Tracking fraction knowledge development using grade-appropriate assessments

Abstract

Fraction knowledge is foundational for later mathematics achievement, yet its developmental organization remains debated and is constrained by the limited availability of grade-appropriate measures with established psychometric properties. We developed and evaluated the Fraction Knowledge Assessment (FKA), a set of grade-specific instruments for 2nd, 3rd, 5th, and 6th grades, and examined changes in fraction knowledge across two adjacent-grade transitions. Using an accelerated longitudinal design, we followed 342 children in two cohorts (2nd–3rd grade; 5th–6th grade). Participants completed standardized mathematics and cognitive measures at baseline and grade-appropriate FKAs across two consecutive years. Confirmatory factor analysis and item response theory were used to examine dimensional structure, item functioning, and to link FKA versions across grades prior to developmental analyses. Results provided no strong evidence for a robust separation between conceptual and procedural fraction knowledge, with one-factor solutions showing comparable fit to two-factor models and highly correlated latent factors. Item-level analyses revealed developmental shifts: fraction magnitude items differentiated ability more strongly in earlier grades, whereas fraction arithmetic items were more discriminative in later grades. Using equated ability scores, significant gains were observed in both cohorts, with larger improvements from 2nd to 3rd grade than from 5th to 6th grade. Baseline mathematics achievement and working memory were longitudinally associated with later fraction knowledge. These findings support an integrated account of fraction knowledge development and illustrate how psychometrically linked, grade-appropriate assessments can be used to characterize developmental change.

Introduction

Fraction knowledge predicts advanced mathematics skills and supports positive academic, occupational, and health outcomes (Bailey et al., 2012; National Mathematics Advisory Panel, 2008). However, fractions remain one of the most challenging topics in elementary school (National Center for Education Statistics, 2017). Children, and even adults, encounter difficulties processing the magnitude of fractions, solving fraction operations (i.e., addition, subtraction, multiplication, and division), and applying fraction concepts to address academic and real-world problems (Neagoy, 2017; Stigler et al., 2010). Characterizing how fraction knowledge changes across grade levels, using reliable and developmentally appropriate instruments, is critical for informing developmental accounts of mathematical cognition and situating educational practices within a developmental framework. In this study, we designed and tested grade-appropriate measures and characterized fraction knowledge development across earlier (2nd grade to 3rd grade) and later elementary school grades (5th grade to 6th grade).

Definition of fraction knowledge

Fraction knowledge has been broadly defined as a blend of conceptual and procedural knowledge (Bailey et al., 2014; Booth & Newton, 2012; Hansen et al., 2015; Lenz et al., 2020). Conceptual mathematics knowledge is typically defined as understanding properties, relations, and principles (i.e., knowing why), whereas procedural mathematics knowledge is typically defined as knowing how to apply rules and algorithms to solve math problems (i.e., knowing how) (Rittle-Johnson & Schneider, 2014).

Specifically for fractions, as defined by Bailey and colleagues (2015), conceptual fraction knowledge is the capacity to comprehend fraction facts and properties, encompassing elements such as fraction notation (e.g., reading and writing fractions in the format numerator/denominator), magnitude (e.g., comparing 3/8 vs. 2/9), equivalence (e.g., identifying that 1/2 = 2/4 = 3/6), and density (e.g., understanding that there are an infinite number of fractions between two fractions). In contrast, procedural fraction knowledge is proficiency in applying a series of algorithms to solve fraction problems, particularly arithmetic operations (e.g., adding fractions with uncommon denominators by finding the common denominator, or applying the “invert and multiply” rule for fraction division) (Bailey et al., 2015; Rittle-Johnson et al., 2001; Rittle-Johnson & Schneider, 2014). Thus, the fraction knowledge construct can be conceptualized as a spectrum of competencies organized around understanding fraction magnitudes and properties, as well as solving diverse problems involving fractions.

Theoretical accounts of fraction knowledge development

How fraction knowledge develops has been debated in the literature. Some developmental frameworks propose that fraction knowledge develops through progressive reorganization and integration of multiple competencies and strategies, which coexist and mutually influence one another over time (e.g., Siegler et al., 2011). In contrast, other models conceptualize fraction knowledge as multicomponent, highlighting the importance of distinguishing between different types of knowledge and processes whose development may not be uniform (e.g., Gilmore, 2023). These frameworks offer distinct perspectives on fraction knowledge development and therefore require psychometrically rigorous measurement approaches capable of testing whether fraction knowledge is best represented as a unified construct, partially separable components, or a structure whose organization changes across development.

The integrated theory of numerical development (Siegler et al., 2011) proposes that fraction knowledge development is grounded in an unified representation of numerical magnitudes. Fraction magnitudes are assumed to be represented on a mental number line that extends from whole numbers to fractions and becomes increasingly precise across development. The theory is grounded in two core assumptions: (1) that understanding of fraction magnitudes builds on prior magnitude representations (e.g., whole-number magnitude processing), and (2) that understanding of fraction magnitudes supports learning and remembering fraction procedures by enabling children to evaluate the solutions to fraction arithmetic problems. This perspective emphasizes the foundational role of magnitude understanding in fraction learning and treats conceptual and procedural fraction knowledge as closely intertwined, with one being highly correlated with the other.

In contrast, the multi-level framework of numerical cognition (Gilmore, 2023) proposes that mathematics knowledge comprises multiple, distinguishable components whose relations and developmental trajectories might differ. Within this framework, specific contents such as fraction concepts and fraction procedures can be treated as separable components of mathematical understanding. Their development may diverge across time and instructional contexts, being influenced by domain-general cognitive skills (e.g., attention, working memory), as well as basic mathematical processes, such as nonsymbolic magnitude processing, number ordering, and place-value understanding. The multi-level framework of numerical cognition thus highlights the importance of analytically distinguishing components of fraction knowledge, and the role of domain-general skills in supporting their development.

How the conceptualization and operationalization of fraction knowledge has influenced developmental research

It remains unclear how developmental trajectories map onto the integrated theory of numerical development (Siegler et al., 2011) and the multi-level framework of numerical cognition (Gilmore, 2023). A central issue concerns the theoretical definition and empirical separability of conceptual and procedural knowledge. While the distinction between conceptual and procedural knowledge is intuitively appealing, its validity has been challenged on both theoretical and empirical grounds (Matthews & Viegut, 2024, p. 519–523). For example, Rittle-Johnson and colleagues (2001) argue that the conceptual/procedural distinction should be thought of as endpoints along a continuum, rather than a strict dichotomy. Furthermore, only 35% of researchers even define what they mean by procedural and conceptual knowledge (Crooks & Alibali, 2014). Empirically, Rittle-Johnson & Alibali (1999) showed that procedural and conceptual knowledge were intimately intertwined: Conceptual instruction improved both conceptual understanding and procedural knowledge; procedural knowledge, conversely, led to improved conceptual understanding. For these reasons, Matthews and Viegut (2024) argue that “the procedural vs. conceptual distinction is arguably not much of a distinction at all.” (p. 520)

Debates around conceptual and procedural fraction knowledge have been reflected in their operationalizations in assessment tools. Some studies have focused on a narrow set of skills, such as fraction operations or magnitude comparisons, treating fraction knowledge as a one-dimensional construct (Bailey et al., 2014; Fazio et al., 2016). Others have adopted a more comprehensive approach by combining multiple skills—such as magnitude processing, property identification, and operations—into tasks assumed to measure a single underlying dimension (Matthews et al., 2016; Schneider & Stern, 2010), or by explicitly distinguishing between conceptual and procedural factors (Jordan et al., 2013; Kalra et al., 2020; Ye et al., 2016). One limitation is that most studies have not used psychometric approaches, such as confirmatory factor analysis (CFA), to test whether their measures had one or multiple factors, leaving it unclear whether conceptual and procedural fraction knowledge are best characterized as unified or distinguishable components.

Understanding whether the structure of conceptual and procedural fraction knowledge differs across development is critical for tracking their growth. Prior empirical studies exploring this distinction have yielded mixed findings, with the strongest evidence for the distinction coming in the oldest students. For example, Lenz et al. (2020) used CFA to demonstrate that conceptual and procedural factors are separable in 8th and 9th grade German students. However, others find that one- and two-dimensional models fit equally well, or that factors are too highly correlated (r>.80) to be considered distinct (U.S. Grades 6 and 8, Hallett et al., 2012; German Grades 5 and 6, Schneider & Stern, 2010; U.S. Grades 3–6, Ye et al., 2016). The high correlation between these factors suggests they might be intertwined, or even that they constitute the same construct. These mixed results suggest that psychometric analyses are essential not only for clarifying the structure of fraction knowledge measures, but also for constraining interpretations of whether fraction knowledge development is best described as unified or organized into multiple components.

The importance of psychometric evidence in fraction knowledge research

The heterogeneous operationalization of fraction knowledge as a unified or multi-factor construct is reflected in the wide variety of assessment tools used across developmental studies. In the absence of a gold-standard measure, most studies have predominantly relied on investigator-created assessments (e.g., McMullen et al., 2015; Siegler et al., 2011) or parts of omnibus standardized tests (e.g., Jordan et al., 2013; Siegler et al., 2012). To index fraction development using these tasks, researchers often rely on changes in raw accuracy (e.g., Bailey et al., 2014; Hansen et al., 2015) or the percentage of correct responses in a task over time (e.g., Hecht et al., 2003; Kalra et al., 2020). However, raw scores and percent of correct responses depend not only on participants' ability, but also on the difficulty of the items (Baker, 2001). For example, a task appropriate for 3rd graders may be too easy for older students, leading to ceiling effects. Failing to calibrate student skill with item difficulty prevents valid developmental comparisons.

Item Response Theory (IRT) provides a refined approach to assess the quality of fraction knowledge measures, since it evaluates item-level performance in relation to students' underlying abilities. For instance, Van Hoof and colleagues (2015) employed IRT to determine which fraction tasks were most difficult for fourth-grade students, based on item difficulty parameters. They found that students struggled more with items assessing fraction density than those targeting fraction size or operations. Thus, IRT can provide valuable information about both the quality of the assessment and the distribution of students' knowledge, accounting for psychometric properties of the test, while also informing about developmental patterns.

To date, most developmental studies using IRT have been cross-sectional and have primarily focused on older students or adults (Chan et al., 2007; Dorri & Rafiepour, 2018; Ölmez & Izsák, 2021). However, as students’ fraction knowledge is closely aligned with their academic experiences, it is important to account for the progression of the curriculum when tracking development. This can be achieved by using grade-specific assessments tailored to students' developmental stages which include cloned items tested and linked via IRT (W.-C. Lee & Ban, 2009). This approach allows determining the probability that a student with a particular ability level (i.e., ability theta, θ) will successfully complete an item of a certain difficulty level (i.e., difficulty level delta, δ). It provides a calibration match between the students' abilities and the difficulty levels of the items, which enables researchers to ensure that observed differences across grades reflect true developmental changes rather than artifacts of test design.

Fraction knowledge development across elementary school years

Most studies on fraction knowledge development have prioritized grades in which the mathematics curriculum explicitly addresses fractions, typically ranging from 3rd to 7th grade. For example, a developmental study examining the trajectory of fraction magnitude comparison and number line estimation tracked children from 4th to 12th grade (Wang & Siegler, 2023). This study showed that children’s representations of fractions are consistent with a number line and closely aligned with the accuracy of their decimal magnitude representations, supporting the integrated theory of numerical development.

The Delaware Longitudinal Study of Fraction Learning, one of the largest investigations in this field, also followed children in more advanced elementary school grades (from the winter of 3rd grade to 6th grade; Jordan et al., 2017). Across several studies, attention and working memory emerged as significant predictors of both conceptual and procedural fraction knowledge. In contrast, whole number magnitude understanding was a significant predictor of conceptual fraction knowledge, whereas fluency in solving arithmetic operations was associated with procedural fraction knowledge (Hansen et al., 2015; Jordan et al., 2013; Ye et al., 2016). When confirmatory factor analysis was applied, conceptual and procedural fraction knowledge measures loaded onto a single latent factor representing general fraction knowledge. However, remaining variance in each measure was not accounted for by the common factor, indicating that each factor captured non-shared knowledge components (Ye et al., 2016). These findings support a hybrid view in which conceptual and procedural fraction knowledge are neither fully separable nor fully interchangeable, with both shared and distinct components.

Since most longitudinal investigations of fraction knowledge development have targeted children in advanced elementary school grades, our understanding of how fraction knowledge changes from before to after formal instruction has been limited. Children may begin developing a foundational understanding of fractions early in development, even before formal instruction in the classroom. For instance, preschool students already demonstrate the ability to compare and solve operations involving nonsymbolic ratio magnitudes, identify and estimate the magnitude of one-half, and recognize some frequently used fractions (Gelman, 1991; Hurst & Cordes, 2018; Mix et al., 1999; Park et al., 2021). Young children's fraction knowledge develops both through informal experiences at home with their caregivers (Eason & Ramani, 2020) and preschool experiences encompassing various mathematical concepts, including whole-number arithmetic, financial literacy, and geometry (Bailey et al., 2014; Viegut et al., 2023). Examining fraction knowledge across earlier grades using psychometric tasks is essential to determine if conceptual and procedural factors are separate, intertwined, or shifting across development.

The Present Study

As discussed above, how fraction knowledge develops remains an open question. Our ability to determine how fraction knowledge develops—whether it develops in an integrated manner (Siegler et al., 2011), separately into conceptual and procedural knowledge (Gilmore, 2023), or in a hybrid way (e.g., Ye et al., 2016)—has been challenged by the limited availability of fraction knowledge measures with tested factor structures and psychometric properties. These limitations make it difficult to determine whether observed integration or distinctions between conceptual and procedural fraction knowledge reflect developmental shifts or measurement artifacts. Moreover, because most longitudinal investigations have focused on students in more advanced elementary grades, it remains unclear whether the separation between conceptual and procedural fraction knowledge emerges, fades, or remains constant throughout elementary school.

In this study, we targeted these gaps by tracking fraction knowledge development with psychometrically tested assessments. Adopting an accelerated longitudinal design, we followed one cohort of students from 2nd to 3rd grade and another from 5th to 6th grade. This design captures the onset of formal instruction and the transition to pre-algebra (National Governors Association, 2010), allowing us to characterize development across key stages. Our goals were twofold: (1) to design and evaluate the psychometric properties of grade-appropriate fraction knowledge measures, minimizing measurement artifacts in developmental analyses; and (2) to examine the relation between conceptual and procedural fraction knowledge, and to characterize changes in fraction knowledge across elementary school grades. Accordingly, we developed the Fraction Knowledge Assessment (FKA), a grade-specific measure assessing a broad range of conceptual and procedural fraction skills aligned with the Mathematics Common Core Standards and drawn from prior research and large-scale assessments (e.g., National Assessment of Educational Progress and the Trends in International Mathematics and Science Study; Carpenter, 1981; Hallett et al., 2012).

We first used confirmatory factor analysis (CFA) to examine the dimensional structure of fraction knowledge within each grade. If fraction knowledge is best characterized as an integrated construct, aligned with the integrated theory of numerical development (Siegler et al., 2011), the conceptual and procedural items should not form fully separable factors. In contrast, if there is a clear separation between conceptual and procedural fraction knowledge, aligned with the multi-level framework of numerical cognition (Gilmore, 2023), distinct conceptual and procedural factors should emerge. Finally, developmental change in structure, aligned with a hybrid approach (Ye et al., 2016), would be reflected in different patterns across grades.

We then employed item response theory (IRT) to evaluate item-level psychometric properties and link scores across adjacent grades (2nd–3rd and 5th–6th). This approach allowed us to identify how items varied in difficulty and discrimination across grades. We also aligned assessments with students’ ability levels, and placed performance on a common metric across grades while accounting for curriculum shifts. Using these calibrated measures, we examined longitudinal changes in fraction knowledge within each cohort, characterized grade-specific challenges, and assessed individual differences in growth.

Finally, we examined longitudinal shifts in fraction knowledge while controlling for students’ age, attention, working memory, math fluency, and calculation skills at baseline. We expected overall improvements in fraction knowledge across both cohorts. However, since prior studies found that longitudinal trajectories differ based on individual differences (Ye et al., 2016), some participants might show similar or even lower performance from baseline to a following grade. From an integrated developmental perspective (Siegler et al., 2011), we would expect that stronger baseline whole number math fluency and calculation skills would predict later fraction knowledge. In addition, we would expect associations between domain-general skills at baseline and fraction knowledge, which is also consistent with the multi-level framework of numerical cognition (Gilmore, 2023).

Methods

Participants

The sample for the current study was drawn from a larger investigation that adopted an accelerated longitudinal design. In the larger study, our team longitudinally followed two cohorts of students, a younger and an older one (total n = 352). The younger cohort was followed from 2^nd to 5^th grade, and the older cohort was followed from 5^th to 8^th grade. This design allowed us to track participants’ development longitudinally, conducting within-cohort comparisons, and cross-sectionally, comparing the two cohorts.

Participants were recruited in two enrollment waves through invitation letters sent to schools in a medium-sized Midwestern city. Participants with a reported diagnosis of a psychiatric, neurological, or learning disability, or a sensory impairment (e.g., blindness or deafness), as identified by parent-report, were excluded from participation. Therefore, our sample did not include students with special educational needs. All participants were able to communicate in English.

In the larger study, a total of 342 participants completed grade-appropriate versions of the FKA. We originally planned for younger participants to complete FKAs in 2^nd, 3^rd, and 5^th grades, and for older participants to complete FKAs in 5^th, 6^th, and 8^th grades. Due to the COVID-19 pandemic, data collection was disrupted in the last year of the longitudinal investigation. Consequently, we do not examine the 8^th-grade FKA in this study. Additionally, although we include data from younger participants' 5^th-grade FKA in the psychometric analyses of the measure, we constraint the developmental analyses to the first and second time points (i.e., from 2^nd to 3^rd grade for younger participants and from 5^th to 6^th grade for older participants).

Table 1 shows demographic characteristics of the samples that completed each FKA version and is included in the current study. Despite our efforts to recruit a diverse sample, most participants self-identified as non-Hispanic (71% non-Hispanic, 11% Hispanic or Latine, 4% other, 14% not reported) and White (68% White, 4% Black or African American, 2% Asian, 8% multiracial, 18% not reported). The proportion of White participants in our sample was comparable to that of public-school students in the state (68% vs. 66.47%), according to U.S. Department of Education data (2014). However, Black and Hispanic students were underrepresented relative to the state population (4% vs. 8.7% for Black students; 11% vs. 13.6% for Hispanic students), and a substantial proportion of participants did not report race or ethnicity. Accordingly, the sample may not be fully representative of state demographic characteristics. Of the 165 families that reported parental education, 92% held a college or university degree, 7% completed partial college or at least one year of specialized training, and 1% listed high school diploma as their most advanced degree.

Table 1.

Participants’ demographic characteristics

FKA Version	N	Age		Gender (n)
		Mean	SD	Male	Female	Non-reported
2nd Grade Form A	89	8.11	0.64	48	33	8
2nd Grade Form B	97	8.01	0.56	55	40	2
3rd Grade	163	9.00	0.87	98	61	4
5th Grade Form A	54	10.99	0.58	25	26	3
5th Grade Form B	118	10.94	0.55	65	51	2
6th Grade	114	11.95	0.53	63	47	4

Note. FKA = Fraction Knowledge Assessment.

General Procedures

The local Institutional Review Board (IRB) approved this study, ensuring that all procedures met ethical guidelines for research with human participants. All participants gave written assent, and their parents or legal guardians signed a consent form. As part of the larger, accelerated longitudinal study, participants were assessed in two one-hour long individual sessions conducted in our laboratory and led by trained research assistants in each year of participation. The first session included standardized measures of domain-general cognitive skills (e.g., attention, working memory), and the second session included measures of fraction knowledge and mathematics achievement. During sessions, participants were offered short breaks to minimize fatigue.

The analytic plan for the present study includes the attention and working memory measures completed in the first session and the fraction and mathematics achievement measures completed in the second session (see Instruments, below, for details). We analyzed attention, working memory, and general mathematics achievement measures collected in Year 1, and fraction knowledge measures collected in Years 1 and 2. The interval between participation in Years 1 and 2 was approximately 12 months (M = 12.52 months, SD = 2.08).

Data availability, reproducibility, and open science

We share the dataset, along with the analysis R code and comprehensive results (including data analyses raw outputs and plots) via our OSF page (<https://osf.io/wfmn8/?view_only=f2dcc45e0bb24805b2340ece5415b557>). We also share all versions of the Fraction Knowledge Assessment (FKA), including original task booklets, grade books, and commented FKAs, to be used in future studies.

Instruments

Fraction Knowledge Assessment (FKA).

The Fraction Knowledge Assessment (FKA) is a pencil-and-paper task designed to assess participants' fraction knowledge. The FKA was originally developed for adults by Matthews and colleagues (2016), and our team adapted it to align with the mathematics curriculum of the targeted grades. The FKAs incorporate items sourced from previous research articles, as well as international, national, and state achievement exams (including Carpenter et al., 1981; Hallett et al., 2012, and editions of the National Assessment of Educational Progress and the Trends in International Mathematics and Science Study).The FKAs items were selected to align with grade-level curriculum expectations and to represent a range of fraction knowledge domains and psychometric characteristics. Detailed information regarding task construction is available on OSF.

We designed six FKA versions: 2^nd Grade Form A (41 items), 2^nd Grade Form B (41 items), 3^rd Grade (40 items), 5^th Grade Form A (41 items), 5^th Grade Form B (39 items), and 6^th Grade (37 items). All versions included items designed to highlight either conceptual or procedural fraction knowledge, with the exception of 2^nd Grade Form A, which only included items intended to assess conceptual fraction knowledge. This initial decision was motivated by developmental considerations, as procedural fraction strategies are not yet consistently taught at this grade level. The FKA items designed to highlight conceptual knowledge investigated abilities related to fraction magnitude understanding, multiplicative reasoning, and fraction density. In contrast, FKA items designed to highlight procedural knowledge tested student competence with fraction operations, including addition, subtraction, multiplication, and division. We show example items in Table 2, and more details in Supplementary Materials.

Table 2.

Example of procedural and conceptual Fraction Knowledge Assessment (FKA) items.

Type of knowledge	2nd Grade A, B, and 3rd Grade FKA	5th Grade A, B, and 6th Grade FKA
Conceptual Knowledge	How many fourths do you need to make one whole?	How many possible fractions are between $\frac{1}{8}$ and $\frac{1}{2}$?
Procedural Knowledge	$1-\frac{5}{12}=$	$2\frac{2}{3}+3\frac{1}{4}=$

Some items remained consistent across FKA versions, serving as anchor items to facilitate equating, whereas others were unique to specific versions. Anchor items were selected to represent fraction knowledge content across grade levels, aligned with Mathematics Common Core State Standards, and to span a range of predicted psychometric characteristics, including various levels of difficulty and discrimination. Participants completed the FKA version tailored for their corresponding grade at the time of the assessment (e.g., a 2^nd-grade student completed the 2^nd grade FKA). Participants took approximately 15 minutes to complete the FKA, which had no time limit.

Many participants who completed FKA 2^nd Grade Form A attained perfect scores, with only a few achieving an accuracy below 50% (Figure 1). Similarly, the distribution of FKA 5^th Grade Form A also had a negative skew. To address this bias, we developed Forms B of 2^nd and 5^th Grade FKAs by maintaining some items from Forms A and introducing others with greater computational complexity. In particular, we increased the complexity of some items intended to measure conceptual knowledge (e.g., comparing probabilities, representing a pictorial fraction as an expression, ordering fractions) in FKA 2^nd Grade Form B, and included procedural fraction knowledge items (e.g., fraction addition and subtraction), which were absent in Form A. In the 5^th Grade FKA Form B, we added more items intended to measure conceptual fraction knowledge (i.e., representing fraction magnitude in a number line and understanding that fractions are relative to the size of the whole) and procedural knowledge (e.g., operations with mixed numbers and fractions with unlike operators) to capture increased demands at later grade levels.

Fig. 1.

Violin plots with box-plots representing the distribution of scores of each FKA version. Each dot represents the score of one participant. No floor effect at the group level was observed. Overall, distributions were left-skewed. FKA versions 2^nd-Grade A and 6^th-Grade had the highest mean scores, and 2^nd-Grade B had the lowest mean score. In the plot, the y-axis was converted to percentage to facilitate comparison across FKA versions.

The 2^nd and 5^th Grade Form A FKAs were completed by participants from the first enrollment wave or the larger study, and the 2^nd and 5^th Grade Form B FKAs were completed by those from the second enrollment wave. All younger participants completed FKA 5^th Grade Form B in their final year of the larger study (i.e., those who began as 2^nd graders and concluded as 5^th graders).

Two independent research assistants graded each FKA, following the same rubric (i.e., double-blind grading), which awarded one point for each correct answer. In cases where a discrepancy arose between the scores assigned by the two assistants, a third research assistant independently assessed the task without prior knowledge of the initial scores. The final score was determined based on the consensus reached by all three judges. A similar procedure was followed during the curation of the FKA database: two independent research assistants entered item-level data from each FKA into our database. In instances where a discrepancy was noted, a third research assistant entered the data to achieve a consensus. We adopted this approach to minimize the likelihood of errors and to ensure fair scores.

Standardized Measures.

We investigated participants’ attention, working memory, and general mathematics achievement as covariates. We selected these skills because, as discussed in the introduction, they have been pointed out as critical predictors of fraction knowledge development.

Attention.

We assessed participants’ attentive behavior in their first year in our study with the Pair-Cancellation subtest from the Woodcock-Johnson Tests of Cognitive Abilities - III (WJ-PC; Woodcock et al., 2001). In this subtest, participants are presented with different images and should find, among them, a pair of target images presented in a specific order. Participants have three minutes to complete this subtest and are encouraged to complete it as fast as they can. In our analyses, we included participants’ raw scores in this subtest. In the Woodcock–Johnson normative sample, this task showed good reliability (Cronbach's alpha >.80). Participants completed this measure in their first year in the study.

Working memory.

We assessed participants’ verbal working memory using the Auditory Working Memory subtest from the Woodcock-Johnson Tests of Cognitive Abilities - III (WJ-PC; Woodcock et al., 2001). In this task, participants hear a series of numbers and words in an intermixed order, and are required to repeat them in a reorganized manner, first repeating the objects in sequence then the numbers in ascending order. In the Woodcock–Johnson normative sample, this task showed good reliability (Cronbach's alpha >.80). Participants completed this measure in their first year in the study.

General mathematics achievement.

Students also completed the Math Fluency (WJ-MF) and Calculation (WJ-Calc) subtests of the Woodcock-Johnson III (Woodcock, 2001) as measures of general mathematics skills. The WJ-MF assesses arithmetic fluency by requiring participants to complete 160 simple calculation problems, including addition, subtraction, and multiplication, within a time limit of 3 minutes. One point is given for each correct answer. The WJ-Calc subtest evaluates written calculation skills by presenting participants with 45 problems of increasing complexity, ranging from basic addition to trigonometry. Participants have no time limit to complete this test, and one point is given for each correct answer. In the Woodcock–Johnson normative sample, these two tasks showed good reliability (Cronbach's alpha >.90) and concurrent validity (correlation with other mathematics tests >.50) (Woodcock, 2001). In the current study, we analyze participants' scores in the WJ-MF and WJ-Calc completed in the first year of the longitudinal study (i.e., 2^nd grade among younger participants and 5^th grade for older participants).

Analyses

All analyses were implemented in R (version 4.3.0) and R Studio (version 2025.05.0). We selected statistical methods suitable to address the twofold nature of our research goals: (1) to assess the psychometric properties of our instrument, and (2) to track the development of fraction knowledge using a developmentally appropriate and psychometrically sound measure. Accordingly, we conducted four primary psychometric analyses (i.e., confirmatory factor analysis, item response theory modeling, reliability analyses, and a score linking procedure) to ensure that the FKAs provided a developmentally interpretable measurement of fraction knowledge. We then investigated the development of fraction knowledge from 2nd to 3rd grade and from 5th to 6th grade.

As a first step, we investigated whether items planned to measure conceptual and procedural fraction knowledge fit this structure by conducting a confirmatory factor analysis (CFA) on each FKA version. We contrasted the fit of a two-factor model (conceptual and procedural fraction knowledge factors) with a one-factor model (i.e., assuming no separation between conceptual and procedural items) for each grade. These analyses were performed using the mirt package (Chalmers et al., 2015).

We then used item response theory (IRT) to examine participants' abilities at the item level for each FKA, with the dual aim of assessing the psychometric properties of the tasks and gaining insight into how students perform across grade levels. Specifically, we implemented both a Rasch model and a two-parameter logistic (2PL) model using the ltm and irtoys packages in R (Partchev et al., 2017; Rizopoulos, 2005). The Rasch model estimates the probability of a correct response as a function of participant ability and item difficulty, assuming equal discrimination across items, whereas the 2PL model allows item discrimination to vary. We compared model fit to determine the most appropriate approach for subsequent analyses, and followed up with reliability analyses. For these analyses, we evaluated measurement precision within the IRT framework using the test information function (TIF). The TIF was computed by summing item information functions across all items, where item information for the 2PL model is defined as

$${\mathrm{I}}_{\mathrm{i}}\left(\mathrm{\theta }\right)={\mathrm{\alpha }}_{\mathrm{i}}^2{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{\theta }\right)[1-{\mathrm{P}}_{\mathrm{i}}\left(\mathrm{\theta }\right)]$$

Conditional standard errors of measurement can be derived from the inverse of the square root of test information:

$$S\mathrm{E}\left(\widehat{\mathrm{\theta }}\right)=\frac{1}{\sqrt{\mathrm{I}\left(\mathrm{\theta }\right)}}$$

In addition, we can calculate an overall marginal reliability index (similar to Cronbach’s alpha, but IRT based) by integrating over the latent proficiency density, i.e,

$${\rho }_{\mathrm{xx}\text{'}}\left(\mathrm{\alpha }\right)=\int \frac{\mathrm{I}\left(\mathrm{\theta }\right)}{\mathrm{I}\left(\mathrm{\theta }\right)+1}\mathrm{g}\left(\mathrm{\theta }\right)\mathrm{d}\left(\mathrm{\theta }\right)$$

where g(θ) is the proficiency density function (i.e., assumed to follow a standard normal distribution) (Cheng et al., 2012; Samejima, 1994).

Fourth, we linked participants' ability scores across grades, as estimated in the 2PL models (which showed a better fit than the Rasch models and provided more information for the linking procedure). In these analyses, implemented with the plink package (Weeks, 2010), we used Stocking-Lord linking constants. This linking procedure was conducted separately to each cohort (i.e., linked participants' performance in 2^nd grade to their performance in 3^rd grade, and participants' performance in 5^th grade to their performance in 6^th grade), based on overlapping items across versions. These items were those that remained identical across 2^nd Grade FKA Forms A and B, and 2^nd Grade FKA Form B and 3^rd Grade FKA for the younger cohort. For the older cohort, they were those overlapping across 5^th Grade FKA Forms A and B, and 5^th Grade FKA Form B and 6^th Grade FKA. This approach enhances the investigation of development by aligning different FKAs to a common metric, despite using distinct versions of the task tailored to specific grade levels.

Finally, we examined fraction knowledge development by leveraging the accelerated longitudinal structure of the dataset. Specifically, we analyzed data from participants who completed the FKA in two consecutive years, separately for each cohort (i.e., 2nd–3rd grades for the younger cohort and 5th–6th grades for the older cohort). These analyses were designed to capture within-individual change in fraction knowledge across adjacent grade levels. To address this goal, we implemented comparison tasks, correlation analyses, and linear mixed-effects models to estimate grade-related changes while accounting for repeated measurements at the participant level. In these models, we also investigated how domain general skills (attention and working memory) and mathematics achievement (math fluency and calculation) predicted fraction knowledge. These models were implemented using the lme4 package (Bates, 2018), with p-values estimated via lmerTest (Kuznetsova et al., 2022).

Given that the study design is not fully longitudinal across the entire developmental span, direct statistical comparisons between cohorts were not used to infer growth trajectories. Instead, results from the younger and older cohorts were contrasted descriptively to evaluate the consistency and direction of developmental patterns across distinct age ranges. Thus, we combined within-cohort longitudinal evidence with cross-cohort comparisons interpreted at the descriptive level.

Results

Descriptive Statistics

Figure 1 illustrates the distribution of FKA scores across grades. Overall, FKA scores were relatively high, even among 2^nd graders, despite fractions not yet being part of their mathematics curriculum. As the numbers of items varied across FKA versions, we converted raw accuracy to percent correct to facilitate descriptive alignment across versions. Participants' mean accuracy was 73% (Interquartile Range (IQR) = [61%, 88%]) on the FKA 2^nd Grade Form A and 52% (IQR = [37%, 66%]) on Form B. For the FKA 3^rd Grade, the mean accuracy was 73% (IQR = [63%, 88%]). The 5^th Grade FKA showed a mean accuracy of 66% for both Form A (IQR = [48%, 80%]) and Form B (IQR = [51%, 82%]). Finally, participants scored an average of 73% (IQR = [62%, 86%]) on the FKA 6^th Grade. We report further details on the distributions, including mean raw scores, standard-deviation, and IQR, in Supplementary Materials.

Some FKA versions showed a negative skew, indicating that most participants were able to successfully solve the items designed for their respective grade levels. This pattern was particularly evident in the FKA 2^nd Grade Form A and FKA 5^th Grade Form A. As noted in the Methods section, we then developed Form B of these versions to increase computational difficulty and minimize ceiling effects. Descriptive analyses showed that the mean and IQR of percent-correct scores were lower in FKA 2^nd Grade Form A than Form B, whereas FKA 5^th Grade Forms A and B showed comparable mean and IQR values. This pattern may suggest that the increase in task difficulty from Form A to Form B was more effective for the 2^nd Grade FKAs than for the 5^th Grade FKAs, which represents a potential limitation of the assessment design. However, because Forms A and B were completed by participants from different enrollment waves, these differences may also reflect cohort variation rather than form difficulty. This result highlights the importance of the IRT analyses, which permit estimation of students’ ability while accounting for item difficulty and enable the horizontal equating of FKA versions across forms and enrollment waves.

Separation of conceptual and procedural fraction knowledge

We used confirmatory factor analysis (CFA) to assess how FKA items would load into a conceptual and a procedural factor across grades. Understanding the factor structure of the FKA is critical to define whether our longitudinal analyses should consider fraction knowledge as a unified construct, more aligned with the integrated theory of numerical development (Siegler et al., 2011), or as a conceptual and a procedural construct, separately, more aligned with a multi-level framework of numerical cognition (Gilmore, 2023). The FKA 2^nd Grade Form A was not included in these analyses since its items were designed to highlight only conceptual fraction knowledge. We broadly describe the factor analyses results here and show details in Supplementary Materials and OSF.

Results did not provide unequivocal evidence for a two-factor structure for the FKAs. One-factor and two-factor models had adequate fit for FKA 2^nd Grade Form B, 3^rd Grade, 5^th Grade Form A, and 5^th Grade Form B (Table 3). All one-factor models had adequate fit (RSMA <.10). Although information criteria tended to favor the two-factor model for the 3rd Grade and 5th Grade FKAs, the magnitude of these differences was modest, and one- and two-factor models showed broadly comparable overall fit across FKA versions. Moreover, when we explored a two-factor solution, the two factors were highly correlated with each other (correlations ranging from .75 to .89, all p < .001; Supplementary Materials), suggesting overlap between constructs.

Table 3.

Results from the Confirmatory Factor Analysis.

FKA Version	Model	RMSEA	AIC	SABIC	BIC	LogLik	χ 2	df	p
2nd Grade Form B	One-dimensional model	0.083	3615.68	3571.96	3808.79	−1732.84	1.117	1	0.29
	Conceptual and Procedural Factors	–	3616.57	3572.26	3812.24	−1732.28
3rd Grade	One-dimensional model	0.054	4900.75	4891.52	5144.74	−2370.38	22.308	1	0
	Conceptual and Procedural Factors	–	4880.45	4871.09	5127.48	−2359.22
5th Grade Form A	One-dimensional model	0.070	2078.99	1987.93	2236.12	−960.50	33.314	1	0
	Conceptual and Procedural Factors	–	2047.68	1955.46	2236.12	−943.84
5th Grade Form B	One-dimensional model	0.064	4081.24	4051.95	4289.04	−1965.62	60.765	1	0
	Conceptual and Procedural Factors	–	4022.47	3992.80	4233.05	−1935.24

Note. FKA = Fraction Knowledge Assessment.

For 6^th Grade FKA, the CFA with a predefined conceptual and procedural factor structure failed to converge. Thus, we conducted an exploratory factor analysis (EFA). When a two-factor model was requested, the resulting exploratory solution diverged from the conceptual and procedural framework. Most importantly, however, the alternative solution performed worse than the one-factor model. Furthermore, following rotation, the factors in the two-factor solution were highly correlated (r = .69, p < .001), providing low evidence for a clear separation of factors in this task.

These results indicate that while a two-factor representation may offer a statistically better fit in some grades, the practical distinction between conceptual and procedural fraction knowledge is limited, with two factors being highly correlated. These findings are more consistent with the integrated theory of numerical development (Siegler et al., 2011), which assumes that conceptual and procedural fraction knowledge are intertwined. They align with previous research on the factor structure of other fraction and rational number tasks, which suggest they might measure a single underlying construct (e.g., Hallett et al., 2012; Schneider & Stern, 2010). Given that all FKA versions demonstrated an acceptable unidimensional structure, and that factors from two-dimensional models were highly correlated, we assumed one-dimensionality for conducting the IRT analyses and linking the FKA versions.

Difficulty and Discrimination of Fraction Tasks Across 2nd to 6th Grade FKAs

After analyzing the factor structure of the FKAs, and finding acceptable fits for one-dimensionality, we proceeded with Item Response Theory (IRT) analyses. These analyses estimate the difficulty and discrimination of each item, as well as students' abilities. In addition to informing on the quality of the items, these analyses also allow for a more precise examination of how students with different fraction knowledge perform across the task items.

We conducted Rasch and 2PL models without including any prior information on individual items. As shown in Table 4, a likelihood ratio test indicated that the Rasch and 2PL models did not fit equally well. Overall, AIC values indicated a better fit for the 2PL model across all FKA versions when compared to the Rasch model. Similarly, log-likelihood analyses favored the 2PL model across all FKA versions compared to the Rasch model. However, BIC values favored the 2PL model only for FKA 3^rd Grade and 5^th Grade Form B, while indicating better fit for the Rasch model in the other FKA versions.

Table 4.

Rasch and 2PL Model fit.

FKA Version	AIC		BIC		Log-Likelihood		LRT (df)
	Rasch	2PL	Rasch	2PL	Rasch	2PL
2nd Grade Form A	3045.82	3026.60	3147.86	3230.66	−1481.91	−1431.30	101.22(41)***
2nd Grade Form B	3723.19	3692.94	3823.60	3893.77	−1822.59	−1768.47	108.24(39)***
3rd Grade	5027.95	4903.80	5149.94	5147.79	−2473.97	−2371.90	204.15(40)***
5th Grade Form A	1995.09	1989.35	2068.68	2136.53	−960.54	−920.67	79.74(37)***
5th Grade Form B	4214.95	4103.95	4323.01	4320.06	−2068.48	−1973.97	189.01(39)***
6th Grade	3705.61	3568.77	3806.85	3765.78	−1815.80	−1712.39	206.83(35)***

Note. FKA = Fraction Knowledge Assessment.

Given that the 2PL model generally demonstrated better fit than the Rasch model across most indices and FKA versions, we selected it for the subsequent analyses. The 2PL model offers greater informational value than the Rasch model, as it estimates both item difficulty and discrimination, whereas the Rasch model considers only item difficulty. In the following paragraphs, we describe the general patterns of item difficulty and discrimination for each FKA version, as identified by the 2PL model. We provide details of the Rasch model analyses in the Supplementary Materials.

Table 5 presents the results of the 2PL model. Following Baker's proposed cutoffs to assess the 2PL indices, we considered difficulty values below zero easy, and those above zero difficult; discrimination values greater than 1.35 were interpreted as indicating high to very high discrimination (Baker, 2001). In general, the FKA versions had more easy than difficult items (mean difficulty index varying from −1.70 to 0.06). In contrast, items had high discrimination (mean discrimination index varying from 1.49 to 1.72). These results suggest that, despite low difficulty levels, many items were sensitive to differences in latent ability, effectively distinguishing between participants with higher and lower levels of fraction knowledge.

Table 5.

FKA versions' mean difficulty and discrimination indices

FKA Version	Difficulty Index		Discrimination Index
	Mean (SD)	Range	Mean	Range
2nd Grade Form A	−1.70 (1.96)	−7.50—0.87	1.54 (0.79)	0.45—3.75
2nd Grade Form B	0.06 (1.91)	−3.56—6.38	1.49 (0.79)	0.16—4.01
3rd Grade	−1.15(1.29)	−5.12—1.76	1.72 (0.83)	0.58—3.73
5th Grade Form A	−0.99 (1.75)	−7.59—1.84	1.57 (0.79)	0.09—3.07
5th Grade Form B	−1.04 (1.45)	−5.47—2.23	1.61 (0.89)	0.31—4.08
6th Grade	−1.05 (1.57)	−3.96—5.37	1.49 (0.74)	−0.56—2.95

Note. FKA = Fraction Knowledge Assessment.

While the primary goal of fitting 2PL models was the psychometric evaluation of the FKAs, differences in item difficulty and discrimination indices across grades also provide insight into developmental patterns of fraction knowledge. In particular, reductions in item difficulty for items measuring similar skills across grades may reflect increasing mastery of specific fraction concepts, whereas changes in discrimination may indicate shifts in how strongly those concepts differentiate students with higher versus lower levels of ability. We briefly summarize these patterns here and provide more extensive details for each FKA version in the Supplementary Materials, with full item-level descriptions available on the OSF page.

Overall, in the FKAs completed by younger participants (i.e., 2nd and 3rd graders), items with higher difficulty and discrimination primarily assessed fraction magnitude understanding. For example, in both 2nd Grade FKA Form A and the 3rd Grade FKA, the items with the highest discrimination required participants to convert a symbolic fraction into a pictorial representation (e.g., “shade the fractional part”). Similarly, in 2nd Grade FKA Form B, the most discriminating item asked participants to identify all fractions equivalent to ½. In contrast, items with lower discrimination at these grade levels tended to assess more basic part–whole relations, such as identifying the number of favorable outcomes relative to the total or counting the number of equal parts in a partitioned whole.

This pattern shifted among older participants (i.e., 5th and 6th graders). In these grades, items with lower discrimination more often involved magnitude-related tasks, such as identifying the fraction represented in a pictorial model or determining which of two fractions was larger. By contrast, items with higher difficulty and discrimination required operations, including computations with mixed numbers and fractions without common denominators. Notably, items assessing fraction division exhibited particularly high difficulty in this group.

Taken together, these item-level patterns suggest that foundational magnitude concepts (e.g., part–whole relations and basic magnitude understanding) differentiate performance earlier, whereas more advanced arithmetic skills differentiate students with higher versus lower levels of mathematical ability in later grades.

Reliability of the Fraction Knowledge Assessments

After fitting the 2PL models, we examined FKAs’ reliability by deriving the marginal reliability index from the Test Information Function (TIF). Differently from methods from Classic Test Theory, such as Cronbach's alpha, this metric does not assume that all items share the same discrimination (i.e., it does not assume tau-equivalence), being well-suited for IRT. As shown in Table 6, the estimated marginal reliability was high for each FKA version. This result indicates good average measurement precision across the latent ability distribution, corroborating the use of FKA for additional longitudinal analyses.

Table 6.

FKA internal consistency

FKA Version	Marginal Reliability
2nd Grade Form A	.89
2nd Grade Form B	.92
3rd Grade	.92
5th Grade Form A	.91
5th Grade Form B	.91
6th Grade	.89

Note. FKA = Fraction Knowledge Assessment.

Fraction Knowledge Development

We then examined developmental changes in fraction knowledge by conducting longitudinal analyses within cohorts. To enable direct comparisons of fraction knowledge across adjacent grade levels (i.e., from the 2nd to the 3rd grade for younger participants, and from the 5th to the 6th grade for older participants), we linked different versions of the FKA to a common latent ability scale.

We performed the linking procedure using only overlapping items common across a reference FKA version and adjacent FKA versions, ensuring that comparisons across grades were based on similar measurement structure. This method allows differences in estimated ability to be attributed to differences in the latent trait rather than to differences in test difficulty or form composition. Analyses were conducted separately for the younger cohort and the older cohort, preventing inappropriate comparisons across non-adjacent developmental stages.

To link the different FKA versions, we used FKA 2^nd Grade Form B and 5^th Grade Form B as references. Specifically, we linked the ability scores of the FKA 2^nd Grade Form B with the ability scores of FKA 2^nd Grade Form A (30 overlapping items) and FKA 3^rd Grade (20 overlapping items). Similarly, we linked the FKA 5^th Grade Form B with those of FKA 5^th Grade Form A (33 overlapping items) and FKA 6^th Grade (13 overlapping items). We selected these versions as references because they contained the largest number of overlapping items with adjacent versions, making them best suited for the linking procedure. Using reference forms with substantial item overlap improves the stability and accuracy of the linking transformation. We applied the Stocking-Lord method, which optimally aligns test characteristic curves in calculating the two linear transformation constants—a slope (A) and an intercept (B)—to align the item parameters across tests and place the calibrations on the base metric. Because this method operates at the level of item parameters rather than raw scores, it is well suited for linking test forms, even when they are administered to non-equivalent samples.

Reference and linked ability scores (in logits) are presented in Table 7. The linked ability scores showed a wider range among younger participants (0.40 to 1.75) than among older participants (0.03 to 0.27). These results reflect differences in the latent trait after accounting for form-specific measurement characteristics, supporting the usefulness for the developmental comparisons.

Table 7.

Participants' Ability Scores Estimated for Each FKA Version

Cohort	FKA Version	θ
		Mean	SD
Younger	2nd Grade Form A – linked ability score	0.40	1.23
	2nd Grade Form B – ability score (reference)	0.08	1.13
	3rd Grade – linked ability score	1.75	1.20
Older	5th Grade Form A – linked ability score	0.03	1.14
	5th Grade Form B – ability score (reference)	−0.09	1.13
	6th Grade – linked ability score	0.27	1.02

Note. FKA = Fraction Knowledge Assessment.

We then analyzed participants' fraction knowledge across grades using the linked ability scores. Since these scores were equated based on common items across adjacent-grade FKAs, they mitigate measurement differences across versions and support comparability over time.

For these longitudinal analyses, we only included participants who completed FKA in both 2^nd and 3^rd grades or both 5^th and 6^th grades and who had completed all standardized measures included in our models (i.e., attention, working memory, math fluency, and calculation). Thus, the sample with 2^nd and 3^rd graders was composed of 138 participants, and the sample with 5^th and 6^th graders was composed of 113 participants.

We predicted overall improvements in students' fraction knowledge from 2nd to 3rd grade, as well as from 5th to 6th grade. However, as prior longitudinal studies have highlighted individual variability in growth, with trajectories ranging from improvement to stagnation or even decline (Ye et al., 2016), we also expected that some participants would show stable or even decreased ability levels at the second assessment. We also predicted that domain-general skills and math skills would be longitudinally associated with fraction knowledge, aligned with predictions from the integrated theory of numerical development (Siegler et al., 2011) and the multi-level framework of numerical cognition (Gilmore, 2023).

Results are illustrated in Figure 2. Out of the 138 participants who completed FKA in 2^nd grade, 132 (96%) showed higher scores in 3^rd grade, suggesting improved fraction knowledge across these grades. In 2^nd grade, participants' mean linked ability score was 0.27 (sd = 1.25, min = −3.53, max = 4.31), whereas in 3^rd grade, their mean linked ability score was 1.78 (sd = 1.21, min = −1.41, max = 5.85). This difference corresponds to an improvement in their fraction knowledge of approximately one and a half standard deviations of the ability scores, t(137) = 18.05, p< .001, d = 1.23. These findings indicate a higher average level of the latent trait being measured in 3^rd grade compared to 2^nd grade.

Fig. 2.

Plots show changes in FKA linked ability scores distribution from 2nd to 3rd grade (Figure 2a) and from 5th to 6th grade (Figure 2b), individual changes in FKA linked ability scores from 2nd to 3rd grade (Figure 2c) and from 5th to 6th grade (Figure 2d), and the correlation between 2nd and 3rd grade FKA linked ability scores (Figure 2e), and 5th and 6th grade FKA linked ability scores (Figure 2f). Plots on the left side represent data from younger participants and plots on the right side represent data from older participants.

Improvements were also observed from 5^th to 6^th grade, but they were less pronounced. Out of the 113 participants who completed FKA in 5^th grade, 78 (69%) showed higher scores in 6^th grade. Participants' mean linked ability scores increased from −0.02 in 5^th grade (sd = 1.19, min = −2.75, max = 3.11) to 0.27 in 6^th grade (sd = 1.02, min = −2.09, max = 3.56), t(112) = 3.50, p = .001, d = 0.26. Although this was a significant improvement, it contrasts with the larger improvement observed from 2^nd to 3^rd grade. These results demonstrate stronger developmental shifts in fraction knowledge in the transition from 2^nd to 3^rd grade and more gradual, continued progress in more advanced grades.

We also examined how participants' fraction knowledge in an earlier grade influenced their fraction knowledge in a subsequent grade. Better FKA linked ability scores in 2^nd grade were associated with better linked ability scores in 3^rd grade, r(136) = .68, p<.001. Similar results were found for 5^th and 6^th grades, r(111) = .69, p<.001 (Figure 2). These findings suggest that participants with stronger fraction knowledge in an earlier grade were more likely to achieve better FKA scores in the subsequent grade.

Finally, we conducted linear mixed models to account for individual differences. In these models, conducted separately for each cohort, we examined changes in participants' FKA scores over time (i.e., from 2nd to 3rd grade, and from 5th to 6th grade) while controlling for age, attention skills (WJ-PC), working memory (WJ-AWM), and overall mathematics performance (i.e., as indexed by the WJ-MF and WJ-Calc tests) in the first year of the study (i.e., fixed effects). A random intercept was included at the subject level.

As shown in Table 8, younger participants' FKA scores improved significantly from 2^nd to 3^rd grade even when accounting for age, and performance in standardized attention, working memory, and mathematics achievement tasks. Improvements in FKA scores were significantly associated with time (i.e., 2^nd vs. 3^rd grade) and performance on the working memory and calculation measures. This finding suggests that students with stronger initial working memory capacity and calculation skills tended to perform better overall.

Table 8.

Linear Mixed Model: How initial age and general math skills predict FKA scores

Cohort	Predictor	Coefficient	95% Confidence Interval	p
Younger (AIC = 757.54)	Time (2nd vs. 3rd grade)	1.51	1.35, 1.68	<.001
	Age (2nd grade)	−0.01	−0.36, 0.35	.980
	Attention (WJ-PC – 2nd grade)	0.01	−0.02, 0.01	.497
	Working memory (WJ-AWM – 2nd grade)	0.07	0.13, 0.23	<.001
	Math fluency (WJ-MF – 2nd grade)	0.00	−0.01, 0.02	.506
	Calculation (WJ-Calc – 2nd grade)	0.18	0.04, 0.10	<.001
Older (AIC = 556.06)	Time (5th vs. 6th grade)	0.29	0.13, 0.45	<.001
	Age (5th grade)	−0.13	−0.36, 0.09	.254
	Attention (WJ-PC – 5th grade)	−0.01	−0.02, 0.01	.383
	Working memory (WJ-AWM – 5th grade)	0.07	0.04, 0.10	<.001
	Math fluency (WJ-MF – 5th grade)	0.01	0.01, 0.02	<.001
	Calculation (WJ-Calc – 5th grade)	0.09	0.06, 0.12	<.001

Note. FKA = Fraction Knowledge Assessment; WJ-PC = Woodcock-Johnson III Pair Cancellation subtest; WJ-AWM = Woodcock-Johnson III Auditory Working Memory subtest; WJ-MF = Woodcock-Johnson III Math Fluency subtest; WJ-Calc = Woodcock-Johnson III Calculation subtest.

We conducted follow-up models investigating interactions between time and age, time and attention, time and working memory, time and math fluency, and time and calculation skills. However, no interaction was significant (see full moderation model described in Supplementary Materials). These findings suggest that the extent of improvement in FKA scores from 2^nd to 3^rd grade was broadly consistent across participants, not varying across levels of their baseline age, attention, working memory, or general mathematics skills.

The linear mixed models indicated that older participants' FKA scores also improved from 5^th to 6^th grade when accounting for age and performance in standardized attention, working memory, and mathematics achievement tasks (see Table 8). Improvements in FKA scores were significantly associated with time (i.e., 5^th vs. 6^th grade), as well as performance in the working memory, math fluency, and calculation measures. However, these effects were weak. This finding suggests that initial working memory, math fluency, and calculation skills were associated with modest gains in fraction knowledge.

We conducted follow-up models investigating interactions between time and age, time and attention, time and working memory, time and math fluency, and time and calculation skills. However, no interaction was significant (see Supplementary Materials). The findings suggest that the extent of improvement in FKA scores from 5^th to 6^th grade was relatively uniform across participants and did not vary depending on their levels of baseline age, attention, working memory, or general math fluency or calculation skills.

These results indicate that fraction knowledge shows substantial stability across adjacent grades while also exhibiting reliable mean-level longitudinal gains. At the same time, individual differences in domain-general and foundational mathematics skills were associated with overall levels of fraction knowledge, aligned with our hypotheses. However, these skills did not moderate developmental change over time, suggesting that fraction knowledge develops in a coordinated manner across students in this sample rather than through diverging trajectories.

Although these longitudinal analyses allow us to examine within-participant changes over time, it does not permit strong causal inferences about the specific sources of fraction knowledge growth. In particular, improvements across grades likely reflect a combination of increased instructional exposure, general cognitive development, and individual differences not fully captured by the scope of included covariates. Accordingly, the reported developmental effects should be interpreted with caution, and future investigations are needed to fully understand the main factors predicting fraction knowledge development.

Discussion

Fraction knowledge is a foundational component of later mathematics achievement, yet how it develops remains debated, with competing accounts proposing either an integrated structure or multiple partially independent components. Progress toward resolving this debate has been limited in part by methodological constraints, including the scarcity of grade-appropriate fraction assessments with detailed, transparent psychometric information and the limited availability of longitudinal studies that target fraction knowledge from earlier grades. To address these gaps, we investigated the development of fraction knowledge in earlier and more advanced elementary school grades using grade-appropriate assessments that we designed and psychometrically tested. In particular, we investigated the structure of fraction knowledge across developmental stages, characterized the difficulty of fraction knowledge problems and how they discriminate between students with higher and lower ability, and investigated how fraction knowledge changes from 2nd to 3rd grade, and 5th to 6th grade, while accounting for baseline domain-general skills and math achievement.

Overall, results provided no strong evidence for separation between conceptual and procedural fraction knowledge across elementary school grades. Even though two-factor CFA solutions tended to favor the separation of these factors, they were not robustly stronger than one-factor solutions, and the two-factors were highly correlated with each other. IRT analyses indicated that magnitude processing better discriminates students with higher and low fraction ability in 2nd and 3rd grades, whereas fraction arithmetic with unlike terms and fraction division showed better discrimination in 5th and 6th grades. These results suggest an important role of magnitude understanding earlier on, and of operation skills in more advanced grades. Finally, we found that fraction knowledge, working memory, and general math skills at baseline were longitudinally associated with fraction knowledge measured one year later. Altogether, these findings can be interpreted under perspectives from the integrated theory of numerical development, which suggests that conceptual and procedural knowledge are intertwined and that prior whole number skills along with a strong magnitude understanding are important predictors of fraction knowledge (Siegler et al., 2011).

Conceptual and procedural fraction knowledge as intertwined skills

There is an ongoing debate regarding the distinction between conceptual and procedural fraction knowledge. In this study, we contributed to this debate by testing the structure of FKAs across 2nd, 3rd, 5th, and 6th grades using CFA. We hypothesized that, across grades, fraction knowledge could fit a one-factor model, aligned with the integrated theory of numerical development (Siegler et al., 2011), a two-factor model, aligned with the multi-level framework of numerical cognition (Gilmore, 2023), or, even, that the best structure could shift across grades, suggesting an hybrid model (Ye et al., 2016).

We observed that the fit of two-factor solutions separating conceptual and procedural factors was comparable to one-factor models. Additionally, when we separated conceptual and procedural factors, they were highly correlated. These findings suggest that conceptual and procedural fraction knowledge are highly intertwined in this age range and measurement context, being more consistent with an integrated account of fraction knowledge development (Siegler et al., 2011).

It is possible that a non-robust separation between conceptual and procedural fraction knowledge within the FKAs is related to how we defined and operationalized these constructs in our instrument. However, we operationalized conceptual fraction knowledge as fraction magnitude understanding and procedural fraction knowledge as fraction operations, following prior studies that have successfully identified a factor structure in fraction knowledge assessments (Hecht et al., 2003; Lenz et al., 2020). Thus, while aspects of our task design may have contributed to the observed factor structure, the limited separability we observed may reflect broader characteristics of fraction knowledge rather than properties unique to the FKA.

Broader methodological factors, such as developmental stage, may also help explain the weak evidence for a separation between conceptual and procedural knowledge in the present study. Prior research reporting psychometric support for this distinction has typically examined older students. For example, using CFA, Lenz and colleagues (2020) identified empirically distinct conceptual and procedural factors in students in the 8th and 9th grades. This raises the possibility that differentiation between conceptual and procedural fraction knowledge emerges later in development, beyond the age range examined here. Similar developmental patterns have been observed in other cognitive domains, where initially unified constructs become increasingly differentiated over time. For instance, research on executive functions indicates that these skills are best characterized as a single, unitary construct in preschool, but later separate into distinct factors (i.e., updating, inhibition, and shifting) during elementary school. Accordingly, future research spanning a broader range of developmental stages is needed to determine whether the one-factor structure observed here reflects a valid unified construct in earlier development, a progression toward greater differentiation across development as students refine their fraction knowledge, or even measurement artifact.

An alternative explanation for the non-separation of conceptual and procedural fraction knowledge in our study may be related to diversity in participants' strategy application when solving the FKAs. As predicted by the integrated theory of numerical development (Siegler et al., 2011), children may adjust, discard, or adopt new problem-solving strategies over time, with distinct strategies coexisting and being strongly connected to one another. In particular, children may assess the accuracy of their solution to fraction arithmetic relying on fraction magnitude understanding. Likewise, mastering fraction arithmetic may refine their magnitude processing skills. We argue that, aligned with this perspective, some of our participants may have used conceptual knowledge (e.g., magnitude understanding) to solve some FKA items designed to measure procedural knowledge, and vice versa. When participants employ a mixture of strategies to solve items intended to target either conceptual or procedural knowledge, a clear factor structure might not emerge, leading to a one-dimensional model with good fit (Rittle-Johnson & Schneider, 2014), as we found for FKA. It is important to note, though, that since we did not collect process-level data (e.g., strategy reports or qualitative analysis of error patterns), this interpretation remains tentative and highlights the need for future work incorporating qualitative or process-tracing methods.

Fraction knowledge development across early and advanced elementary school grades

Our developmental analyses indicated that 2^nd graders, who had received little or no formal fraction instruction in classrooms, did not show floor effects in the FKAs. These participants demonstrated high proficiency in solving problems involving magnitude comparison and estimation, as well as in mapping nonsymbolic to symbolic representations of halves and fourths. These results corroborate prior evidence showing that fraction magnitude understanding—particularly the ability to process different representations of “half”—emerges before formal fraction instruction, being observed in preschoolers and first-graders (Barth et al., 2009; Mix et al., 1999; Park et al., 2021; Spinillo & Bryant, 1991; Viegut et al., 2023).

In the absence of, or with minimal formal fraction instruction in classrooms, children may acquire fraction skills via informal experiences (i.e., those that take place outside the elementary school setting, or that are linked to playful activities). Prior research indicates that children interact with fractions in home and preschool settings. For instance, Eason and Ramani (2020) showed that parent–child guided play provides meaningful opportunities for fraction learning among 4- and 5-year-old children. Similarly, Viegut and colleagues (2023) found that children exhibit emerging fraction knowledge as early as the beginning of first grade, suggesting an important role of preschool in shaping these skills. Even though our study did not track students' informal learning experiences to understand how they shape fraction knowledge progression from preschool to elementary school grades, our results emphasize that a comprehensive account of the developmental trajectory of fraction knowledge must begin in earlier grades, even before fractions are formally introduced in the mathematics curriculum.

Our longitudinal analyses showed significant improvements in fraction knowledge from the 2nd to the 3rd grade (more than 1 SD), when fractions are then introduced in the school curriculum. They also indicated significant improvements from the 5th to the 6th grade, after students have received a few years of fraction instruction, even though they were less robust (less than 0.5 SD) than those observed in earlier grades. This distinct improvement rate across younger and older participants may be explained by various factors, such as cohort effects or contextual aspects not targeted by our measures, such as curriculum emphasis and instructional strategies. However, it may also reflect nonlinear developmental trajectories. Like our findings, prior research has found more substantial improvements in fraction knowledge across earlier grades, when formal fraction instruction begins, than across later grades. For example, Hansen and colleagues (2017) reported larger differences in raw scores on measures of fraction conceptual knowledge between 3^rd and 4^th graders than between either 4^th and 5^th graders or 5^th and 6^th graders. It is possible that younger students have more room to improve their fraction skills when receiving formal instruction for the first time, whereas gains may become more incremental after years of instruction and experience with fractions. This possibility should be investigated in longitudinal studies following the same cohort from earlier to more advanced grades.

Beyond differences in developmental pace across cohorts, we observed shifts in which fraction knowledge tasks best discriminated between students with lower and higher ability. The IRT analyses revealed an important role of fraction magnitude understanding in discriminating ability levels in earlier grades and an important role of fraction arithmetic in more advanced grades. This observed pattern offers insight into a potential hierarchy of fraction knowledge. In earlier grades, individual differences may be primarily driven by magnitude representations, suggesting that establishing robust fraction magnitude understanding is a foundational step. In later grades, performance may be increasingly differentiated by complex arithmetic operations, particularly operations with unlike terms and fraction division.

We speculate that this progression implies increasing cognitive complexity, from representational understanding to arithmetic procedures. These item-level patterns indicate a developmentally sensitive map of fraction competence, highlighting how different skills become bottlenecks at different stages.

One possible interpretation for this finding is that it is tied to contextual factors, such as the mathematics school curriculum progression. The Mathematics Common Core Standards (National Governors Association, 2010) emphasize fraction magnitude understanding starting in the 3^rd grade, whereas fraction operations are central in later grades. However, this pattern may also reflect underlying developmental processes.

Both the integrated theory of numerical development (Siegler et al., 2011) and the multi-level framework of numerical cognition (Gilmore, 2023) assume that magnitude processing understanding is a foundational skill in mathematics development. Importantly, we found that working memory and general mathematics skills (as measured by standardized tasks that rely largely on whole-number skills) were longitudinally associated with fraction knowledge in both cohorts. These findings are also aligned with both the integrated theory of numerical development (Siegler et al., 2011) and the multi-level framework of numerical cognition (Gilmore, 2023), and corroborates previous studies showing that early working memory and mathematics skills predict later fraction performance (Jordan et al., 2013; Hansen et al., 2016; Schadl & Ufer, 2023; Siegler et al., 2011; Viegut et al., 2023; Xu et al., 2025). Together, these results suggest that fraction knowledge builds upon foundational numerical competencies while being supported by domain-general cognitive resources.

While differential item discrimination across grades could be interpreted as evidence for distinct components of fraction knowledge, consistent with the multi-level framework of numerical cognition (Gilmore, 2023), we argue that the overall pattern of findings more strongly supports the integrated theory of numerical development (Siegler et al., 2011; Siegler, 2016). This framework proposes that magnitude understanding serves as a foundational representational system upon which more advanced mathematical competencies, including fraction arithmetic, are built. Working memory is thought to support this process by enabling children to maintain, compare, and manipulate numerical representations during learning. Consistent with this account, magnitude understanding played a central role in discriminating ability levels in earlier grades, while arithmetic skills became more discriminative at later stages, reflecting the progressive extension of magnitude representations to more complex procedures. Moreover, the CFA results supported a unified latent structure, and working memory and general mathematics skills longitudinally predicted fraction knowledge across cohorts. Together, these converging findings—including unified latent structure, developmental shifts in item discrimination, and shared cognitive predictors—provide strong support for an integrated developmental account in which magnitude understanding forms the foundation for subsequent fraction learning.

Implications for theory and practice

Our findings have implications for both theoretical frameworks and educational practice. From a theoretical perspective, the lack of robust separation between conceptual and procedural fraction knowledge, together with their high intercorrelations across grades, supports integrated models of numerical development (Siegler et al., 2011). Moreover, IRT analyses revealed systematic developmental shifts in which skills best differentiated students with lower and higher ability, suggesting a progression from magnitude-based representations in earlier grades to arithmetic operations in later grades. These findings illustrate how psychometric information about factor structure and item difficulty and discrimination can inform developmental frameworks, highlighting the value of integrating measurement and theory in studies of fraction knowledge.

From an educational perspective, our results suggest that practices targeting the consolidation of mathematical skills from the earliest grades are critical, as they may help shape students’ developmental trajectories in fraction knowledge. In particular, given that 2nd graders already demonstrated foundational fraction understanding, expanding learning opportunities from preschool onward may have long-term benefits for fraction skills. In addition, because general mathematics skills were longitudinally associated with fraction knowledge, strengthening basic whole-number competencies may support the acquisition and mastery of fractions.

Another contribution of this study is the development and psychometric testing of the Fraction Knowledge Assessments (FKAs). Currently available measures often lack comprehensive coverage across grades and detailed reporting of psychometric properties (Rodrigues et al., 2019; Viegut et al., 2023). Our findings indicate that the FKAs achieve a balance between task difficulty and student ability, contributing to reliable measurement across grades, particularly when linking procedures are applied. These measures and the associated linking code are shared via OSF for direct use or adaptation. To the best of our knowledge, the FKA is the most comprehensive tool currently available for tracking the development of fraction knowledge while accounting for both mathematics curriculum progression and psychometric properties.

For practical applications, the FKA could be refined by retaining the most informative items to create efficient versions suitable for brief administration, allowing students to be placed on a vertically scaled ability map. Items could also be incorporated into adaptive computer-based tasks, enabling dynamic assessment tailored to individual ability. Given its potential significance and versatility, the FKA represents a valuable tool for researchers, educators, curriculum developers, and policymakers, supporting concurrent and longitudinal assessment of fraction knowledge to inform diagnosis, interventions, and academic placement decisions.

Limitations and Future Directions

This study is not without limitations. Despite efforts to recruit a representative sample, participants were predominantly non-Hispanic, White, and from middle-class backgrounds, with most coming from families with high educational attainment. Prior research indicates that socioeconomic factors are associated with early mathematics development. For example, Dearing and colleagues (2022) found that children whose parents have higher educational attainment and income demonstrate stronger early math skills than those from lower socioeconomic backgrounds. Given these demographic characteristics, participants in the present study may have had greater access to learning opportunities that support cognitive skills relevant to fraction knowledge, as well as earlier exposure to fractions. Consequently, performance on the FKAs may not generalize to broader populations, particularly to younger children from lower socioeconomic backgrounds. To strengthen the generalizability of our findings, future research should examine fraction knowledge development and the validity of FKA in more diverse samples.

It is also important to consider limitations related to our measures. During FKA development, we observed left-skewed score distributions in the 2nd- and 5th-grade versions. To address this issue, we created Form B versions for these grades with increased conceptual and computational complexity. However, this manipulation proved more effective in 2nd grade than in 5th grade. IRT analyses indicated that 2nd Grade FKA Form B had higher mean difficulty than Form A, while discrimination remained comparable. In contrast, although 5th Grade FKA Form B showed higher mean discrimination than Form A, both forms exhibited similar difficulty levels. In the longitudinal analyses, these differences were addressed through equating and linking procedures. Nevertheless, the relatively low difficulty of both Form A and B of the 5th Grade FKA represents a limitation of the current study and should be addressed in future iterations of the assessment.

Another limitation concerns the grade levels covered by the FKA. In the current study, we targeted fraction knowledge in 2^nd, 3^rd, 5^th, and 6^th grades. Although we initially aimed to include 8^th-grade FKA in this analysis, COVID-19 disruptions prevented data collection when participants reached this grade. Moving forward, future research should expand coverage both earlier (e.g., preschool) and later (e.g., high school, college) to provide a more comprehensive picture of fraction knowledge development. The FKAs can be adapted to different curricular levels, enabling comprehensive developmental studies, including but not limited to latent growth, cross-lagged panels, and structural equation models.

Finally, when examining factors associated with fraction knowledge development, we included standardized measures of attention (WJ–Pair Cancellation), working memory (WJ–Auditory Working Memory), math fluency (WJ–Math Fluency), and calculation (WJ–Calculation). In our analyses, we relied primarily on published normative data for these measures without evaluating their reliability within our sample. Because reliability can vary across samples (Taban, 2011), and we did not directly assess it, our developmental models may partially reflect instrument-related biases.

In addition, our regression models did not control for a series of contextual and cognitive factors that may support fraction knowledge, such as socioeconomic status, language skills, or inhibitory control (Hansen et al., 2016; Leib et al., 2023). Consequently, beyond the effects of one additional year of schooling and baseline working memory, attention, and mathematics achievement, longitudinal improvements in fraction knowledge may reflect influences from children’s home or school environments, as well as cognitive or emotional factors not captured by our measures. These longitudinal improvements may also partially reflect measurement error in the FKAs. Thus, further research is needed to more fully characterize fraction knowledge development across early and advanced elementary school grades. Moving forward, future research should employ grade-appropriate measures to track developmental shifts in fraction knowledge across elementary, middle, and high school, while also investigating the interplay between fraction knowledge, general math achievement, as well as contextual, cognitive, and emotional factors.

Conclusion

Whether fraction knowledge develops in an integrated or multi-component manner remains an open question. Reliable and valid tools are essential for clarifying its structure and tracking its development over time. In this study, we designed and psychometrically tested grade-appropriate measures of fraction knowledge for 2nd, 3rd, 5th, and 6th grades, and examined longitudinal change from 2nd to 3rd grade and from 5th to 6th grade. Our results suggest that, in this age range, fraction knowledge is better characterized as a unidimensional construct. Item-level analyses further revealed developmental shifts in which skills best differentiated students: magnitude understanding played a stronger role in distinguishing lower versus higher ability in earlier grades, whereas fraction arithmetic was more discriminative in later grades. Although significant gains in fraction knowledge were observed across both cohorts, improvements were substantially larger from 2nd to 3rd grade than from 5th to 6th grade. In addition, stronger baseline mathematics achievement and working memory were longitudinally associated with better fraction knowledge. Together, these findings align with the integrated theory of numerical development (Siegler et al., 2011), highlighting the importance of early mathematics skills, domain-general cognitive resources, and magnitude understanding in supporting fraction learning. Beyond advancing our understanding of fraction knowledge across early and later elementary school years, this study also contributes to its measurement. While we remain mindful of the ongoing work in this area, to our knowledge, the FKAs represent the first tool psychometrically tested across multiple grades, which offers a robust resource for tracking fraction knowledge development. To support broader research and practice, we share the FKAs freely, enabling researchers, educators, curriculum developers, and policymakers to use or adapt these tools to assess children’s developing fraction knowledge.

Highlights.

• Investigated fraction knowledge from 2^nd to 3^rd grade and from 5^th to 6^th grade

• Used grade-appropriate Fraction Knowledge Assessments (FKA)

• Placed students on an ability map scaled vertically with an interval-level metric

• Observed steeper improvements from 2^nd to 3^rd grade than from 5^th to 6^th grade

• Early fraction and math skills predicted future fraction skills