouw et al. (
2014) suggested that manipulatives offer experiences that can support motor-based strategies and embodied understanding (see also
Glenberg, 2008), whereas virtual ones emphasize visual cues and patterns. It is worth noting that we do not discourage the use of physical LEGO materials in the classroom; however, more research is needed to examine whether the same results based on images of LEGO bricks can be extended to concrete instantiations of those images.
Finally, the findings support the notion that students’ representational competence–that is, their ability to select, interpret, and use visual representations to reason through mathematical ideas, solve problems, and communicate thinking (
Rau, 2017a)—is a consideration for effective mathematics instruction. Explicit teaching of why, for instance, a specific LEGO brick represents one-sixteenth or why a 2 × 2 brick should be counted instead of its individual studs can facilitate the development of representational competence, allowing learners to shift the external images of LEGO bricks into visualizations that can be used to enhance problem-solving performance (see
Daniel et al., 2018). Such scaffolding is likely to support successful problem solving in fraction division but may also promote more flexible representational thinking that will equip students to work with a wide range of mathematical representations more effectively (
Rau, 2017c).