From quantity-based to operation-based errors: Developmental shifts in one-step additive word problem solving

From quantity-based to operation-based errors: Developmental shifts in one-step additive word problem solving | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article From quantity-based to operation-based errors: Developmental shifts in one-step additive word problem solving Maria Santagueda-Villanueva, Emilia López-Iñesta, Daniel García, and 2 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8648711/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study investigates developmental changes in the patterns of errors produced by primary school students when solving one-step additive arithmetic word problems. A sample of 3,111 students from Grades 1 to 6 completed a battery of Change, Combine, and Compare problems presented in a multiple-choice format with theoretically motivated distractors. These distractors captured three recurrent error types: selection of the higher quantity, selection of the lower quantity, and application of the inverse operation. Results show clear grade-related shifts in the distribution of error types. In the early grades, students’ incorrect responses are dominated by quantity-based choices, indicating a strong reliance on salient numerical values in the problem statement. From approximately Grade 3 onward, quantity-based errors decrease substantially, while inverse-operation errors account for an increasing proportion of remaining errors, particularly in Change and Compare problems. Importantly, these developmental trajectories are not uniform across semantic families: Combine problems exhibit comparatively stable error profiles, whereas Change and Compare problems show pronounced reorganization in the nature of errors across grades. Rather than focusing exclusively on accuracy, the findings highlight how developmental progress in additive word problem solving is reflected in systematic changes in error profiles that are sensitive to semantic structure. By treating errors as diagnostically meaningful outcomes, this study provides a fine-grained account of how students’ interpretations of additive situations evolve across primary education and offers insights relevant for both research and instructional design in mathematics education. arithmetic word problems additive reasoning error analysis semantic structure primary mathematics education Figures Figure 1 Figure 2 Figure 3 1. Introduction Arithmetic word problems (AWPs) constitute a central component of primary mathematics education, as they require students to coordinate numerical knowledge with the comprehension of a verbally described situation. Solving such problems goes beyond executing calculations: it involves constructing a representation of the situation, identifying relevant quantitative relations, and mapping these relations onto an appropriate arithmetic operation (Kintsch & Greeno, 1985 ; Verschaffel et al., 2020 ). Despite their apparent simplicity, even one-step additive AWPs pose persistent difficulties for many learners across the primary years. A substantial body of research has shown that children’s success in AWPs is strongly influenced by the semantic structure of the problem (Riley et al., 1983 ; Carpenter et al., 1999 ; De Corte & Verschaffel, 1987 ). Classic classifications distinguish among Change, Combine, and Compare problems, each imposing distinct relational demands. Empirical studies have consistently documented that these semantic differences affect difficulty, strategy choice, and error rates, with Compare problems in particular remaining challenging well beyond the early grades (Arendasy et al., 2005 ; Verschaffel et al., 2020 ). More recent syntheses confirm that linguistic and structural task characteristics, such as the position of the unknown or lexical consistency, systematically shape students’ performance in elementary-school AWPs (Daroczy et al., 2015 ; Vessonen et al., 2024 ). Importantly, research has repeatedly emphasized that incorrect responses in AWPs are not merely random mistakes but provide valuable information about students’ underlying representations and reasoning processes. Early work on error analysis demonstrated that children’s wrong answers often correspond to coherent, theoretically interpretable strategies or partial models (Jaspers & van Lieshout, 1994 ; Jitendra & Kameenui, 1996 ). In the context of additive AWPs, many errors reflect surface-based approaches in which students select salient numbers from the problem statement rather than constructing a relational model of the situation (González-Pienda et al., 1999 ; Pape, 2004 ). Such quantity-selection errors are particularly frequent in the early grades and have been linked to direct translation strategies and shallow encodings of the problem text. With increasing age and schooling, children tend to rely less on pure quantity selection, yet they continue to make systematic errors related to the choice of arithmetic operation. A well-documented example is the tendency to apply the inverse operation when relational language or familiar heuristics are misleading (e.g., adding when subtraction is required, or vice versa). This phenomenon has been extensively studied in relation to language consistency and reversal errors, especially in Compare and Change problems (Verschaffel, 1994 ; Pape, 2003 ; Leong & Jerred, 2001 ). Such inverse-operation errors suggest that students are engaging with relational information but struggle to correctly map that information onto an arithmetic model, often due to the influence of linguistic cues or overlearned rules. Developmental research further indicates that these error types are not uniformly distributed across grades or problem families. Instead, children’s profiles evolve as their mathematical knowledge, language skills, and domain-general cognitive resources develop (Kail & Hall, 1999 ; Bjorn et al., 2016; Tolar et al., 2012 ). Studies focusing on executive functions highlight that inhibition and working memory updating play a critical role in resolving semantically incongruent AWPs, particularly when misleading heuristics must be suppressed (Lubin et al., 2013 ; Passolunghi et al., 2022 ). From this perspective, shifts in error patterns may reflect not only growing conceptual understanding but also changes in the cognitive demands that different problem types impose at different ages. Despite this extensive literature, relatively few large-scale studies have traced systematic developmental changes in distinct, theoretically motivated error types across the full span of primary education while simultaneously considering semantic problem families. Many investigations focus on accuracy, on specific grades, or on isolated problem types, making it difficult to describe how children’s interpretations of additive situations evolve from early to later primary school years. Moreover, error analyses are often based on open responses, which, while rich, limit scalability and comparability across large samples. The present study addresses this gap by examining grade-related changes in children’s error patterns when solving one-step additive AWPs from Grades 1 to 6. Using a large sample of Spanish primary-school students (N = 3,111), we administered a battery of Change, Combine, and Compare problems in a multiple-choice format in which distractors were deliberately constructed to capture three theoretically meaningful error types: selection of the higher quantity, selection of the lower quantity, and application of the inverse operation. By analyzing how the distribution of these error categories varies across grades and semantic families, we aim to document systematic developmental differences in students’ error profiles when interpreting additive situations. Specifically, the study addresses the following research questions: How do the frequencies of higher-quantity, lower-quantity, and inverse-operation errors change across Grades 1 to 6 in one-step additive AWPs? Do these developmental trajectories differ across Change, Combine, and Compare problem families? Beyond documenting grade-related differences in performance, the present study contributes to research on arithmetic word problem solving in three main ways. First, it adopts an error-centered perspective in which incorrect responses are treated as diagnostically meaningful indicators of students’ interpretations, rather than as undifferentiated failures. By systematically distinguishing between quantity-based and inverse-operation errors, the study provides a fine-grained account of how students’ difficulties are distributed across semantic structures and developmental stages. Second, by combining a large-scale design with theoretically motivated distractors embedded in a controlled multiple-choice format, the study allows for the identification of developmental regularities that are difficult to capture through accuracy-based or small-sample analyses alone. Third, the explicit comparison of Change, Combine, and Compare problem families reveals that developmental change in additive word problem solving is not uniform, but closely tied to the representational demands of different semantic structures. By focusing on the structure of incorrect responses, the study seeks to contribute to a more fine-grained understanding of children’s evolving interpretations of additive situations and to inform both research and instruction on how and why students struggle with seemingly simple word problems throughout primary education. 2. Theoretical Framework AWPs constitute a central locus for studying how learners coordinate linguistic information with quantitative relations to produce a mathematical model. A consistent finding across decades of research is that success in AWPs depends not only on calculation skills but also on the construction of adequate representations from the text, and on the ability to select and justify operations that match the underlying situation structure (Kintsch & Greeno, 1985 ; Pape, 2004 ; Verschaffel et al., 2020 ). In this sense, developmental progress in AWP solving can be understood as a gradual shift from surface-driven processing—where salient numbers or keywords guide responses—toward more structurally informed situation models that support relational reasoning and operation selection (Okamoto, 1996 ; Christou & Philippou, 1998 ; Verschaffel & De Corte, 1993 ). 2.1. Situation models, problem models, and semantic structures Classical frameworks emphasize that understanding a AWP involves building a text base and integrating it with prior knowledge to form a situation model that captures the relevant quantitative relations (Kintsch & Greeno, 1985 ). In school contexts, however, learners often rely on direct translation heuristics, mapping lexical cues to arithmetic operations without fully representing the situation (Pape, 2004 ; González-Pienda et al., 1999 ). This tendency is especially consequential for additive problems, where the same operation family (addition/subtraction) supports multiple semantic structures with distinct relational demands (Riley et al., 1983 ; Carpenter et al., 1999 ; Verschaffel et al., 2020 ). Research based on semantic taxonomies shows that task difficulty and solution behavior systematically vary across families such as Change, Combine, and Compare, and that these differences are already present in early grades (De Corte & Verschaffel, 1987 ; Christou & Philippou, 1998 ). A key implication is that learners may generate answers that are coherent with an incomplete or biased representation of the described situation. This includes cases where the qualitative situation model (everyday interpretation) appears to conflict with the quantitative problem model (formal relations), which can reduce performance when schema activation is not yet automatized (Coquin-Viennot & Moreau, 2007 ). Such representational tensions provide a principled basis for analyzing wrong answers as informative outcomes rather than random noise: error patterns can index which aspects of the situation structure have been encoded, which relations have been ignored, and which procedures have been misapplied (Jaspers & Van Lieshout, 1994 ; Kingsdorf & Krawec, 2014 ). 2.2. Semantic congruence, consistency effects, and operation reversal A prominent line of work has shown that performance in compare problems is sensitive to whether the relational language is consistent with the required operation. In “inconsistent language” formats, relational terms can cue an operation that is opposite to the mathematically required one, leading to systematic reversal (inverse-operation) errors and distortions in recall (Pape, 2003 ). Evidence from retelling methods indicates that when relational statements mismatch solvers’ preferred formats, representational difficulties increase and errors become more likely (Verschaffel, 1994 ). Eye-movement research further suggests that less successful solvers show distinctive processing costs under inconsistency, modulated by linguistic markedness (e.g., “less than” vs. “more than”) (van der Schoot et al., 2009 ). These findings support the view that operation selection is not purely computational but is strongly shaped by semantic cues that steer interpretation. More recent theoretical developments propose that AWP solving is constrained by semantic congruence between world meanings and mathematical procedures. The Semantic Congruence (SECO) model argues that world semantics can bias the encoding and recoding of problems, such that semantically incongruent representations require deliberate recoding that learners often fail to complete (Gros et al., 2020 ). In this perspective, operation errors are not merely “wrong operations” but manifestations of incomplete recoding between an initially compelling situation-based representation and the formal arithmetic structure. Empirical work manipulating linguistic and numerical factors likewise indicates that AWP difficulty emerges from the interaction of these components rather than from either domain alone (Daroczy et al., 2020 ). Taken together, these approaches justify analyzing inverse-operation errors as theoretically meaningful outcomes, especially in problem families (notably Compare and certain Change formats) where semantic cues can conflict with the required transformation. 2.3. Cognitive constraints: executive functions, inhibition, and reading-related resources Beyond linguistic-semantic factors, individual differences in domain-general cognition contribute substantially to AWP performance. Research consistently shows that reading comprehension is a strong predictor of success, even when controlling for calculation ability and other skills (Björn et al., 2016 ; Boonen et al., 2013 ; Passolunghi et al., 2022 ). In addition, executive functions—particularly inhibition and working memory updating—play a distinct role when students must suppress misleading heuristics and maintain intermediate representations (Passolunghi et al., 1999 ; Passolunghi & Siegel, 2001 ). Developmental evidence from negative priming paradigms indicates that solving inconsistent compare problems requires inhibiting default strategies such as “add if more, subtract if less,” and that this inhibitory demand is present across age groups (Lubin et al., 2013 ). Complementarily, process-oriented studies suggest that executive functions support different phases of solving, including selecting relevant information, coordinating relations, and planning the appropriate operation sequence (Viterbori et al., 2017 ). These findings align with developmental accounts proposing that, as students gain experience, difficulties may shift from identifying salient quantities toward selecting and inhibiting operations. For instance, children may initially commit quantity-selection errors (e.g., choosing one of the numbers in the statement), but later increasingly exhibit operation-based errors when their representations become more structured yet still vulnerable to semantic biases (Kail & Hall, 1999 ; Degrande et al., 2019 ). This developmental reorientation is also consistent with models emphasizing increasing flexibility in dealing with additive situations during the first years of schooling (Ufer et al., 2024 ) and with observations that young learners’ strategy choices are shaped by semantic structure, cognitive economy, and number relations (Sevinc et al., 2024 ). 2.4. Task characteristics, stimulus control, and the diagnostic value of distractors Because error patterns are shaped by both representation and task design, methodological control of item characteristics is essential. Reviews and meta-analyses show that AWP difficulty is influenced by multiple linguistic features (e.g., lexical consistency, position of the unknown, irrelevant information) as well as numerical features (e.g., number of operations), with particularly strong effects when realism considerations or lexical inconsistency are involved (Daroczy et al., 2015 ; Vessonen et al., 2024 ). Experimental evidence further supports that combining inconsistency with other disruptive features (such as irrelevant numbers) can change reasoning and increase performance drops (Vondrová, 2022 ). Therefore, constructing response options that map onto theoretically motivated misconceptions provides a principled approach to diagnosing students’ representations. A productive strategy is to embed distractors that correspond to specific error types. In additive contexts, three recurrent patterns are (a) selecting the higher quantity in the statement, (b) selecting the lower quantity, and (c) applying the inverse operation. Such distractors operationalize distinct failure modes: quantity-based choices reflect surface-driven selection of salient values, whereas inverse-operation choices reflect representational or recoding failures at the operation-selection stage (Pape, 2003 ; Gros et al., 2020 ; Verschaffel et al., 2020 ). This error-sensitive design aligns with research showing that wrong answers can be systematically diagnosed and linked to underlying procedures rather than treated as random guessing (Chval et al., 2021 ; Jaspers & Van Lieshout, 1994 ). 3. Methodology 3.1. Research design and Participants This study employed a quantitative, cross-sectional, non-experimental design to examine how primary-school students respond to one-step additive AWPs. Because the purpose was to document grade-related differences in erroneous responses—rather than evaluate instructional interventions—an cross-sectional observational design approach was appropriate. This design is consistent with prior research on the semantic structures of AWPs (Riley et al., 1983 ; Carpenter et al., 1999 ; Verschaffel et al., 2020 ). Nineteen public primary schools from a Spanish autonomous community were randomly sampled from the official registry provided by the regional Ministry of Education. All students in Grades 1–6 attending school on the testing day were invited to participate, yielding a final sample of N = 3,111 (1,539 girls and 1,572 boys). Table 1 presents the distribution by grade and sex. This sample size ensures a maximum sampling error below 5% at the 95% confidence level, offering stable estimates for grade-level comparisons. Table 1 Distribution of participants by grade level and sex. 1st 2nd 3rd 4th 5th 6th Total Girls 236 230 291 260 272 250 1539 Boys 219 276 233 276 267 301 1572 Total 455 506 524 536 539 551 3111 All participating schools follow the national mathematics curriculum of Spain, which establishes common learning goals, content progressions, and competency-based outcomes across all autonomous communities (Real Decreto 157/2022). Although the Valencian Community develops its own curricular guidelines, these must remain aligned with the national framework, resulting in substantial uniformity in the teaching and learning of mathematics throughout the country. This coherence ensures that the developmental patterns observed in the present study reflect typical learning trajectories within the Spanish primary curriculum, which itself is consistent with broader European orientations toward competency-based mathematics education (Eurydice Network, 2025 ). 3.2. Instruments The task battery comprised 14 one-step additive AWPs classified into Change (6 items), Combine (2 items), and Compare (6 items) categories, following the canonical semantic taxonomy of Riley et al. ( 1983 ). Each item was designed to capture students’ interpretation of the underlying situation structure and their tendency to rely on surface-level numerical cues. Each problem included four multiple-choice options: the correct answer, two quantities given in the statement (the higher and the lower value), and one distractor generated by applying the inverse operation. These distractors correspond to three error types widely documented in AWP research—Higher-quantity, Lower-quantity, and Inverse-operation errors—which serve as indicators of students’ conceptual models when interpreting additive situations (Kintsch & Greeno, 1985 ; Carpenter et al., 1999 ; Verschaffel et al., 2020 ). Table 2 presents the full set of items, grouped by semantic family, along with their response options. Table 2 Battery of one-step additive AWPs by type, problem statement, and solution options. Type Statement Solutions Change CH1. Joe tiene 3 canicas. Luego Tom le dio 5 canicas más. ¿Cuántas canicas tiene Joe ahora? a. 3 b. 5 c. 8 d. 2 CH2. Joe tenía 8 canicas. Luego le dio 5 canicas a Tom. ¿Cuántas canicas tiene Joe ahora? a. 3 b. 5 c. 8 d. 13 CH3. Joe tenía 5 canicas. Luego Tom le dio algunas canicas más. Ahora Joe tiene 8 canicas. ¿Cuántas canicas le dio Tom? a. 3 b. 5 c. 8 d. 13 CH4. Joe tenía 8 canicas. Luego le dio algunas canicas a Tom. Ahora Joe tiene 3 canicas. ¿Cuántas canicas le dio a Tom? a. 3 b. 5 c. 8 d. 11 CH5. Joe tenía algunas canicas. Luego Tom le dio 5 canicas más. Ahora Joe tiene 8 canicas. ¿Cuántas canicas tenía Joe al principio? a. 3 b. 5 c. 8 d. 13 CH6. Joe tenía algunas canicas. Luego él le dio 5 canicas a Tom. Joe tiene 3 canicas. ¿Cuántas canicas tenía Joe al principio? a. 3 b. 5 c. 8 d. 2 Combine CB1. Joe tiene 3 canicas. Tom tiene 5 canicas. ¿Cuántas canicas tienen en total? a. 3 b. 5 c. 8 d. 2 CB2. Joe y Tom tienen 8 canicas en total. Joe tiene 3 canicas. ¿Cuántas canicas tiene Tom? a. 8 b. 3 c. 5 d. 11 Compare CP1. Joe tiene 8 canicas. Tom tiene 5 canicas. ¿Cuántas canicas tiene Joe más que Tom? a. 8 b. 5 c. 3 d. 13 CP2. Joe tiene 8 canicas. Tom tiene 5 canicas. ¿Cuántas canicas tiene Tom menos que Joe? a. 8 b. 5 c. 3 d. 13 CP3. Joe tiene 3 canicas. Tom tiene 5 canicas más que Joe. ¿Cuántas canicas tiene Tom? a. 3 b. 5 c. 8 d. 2 CP4. Joe tiene 8 canicas. Tom tiene 5 canicas menos que Joe. ¿Cuántas canicas tiene Tom? a. 8 b. 5 c. 3 d. 13 CP5. Joe tiene 8 canicas. Él tiene 5 canicas más que Tom. ¿Cuántas canicas tiene Tom? a. 8 b. 5 c. 3 d. 13 CP6. Joe tiene 3 canicas. Él tiene 5 canicas menos que Tom. ¿Cuántas canicas tiene Tom? a. 3 b. 5 c. 8 d. 2 Item wording and numerical values were linguistically adapted to Spanish. Readability and syntactic balance across items were refined through linguistic-masking techniques described in Álvarez et al. (2024), ensuring that: lexical or structural cues did not privilege any distractor, numerical information was comparable across problems, and variability in difficulty was attributable to semantic structure, not language. All items were administered through AWPSolver (Álvarez et al., 2024), a web-based research platform that: (a) displays problem statements in masked segments, requiring students to reveal each part sequentially; (b) provides a virtual scratch-pad and basic calculator (optional); and (c) records chosen answers, reading-completeness logs, and other interaction data. For the purposes of the present study, only answer choices and reading-completeness indicators were used in the analyses. 3.3. Procedure Data were collected during regular mathematics lessons in the 2023–2024 school year. After obtaining school and parental consent, each class received a 10-minute familiarisation session with AWPSolver. The full test was administered digitally and lasted approximately 45 minutes. Two exclusion criteria ensured that each recorded response reflected genuine task engagement: Incomplete reading: If any segment of a problem statement was skipped (verified through AWPSolver’s masking-navigation logs), the item was discarded for that pupil. Missing response: Items with no answer selected were also discarded, because comprehension in our framework requires both reading and responding. Table 3 shows the number of valid responses per item and grade after applying these criteria. Table 3 Number of students who completed each problem type. 1st 2nd 3rd 4th 5th 6th CH1 384 461 490 510 519 540 CH2 331 414 462 484 493 530 CH3 308 433 474 514 508 529 CH4 284 402 448 472 478 494 CH5 291 434 469 499 493 524 CH6 279 413 462 482 495 522 CB1 297 433 498 517 522 534 CB2 275 430 489 518 521 534 CP1 278 425 482 508 519 539 CP2 272 426 480 507 520 533 CP3 238 413 472 501 512 531 CP4 237 384 460 476 496 516 CP5 225 391 457 480 501 524 CP6 219 373 446 476 488 517 3.4. Data Analysis All analyses were conducted in R 4.3.2 (R Core Team, 2023 ). We began by computing descriptive statistics for each item and grade level, including the frequencies and percentages of the three error categories—Higher-quantity, Lower-quantity, and Inverse-operation errors. Ninety-five percent confidence intervals were calculated using Wilson’s method (Brown, et al., 2001 ), which provides accurate coverage for binomial proportions. To examine whether error distributions differed across grade levels, we performed χ² tests of independence. When expected cell frequencies fell below 5, Monte Carlo p-values based on 10,000 replicates were reported. Statistical significance was set at α = .05, and pairwise contrasts were evaluated using the Holm sequential correction (Holm, 1979 ) to control for multiple comparisons. Because students’ responses were nested within schools and items, we additionally estimated mixed-effects logistic regression models as robustness checks, using the lme4 package (Bates et al., 2015 ). These models included random intercepts for schools and students, while grade level and problem family were entered as fixed effects. The mixed-effects models yielded conclusions fully consistent with those obtained from the χ² analyses; therefore, the inferential results presented in the article focus on the χ² and Holm-adjusted comparisons. 4. Results 4.1 Descriptive Error Patterns Figure 1 displays the distribution of response outcomes (correct responses and three error types) across Grades 1–6. Accuracy rises sharply from 43% correct in Grade 1 to 60% (Grade 2), 75% (Grade 3), 83% (Grade 4), 86% (Grade 5), and 88% (Grade 6). As a share of all responses, each error type decreases with grade level: larger-quantity errors drop from 22% (Grade 1) to 14% (Grade 2), 8% (Grade 3), 5% (Grade 4), 4% (Grade 5), and 3% (Grade 6); smaller-quantity errors decline from 19% to 13%, 8%, 5%, 4%, and 3%, respectively; and inverse-operation errors decrease from 16% to 13%, 9%, 7%, 6%, and 5%. Importantly, although inverse-operation errors shrink in absolute terms (because overall correctness increases), they represent an increasing share of the remaining errors in the upper grades (e.g., about 28% of errors in Grade 1 vs. about 42% in Grade 6), suggesting that once students largely master magnitude selection, the main residual difficulty concerns choosing the appropriate arithmetic operation. Figure 2 presents the distribution of response outcomes by grade within each problem family (Change, Combine, and Compare). In Change problems (Fig. 2a), accuracy increases from 45% in Grade 1 to 61% in Grade 2, 74% in Grade 3, 82% in Grade 4, 84% in Grade 5, and 85% in Grade 6. Although all error types decrease in absolute terms, inverse-operation errors remain comparatively persistent across grades (17%, 15%, 11%, 9%, 8%, and 8%), whereas larger-quantity errors drop from 21% to 3% and smaller-quantity errors from 17% to 3%. In Combine problems (Fig. 2b), accuracy is consistently higher, rising from 54% (Grade 1) to 70%, 83%, 90%, 94%, and 96% by Grade 6. Larger-quantity errors dominate incorrect responses in the first grades (19% in Grade 1 and 13% in Grade 2) but decrease sharply thereafter (7% in Grade 3 and ≤ 4% from Grade 4 onwards); smaller-quantity and inverse-operation errors follow similar declining trajectories, falling to 2% or less in the upper grades. In Compare problems (Fig. 2c), accuracy starts lower (37% in Grade 1; 54% in Grade 2) and improves steadily to 73% (Grade 3), 82% (Grade 4), 86% (Grade 5), and 89% (Grade 6). Early grades show high proportions of quantity-based errors—larger-quantity (26% and 18%) and smaller-quantity (24% and 16%)—which diminish substantially across grades, reaching 4–5% by Grades 5 and 6, while inverse-operation errors decrease from 13–12% in Grades 1–2 to 4% in Grade 6. Item-level patterns shown in Fig. 3 provide a more fine-grained view of the developmental shift identified at the aggregate level. Across Change problems, several items (notably CH2, CH3, and CH4) show a progressive concentration of inverse-operation errors as grade level increases, indicating that difficulties increasingly stem from selecting the appropriate operation rather than from magnitude comparison. In contrast, Combine problems (CB1 and CB2) exhibit a markedly different profile: higher-quantity errors dominate across grades, with relatively limited diversification toward inverse-operation errors, suggesting a more persistent reliance on salient numerical values in this problem family. Compare problems display the clearest strategic transition at the item level. For several items (e.g., CP3 and CP4), errors in Grade 1 are predominantly associated with higher-quantity selection, but from Grade 3 onwards the dominant error pathway shifts toward inverse-operation errors, mirroring the broader reorientation in students’ problem-solving strategies observed across grades. 4.2 Statistical Testing and Localization of Grade-Level Differences 4.2.1 Global Association Tests (χ²) To examine whether the distribution of error types varied across grades, a global χ² test was conducted on the full matrix of 9,216 errors. The association was statistically significant, χ²(10, N = 9,216) = 165.81, p < .001, with Cramer’s V = .095, indicating a small but reliable relationship between grade level and error type. When the analysis was disaggregated by problem family, similar patterns emerged. Change problems yielded χ²(10) = 134.20, p < .001, V = .126; Combine problems yielded χ²(10) = 34.35, p < .001, V = .139; and Compare problems yielded χ²(10) = 78.89, p < .001, V = .098. Although the effect sizes fall within the small range, the consistency of these associations across families suggests systematic developmental differences in the types of errors students produce. These global tests therefore justify a finer-grained examination of which specific grade transitions account for the observed variation. 4.2.2 Pairwise Grade Comparisons Holm-adjusted pairwise comparisons were conducted for every grade combination (1st–6th) for each error type within each problem family. Only statistically significant contrasts (adjusted p < .05) are summarised here; full results are available in Online Appendix A. In Change problems, the proportion of higher-quantity errors is markedly higher in Grade 1 than in any other grade (p ≤ .001), and an additional difference appears between Grades 2 and 4 (p = .011). Lower-quantity errors remain relatively stable until Grade 6, where their frequency becomes significantly lower than in Grades 1, 2, and 3 (p ≈ .002). By contrast, inverse-operation errors increase steadily with grade: all comparisons pairing Grades 1 or 2 with Grades 4, 5, or 6 reach significance (p < .001), as does the comparison between Grades 3 and 6 (p = .001). The pattern observed in Combine problems is less variable. Higher-quantity errors do not differ significantly across grades. Lower-quantity errors, however, show a moderate decline from Grade 1 to Grades 3, 4, and 5 (p values ranging from .017 to .003). Inverse-operation errors vary minimally, with a single significant contrast appearing between Grades 1 and 5 (p = .010). Overall, the Combine family presents a comparatively stable error profile across primary schooling. In Compare problems, the proportion of higher-quantity errors decreases after the first year, with significant differences between Grades 1 and 4 (p = .002), Grades 1 and 5 (p = .027), and Grades 2 and 4 (p = .027). Lower-quantity errors show one significant decline, from Grade 1 to Grade 4 (p = .014). The most pronounced developmental change involves inverse-operation errors, where all contrasts comparing Grades 1–2 with Grades 4–6 reach significance (p < .001–.009), as does the transition from Grade 3 to Grade 4 (p = .009). 5. Discussion The present study set out to examine how primary school students’ errors in one-step additive AWPs evolve across Grades 1 to 6, with particular attention to the role of semantic structure and the nature of the erroneous responses produced. Rather than treating errors as undifferentiated indicators of failure, the analysis focused on theoretically motivated error categories that reflect distinct ways of interpreting the problem situation. This approach allows for a more fine-grained understanding of developmental change in AWP solving than accuracy-based measures alone. Beyond documenting age-related differences in accuracy, this study makes a specific contribution to research on arithmetic word problem solving by characterizing developmental change in terms of qualitatively distinct error patterns across semantic structures. Whereas much previous work has focused on whether students succeed or fail, or on isolated problem types or age groups, the present analysis shows how the nature of students’ errors systematically evolves across the entire span of primary education. By combining a large-scale design with theoretically motivated distractors, the study provides empirical evidence of a developmental reorganization in error profiles, characterized by a decreasing reliance on surface-based quantity selection and an increasing prominence of operation-based errors that are sensitive to semantic structure. This error-focused perspective offers a complementary lens for understanding conceptual development in word problem solving that cannot be captured by accuracy measures alone. Across grades, the results reveal a clear developmental shift in the dominant types of errors students produce. In the early years of primary school, errors are largely driven by the selection of salient quantities mentioned in the problem statement, especially the larger numerical value. As students progress through the grades, these quantity-based errors gradually decrease, while errors involving the use of the inverse operation become increasingly frequent. Importantly, this shift does not occur uniformly across problem types, but interacts systematically with the semantic structure of the problems, yielding distinct developmental trajectories for Change, Combine, and Compare problems. 5.1. From surface-based responses to relational reasoning The predominance of higher-quantity errors in Grades 1 and 2 suggests that younger students often rely on surface features of the problem text, such as the magnitude of the numbers involved, rather than on an explicit representation of the underlying quantitative relations. This pattern is consistent with previous findings showing that novice problem solvers tend to adopt direct translation strategies, mapping numbers and keywords onto operations without constructing a coherent situation model (Kintsch & Greeno, 1985 ; Pape, 2004 ). From Grade 3 onwards, however, the error profile begins to change. The decline in quantity-based errors coincides with a steady increase in inverse-operation errors, indicating that students are no longer merely selecting numbers based on salience, but are actively attempting to model the relations described in the problem. From a developmental perspective, this transition can be interpreted as a shift in the types of processing students attempt, from surface-driven responses toward forms of reasoning that increasingly engage relational information. Rather than reflecting a regression or persistent misunderstanding, the rise of inverse-operation errors suggests that students are engaging with the structure of the situation, albeit in ways that remain fragile or incomplete. This interpretation aligns with developmental accounts that view learning as a process of progressive reorganization, in which emerging representations coexist and sometimes compete with one another (Okamoto, 1996 ; Kail & Hall, 1999 ). In this sense, the persistence of errors does not necessarily signal a lack of conceptual growth, but may instead reflect the cognitive costs associated with coordinating multiple sources of information—linguistic, numerical, and relational—during problem solving. 5.2. The role of semantic structure in shaping developmental trajectories A central contribution of the study lies in demonstrating that developmental change in error patterns is strongly conditioned by the semantic family of the problems. Although all items involved simple additive relations, Change, Combine, and Compare problems elicited markedly different profiles of errors across grades. Change problems showed the most pronounced developmental shift, with inverse-operation errors becoming dominant in the upper grades. These problems often require students to reason backward from a final state to an initial or intermediate quantity, a demand that has long been recognized as particularly challenging for learners (Brissiaud, 1994 ; Riley et al., 1983 ). The increasing prevalence of inverse-operation errors in these items suggests that, as students gain experience, they attempt to coordinate the temporal and relational aspects of the situation, but may struggle to inhibit an initially activated but inappropriate operation. By contrast, Combine problems exhibited a comparatively stable error profile across grades, with higher-quantity errors remaining prevalent even among older students. This stability may reflect the relative transparency of part–whole relations in this family, which encourages a straightforward aggregation strategy and offers fewer opportunities for representational conflict. The results thus suggest that some semantic structures may foster early procedural success without necessarily promoting deeper representational flexibility. Compare problems occupied an intermediate position. In the early grades, higher- and lower-quantity errors were similarly frequent, indicating uncertainty about how to interpret comparative relations. From Grade 3 onward, inverse-operation errors became increasingly prominent, especially in problems involving marked or linguistically inconsistent relational terms. This pattern echoes a substantial body of research documenting the particular difficulty of comparison problems and the role of linguistic consistency in shaping students’ interpretations (Verschaffel, 1994 ; Pape, 2003 ; van der Schoot et al., 2009 ). Taken together, these findings reinforce the view that AWPs cannot be treated as a homogeneous category. Even within the domain of one-step additive problems, different semantic structures place distinct demands on students’ representational resources and give rise to qualitatively different developmental paths. 5.3. Inverse-operation errors as indicators of representational conflict One of the most theoretically informative results of the study concerns the developmental status of inverse-operation errors. Traditionally, such errors have often been interpreted as evidence of confusion about arithmetic operations or insufficient mastery of inverse relations. The present findings suggest a more nuanced interpretation. The increase in inverse-operation errors in later grades indicates that these errors emerge precisely when students begin to engage more deeply with the relational structure of the problem. Rather than reflecting a simple procedural mistake, inverse-operation errors can be understood as manifestations of a conflict between competing representations: a situation model that is semantically plausible and a problem model that requires a different operation for its numerical resolution. This interpretation resonates with accounts emphasizing the tension between situation-based strategies and formal mathematical models (Coquin-Viennot & Moreau, 2007 ; Brissiaud & Sander, 2010 ), as well as with more recent theoretical frameworks highlighting semantic congruence and recoding processes in word problem solving (Gros et al., 2020 ). From this perspective, inverse-operation errors signal that students are no longer operating at a purely surface level, but are attempting to integrate linguistic cues, quantitative relations, and arithmetic knowledge. The difficulty lies not in the absence of understanding, but in the need to inhibit an initially activated operation and to recode the problem representation accordingly. This interpretation is further supported by research linking performance on inconsistent AWPs to executive functions such as inhibition and updating (Lubin et al., 2013 ; Passolunghi et al., 2022 ). Viewing inverse-operation errors as indicators of representational conflict rather than mere failure has important implications for how developmental progress in word problem solving is conceptualized. It shifts the focus from eliminating errors to understanding the cognitive processes that give rise to them, and highlights the transitional nature of many errors observed in the upper primary grades. 5.4. Errors as windows into representational development A distinctive feature of the present study is its systematic use of error categories as theoretically meaningful units of analysis. By examining how different types of errors distribute across grades and semantic structures, the study reveals developmental regularities that would remain invisible if only correct responses were considered. This approach supports a reconceptualization of errors as windows into students’ evolving representations of quantitative situations. The observed shift from quantity-based to inverse-operation errors reflects a qualitative reorganization in how students approach the task, from focusing on isolated numerical elements to attempting to coordinate relations between quantities. Importantly, this shift does not imply a linear progression toward correctness, but rather a period of instability in which more sophisticated forms of reasoning coexist with persistent difficulties. Moreover, the differential patterns observed across semantic families underscore the importance of considering the structure of the situation when analyzing students’ reasoning. Errors are not random or idiosyncratic, but systematically related to the representational demands of the task. This finding lends empirical support to long-standing theoretical claims about the centrality of semantic structure in word problem solving (Riley et al., 1983 ; Verschaffel et al., 2020 ), while extending them by showing how these structures interact with development over the entire span of primary education. 5.5. Implications for research on AWP solving Although the data were collected within a single national curriculum, the semantic structures examined in this study—Change, Combine, and Compare problems, as well as consistency effects in operation selection—are core constructs in international research on arithmetic word problem solving. As such, the observed developmental trajectories speak to instructional and theoretical challenges that are likely to be relevant across educational contexts beyond the Spanish setting. The findings of this study have several implications for future research on AWP solving. First, they highlight the value of incorporating theoretically grounded error analyses into developmental studies. Accuracy-based measures alone may obscure meaningful changes in students’ reasoning, particularly when different types of errors reflect distinct underlying cognitive processes. An error-centered perspective allows researchers to capture how students’ interpretations of additive situations reorganize over time, even when overall performance improves. Second, the results underscore the need for research designs that explicitly account for semantic structure. Treating arithmetic word problems as a unitary class risks overlooking important sources of variability in students’ performance and development. Future studies could examine how instructional interventions or curricular emphases differentially affect error profiles across problem families, and whether improvements in accuracy are accompanied by systematic changes in the nature of students’ errors. Third, the prominence of inverse-operation errors in the upper primary grades points to the importance of integrating models of executive control into theories of word problem solving. Understanding how students manage conflicts between competing representations—such as suppressing an initially salient but inappropriate operation—may be crucial for explaining why certain difficulties persist even after years of instruction. This opens promising avenues for research at the intersection of mathematical cognition, language processing, and executive functioning. From an instructional perspective, the differentiated error profiles identified across grades and semantic families suggest that diagnostic tasks can be strategically aligned with students’ developmental stages. For example, the increasing prominence of inverse-operation errors from approximately Grades 3–4 onwards indicates a critical period for engaging students with linguistically inconsistent Change and Compare problems that require explicit attention to relational meanings and operation choice. Conversely, distractors based on quantity selection may serve as effective diagnostic tools for identifying surface-driven strategies in the early grades. Finally, from a methodological standpoint, the use of a digital environment that ensures complete reading of the problem statements strengthens the interpretability of the results. By reducing the likelihood that observed error patterns are driven by skipped information, such designs support more confident inferences about students’ representational choices rather than omissions in text processing. Future research may benefit from combining similar environments with process-oriented measures, such as response times or eye movements, to further illuminate the dynamics of arithmetic word problem solving. 6. Conclusions This study examined developmental changes in primary school students’ errors when solving one-step additive AWPs, with a particular focus on how these changes interact with the semantic structure of the problems. By moving beyond accuracy-based analyses and treating errors as theoretically meaningful indicators of students’ representations, the study provides a more nuanced picture of development in AWP solving. The results reveal a systematic shift across primary education: early reliance on salient numerical quantities gradually gives way to errors involving inverse operations, especially in problem types that require more complex relational reasoning. Crucially, this shift is not uniform across semantic families. Change, Combine, and Compare problems exhibit distinct developmental trajectories, underscoring the central role of semantic structure in shaping how students interpret and solve word problems. Rather than viewing inverse-operation errors as simple indicators of misunderstanding, the findings suggest that these errors reflect a transitional stage in which students actively engage with relational representations but struggle to resolve conflicts between competing interpretations. From this perspective, errors are not merely obstacles to be eliminated, but windows into the evolving coordination of linguistic, numerical, and relational knowledge. By documenting how error patterns evolve across grades and semantic structures, the present study contributes to theoretical models of word problem solving that emphasize representation, meaning-making, and developmental change. It also highlights the importance of designing research and assessment tools that capture not only whether students succeed or fail, but how they reason about quantitative situations. More broadly, the findings invite a reconceptualization of progress in AWP solving—not as a smooth trajectory toward correctness, but as a process marked by qualitative changes in the nature of students’ reasoning. Understanding these changes is essential for advancing theory and for designing learning environments that support students as they move from surface-based responses toward more flexible and structurally grounded forms of mathematical thinking. Declarations Funding This work was supported by the Generalitat Valenciana of Comunidad Valenciana of Spain under Grant GV/2021/110 and under Grant CIAICO 2022/154. Declaration of interest statement The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article. Ethical approval and consent to participate All relevant ethical guidelines and principles were carefully considered during the preparation of this article. The participants were voluntary, and informed consent was obtained from all participants. Ethical review and approval were required by University of Valencia (2024-MAGPED-3257471). Data availability statement The data that support the findings of this study are available from the corresponding author, [M.T.S.], upon reasonable request. Author Contribution Conceptualization, M.T.S. and F. G.; data curation, E.L.-I., D.G.-C., F.G. and M.T.S.; writing—original draft preparation, E.L.-I. and D. G-C; writing—review and editing, M.T.S. and F.G.; supervision, M.T.S. and F.G.; project administration, F.G.; funding acquisition, F.G and M.T.S. All authors have read and agreed to the published version of the manuscript. References Arendasy, M., Sommer, M., & Ponocny, I. (2005). Psychometric approaches help resolve competing cognitive models: When less is more than it seems. Cognition and Instruction , 23 (4), 503–521. https://doi.org/10.1207/s1532690xci2304_3 Álvarez-García, E., García-Costa, D., Paniagua, S., Sanz, M. T., Santagueda, M., Mocholí, P., López-Iñesta, E., & Grimaldo, F. (2024). A Tool for Reading Evaluation in Arithmetic Word Problems. Frontiers in Artificial Intelligence and Applications , 390 , 325–332. https://doi.org/10.3233/FAIA240455 Bates, D., Mächler, M., Bolker, B., & Walker, S. (2015). Fitting linear mixed-effects models using lme4. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8648711","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":583813253,"identity":"8ca362cc-f968-4c2d-8375-634c73685ab1","order_by":0,"name":"Maria Santagueda-Villanueva","email":"","orcid":"","institution":"Jaume I University","correspondingAuthor":false,"prefix":"","firstName":"Maria","middleName":"","lastName":"Santagueda-Villanueva","suffix":""},{"id":583813254,"identity":"fc68301d-d0f7-42a6-b743-f50ac1438bed","order_by":1,"name":"Emilia López-Iñesta","email":"","orcid":"","institution":"University of Valencia","correspondingAuthor":false,"prefix":"","firstName":"Emilia","middleName":"","lastName":"López-Iñesta","suffix":""},{"id":583813255,"identity":"d35b993f-e6f2-4a23-b142-c0c52da0010c","order_by":2,"name":"Daniel García","email":"","orcid":"","institution":"Jaume I University","correspondingAuthor":false,"prefix":"","firstName":"Daniel","middleName":"","lastName":"García","suffix":""},{"id":583813256,"identity":"da5dfb9f-9e5c-410f-9240-4325a613107c","order_by":3,"name":"Francisco Grimaldo","email":"","orcid":"","institution":"University of Valencia","correspondingAuthor":false,"prefix":"","firstName":"Francisco","middleName":"","lastName":"Grimaldo","suffix":""},{"id":583813257,"identity":"5a425255-74e5-4d23-b4d0-5584ff32d5c3","order_by":4,"name":"Maria T. 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Introduction","content":"\u003cp\u003eArithmetic word problems (AWPs) constitute a central component of primary mathematics education, as they require students to coordinate numerical knowledge with the comprehension of a verbally described situation. Solving such problems goes beyond executing calculations: it involves constructing a representation of the situation, identifying relevant quantitative relations, and mapping these relations onto an appropriate arithmetic operation (Kintsch \u0026amp; Greeno, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1985\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Despite their apparent simplicity, even one-step additive AWPs pose persistent difficulties for many learners across the primary years.\u003c/p\u003e \u003cp\u003eA substantial body of research has shown that children\u0026rsquo;s success in AWPs is strongly influenced by the semantic structure of the problem (Riley et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e; Carpenter et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; De Corte \u0026amp; Verschaffel, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1987\u003c/span\u003e). Classic classifications distinguish among Change, Combine, and Compare problems, each imposing distinct relational demands. Empirical studies have consistently documented that these semantic differences affect difficulty, strategy choice, and error rates, with Compare problems in particular remaining challenging well beyond the early grades (Arendasy et al., \u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e2005\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). More recent syntheses confirm that linguistic and structural task characteristics, such as the position of the unknown or lexical consistency, systematically shape students\u0026rsquo; performance in elementary-school AWPs (Daroczy et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Vessonen et al., \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eImportantly, research has repeatedly emphasized that incorrect responses in AWPs are not merely random mistakes but provide valuable information about students\u0026rsquo; underlying representations and reasoning processes. Early work on error analysis demonstrated that children\u0026rsquo;s wrong answers often correspond to coherent, theoretically interpretable strategies or partial models (Jaspers \u0026amp; van Lieshout, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Jitendra \u0026amp; Kameenui, \u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e1996\u003c/span\u003e). In the context of additive AWPs, many errors reflect surface-based approaches in which students select salient numbers from the problem statement rather than constructing a relational model of the situation (Gonz\u0026aacute;lez-Pienda et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Pape, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2004\u003c/span\u003e). Such quantity-selection errors are particularly frequent in the early grades and have been linked to direct translation strategies and shallow encodings of the problem text.\u003c/p\u003e \u003cp\u003eWith increasing age and schooling, children tend to rely less on pure quantity selection, yet they continue to make systematic errors related to the choice of arithmetic operation. A well-documented example is the tendency to apply the inverse operation when relational language or familiar heuristics are misleading (e.g., adding when subtraction is required, or vice versa). This phenomenon has been extensively studied in relation to language consistency and reversal errors, especially in Compare and Change problems (Verschaffel, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Pape, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Leong \u0026amp; Jerred, \u003cspan citationid=\"CR26\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Such inverse-operation errors suggest that students are engaging with relational information but struggle to correctly map that information onto an arithmetic model, often due to the influence of linguistic cues or overlearned rules.\u003c/p\u003e \u003cp\u003eDevelopmental research further indicates that these error types are not uniformly distributed across grades or problem families. Instead, children\u0026rsquo;s profiles evolve as their mathematical knowledge, language skills, and domain-general cognitive resources develop (Kail \u0026amp; Hall, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Bjorn et al., 2016; Tolar et al., \u003cspan citationid=\"CR38\" class=\"CitationRef\"\u003e2012\u003c/span\u003e). Studies focusing on executive functions highlight that inhibition and working memory updating play a critical role in resolving semantically incongruent AWPs, particularly when misleading heuristics must be suppressed (Lubin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Passolunghi et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). From this perspective, shifts in error patterns may reflect not only growing conceptual understanding but also changes in the cognitive demands that different problem types impose at different ages.\u003c/p\u003e \u003cp\u003eDespite this extensive literature, relatively few large-scale studies have traced systematic developmental changes in distinct, theoretically motivated error types across the full span of primary education while simultaneously considering semantic problem families. Many investigations focus on accuracy, on specific grades, or on isolated problem types, making it difficult to describe how children\u0026rsquo;s interpretations of additive situations evolve from early to later primary school years. Moreover, error analyses are often based on open responses, which, while rich, limit scalability and comparability across large samples.\u003c/p\u003e \u003cp\u003eThe present study addresses this gap by examining grade-related changes in children\u0026rsquo;s error patterns when solving one-step additive AWPs from Grades 1 to 6. Using a large sample of Spanish primary-school students (N\u0026thinsp;=\u0026thinsp;3,111), we administered a battery of Change, Combine, and Compare problems in a multiple-choice format in which distractors were deliberately constructed to capture three theoretically meaningful error types: selection of the higher quantity, selection of the lower quantity, and application of the inverse operation. By analyzing how the distribution of these error categories varies across grades and semantic families, we aim to document systematic developmental differences in students\u0026rsquo; error profiles when interpreting additive situations.\u003c/p\u003e \u003cp\u003eSpecifically, the study addresses the following research questions:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eHow do the frequencies of higher-quantity, lower-quantity, and inverse-operation errors change across Grades 1 to 6 in one-step additive AWPs?\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eDo these developmental trajectories differ across Change, Combine, and Compare problem families?\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eBeyond documenting grade-related differences in performance, the present study contributes to research on arithmetic word problem solving in three main ways. First, it adopts an error-centered perspective in which incorrect responses are treated as diagnostically meaningful indicators of students\u0026rsquo; interpretations, rather than as undifferentiated failures. By systematically distinguishing between quantity-based and inverse-operation errors, the study provides a fine-grained account of how students\u0026rsquo; difficulties are distributed across semantic structures and developmental stages. Second, by combining a large-scale design with theoretically motivated distractors embedded in a controlled multiple-choice format, the study allows for the identification of developmental regularities that are difficult to capture through accuracy-based or small-sample analyses alone. Third, the explicit comparison of Change, Combine, and Compare problem families reveals that developmental change in additive word problem solving is not uniform, but closely tied to the representational demands of different semantic structures. By focusing on the structure of incorrect responses, the study seeks to contribute to a more fine-grained understanding of children\u0026rsquo;s evolving interpretations of additive situations and to inform both research and instruction on how and why students struggle with seemingly simple word problems throughout primary education.\u003c/p\u003e"},{"header":"2. Theoretical Framework","content":"\u003cp\u003eAWPs constitute a central locus for studying how learners coordinate linguistic information with quantitative relations to produce a mathematical model. A consistent finding across decades of research is that success in AWPs depends not only on calculation skills but also on the construction of adequate representations from the text, and on the ability to select and justify operations that match the underlying situation structure (Kintsch \u0026amp; Greeno, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1985\u003c/span\u003e; Pape, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In this sense, developmental progress in AWP solving can be understood as a gradual shift from surface-driven processing\u0026mdash;where salient numbers or keywords guide responses\u0026mdash;toward more structurally informed situation models that support relational reasoning and operation selection (Okamoto, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Christou \u0026amp; Philippou, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1998\u003c/span\u003e; Verschaffel \u0026amp; De Corte, \u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e1993\u003c/span\u003e).\u003c/p\u003e \u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Situation models, problem models, and semantic structures\u003c/h2\u003e \u003cp\u003eClassical frameworks emphasize that understanding a AWP involves building a text base and integrating it with prior knowledge to form a situation model that captures the relevant quantitative relations (Kintsch \u0026amp; Greeno, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1985\u003c/span\u003e). In school contexts, however, learners often rely on direct translation heuristics, mapping lexical cues to arithmetic operations without fully representing the situation (Pape, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2004\u003c/span\u003e; Gonz\u0026aacute;lez-Pienda et al., \u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e1999\u003c/span\u003e). This tendency is especially consequential for additive problems, where the same operation family (addition/subtraction) supports multiple semantic structures with distinct relational demands (Riley et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e; Carpenter et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Research based on semantic taxonomies shows that task difficulty and solution behavior systematically vary across families such as Change, Combine, and Compare, and that these differences are already present in early grades (De Corte \u0026amp; Verschaffel, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e1987\u003c/span\u003e; Christou \u0026amp; Philippou, \u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e1998\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eA key implication is that learners may generate answers that are coherent with an incomplete or biased representation of the described situation. This includes cases where the qualitative situation model (everyday interpretation) appears to conflict with the quantitative problem model (formal relations), which can reduce performance when schema activation is not yet automatized (Coquin-Viennot \u0026amp; Moreau, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2007\u003c/span\u003e). Such representational tensions provide a principled basis for analyzing wrong answers as informative outcomes rather than random noise: error patterns can index which aspects of the situation structure have been encoded, which relations have been ignored, and which procedures have been misapplied (Jaspers \u0026amp; Van Lieshout, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Kingsdorf \u0026amp; Krawec, \u003cspan citationid=\"CR24\" class=\"CitationRef\"\u003e2014\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Semantic congruence, consistency effects, and operation reversal\u003c/h2\u003e \u003cp\u003eA prominent line of work has shown that performance in compare problems is sensitive to whether the relational language is consistent with the required operation. In \u0026ldquo;inconsistent language\u0026rdquo; formats, relational terms can cue an operation that is opposite to the mathematically required one, leading to systematic reversal (inverse-operation) errors and distortions in recall (Pape, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2003\u003c/span\u003e). Evidence from retelling methods indicates that when relational statements mismatch solvers\u0026rsquo; preferred formats, representational difficulties increase and errors become more likely (Verschaffel, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e1994\u003c/span\u003e). Eye-movement research further suggests that less successful solvers show distinctive processing costs under inconsistency, modulated by linguistic markedness (e.g., \u0026ldquo;less than\u0026rdquo; vs. \u0026ldquo;more than\u0026rdquo;) (van der Schoot et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2009\u003c/span\u003e). These findings support the view that operation selection is not purely computational but is strongly shaped by semantic cues that steer interpretation.\u003c/p\u003e \u003cp\u003eMore recent theoretical developments propose that AWP solving is constrained by semantic congruence between world meanings and mathematical procedures. The Semantic Congruence (SECO) model argues that world semantics can bias the encoding and recoding of problems, such that semantically incongruent representations require deliberate recoding that learners often fail to complete (Gros et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). In this perspective, operation errors are not merely \u0026ldquo;wrong operations\u0026rdquo; but manifestations of incomplete recoding between an initially compelling situation-based representation and the formal arithmetic structure. Empirical work manipulating linguistic and numerical factors likewise indicates that AWP difficulty emerges from the interaction of these components rather than from either domain alone (Daroczy et al., \u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). Taken together, these approaches justify analyzing inverse-operation errors as theoretically meaningful outcomes, especially in problem families (notably Compare and certain Change formats) where semantic cues can conflict with the required transformation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e2.3. Cognitive constraints: executive functions, inhibition, and reading-related resources\u003c/h2\u003e \u003cp\u003eBeyond linguistic-semantic factors, individual differences in domain-general cognition contribute substantially to AWP performance. Research consistently shows that reading comprehension is a strong predictor of success, even when controlling for calculation ability and other skills (Bj\u0026ouml;rn et al., \u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e2016\u003c/span\u003e; Boonen et al., \u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Passolunghi et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). In addition, executive functions\u0026mdash;particularly inhibition and working memory updating\u0026mdash;play a distinct role when students must suppress misleading heuristics and maintain intermediate representations (Passolunghi et al., \u003cspan citationid=\"CR31\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Passolunghi \u0026amp; Siegel, \u003cspan citationid=\"CR32\" class=\"CitationRef\"\u003e2001\u003c/span\u003e). Developmental evidence from negative priming paradigms indicates that solving inconsistent compare problems requires inhibiting default strategies such as \u0026ldquo;add if more, subtract if less,\u0026rdquo; and that this inhibitory demand is present across age groups (Lubin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2013\u003c/span\u003e). Complementarily, process-oriented studies suggest that executive functions support different phases of solving, including selecting relevant information, coordinating relations, and planning the appropriate operation sequence (Viterbori et al., \u003cspan citationid=\"CR45\" class=\"CitationRef\"\u003e2017\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eThese findings align with developmental accounts proposing that, as students gain experience, difficulties may shift from identifying salient quantities toward selecting and inhibiting operations. For instance, children may initially commit quantity-selection errors (e.g., choosing one of the numbers in the statement), but later increasingly exhibit operation-based errors when their representations become more structured yet still vulnerable to semantic biases (Kail \u0026amp; Hall, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Degrande et al., \u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e2019\u003c/span\u003e). This developmental reorientation is also consistent with models emphasizing increasing flexibility in dealing with additive situations during the first years of schooling (Ufer et al., \u003cspan citationid=\"CR39\" class=\"CitationRef\"\u003e2024\u003c/span\u003e) and with observations that young learners\u0026rsquo; strategy choices are shaped by semantic structure, cognitive economy, and number relations (Sevinc et al., \u003cspan citationid=\"CR37\" class=\"CitationRef\"\u003e2024\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e2.4. Task characteristics, stimulus control, and the diagnostic value of distractors\u003c/h2\u003e \u003cp\u003eBecause error patterns are shaped by both representation and task design, methodological control of item characteristics is essential. Reviews and meta-analyses show that AWP difficulty is influenced by multiple linguistic features (e.g., lexical consistency, position of the unknown, irrelevant information) as well as numerical features (e.g., number of operations), with particularly strong effects when realism considerations or lexical inconsistency are involved (Daroczy et al., \u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e2015\u003c/span\u003e; Vessonen et al., \u003cspan citationid=\"CR44\" class=\"CitationRef\"\u003e2024\u003c/span\u003e). Experimental evidence further supports that combining inconsistency with other disruptive features (such as irrelevant numbers) can change reasoning and increase performance drops (Vondrov\u0026aacute;, \u003cspan citationid=\"CR46\" class=\"CitationRef\"\u003e2022\u003c/span\u003e). Therefore, constructing response options that map onto theoretically motivated misconceptions provides a principled approach to diagnosing students\u0026rsquo; representations.\u003c/p\u003e \u003cp\u003eA productive strategy is to embed distractors that correspond to specific error types. In additive contexts, three recurrent patterns are (a) selecting the higher quantity in the statement, (b) selecting the lower quantity, and (c) applying the inverse operation. Such distractors operationalize distinct failure modes: quantity-based choices reflect surface-driven selection of salient values, whereas inverse-operation choices reflect representational or recoding failures at the operation-selection stage (Pape, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; Gros et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2020\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e). This error-sensitive design aligns with research showing that wrong answers can be systematically diagnosed and linked to underlying procedures rather than treated as random guessing (Chval et al., \u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e2021\u003c/span\u003e; Jaspers \u0026amp; Van Lieshout, \u003cspan citationid=\"CR21\" class=\"CitationRef\"\u003e1994\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Methodology","content":"\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e3.1. Research design and Participants\u003c/h2\u003e \u003cp\u003eThis study employed a quantitative, cross-sectional, non-experimental design to examine how primary-school students respond to one-step additive AWPs. Because the purpose was to document grade-related differences in erroneous responses\u0026mdash;rather than evaluate instructional interventions\u0026mdash;an cross-sectional observational design approach was appropriate. This design is consistent with prior research on the semantic structures of AWPs (Riley et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e; Carpenter et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eNineteen public primary schools from a Spanish autonomous community were randomly sampled from the official registry provided by the regional Ministry of Education. All students in Grades 1\u0026ndash;6 attending school on the testing day were invited to participate, yielding a final sample of N\u0026thinsp;=\u0026thinsp;3,111 (1,539 girls and 1,572 boys). Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e presents the distribution by grade and sex. This sample size ensures a maximum sampling error below 5% at the 95% confidence level, offering stable estimates for grade-level comparisons.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eDistribution of participants by grade level and sex.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"8\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c8\" colnum=\"8\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1st\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2nd\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3rd\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4th\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5th\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6th\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c8\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eGirls\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e236\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e230\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e291\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e260\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e272\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e250\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1539\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eBoys\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e219\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e276\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e233\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e276\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e267\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e301\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e1572\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eTotal\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e455\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e506\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e524\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e536\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e539\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e551\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c8\"\u003e \u003cp\u003e3111\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eAll participating schools follow the national mathematics curriculum of Spain, which establishes common learning goals, content progressions, and competency-based outcomes across all autonomous communities (Real Decreto 157/2022). Although the Valencian Community develops its own curricular guidelines, these must remain aligned with the national framework, resulting in substantial uniformity in the teaching and learning of mathematics throughout the country. This coherence ensures that the developmental patterns observed in the present study reflect typical learning trajectories within the Spanish primary curriculum, which itself is consistent with broader European orientations toward competency-based mathematics education (Eurydice Network, \u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e2025\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.2. Instruments\u003c/h2\u003e \u003cp\u003eThe task battery comprised 14 one-step additive AWPs classified into Change (6 items), Combine (2 items), and Compare (6 items) categories, following the canonical semantic taxonomy of Riley et al. (\u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e). Each item was designed to capture students\u0026rsquo; interpretation of the underlying situation structure and their tendency to rely on surface-level numerical cues.\u003c/p\u003e \u003cp\u003eEach problem included four multiple-choice options:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003ethe correct answer,\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003etwo quantities given in the statement (the higher and the lower value), and\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eone distractor generated by applying the inverse operation.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eThese distractors correspond to three error types widely documented in AWP research\u0026mdash;Higher-quantity, Lower-quantity, and Inverse-operation errors\u0026mdash;which serve as indicators of students\u0026rsquo; conceptual models when interpreting additive situations (Kintsch \u0026amp; Greeno, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1985\u003c/span\u003e; Carpenter et al., \u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e1999\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e presents the full set of items, grouped by semantic family, along with their response options.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eBattery of one-step additive AWPs by type, problem statement, and solution options.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eType\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eStatement\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSolutions\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"5\" rowspan=\"6\"\u003e \u003cp\u003eChange\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH1. Joe tiene 3 canicas. Luego Tom le dio 5 canicas m\u0026aacute;s. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Joe ahora?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH2. Joe ten\u0026iacute;a 8 canicas. Luego le dio 5 canicas a Tom. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Joe ahora?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH3. Joe ten\u0026iacute;a 5 canicas. Luego Tom le dio algunas canicas m\u0026aacute;s. Ahora Joe tiene 8 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas le dio Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH4. Joe ten\u0026iacute;a 8 canicas. Luego le dio algunas canicas a Tom. Ahora Joe tiene 3 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas le dio a Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 11\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH5. Joe ten\u0026iacute;a algunas canicas. Luego Tom le dio 5 canicas m\u0026aacute;s. Ahora Joe tiene 8 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas ten\u0026iacute;a Joe al principio?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCH6. Joe ten\u0026iacute;a algunas canicas. Luego \u0026eacute;l le dio 5 canicas a Tom. Joe tiene 3 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas ten\u0026iacute;a Joe al principio?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003eCombine\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCB1. Joe tiene 3 canicas. Tom tiene 5 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas tienen en total?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCB2. Joe y Tom tienen 8 canicas en total. Joe tiene 3 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 8 b. 3 c. 5 d. 11\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"5\" rowspan=\"6\"\u003e \u003cp\u003eCompare\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP1. Joe tiene 8 canicas. Tom tiene 5 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Joe m\u0026aacute;s que Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 8 b. 5 c. 3 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP2. Joe tiene 8 canicas. Tom tiene 5 canicas. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom menos que Joe?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 8 b. 5 c. 3 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP3. Joe tiene 3 canicas. Tom tiene 5 canicas m\u0026aacute;s que Joe. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP4. Joe tiene 8 canicas. Tom tiene 5 canicas menos que Joe. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 8 b. 5 c. 3 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP5. Joe tiene 8 canicas. \u0026Eacute;l tiene 5 canicas m\u0026aacute;s que Tom. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 8 b. 5 c. 3 d. 13\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eCP6. Joe tiene 3 canicas. \u0026Eacute;l tiene 5 canicas menos que Tom. \u0026iquest;Cu\u0026aacute;ntas canicas tiene Tom?\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003ea. 3 b. 5 c. 8 d. 2\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eItem wording and numerical values were linguistically adapted to Spanish. Readability and syntactic balance across items were refined through linguistic-masking techniques described in \u0026Aacute;lvarez et al. (2024), ensuring that:\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003elexical or structural cues did not privilege any distractor,\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003enumerical information was comparable across problems, and\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003evariability in difficulty was attributable to semantic structure, not language.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eAll items were administered through AWPSolver (\u0026Aacute;lvarez et al., 2024), a web-based research platform that:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e(a) displays problem statements in masked segments, requiring students to reveal each part sequentially;\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e(b) provides a virtual scratch-pad and basic calculator (optional); and\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003e(c) records chosen answers, reading-completeness logs, and other interaction data.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eFor the purposes of the present study, only answer choices and reading-completeness indicators were used in the analyses.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.3. Procedure\u003c/h2\u003e \u003cp\u003eData were collected during regular mathematics lessons in the 2023\u0026ndash;2024 school year. After obtaining school and parental consent, each class received a 10-minute familiarisation session with AWPSolver. The full test was administered digitally and lasted approximately 45 minutes.\u003c/p\u003e \u003cp\u003eTwo exclusion criteria ensured that each recorded response reflected genuine task engagement:\u003c/p\u003e \u003cp\u003e \u003col\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eIncomplete reading: If any segment of a problem statement was skipped (verified through AWPSolver\u0026rsquo;s masking-navigation logs), the item was discarded for that pupil.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003cspan\u003e \u003cli\u003e \u003cp\u003eMissing response: Items with no answer selected were also discarded, because comprehension in our framework requires both reading and responding.\u003c/p\u003e \u003c/li\u003e \u003c/span\u003e \u003c/ol\u003e \u003c/p\u003e \u003cp\u003eTable\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e shows the number of valid responses per item and grade after applying these criteria.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eNumber of students who completed each problem type.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"7\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c5\" colnum=\"5\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c6\" colnum=\"6\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c7\" colnum=\"7\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e1st\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2nd\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3rd\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c5\"\u003e \u003cp\u003e4th\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c6\"\u003e \u003cp\u003e5th\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c7\"\u003e \u003cp\u003e6th\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e384\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e461\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e490\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e510\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e519\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e540\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e331\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e414\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e462\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e484\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e530\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e308\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e433\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e474\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e514\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e508\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e529\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e284\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e402\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e448\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e472\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e478\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e494\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e291\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e434\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e469\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e499\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e493\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e524\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCH6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e279\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e413\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e462\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e482\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e495\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e522\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCB1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e297\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e433\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e498\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e517\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e534\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCB2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e275\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e430\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e489\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e518\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e521\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e534\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e278\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e425\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e482\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e508\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e519\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e539\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e272\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e426\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e480\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e507\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e520\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e533\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e238\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e413\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e472\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e501\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e512\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e531\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e237\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e384\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e460\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e476\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e496\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e516\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e225\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e391\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e457\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e480\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e501\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e524\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eCP6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e219\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e373\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c4\"\u003e \u003cp\u003e446\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c5\"\u003e \u003cp\u003e476\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c6\"\u003e \u003cp\u003e488\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c7\"\u003e \u003cp\u003e517\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.4. Data Analysis\u003c/h2\u003e \u003cp\u003eAll analyses were conducted in R 4.3.2 (R Core Team, \u003cspan citationid=\"CR34\" class=\"CitationRef\"\u003e2023\u003c/span\u003e). We began by computing descriptive statistics for each item and grade level, including the frequencies and percentages of the three error categories\u0026mdash;Higher-quantity, Lower-quantity, and Inverse-operation errors. Ninety-five percent confidence intervals were calculated using Wilson\u0026rsquo;s method (Brown, et al., \u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e2001\u003c/span\u003e), which provides accurate coverage for binomial proportions.\u003c/p\u003e \u003cp\u003eTo examine whether error distributions differed across grade levels, we performed χ\u0026sup2; tests of independence. When expected cell frequencies fell below 5, Monte Carlo p-values based on 10,000 replicates were reported. Statistical significance was set at α\u0026thinsp;=\u0026thinsp;.05, and pairwise contrasts were evaluated using the Holm sequential correction (Holm, \u003cspan citationid=\"CR20\" class=\"CitationRef\"\u003e1979\u003c/span\u003e) to control for multiple comparisons.\u003c/p\u003e \u003cp\u003eBecause students\u0026rsquo; responses were nested within schools and items, we additionally estimated mixed-effects logistic regression models as robustness checks, using the lme4 package (Bates et al., \u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e2015\u003c/span\u003e). These models included random intercepts for schools and students, while grade level and problem family were entered as fixed effects. The mixed-effects models yielded conclusions fully consistent with those obtained from the χ\u0026sup2; analyses; therefore, the inferential results presented in the article focus on the χ\u0026sup2; and Holm-adjusted comparisons.\u003c/p\u003e \u003c/div\u003e"},{"header":"4. Results","content":"\u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e4.1 Descriptive Error Patterns\u003c/h2\u003e \u003cp\u003eFigure \u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e displays the distribution of response outcomes (correct responses and three error types) across Grades 1\u0026ndash;6. Accuracy rises sharply from 43% correct in Grade 1 to 60% (Grade 2), 75% (Grade 3), 83% (Grade 4), 86% (Grade 5), and 88% (Grade 6). As a share of all responses, each error type decreases with grade level: larger-quantity errors drop from 22% (Grade 1) to 14% (Grade 2), 8% (Grade 3), 5% (Grade 4), 4% (Grade 5), and 3% (Grade 6); smaller-quantity errors decline from 19% to 13%, 8%, 5%, 4%, and 3%, respectively; and inverse-operation errors decrease from 16% to 13%, 9%, 7%, 6%, and 5%. Importantly, although inverse-operation errors shrink in absolute terms (because overall correctness increases), they represent an increasing share of the remaining errors in the upper grades (e.g., about 28% of errors in Grade 1 vs. about 42% in Grade 6), suggesting that once students largely master magnitude selection, the main residual difficulty concerns choosing the appropriate arithmetic operation.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eFigure 2 presents the distribution of response outcomes by grade within each problem family (Change, Combine, and Compare). In Change problems (Fig.\u0026nbsp;2a), accuracy increases from 45% in Grade 1 to 61% in Grade 2, 74% in Grade 3, 82% in Grade 4, 84% in Grade 5, and 85% in Grade 6. Although all error types decrease in absolute terms, inverse-operation errors remain comparatively persistent across grades (17%, 15%, 11%, 9%, 8%, and 8%), whereas larger-quantity errors drop from 21% to 3% and smaller-quantity errors from 17% to 3%. In Combine problems (Fig.\u0026nbsp;2b), accuracy is consistently higher, rising from 54% (Grade 1) to 70%, 83%, 90%, 94%, and 96% by Grade 6. Larger-quantity errors dominate incorrect responses in the first grades (19% in Grade 1 and 13% in Grade 2) but decrease sharply thereafter (7% in Grade 3 and \u0026le;\u0026thinsp;4% from Grade 4 onwards); smaller-quantity and inverse-operation errors follow similar declining trajectories, falling to 2% or less in the upper grades. In Compare problems (Fig.\u0026nbsp;2c), accuracy starts lower (37% in Grade 1; 54% in Grade 2) and improves steadily to 73% (Grade 3), 82% (Grade 4), 86% (Grade 5), and 89% (Grade 6). Early grades show high proportions of quantity-based errors\u0026mdash;larger-quantity (26% and 18%) and smaller-quantity (24% and 16%)\u0026mdash;which diminish substantially across grades, reaching 4\u0026ndash;5% by Grades 5 and 6, while inverse-operation errors decrease from 13\u0026ndash;12% in Grades 1\u0026ndash;2 to 4% in Grade 6.\u003c/p\u003e\u003cp\u003eItem-level patterns shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e3\u003c/span\u003e provide a more fine-grained view of the developmental shift identified at the aggregate level. Across Change problems, several items (notably CH2, CH3, and CH4) show a progressive concentration of inverse-operation errors as grade level increases, indicating that difficulties increasingly stem from selecting the appropriate operation rather than from magnitude comparison. In contrast, Combine problems (CB1 and CB2) exhibit a markedly different profile: higher-quantity errors dominate across grades, with relatively limited diversification toward inverse-operation errors, suggesting a more persistent reliance on salient numerical values in this problem family. Compare problems display the clearest strategic transition at the item level. For several items (e.g., CP3 and CP4), errors in Grade 1 are predominantly associated with higher-quantity selection, but from Grade 3 onwards the dominant error pathway shifts toward inverse-operation errors, mirroring the broader reorientation in students\u0026rsquo; problem-solving strategies observed across grades.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e4.2 Statistical Testing and Localization of Grade-Level Differences\u003c/h2\u003e \u003cdiv id=\"Sec15\" class=\"Section3\"\u003e \u003ch2\u003e4.2.1 Global Association Tests (χ\u0026sup2;)\u003c/h2\u003e \u003cp\u003eTo examine whether the distribution of error types varied across grades, a global χ\u0026sup2; test was conducted on the full matrix of 9,216 errors. The association was statistically significant,\u003c/p\u003e \u003cp\u003eχ\u0026sup2;(10, N\u0026thinsp;=\u0026thinsp;9,216)\u0026thinsp;=\u0026thinsp;165.81, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, with Cramer\u0026rsquo;s V\u0026thinsp;=\u0026thinsp;.095, indicating a small but reliable relationship between grade level and error type.\u003c/p\u003e \u003cp\u003eWhen the analysis was disaggregated by problem family, similar patterns emerged.\u003c/p\u003e \u003cp\u003e \u003cul\u003e \u003cli\u003e \u003cp\u003eChange problems yielded χ\u0026sup2;(10)\u0026thinsp;=\u0026thinsp;134.20, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, V\u0026thinsp;=\u0026thinsp;.126;\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eCombine problems yielded χ\u0026sup2;(10)\u0026thinsp;=\u0026thinsp;34.35, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, V\u0026thinsp;=\u0026thinsp;.139; and\u003c/p\u003e \u003c/li\u003e \u003cli\u003e \u003cp\u003eCompare problems yielded χ\u0026sup2;(10)\u0026thinsp;=\u0026thinsp;78.89, p\u0026thinsp;\u0026lt;\u0026thinsp;.001, V\u0026thinsp;=\u0026thinsp;.098.\u003c/p\u003e \u003c/li\u003e \u003c/ul\u003e \u003c/p\u003e \u003cp\u003eAlthough the effect sizes fall within the small range, the consistency of these associations across families suggests systematic developmental differences in the types of errors students produce. These global tests therefore justify a finer-grained examination of which specific grade transitions account for the observed variation.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section3\"\u003e \u003ch2\u003e4.2.2 Pairwise Grade Comparisons\u003c/h2\u003e \u003cp\u003eHolm-adjusted pairwise comparisons were conducted for every grade combination (1st\u0026ndash;6th) for each error type within each problem family. Only statistically significant contrasts (adjusted p\u0026thinsp;\u0026lt;\u0026thinsp;.05) are summarised here; full results are available in Online Appendix A.\u003c/p\u003e \u003cp\u003eIn Change problems, the proportion of higher-quantity errors is markedly higher in Grade 1 than in any other grade (p\u0026thinsp;\u0026le;\u0026thinsp;.001), and an additional difference appears between Grades 2 and 4 (p\u0026thinsp;=\u0026thinsp;.011). Lower-quantity errors remain relatively stable until Grade 6, where their frequency becomes significantly lower than in Grades 1, 2, and 3 (p\u0026thinsp;\u0026asymp;\u0026thinsp;.002). By contrast, inverse-operation errors increase steadily with grade: all comparisons pairing Grades 1 or 2 with Grades 4, 5, or 6 reach significance (p\u0026thinsp;\u0026lt;\u0026thinsp;.001), as does the comparison between Grades 3 and 6 (p\u0026thinsp;=\u0026thinsp;.001).\u003c/p\u003e \u003cp\u003eThe pattern observed in Combine problems is less variable. Higher-quantity errors do not differ significantly across grades. Lower-quantity errors, however, show a moderate decline from Grade 1 to Grades 3, 4, and 5 (p values ranging from .017 to .003). Inverse-operation errors vary minimally, with a single significant contrast appearing between Grades 1 and 5 (p\u0026thinsp;=\u0026thinsp;.010). Overall, the Combine family presents a comparatively stable error profile across primary schooling.\u003c/p\u003e \u003cp\u003eIn Compare problems, the proportion of higher-quantity errors decreases after the first year, with significant differences between Grades 1 and 4 (p\u0026thinsp;=\u0026thinsp;.002), Grades 1 and 5 (p\u0026thinsp;=\u0026thinsp;.027), and Grades 2 and 4 (p\u0026thinsp;=\u0026thinsp;.027). Lower-quantity errors show one significant decline, from Grade 1 to Grade 4 (p\u0026thinsp;=\u0026thinsp;.014). The most pronounced developmental change involves inverse-operation errors, where all contrasts comparing Grades 1\u0026ndash;2 with Grades 4\u0026ndash;6 reach significance (p\u0026thinsp;\u0026lt;\u0026thinsp;.001\u0026ndash;.009), as does the transition from Grade 3 to Grade 4 (p\u0026thinsp;=\u0026thinsp;.009).\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"5. Discussion","content":"\u003cp\u003eThe present study set out to examine how primary school students\u0026rsquo; errors in one-step additive AWPs evolve across Grades 1 to 6, with particular attention to the role of semantic structure and the nature of the erroneous responses produced. Rather than treating errors as undifferentiated indicators of failure, the analysis focused on theoretically motivated error categories that reflect distinct ways of interpreting the problem situation. This approach allows for a more fine-grained understanding of developmental change in AWP solving than accuracy-based measures alone.\u003c/p\u003e \u003cp\u003eBeyond documenting age-related differences in accuracy, this study makes a specific contribution to research on arithmetic word problem solving by characterizing developmental change in terms of qualitatively distinct error patterns across semantic structures. Whereas much previous work has focused on whether students succeed or fail, or on isolated problem types or age groups, the present analysis shows how the nature of students\u0026rsquo; errors systematically evolves across the entire span of primary education. By combining a large-scale design with theoretically motivated distractors, the study provides empirical evidence of a developmental reorganization in error profiles, characterized by a decreasing reliance on surface-based quantity selection and an increasing prominence of operation-based errors that are sensitive to semantic structure. This error-focused perspective offers a complementary lens for understanding conceptual development in word problem solving that cannot be captured by accuracy measures alone.\u003c/p\u003e \u003cp\u003eAcross grades, the results reveal a clear developmental shift in the dominant types of errors students produce. In the early years of primary school, errors are largely driven by the selection of salient quantities mentioned in the problem statement, especially the larger numerical value. As students progress through the grades, these quantity-based errors gradually decrease, while errors involving the use of the inverse operation become increasingly frequent. Importantly, this shift does not occur uniformly across problem types, but interacts systematically with the semantic structure of the problems, yielding distinct developmental trajectories for Change, Combine, and Compare problems.\u003c/p\u003e \u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e5.1. From surface-based responses to relational reasoning\u003c/h2\u003e \u003cp\u003eThe predominance of higher-quantity errors in Grades 1 and 2 suggests that younger students often rely on surface features of the problem text, such as the magnitude of the numbers involved, rather than on an explicit representation of the underlying quantitative relations. This pattern is consistent with previous findings showing that novice problem solvers tend to adopt direct translation strategies, mapping numbers and keywords onto operations without constructing a coherent situation model (Kintsch \u0026amp; Greeno, \u003cspan citationid=\"CR25\" class=\"CitationRef\"\u003e1985\u003c/span\u003e; Pape, \u003cspan citationid=\"CR30\" class=\"CitationRef\"\u003e2004\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFrom Grade 3 onwards, however, the error profile begins to change. The decline in quantity-based errors coincides with a steady increase in inverse-operation errors, indicating that students are no longer merely selecting numbers based on salience, but are actively attempting to model the relations described in the problem. From a developmental perspective, this transition can be interpreted as a shift in the types of processing students attempt, from surface-driven responses toward forms of reasoning that increasingly engage relational information. Rather than reflecting a regression or persistent misunderstanding, the rise of inverse-operation errors suggests that students are engaging with the structure of the situation, albeit in ways that remain fragile or incomplete.\u003c/p\u003e \u003cp\u003eThis interpretation aligns with developmental accounts that view learning as a process of progressive reorganization, in which emerging representations coexist and sometimes compete with one another (Okamoto, \u003cspan citationid=\"CR28\" class=\"CitationRef\"\u003e1996\u003c/span\u003e; Kail \u0026amp; Hall, \u003cspan citationid=\"CR23\" class=\"CitationRef\"\u003e1999\u003c/span\u003e). In this sense, the persistence of errors does not necessarily signal a lack of conceptual growth, but may instead reflect the cognitive costs associated with coordinating multiple sources of information\u0026mdash;linguistic, numerical, and relational\u0026mdash;during problem solving.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e5.2. The role of semantic structure in shaping developmental trajectories\u003c/h2\u003e \u003cp\u003eA central contribution of the study lies in demonstrating that developmental change in error patterns is strongly conditioned by the semantic family of the problems. Although all items involved simple additive relations, Change, Combine, and Compare problems elicited markedly different profiles of errors across grades.\u003c/p\u003e \u003cp\u003eChange problems showed the most pronounced developmental shift, with inverse-operation errors becoming dominant in the upper grades. These problems often require students to reason backward from a final state to an initial or intermediate quantity, a demand that has long been recognized as particularly challenging for learners (Brissiaud, \u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Riley et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e). The increasing prevalence of inverse-operation errors in these items suggests that, as students gain experience, they attempt to coordinate the temporal and relational aspects of the situation, but may struggle to inhibit an initially activated but inappropriate operation.\u003c/p\u003e \u003cp\u003eBy contrast, Combine problems exhibited a comparatively stable error profile across grades, with higher-quantity errors remaining prevalent even among older students. This stability may reflect the relative transparency of part\u0026ndash;whole relations in this family, which encourages a straightforward aggregation strategy and offers fewer opportunities for representational conflict. The results thus suggest that some semantic structures may foster early procedural success without necessarily promoting deeper representational flexibility.\u003c/p\u003e \u003cp\u003eCompare problems occupied an intermediate position. In the early grades, higher- and lower-quantity errors were similarly frequent, indicating uncertainty about how to interpret comparative relations. From Grade 3 onward, inverse-operation errors became increasingly prominent, especially in problems involving marked or linguistically inconsistent relational terms. This pattern echoes a substantial body of research documenting the particular difficulty of comparison problems and the role of linguistic consistency in shaping students\u0026rsquo; interpretations (Verschaffel, \u003cspan citationid=\"CR41\" class=\"CitationRef\"\u003e1994\u003c/span\u003e; Pape, \u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e2003\u003c/span\u003e; van der Schoot et al., \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e2009\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eTaken together, these findings reinforce the view that AWPs cannot be treated as a homogeneous category. Even within the domain of one-step additive problems, different semantic structures place distinct demands on students\u0026rsquo; representational resources and give rise to qualitatively different developmental paths.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e5.3. Inverse-operation errors as indicators of representational conflict\u003c/h2\u003e \u003cp\u003eOne of the most theoretically informative results of the study concerns the developmental status of inverse-operation errors. Traditionally, such errors have often been interpreted as evidence of confusion about arithmetic operations or insufficient mastery of inverse relations. The present findings suggest a more nuanced interpretation.\u003c/p\u003e \u003cp\u003eThe increase in inverse-operation errors in later grades indicates that these errors emerge precisely when students begin to engage more deeply with the relational structure of the problem. Rather than reflecting a simple procedural mistake, inverse-operation errors can be understood as manifestations of a conflict between competing representations: a situation model that is semantically plausible and a problem model that requires a different operation for its numerical resolution. This interpretation resonates with accounts emphasizing the tension between situation-based strategies and formal mathematical models (Coquin-Viennot \u0026amp; Moreau, \u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e2007\u003c/span\u003e; Brissiaud \u0026amp; Sander, \u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e2010\u003c/span\u003e), as well as with more recent theoretical frameworks highlighting semantic congruence and recoding processes in word problem solving (Gros et al., \u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e2020\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eFrom this perspective, inverse-operation errors signal that students are no longer operating at a purely surface level, but are attempting to integrate linguistic cues, quantitative relations, and arithmetic knowledge. The difficulty lies not in the absence of understanding, but in the need to inhibit an initially activated operation and to recode the problem representation accordingly. This interpretation is further supported by research linking performance on inconsistent AWPs to executive functions such as inhibition and updating (Lubin et al., \u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e2013\u003c/span\u003e; Passolunghi et al., \u003cspan citationid=\"CR33\" class=\"CitationRef\"\u003e2022\u003c/span\u003e).\u003c/p\u003e \u003cp\u003eViewing inverse-operation errors as indicators of representational conflict rather than mere failure has important implications for how developmental progress in word problem solving is conceptualized. It shifts the focus from eliminating errors to understanding the cognitive processes that give rise to them, and highlights the transitional nature of many errors observed in the upper primary grades.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e5.4. Errors as windows into representational development\u003c/h2\u003e \u003cp\u003eA distinctive feature of the present study is its systematic use of error categories as theoretically meaningful units of analysis. By examining how different types of errors distribute across grades and semantic structures, the study reveals developmental regularities that would remain invisible if only correct responses were considered.\u003c/p\u003e \u003cp\u003eThis approach supports a reconceptualization of errors as windows into students\u0026rsquo; evolving representations of quantitative situations. The observed shift from quantity-based to inverse-operation errors reflects a qualitative reorganization in how students approach the task, from focusing on isolated numerical elements to attempting to coordinate relations between quantities. Importantly, this shift does not imply a linear progression toward correctness, but rather a period of instability in which more sophisticated forms of reasoning coexist with persistent difficulties.\u003c/p\u003e \u003cp\u003eMoreover, the differential patterns observed across semantic families underscore the importance of considering the structure of the situation when analyzing students\u0026rsquo; reasoning. Errors are not random or idiosyncratic, but systematically related to the representational demands of the task. This finding lends empirical support to long-standing theoretical claims about the centrality of semantic structure in word problem solving (Riley et al., \u003cspan citationid=\"CR36\" class=\"CitationRef\"\u003e1983\u003c/span\u003e; Verschaffel et al., \u003cspan citationid=\"CR43\" class=\"CitationRef\"\u003e2020\u003c/span\u003e), while extending them by showing how these structures interact with development over the entire span of primary education.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e5.5. Implications for research on AWP solving\u003c/h2\u003e \u003cp\u003eAlthough the data were collected within a single national curriculum, the semantic structures examined in this study\u0026mdash;Change, Combine, and Compare problems, as well as consistency effects in operation selection\u0026mdash;are core constructs in international research on arithmetic word problem solving. As such, the observed developmental trajectories speak to instructional and theoretical challenges that are likely to be relevant across educational contexts beyond the Spanish setting.\u003c/p\u003e \u003cp\u003eThe findings of this study have several implications for future research on AWP solving. First, they highlight the value of incorporating theoretically grounded error analyses into developmental studies. Accuracy-based measures alone may obscure meaningful changes in students\u0026rsquo; reasoning, particularly when different types of errors reflect distinct underlying cognitive processes. An error-centered perspective allows researchers to capture how students\u0026rsquo; interpretations of additive situations reorganize over time, even when overall performance improves.\u003c/p\u003e \u003cp\u003eSecond, the results underscore the need for research designs that explicitly account for semantic structure. Treating arithmetic word problems as a unitary class risks overlooking important sources of variability in students\u0026rsquo; performance and development. Future studies could examine how instructional interventions or curricular emphases differentially affect error profiles across problem families, and whether improvements in accuracy are accompanied by systematic changes in the nature of students\u0026rsquo; errors.\u003c/p\u003e \u003cp\u003eThird, the prominence of inverse-operation errors in the upper primary grades points to the importance of integrating models of executive control into theories of word problem solving. Understanding how students manage conflicts between competing representations\u0026mdash;such as suppressing an initially salient but inappropriate operation\u0026mdash;may be crucial for explaining why certain difficulties persist even after years of instruction. This opens promising avenues for research at the intersection of mathematical cognition, language processing, and executive functioning.\u003c/p\u003e \u003cp\u003eFrom an instructional perspective, the differentiated error profiles identified across grades and semantic families suggest that diagnostic tasks can be strategically aligned with students\u0026rsquo; developmental stages. For example, the increasing prominence of inverse-operation errors from approximately Grades 3\u0026ndash;4 onwards indicates a critical period for engaging students with linguistically inconsistent Change and Compare problems that require explicit attention to relational meanings and operation choice. Conversely, distractors based on quantity selection may serve as effective diagnostic tools for identifying surface-driven strategies in the early grades.\u003c/p\u003e \u003cp\u003eFinally, from a methodological standpoint, the use of a digital environment that ensures complete reading of the problem statements strengthens the interpretability of the results. By reducing the likelihood that observed error patterns are driven by skipped information, such designs support more confident inferences about students\u0026rsquo; representational choices rather than omissions in text processing. Future research may benefit from combining similar environments with process-oriented measures, such as response times or eye movements, to further illuminate the dynamics of arithmetic word problem solving.\u003c/p\u003e \u003c/div\u003e"},{"header":"6. Conclusions","content":"\u003cp\u003eThis study examined developmental changes in primary school students\u0026rsquo; errors when solving one-step additive AWPs, with a particular focus on how these changes interact with the semantic structure of the problems. By moving beyond accuracy-based analyses and treating errors as theoretically meaningful indicators of students\u0026rsquo; representations, the study provides a more nuanced picture of development in AWP solving.\u003c/p\u003e \u003cp\u003eThe results reveal a systematic shift across primary education: early reliance on salient numerical quantities gradually gives way to errors involving inverse operations, especially in problem types that require more complex relational reasoning. Crucially, this shift is not uniform across semantic families. Change, Combine, and Compare problems exhibit distinct developmental trajectories, underscoring the central role of semantic structure in shaping how students interpret and solve word problems.\u003c/p\u003e \u003cp\u003eRather than viewing inverse-operation errors as simple indicators of misunderstanding, the findings suggest that these errors reflect a transitional stage in which students actively engage with relational representations but struggle to resolve conflicts between competing interpretations. From this perspective, errors are not merely obstacles to be eliminated, but windows into the evolving coordination of linguistic, numerical, and relational knowledge.\u003c/p\u003e \u003cp\u003eBy documenting how error patterns evolve across grades and semantic structures, the present study contributes to theoretical models of word problem solving that emphasize representation, meaning-making, and developmental change. It also highlights the importance of designing research and assessment tools that capture not only whether students succeed or fail, but how they reason about quantitative situations.\u003c/p\u003e \u003cp\u003eMore broadly, the findings invite a reconceptualization of progress in AWP solving\u0026mdash;not as a smooth trajectory toward correctness, but as a process marked by qualitative changes in the nature of students\u0026rsquo; reasoning. Understanding these changes is essential for advancing theory and for designing learning environments that support students as they move from surface-based responses toward more flexible and structurally grounded forms of mathematical thinking.\u003c/p\u003e"},{"header":"Declarations","content":"\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThis work was supported by the Generalitat Valenciana of Comunidad Valenciana of Spain under Grant GV/2021/110 and under Grant CIAICO 2022/154.\u003c/p\u003e \u003cp\u003eDeclaration of interest statement\u003c/p\u003e \u003cp\u003eThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.\u003c/p\u003e \u003cp\u003eEthical approval and consent to participate\u003c/p\u003e \u003cp\u003e All relevant ethical guidelines and principles were carefully considered during the preparation of this article. The participants were voluntary, and informed consent was obtained from all participants. Ethical review and approval were required by University of Valencia (2024-MAGPED-3257471).\u003c/p\u003e \u003cp\u003eData availability statement\u003c/p\u003e \u003cp\u003eThe data that support the findings of this study are available from the corresponding author, [M.T.S.], upon reasonable request.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eConceptualization, M.T.S. and F. G.; data curation, E.L.-I., D.G.-C., F.G. and M.T.S.; writing\u0026mdash;original draft preparation, E.L.-I. and D. G-C; writing\u0026mdash;review and editing, M.T.S. and F.G.; supervision, M.T.S. and F.G.; project administration, F.G.; funding acquisition, F.G and M.T.S. All authors have read and agreed to the published version of the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eArendasy, M., Sommer, M., \u0026amp; Ponocny, I. (2005). 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The effect of an irrelevant number and language consistency in a word problem on pupils' achievement and reasoning. \u003cem\u003eInternational Journal of Mathematical Education in Science and Technology\u003c/em\u003e, \u003cem\u003e53\u003c/em\u003e(4), 807\u0026ndash;826. \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003ehttps://doi.org/10.1080/0020739X.2020.1782497\u003c/span\u003e\u003cspan address=\"10.1080/0020739X.2020.1782497\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"arithmetic word problems, additive reasoning, error analysis, semantic structure, primary mathematics education","lastPublishedDoi":"10.21203/rs.3.rs-8648711/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8648711/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates developmental changes in the patterns of errors produced by primary school students when solving one-step additive arithmetic word problems. A sample of 3,111 students from Grades 1 to 6 completed a battery of Change, Combine, and Compare problems presented in a multiple-choice format with theoretically motivated distractors. These distractors captured three recurrent error types: selection of the higher quantity, selection of the lower quantity, and application of the inverse operation. Results show clear grade-related shifts in the distribution of error types. In the early grades, students\u0026rsquo; incorrect responses are dominated by quantity-based choices, indicating a strong reliance on salient numerical values in the problem statement. From approximately Grade 3 onward, quantity-based errors decrease substantially, while inverse-operation errors account for an increasing proportion of remaining errors, particularly in Change and Compare problems. Importantly, these developmental trajectories are not uniform across semantic families: Combine problems exhibit comparatively stable error profiles, whereas Change and Compare problems show pronounced reorganization in the nature of errors across grades. Rather than focusing exclusively on accuracy, the findings highlight how developmental progress in additive word problem solving is reflected in systematic changes in error profiles that are sensitive to semantic structure. By treating errors as diagnostically meaningful outcomes, this study provides a fine-grained account of how students\u0026rsquo; interpretations of additive situations evolve across primary education and offers insights relevant for both research and instructional design in mathematics education.\u003c/p\u003e","manuscriptTitle":"From quantity-based to operation-based errors: Developmental shifts in one-step additive word problem solving","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-02 13:27:46","doi":"10.21203/rs.3.rs-8648711/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"7f0269b3-dc9f-46ae-9ec2-2a01eb07b7e6","owner":[],"postedDate":"February 2nd, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T21:23:11+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-02 13:27:46","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8648711","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8648711","identity":"rs-8648711","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}