Examining Fraction Performance and Learning Trajectories in Students with Learning Disabilities: Effects of Whole-Class Intervention

Abstract

Digital game-based interventions have shown promise in improving STEM engagement and conceptual understanding among elementary students. This study focuses on how a game-enhanced, whole-class mathematics intervention influences the fraction performance and conceptual development of students with learning disabilities (LDs). Using a triangulated mixed methods design, we analyzed pre- and post-intervention data from students in intervention and comparison classrooms. Students participating in the game-based instruction demonstrated medium, positive gains in both performance and conceptual reasoning, in contrast to declines observed in the comparison group. The findings illustrate how equitable design principles can support mathematical growth among students with LDs, with implications for research and practice.

1. Introduction and Review of Literature

Broadening participation in STEM and ICT-related careers begins with inclusive, high-quality mathematics instruction in elementary school. However, students with learning disabilities (LDs) frequently encounter instruction that lacks relevance, multimodal engagement, or conceptual rigor. Prior research highlights the potential of students’ mathematical reasoning when structural barriers are removed (see Hunt & Empson, 2015; Hunt et al., 2019a, 2019b; Silva et al., 2022). This study builds on that work and concentrates on conceptual access in fraction learning for students with LD, investigating the impact of whole-class, game-enhanced instruction on students’ learning and conceptual growth.

Fractions remain a challenging topic within mathematics education, with many students with LDs struggling to gain the needed conceptual underpinnings (Siegler et al., 2013). Such struggle is not just a phase in educational growth but can lead to a continuous cycle of mathematical difficulties and avoidance (Shin & Bryant, 2015). The complexity of fractions, involving the iterative development of both conceptual and procedural knowledge, often poses significant challenges for students, leading to a lack of confidence and interest in mathematics (Makhubele, 2021). The struggle with fractions has been linked to future difficulties in mathematics and other STEM fields (Powell & Nelson, 2021).

Mathematics intervention programs that contain serious games may offer a promising approach to promoting access, engagement, and learning outcomes in elementary STEM education, including fraction content (Jensen & Skott, 2022). Serious games provide an interactive and immersive learning environment where students can actively participate in challenges and scenarios that require them to learn and apply mathematical concepts and skills in practical contexts (Cecotti & Callaghan, 2021). This hands-on approach often incorporates multiple representations, tools, and solution strategies, allowing students to approach problems from various angles (CAST, 2024). Furthermore, contextualizing mathematical concepts within relevant and engaging scenarios motivates and reinforces the relevance of mathematics in their lives and future career pathways (Malaluan & Andrade, 2023). Yet, the potential of serious games as a whole-class intervention and their impact on learning for students with LDs are not yet well understood. Moreover, there exists a gap in longitudinal research on classroom-based intervention designs, in general and in for students with LDs in particular.

This study focuses on teacher implementation of the 36-lesson Model Mathematics Education (ModelME) program, which integrates a sandbox-based game across multiple virtual worlds. The program is designed to be either a whole-class or supplemental intervention to increase student engagement, fraction knowledge and STEM/ICT career interest and is framed in Universal Design for Learning (UDL), an efficacious design framework for accessible and equitable instructional materials (CAST, 2024). A triangulation design for validating quantitative data mixed methods design is employed to investigate the impacts of the program on the fraction performance and learning trajectories of students with LD. The research questions are:

• To what extent did students with LD who did and did not participate in a whole-class game-enhanced intervention improve their fraction performance and their fraction schemes?
• What operational development is evident in students’ learning strategies who did and did not participate in the whole-class game-enhanced intervention?

1.1. Learning Disabilities and Fractions

Students have traditionally been identified and labeled as LD in the United States (US) based on a perceived gap between their performance on achievement tests and their scores on IQ assessments. However, this approach has faced mounting challenges in recent years in the US, primarily due to deficit assumptions resulting from the label (Kashikar et al., 2023), a lack of empirical support for its validity (Kovaleski et al., 2022), and, as a result, revisions to the US Individuals with Disabilities Education Act (IDEA). As an alternative, we use the US IDEA language to define in this study as “a disorder in one or more of the basic psychological processes involved in understanding or in using language, spoken or written, that may manifest itself in the imperfect ability to listen, think, speak, read, write, spell, or to do mathematical calculations, including conditions such as perceptual disabilities, brain injury, minimal brain dysfunction, dyslexia, and developmental aphasia. Specific learning disability does not include learning problems that are primarily the result of visual, hearing, or motor disabilities, of intellectual disability, of emotional disturbance, or of environmental, cultural, or economic disadvantage” (IDEA, Sec 300.8 (c) (10)).

Over the past decade, research into students with LD and their fraction understanding has explored how students perceive fractions at fixed points over time, consistently revealing enduring challenges (e.g., Hansen et al., 2017; Jordan et al., 2017; Namkung et al., 2018). For instance, Hansen et al. (2017) tracked the evolution of fraction understanding among a cohort of students from third to sixth grade. They observed a subgroup of students, particularly those with LDs, grappling with fractions longitudinally, often treating fractions with different denominators as distinct whole numbers rather than unified entities (e.g., viewing two thirds as two over three as opposed to two iterations of the quantity one-third). These persistent issues, echoed in studies by Jordan et al. (2017) and Namkung et al. (2018), underscore the inherent difficulties inherent in the bipart nature of rational numbers, hindering the connection of whole number unit understanding to fractional unit understanding, which has been found true for all students, regardless of disability (National Mathematics Advisory Panel, 2008).

1.2. Traditional Fraction Instruction and Whole-Class Interventions

Traditional instruction for fractions involves a progression of concepts and activities aimed at developing students’ concept of fractions as parts of a whole. Students typically begin by taking part in instruction that presents and represents fractions as equal parts of a whole or a group. Next, students learn to read and write fractions using fraction notation (e.g., 1/2, 3/4) and understand that the numerator represents the number of equal parts being considered, and the denominator represents the total number of equal parts in the whole. Teachers often use visual aids such as fraction bars, circles, or rectangles to help students visualize fractions. Students may work with manipulatives like fraction strips or fraction circles to understand concepts concretely.

Next, students learn to compare fractions using symbols such as “<”, “>”, or “=” to determine which fraction is greater or less than another. They also learn strategies like finding a common denominator to compare fractions. They learn strategies for finding equivalent fractions, such as multiplying or dividing both the numerator and denominator by the same number. Basic operations with fractions are introduced next, including addition, subtraction, multiplication, and division. Initially, these operations may involve fractions with like denominators and, later, progress to fractions with unlike denominators. Throughout this instruction, students may solve word problems involving fractions to apply their understanding of fraction concepts in real-world contexts after learning and practicing strategies. Unfortunately, this approach has typically failed to produce positive student outcomes for many students (Fuchs et al., 2017), including students with LDs, as evidenced by the US National Assessment of Educational Progress (NAEP; U.S. Department of Education [DOE], 2024).

Conversely, significant strides have been made in interventions aimed at strengthening students’ grasp of fractions. For example, employing fractions as measurements on a number line (e.g., Fuchs et al., 2017; Jordan et al., 2017) can relate the concept of whole numbers with that of rational numbers as different units of measure, bridging conceptual divides. Interventions highlight key elements of effective instruction, such as student articulation of teacher-modeled strategies or reasoning, learning to compare fractions, and placing fractions on number lines through teacher guidance. However, criticisms of these approaches have surfaced, such as the argument that instructional approaches that heavily rely on teacher-modeled instruction can overlook or deny access to students’ individual reasoning processes or not provide relevant and/or timely feedback on students’ own strategies and reasoning processes (Hunt et al., 2016a, 2016b, 2016c, 2019a, 2019b) preventing conceptual change.

1.3. Games as Innovative, Whole-Class Interventions

The potential of games to improve accessibility to STEM content, foster individual and collaborative problem-solving, and facilitate innovative exploration of mathematical concepts is highly promising. Over the last decade, digital game-enhanced instruction in mathematics has emerged as a strategy to enhance student outcomes, including motivation, engagement, and learning, across a diverse student population (Clark et al., 2016). Games can serve as valuable tools in elementary mathematics classrooms by complementing core instructional content, providing unique learning experiences in various STEM subjects, and offering additional support for students. For instance, some games provide students with sandbox play and low-stakes ways to solve problems, offering multiple ways to engage and receive prompts or feedback (Alghamdi & Holland, 2017).

Despite these advancements, the literature still lacks comprehensive insights into integrating game-enhanced learning within broader mathematics education frameworks, particularly for fractions and even more so for intervention programs (Thomas, 2016). As a result, there is a need for more extensive empirical studies to evaluate the effectiveness and implementation strategies of game-enhanced learning for fractions (Clark et al., 2016). Recent research by Hunt et al. (2022, 2023a, 2023b, 2025) underscores the positive impact of game-enhanced supplemental fraction curriculum on student engagement, fraction knowledge, and STEM interest, highlighting the untapped potential of this innovative approach in mathematics education and special education. However, specific impacts on fraction performance and learning for students with LD have yet to be reported in this line of research.

1.4. Study Framing

The current study is framed through two conceptual underpinnings, or areas, that relate specifically to access and achievement. The first area is learning trajectories (LTs), which are thought to consist of a learning goal, learning materials, tasks, and the developmental progressions that students progress through as they learn the target concept or skill, are documented in research on student thinking and learning, and often begin as hypothetical (M. Simon, 2020). They are refined and confirmed over time through research that involves observation of students’ activity with the tasks and other learning materials, students’ evolving mathematical development, and corresponding refinements to the hypothesized LT and associated learning experiences (M. Simon, 2020). LTs for fractions have been hypothesized and confirmed (e.g., Confrey & Maloney, 2010; S. Empson, 1999; S. B. Empson et al., 2005; Olive & Steffe, 2002; Tzur, 2007) in mathematics education research over a span of fifty years for various samples of populations of students and, more recently, in cross-disciplinary research involving samples of students with LDs (Hunt et al., 2016a, 2016b, 2016c, 2019a, 2019b, 2023a; Hunt & Silva, 2020; Crawford, 2022). The confirmations across research form the basis for the delineation of instructional trajectories (Hunt et al., 2023a; Daro et al., 2011) for use as a basis for instruction.

The second area is students’ mathematical schemes, or their abstracted mental concepts (Norton & Wilkins, 2009; Tzur, 2007). Grounded in theories of radical constructivism (Von Glasersfeld, 2013), schemes are thought to be built up and solidified through (1) students’ goal-driven activity (e.g., actions, such as partitioning or repeating units of parts of units; spoken words or gestures; construction of representations) upon some object (e.g., a mathematical task) and (2) students noticing and reflecting upon the results of that activity. As students act in a mathematical task, for example, they engage in periods of noticing and reflecting upon the results of their activity, sorting between actions that led them closer to or further away from their goal (M. A. Simon et al., 2004). Over time, a student’s processes of acting, reflecting, and sorting support their awareness of not only patterns of activity that lead toward a desired result, but also the logical necessity of those patterns. Realization of the logical necessity of activity patterns leads to a mental coordination of those actions and, over time, abstraction of the student’s scheme (M. Simon et al., 2010).

Fraction LT and Schemes—Students with LD

In this study, we adopt Hunt et al.’s (2016a, 2016b, 2016c, 2019a, 2019b, 2023a) LTs of students with LD as they develop their understanding of fractions as quantities, or numbers, alongside the corresponding instructional trajectories (Hunt et al., 2023a). Hunt’s LTs are grounded in a synthesis of the prior work of several researchers (S. Empson, 1999; S. B. Empson et al., 2005; Hackenberg, 2013; Steffe & Olive, 2009; Tzur & Hunt, 2022; Wilkins & Norton, 2018); the corresponding instructional trajectories are based in the learning goal of “fractions are numeric quantities that involve a multiplicative coordination of the numerator with the denominator”. The LTs involve students’ unit fractions, partition fractions, iterative fractions, and reversible fractions (there are two others—multiplicative fraction and distributive fraction schemes—that we do not address in this study).

Table 1. Learning trajectory.

Concept/ Scheme	Definition
Unit fractions (1/n)	A measure of the whole that fits n times within the whole such that the whole is n times as much of 1/n. Operations: Partitioning, iterating
Partitive Fractions (m/n)	Iterating a given unit fraction (1/n) a few (m) times, not exceeding the n/n whole (i.e., m ≤ n), yields a composite fraction that is m times as much as 1/n. Operations: Iterating, partitioning
Iterative Fractions (m/n)	Iterating a given unit fraction (1/n) a few (m) times exceeding the n/n whole (i.e., m ≤ n), yields a composite fraction that is m times as much as 1/n. Operations: Iterating and partitioning together
Reversible Fractions (m/n → 1/n → n/n)	Reversing the iteration supposedly used to create a composite fraction m/n by partitioning it into m parts to create 1/n and then “undo” the initial partitioning of the whole, which created 1/n, by iterating n times to make the n/n whole. Operations: Partitioning and iterating together

lists the defining features of each scheme implicated in this study along the developmental progression, activity that would indicate its presence, and tasks and other learning materials that can support it.

Because schemes are mental structures, they cannot be observed. However, students’ enacted activity (the actions they take in a task, the words they say, the pictures they draw, the models they construct) can be observed in tasks and, over time, their schemes can be inferred via repeated observation.

Present across all schemes in the aforementioned LTs are the actions of partitioning and iterating. Also referred to as operations, partitioning and iterating are used in earlier schemes (e.g., unit fraction scheme, partition fraction scheme) in a sequential manner (e.g., partitioning a whole unit to create 1/n, then iterating 1/n to confirm its accuracy against the whole [n/n]; partitioning a whole unit to create 1/n, then iterating 1/n m times to confirm m/n). Later, partitioning and iterating become coordinated, or used together as a single operation, to support reversibility and develop more sophisticated schemes, such as multiplicative and distributive fraction schemes.

1.5. The Current Study

The current exploratory study evaluates how a whole-class game-enhanced fraction intervention impacts performance and scheme development of students with LD. The intervention program integrates into participating students’ whole-class mathematics instructional time three times per week, with each session lasting 35 min. The intervention is based on Hunt et al.’s (2016a, 2016b, 2016c, 2019a, 2019b, 2023a) prior work in LTs and has four core components. The first component is the inclusion of various means of expression, representation, and engagement in the design of the game tasks and user interface. The second component is the use of carefully selected and sequenced tasks upon the LTs of unit fractions, partitive fractions, iterative fractions, reversible fractions, multiplicative fractions, and distributive fractions to inform the instructional trajectory. The third component is the use of cognitive supports, such as nudges, action adaptive promoting, hints, and reflective feedback to support students as they build and internalize schemes. The final component is the after-game tasks, which support students to share, compare, and contrast their gameplay strategies to explicate key developmental ideas (M. A. Simon, 2006).

More specifically, the game’s interface, tasks, and tools offer students access to different means to solve problems, varied tools, and different ways to express solutions, allowing students to choose based on their preferences. Through sandbox play, students can access their prior knowledge, noticing and reflecting upon their strategies without high-stakes repercussions (Hunt et al., 2022, 2023b, 2023c). The tasks presented in the game are structured and sequenced in clusters that support access to and advancement of established schemes for understanding fractions (see Table 1) in both leveled and non-leveled gameplay. Prompts and feedback within the game not only provide hints but actively assist students in setting goals, monitoring progress, and noticing and reflecting on outcomes, aspects of instruction that are often neglected in traditional math instruction. Furthermore, the intervention integrates gameplay with opportunities for students to step out of the game and explain and justify their mathematical reasoning using pedagogical routines such as game replays or number strings. For a fuller discussion of the intervention’s design, see Hunt et al. (2023b, 2023c) and Hunt et al. (2022).

2. Materials and Methods

2.1. Participants and Setting

Data for this study were part of a larger experimental study of the impact of the program on whole classes of elementary aged students compared to matched comparisons (see Hunt et al., 2025). Six fourth-grade teachers and 135 students in two different schools in the southeast United States participated in the study. Each school was in a rural setting and included students with intersecting identities in terms of race, language, and disability. Nine students with LD, who were a part of the larger student sample and their representative teachers’ whole-class mathematics instruction, are the participants for this study. Five of the students were in the intervention group and four of the students were in the comparison group. While the total sample size is low, it is in line with prevalence rates of students with LD in the general population (U.S. National Center for Education Statistics (NCES), 2024), which is generally thought to be 6–8.

The supplemental curriculum was administered by the teachers in their core mathematics classrooms, which included approximately 15–25 students with each teacher. The program took place over 12 weeks, which is considered best practice in terms of time period for technology-based interventions (Gersten & Edyburn, 2007). Prior to the study, informed consent and assent were gathered from teacher and student participants using Institutional Review Board (IRB) approved documents. The consent form for participation was distributed to all participants and signed. Demographic information for the nine students with LD is given in

Table 2. Student demographics.

ModelME Group			Comparison Group
Gender	Race	Disability	Gender	Race	Disability Status
Female (4)	Black (2)	LD (5)	Female (2)	Black (1)	LD (4)
Male (1)	White (2)		Male (2)	White (3)
	Vietnamese (1)			Vietnamese (0)

.

2.2. Research Design

A validating quantitative data model triangulation design is adopted in this study to better understand the impact of the game-enhanced intervention on fraction performance and learning of students with LD. A validating quantitative data model is a mixed methods design that requires using both qualitative and quantitative types of data within one instrument (e.g., survey, test measure). The measures for this study (described below) include both quantitative (i.e., correct, partially correct, or incorrect solutions) and qualitative (i.e., observable use of operations, such as partitioning, iterating) aspects. In this model, the qualitative data is used to validate and expand on the findings of quantitative data. Our rationale for this approach is that the quantitative data and their subsequent analysis provide a general understanding of the research questions in terms of students’ changes in performance and fraction schemes, while the qualitative data analysis adds a level of refinement by exploring participating students enacted operations in more depth (Creswell, 1999).

2.3. Data Sources and Measures

Student understanding was evaluated using established measures of fraction performance and conceptions—the Trajectory Aligned Fraction Assessment (TEFA), hereafter called “FractionCBM” and the Test of Fraction Schemes (Wilkins et al., 2013). See Hunt et al. (2025), for fuller descriptions of these assessments. Both tests were group-administered before and after the intervention by the teacher and were paper-and-pencil-based.

The TAFA is aligned with elementary US fraction standards and the tasks taught in the game worlds. Items were scored as correct (viable drawing, correct quantification), partially correct (viable drawing OR correct quantification), or incorrect (no viable drawing, no correct quantification). Construct validity with the Test of Fraction Schemes (see below) was 0.71. Test–retest reliability was 0.86; internal consistency reliability for the test was 0.75. Figure 1 shows sample items for the assessment.

The Test of Fraction Schemes (Wilkins et al., 2013) included six items from the original test (i.e., two items on partitioning, two items on iterating, and two on splitting) and six items on overall fraction concepts (e.g., unit fraction, partitive fraction, iterative fraction). Items were aligned with Hunt’s LT and piloted in a previous study (see Hunt et al., 2025). Items were scored as correct (viable drawing, correct quantification), partially correct (viable drawing OR correct quantification), or incorrect (no viable drawing, no correct quantification). Internal consistency reliability for the test was reported by the authors as 0.70; criterion-related validity was 0.58 (p < 0.01). Figure 2 shows sample items from the assessment.

Qualitative measurement of students’ operations and schematic change was gained from both outcome measures. Researchers examined both tests for evidence of enacted partitioning and iterating as well as any written words or explanations students jotted down as they worked. For each item, researchers recorded evidence of each operation and scheme with deductive codes related to developmental stages of partitioning and iterating from Hunt et al.’s (2023a) prior research.

2.4. Procedures

2.4.1. Intervention Procedures

Teachers implementing the whole-class intervention adhered to a structured schedule and lesson guides, delivering the supplemental program according to the lesson guides over a span of 12 consecutive weeks, three times a week. Each session lasted 35 min. Each lesson began with a five-minute preview, followed by 10–15 min of student gameplay, and concluded with a 15–20 min post-game task, such as a number string or a game replay. Previews were supplemented with videos and often included discussion prompts to engage students. Topics encouraged to explore STEM or ICT career possibilities depicted in the game, such as exploring the prevalence of windmills in the US nationwide to the mechanics of wind turbine systems in the first game world.

During gameplay, students interacted with the universally designed video game, tackling fraction challenges across six game worlds using sandbox-style play. In the interactive game, students embody a character navigating fraction challenges aligned with developmental learning trajectories. Challenges varied in complexity, some offering specific objectives while others encouraged diverse problem-solving approaches. Progression through the game was generally contingent on successfully completing each challenge, with individualized gameplay saving student progress. However, students and teachers could navigate to specific challenges as needed to align with after-game discussion.

Following gameplay, students participated in structured discussions designed to surface and reflect on fraction strategies. This discussion incorporated number strings and either a game replay or additional number strings. Game replays encourage students to recreate strategies employed within the game challenges. Initially, students independently revisit their approaches, with teachers probing students’ thinking with questions like, “Could you explain your illustration?” and/or “What’s the significance of the rectangles in your picture? What do they represent?” This individual phase was followed by paired discussions centered on questions that support students to explain their strategy, its presentation, and the resulting fractional quantity. Throughout this exchange, teachers offered scaffolding with structured sentence stems (e.g., “The approach I used was _______”), facilitating dialogue. Finally, the teacher directed a whole-class discussion using scripting from the program, including talk moves (O’Connor & Michaels, 2019) to foster peer-to-peer and student-to-teacher communication.

Number strings comprised a series of related problems aimed at fostering specific reasoning skills, such as doubling and halving. Each problem was presented sequentially, with students signaling readiness to share their solutions nonverbally. Teachers then selected students to articulate their thought processes, utilizing core representations to illustrate strategies. Encouraged to draw upon previously shared strategies, students navigated through the problem strings, promoting procedural fluency and conceptual understanding aligned with the trajectory of the program.

2.4.2. Data Collection Procedures

Consenting students took the measures of engagement, conceptual understanding, and STEM/ICT interests in their mathematics classrooms in three consecutive 30-min class sessions immediately before and immediately after the intervention. Teachers administered all measures after being trained on implementation. Measures were given to students as a group. Teachers told students to answer questions about their perspectives and interest in STEM and fraction knowledge and to do their best. Teachers gave no other direction. Items were read aloud to students who required assistance; in some cases, a scribe was used to write numeric answers for students who required support. Post-test procedures mirrored pretest procedures.

2.5. Data Analysis

2.5.1. Quantitative Analysis

To address the first research question, we calculated normalized learning gains to understand the extent to which students’ fraction performance and schemes changed after participating in the game-enhanced intervention. Normalized learning gain (NLG), a common measure for evaluating conceptual change (Hake, 1998), was used to track student growth in fraction understanding The metric helps create a fair comparison among students of different measured prior knowledge, as students who scored high on the pre-test were still capable of achieving high NLG scores since their maximum possible amount of increase is comparably lower to a student who scored low on the pre-test. NLGs were calculated for each student as well as for the group (i.e., intervention group and comparison group).

To determine the statistical significance of normalized learning gains for individual students, we calculated a z-score for each student’s normalized gain and compared it to the critical z-value corresponding to a significance level of 0.05. This approach helps assess whether a student’s individual learning gain is significantly different from the average or expected gain.

2.5.2. Qualitative Analysis

To address the second research question (i.e., to delineate students’ observable operations and the schemes the observed operations support), the research team (one expert in math education and special education, two experts in special education, and one expert in the learning sciences) utilized constant comparison methods (Flick, 2014) using a deductive framework based on the literature review. This process began from a scrutiny of written data for two students, where a collaborative effort among the research team was made to name and describe nuances in operations and the schemes observed in students’ written work assessments. For each task, researchers examined (a) the way in which students solved the problem, representations used, or models drawn and (b) observable operations employed. We then gave each element of students’ thinking an initial code.

As more tasks were coded, we carefully compared each new chunk of data (i.e., each task) with data coded previously and searched for confirming and disconfirming evidence of the codes to ensure consistency and validity (Leech & Onwuegbuzie, 2009). Next, each researcher independently coded three more students’ pre- and post-exams using the initial codebook. Codes were then compared using peer debriefing and collaborative work (Brantlinger et al., 2005). The iterative process of coding, comparing, and refining continued through three additional rounds of independent coding until all tasks in all students’ pre- and post-exams were coded. Codes for each student’s exams were then reexamined across associated tasks to (a) delineate dominant ways of operating for each student and (b) infer whether the observable evidence aligned with a fraction scheme or schemes. The dominant ways of operating for each student were named for each student pre- and post-test and then tabulated for each group. Finally, positive evidence of a scheme across tasks was for each student given a name of the relevant scheme(s); dominant schemes were also considered for each student and for each group at pre- and post-test.

Next, a classical content analysis was conducted to quantify the frequency of different operations and schemes exhibited by individual students as well as by each student and for students as a group (intervention and comparison groups) across their written work. These insights enriched the initial comparative analyses and provided a way to report on the development of each group holistically. Individual student operational trajectories were also shown visually.

2.6. Merging and Final Interpretation

To merge and compare the data, trends were identified. For example, results of the classical content analysis depicting trajectories of operational growth were merged with each student’s fraction performance NLGs and scheme NLGs. Commonalities or divergences across these quantitative and qualitative analyzed data were compared and summarized.

3. Results

3.1. Student Performance and Scheme Change

Our first research question addressed the extent to which students’ fraction performance and schemes changed as a percentage of items correct on each measure after participating in the game-enhanced intervention. To address it, we assessed students’ fraction performance and schemes’ normalized learning gains (NLGs). We will first discuss students’ NLG across all items on the TAFA across all students in both the intervention and comparison groups. The mean gain across all students was 8% percent (standard deviation of 34.2%), with a minimum gain of −52.63% and a maximum gain of 54.26%. Figure 3 illustrates NLG broken down by intervention and comparison group for the TAFA. As shown, seven students had positive NLG (15%, 55%, 33%, 22%, 22%, 15%, and 5%, respectively), while two students had negative (−40% and −53%, respectively) gains, with the intervention group containing small to medium gains and the comparison group containing small gains along with medium to large losses.

Z-scores in terms of performance for each student and their significance from the mean are shown in

Table 3. Student Z-scores, performance.

	Student 1	Student 2	Student 3	Student 4	Student 5	Student 6	Student 7	Student 8	Student 9
Z score	0.725	0.404 *	0.404 *	0.199	−0.094	0.199	−0.094	−1.41 *	−1.79 *

* significant at p < 0.05.

. They reflect that performance losses for two students, both in the intervention group, were significantly lower than the group mean, while performance gains for two students in the comparison group were significantly higher than the group mean. No other NLGs were significantly different from the mean.

Next, we will discuss NLG for students tested fraction schemes. The mean gain across all students was −0.76% (standard deviation 18.26%) with a minimum gain of −37.50% and a maximum gain of 21.43%. Figure 4 illustrates NLG for fraction schemes broken down by intervention and comparison groups.

As shown, five students had positive NLG (12.5%, 21.43%, 13.64%, 12.5% and 4.55%, respectively), while four students had negative gains (−4.55%, −4.55%, −15.78%, and −37.5%, respectively). In this measure, gains and losses were distributed across intervention and comparison, with the largest gain shown in the intervention group and the largest loss shown in the comparison group.

Z-scores for each student in terms of scheme change and their significance from the mean are shown in

Table 4. Student Z-scores, schemes.

	Student 1	Student 2	Student 3	Student 4	Student 5	Student 6	Student 7	Student 8	Student 9
Z score	0.726 *	1.215 *	0.789 *	−0.291	−0.91 *	−2.09 *	−0.291	0.726 *	−0.291

* significant at p < 0.05.

. They reflect that conceptual gains for five students, three of which were in the intervention group and two of which were in the comparison group, were all considered to be significantly higher than the group mean. Two conceptual losses, one in the intervention group and one in the comparison group, were found to be significantly different from the mean. No other NLGs were significantly different from the mean.

3.2. Students’ Learning Trajectories

The second research question addressed the operational development evident in students’ written work from their pre- to post-tests along with scheme change. We present the results in terms of students’ partitioning, iterating, and scheme change below.

3.2.1. Partitioning

Three nuances in partitioning across the two measures were coded: (a) halving, (b) in action-linked, and (c) before action-linked partitioning. Halving was coded when students partitioned items into halves, even though the strategy did not work with the numbers in the problem. In action-linked partitioning was coded when students engaged in partitioning linked to the number of sharers as an afterthought during solving problems; this strategy was often evidenced by erasures that indicated an initial partitioning plan into a number of shares that was not connected with the number of sharers. Before action-linked partitioning was coded when students’ strategy evidenced partitioning into a number of parts that was related to the number of sharers, a lack of erasures, and accuracy of size of created parts indicate a partitioning plan before solving the problem.

Table 5. Students’ ways of partitioning and iterating before and after intervention.

	Partitioning			Iterating
	Halving	In Action	Before Action	Trial and Error	Planned Adjustment	Iterating to Partition
Intervention Group, Pre	2	3	0	5	0	0
Intervention Group, Post	0	4	1	0	1	4
Comparison Group, Pre	3	1	0	3	1	0
Comparison Group, Post	2	2	0	2	2	0

shows students’ ways of partitioning (and iterating, see below) before and after intervention for both groups.

Before the intervention program, two out of five students in the intervention group and three out of four students in the comparison group used halving-based partitioning. Three out of five students in the comparison group and one student in the comparison group used before action-linked partitioning. No students used in action-linked partitioning. After the intervention program, no students in the intervention group employed halving as their dominant means of partitioning (versus two out of four students in the comparison group). Four out of five students in the intervention group used in action-linked partitioning at post-test (versus two out of four students in the comparison group). Finally, one out of five students in the intervention group used before-action partitioning (versus no students in the comparison group).

3.2.2. Iterating

Students’ coded ways of iteration across the two measures included (a) in-action trial and error of a created or supplied fraction unit (i.e., students iterated yet did not iterate the needed number of times or did not keep the size of the part consistent when iterating), (b) planned, in-action adjustment (i.e., students created an equal share equivalent to the one given and corrected iterating the number of times) and (c) planned iteration to partition (i.e., students created a fractional part and used iteration of the part to partition a whole or wholes or create another fractional quantity).

Before the intervention program, five out of five students in the intervention group and three out of four students in the comparison group used in-action, trial and error iteration. One out of four students in the comparison group used planned, in-action adjustment. No students in either group used planned iteration to partition. On the post-tests, only one student in the intervention group used in-action adjustment (versus half of the students in the comparison group). Two out of four students in the comparison group used trial and error to partition at the post-test. Four out of five students in the intervention group used planned iteration to partition at the post-test (versus no students in the comparison group).

3.3. Merging and Interpretation

Individual Trajectories

Figure 5 and Figure 6 present visualizations of operational change over the course of time for each student in the intervention and comparison groups.

As shown in the graphs, students display varying levels of advancement in terms of their partitioning and iterating operations. In the intervention group, Student 1 and 2 advanced both their partitioning and iterating operations: one- and two-level advances in partitioning and a two-level advance in iterating, respectively. Students 3, 4, and 5 did not advance their partitioning at all, while all students advanced their iterating to one (i.e., student 5) to two (i.e., students 1, 2, 3, and 4) levels. In the control group, only Student 8 advanced their partitioning one level, and only student 9 advanced their iterating operations one level. All other students did not experience a change in their operations.

Next, we discuss the merging of our classical content analysis depicting trajectories of operational growth with each student’s fraction performance NLGs and scheme NGLs.

To interpret the findings, we discuss commonalities or divergences across these quantitative and qualitative data across the two groups for each student. Across the two groups, students 1, 2, 3, 8 and 9 experienced positive gains in terms of scheme change, while students 4, 5, 6, and 7 experienced negative gains in terms of scheme change. When we compare those results with their operational development, we see that students who experienced positive changes in their schemes are the same students who experienced changes in their operational development. Students who experienced change were students who also jumped one (i.e., students 5 and 8) to two (i.e., students 1, 2, 3, and 4) levels in iterating and a one (i.e., student 1) and two (i.e., student 2) level change in partitioning. In the same way, students who did not experience operational change (i.e., students 3, 4, 5, and 6) also did not experience schematic change. The convergence of the data, therefore, support and inform each other in terms of the relationship between operational and conceptual development.

In terms of performance across the two groups, alignment with operational and scheme change is mixed. While most students (i.e., students 1, 2, 3, 4, 5, 8 and 9) operational, schematic, and performance advances were in line directionally, two students (i.e., students 7 and 8) experienced increased performance concurrently with decreased operational and schematic change. Nonetheless, for the majority of students, the interpretation of operational change informed not only conceptual change but fraction performance in this study.

4. Discussion

The purpose of this study was to explore the fraction performance and learning trajectories of 4th grade students with LD within a game-enhanced supplemental curriculum for fractions called ModelME. In terms of research question one, we learned that performance improved for all those in the ModelME intervention. At the same time, performance decreased among two students with LD in the comparison group. This outcome shows the potential of the program to increase students’ access to improved performance in fractions, which is historically an area where we see inequities for this population in terms of access to STEM learning and job opportunities (Kalra et al., 2020; Waber et al., 2022). In terms of schematic change, three students in the ModelME group did show increased conceptions in the form of scheme change. Yet, we were surprised to learn that students in both groups expanded and/or contracted their conceptual understanding. Three students in the intervention groups significantly changed their fraction conceptions, while one student’s conception significantly decreased. In the same way, one student in the comparison group significantly increased their fraction schemes, while another student in that group experienced significantly decreased schemes. The mixed results warrant further investigation to understand individualistic factors that led toward or away from conceptual change for students in both groups.

For the second research question, we were interested in learning about the operational changes students experienced as reflected in the pre- and post-test problem solutions. Results reflected an increase in students’ use of more sophisticated partitioning and iterating operations to varying degrees in both groups. For example, an increased use of more sophisticated forms of partitioning alongside the decrease in more rudimentary forms of partitioning showcases operational growth in both groups. Yet, the use of more sophisticated partitioning in the intervention group was more prevalent for before-action and in-action-linked partitioning in the intervention group. In the same way, the increased use of more sophisticated forms of iteration alongside the decrease in more rudimentary forms showcases students’ growth in both groups yet favors the intervention group in terms of the use of partitioning to iterate, such suggests for three students a coordination of those actions. That is, iteration combined with partitioning appeared as a prevalent driver of conceptual change in the intervention group (Tzur, 2007) and led toward significantly changed conceptions for several students in that group.

Conceptual change is also apparent in the scheme change, or lack thereof, evidenced by both groups. For example, as evidenced by the test, three students in the intervention group advanced from the unit fraction scheme to the partitive fraction scheme—a significant change—and two out of five students advanced to the unit fraction scheme—also a significant change. For these students, advances in partitioning processes appeared to occur alongside advances in iterating processes, suggesting a coordination of operations. For students in the comparison group, one out of four students advanced from the unit fraction scheme to the partitive fraction scheme—a significant change. Two students in the comparison did not advance their pretest schemes. For students in the intervention group, advances in iterating processes appeared to lead to development and, arguably, scheme change for those students. These findings suggest that learning was occurring, albeit at differential rates, for students with LD (e.g., Hunt et al., 2023a; Crawford, 2022), and more so for students in the intervention group. That is, operational development precedes schematic change (Wilkins & Norton, 2018), so the fact that the students’ operations were progressing suggests conceptual change that may not yet be reflected in their mental schemes. More intricate analysis is needed to understand the user experience in ModelMe group to understand the differential effects of the program on scheme change.

In all, the findings suggest that the game was effective for some students in the intervention group in terms of advancing their conceptual notions of fractions through the problem challenges, nudges, and/or after game discussions, and all students in terms of advancing their performance. Working from what students do know, as opposed to what they do not, may work to address inequities, challenge deficit depictions of students with LD, and build and connect their identities to mathematics. More work is needed to further test this assertion.

Limitations and Future Research

The results show the potential of the program to improve students’ access and achievement in STEM education. Yet, limitations and areas for potential future research exist. More research is needed with larger groups of students with LD to further understand the program’s affordances and limitations within and beyond mainstream areas of equity. That is, more research is needed to find out the program’s impact on other outcomes important to equity, such as identity and empowerment in students’ math and math learning. Development of additional content and spaces to connect to students’ identities and empowerment are warranted. Additionally, we did not collect or analyze students’ real-time game play or operations apparent as they played the game. Thus, the pre- and post-tests are approximations of operational change. More work is needed using students’ real-time gameplay to understand individual effects of the program on students’ conceptual change and performance.

Several other limitations need to be acknowledged in this study. First, the sample size, although reflective of the prevalence rates of LD in the general population, is small. Second, there is substantial variability among students with LD. Thus, the reported findings are limited to the current sample. Finally, differences in cognitive functioning, complexity of LD, and/or prior math knowledge were not measured or reflected in the analysis. Future work should address these limitations with larger sample sizes and intricate analyses of students’ disability and prior knowledge in relation to their learning in the game.

5. Conclusions

In this study, we learned that the ModelME program has the potential to significantly impact students with LD in terms of their performance in fractions, increasing their access to STEM. More research is needed to better understand the differential effects of the program on students’ achievement as well as opportunities for the program to better connect to students’ longer-term growth and empowerment in mathematics.