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СпутникиРНSpaceXРоскосмосNASABackground The relationship between what students believe about mathematics and how well they actually solve problems has long been a focus of educational research, yet the gap between high beliefs and low performance remains underexplored among elementary school students. Methods This study used an explanatory sequential mixed-methods design (Creswell & Plano Clark, 2018). The quantitative phase involved 50 fifth-grade students from three elementary schools, who completed a mathematical beliefs questionnaire (44 items; α = 0.886; 37 valid items) and a problem-solving test (4 items; KR-20 = 0.78) covering integers, greatest common factor, least common multiple, and decimals. Belief-performance gap categories were identified using a median split. The qualitative phase involved semi-structured interviews with 8 purposively selected students (4 High-Belief-Low-Performance [HBLP], 4 Low-Belief-High-Performance [LBHP]), analyzed thematically following Braun and Clarke (2006). Results Pearson correlation indicated a significant, moderate positive relationship between mathematical beliefs and problem-solving ability (r = 0.52; p
Research Article
[version 1; peer review: 1 approved with reservations]
https://orcid.org/0000-0002-3906-1164
1, Ayu Fitri2, Retno Andriani3, [...] Nurdin -4, Fahrudin -4, Indra Suhendra4, Dwi Nanda Akhmad Romadhonhttps://orcid.org/0009-0006-7276-8440
4https://orcid.org/0000-0002-3906-1164
1, Ayu Fitri2, [...] Retno Andriani3, Nurdin -4, Fahrudin -4, Indra Suhendra4, Dwi Nanda Akhmad Romadhonhttps://orcid.org/0009-0006-7276-8440
41 Universitas Borneo Tarakan, Tarakan, East Kalimantan, Indonesia
2 Universitas Buana Perjuangan Karawang, Karawang Regency, West Java, Indonesia
3 Universitas Muhammadiyah Tangerang, Tangerang, Banten, Indonesia
4 Universitas Pendidikan Indonesia, Bandung, West Java, Indonesia
Aras Irianto
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Writing – Original Draft Preparation, Writing – Review & Editing
Ayu Fitri
Roles: Data Curation, Investigation, Writing – Review & Editing
Retno Andriani
Roles: Formal Analysis, Investigation, Writing – Review & Editing
Nurdin -
Roles: Validation, Writing – Review & Editing
Fahrudin -
Roles: Supervision, Writing – Review & Editing
Indra Suhendra
Roles: Supervision, Writing – Review & Editing
Dwi Nanda Akhmad Romadhon
Roles: Formal Analysis, Writing – Review & Editing
OPEN PEER REVIEW
REVIEWER STATUS
Corresponding author: Aras Irianto Competing interests: No competing interests were disclosed.
Grant information: This research was supported by the Indonesian Education Scholarship (Beasiswa Pendidikan Indonesia, BPI), administered by the Center for Higher Education Funding and Assessment (Pusat Pembiayaan dan Pendanaan Pendidikan Tinggi, PPAPT), Ministry of Higher Education, Science, and Technology of the Republic of Indonesia, and by the Indonesian Endowment Fund for Education (Lembaga Pengelola Dana Pendidikan, LPDP).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Copyright: © 2026 Irianto A et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. How to cite: Irianto A, Fitri A, Andriani R et al. Exploring the Gap Between Mathematical Beliefs and Problem Solving in Elementary School: A Mixed Methods Study [version 1; peer review: 1 approved with reservations]. F1000Research 2026, 15:1350 (https://doi.org/10.12688/f1000research.187341.1) First published: 11 Aug 2026, 15:1350 (https://doi.org/10.12688/f1000research.187341.1) Latest published: 11 Aug 2026, 15:1350 (https://doi.org/10.12688/f1000research.187341.1)
Mathematical problem solving constitutes a foundational competency in elementary education. The National Council of Teachers of Mathematics (NCTM, 2000) positions it as the cornerstone of mathematics learning, and Indonesia’s Merdeka Curriculum prioritizes reasoning and problem solving as key outcomes at the primary level (Kemendikbudristek, 2022). Despite this curricular emphasis, empirical evidence points to persistent difficulties among Indonesian students: PISA data consistently place Indonesian mathematics performance well below the OECD average (OECD, 2023), and national assessments indicate that a considerable proportion of elementary students require targeted instructional support (Kemendikbudristek, 2024). Historically, interventions have emphasized cognitive factors, yet a growing body of research demonstrates that students’ mathematical beliefs exert an equally influential role in shaping learning outcomes (Kloosterman et al., 1996; Muis, 2004).
The present study draws on two complementary theoretical perspectives. Op’t Eynde, De Corte and Verschaffel (2002) articulated a tripartite model encompassing views about the nature of mathematics, beliefs regarding one’s own mathematical capability, and perceptions of the social context of learning; each dimension shapes how students approach unfamiliar problem-solving tasks. Bandura (1997) established self-efficacy as a robust predictor of academic persistence. Schoenfeld (1985, 1992) proposed a four-component model of problem solving comprising resources, heuristics, metacognitive control, and belief systems. Within this framework, deficits in metacognitive control help explain why students with adequate procedural knowledge nevertheless fail to solve problems successfully; longitudinal evidence confirms that metacognitive control over arithmetic errors correlates with calculation accuracy across elementary grade levels (Jacobs et al., 2024).
Metacognitive awareness enables students to regulate their own cognitive processes (Flavell, 1979). Comprehensive syntheses indicate that word-problem interventions produce educationally meaningful improvements, with treatment fidelity as a critical moderator (Vessonen et al., 2025). Teacher beliefs represent an equally consequential factor: personal beliefs about problem solving strongly predict classroom decisions, although external constraints often limit the adoption of reform-oriented practices (Hanin & Van Nieuwenhoven, 2019).
Cross-national research documents a persistent misalignment between beliefs and performance. Large-scale investigations confirm the presence of the Dunning–Kruger effect in mathematics education, whereby low-achieving students systematically overestimate their competence (Yang Hansen et al., 2024); Dunning (2011) attributes this overconfidence to fundamental deficits in metacognitive skill. The literature reveals two distinct gap profiles: high beliefs with low performance (classic overconfidence) and low beliefs with high performance (underconfidence with compensation). Clance and Imes (1978) introduced the impostor phenomenon to describe high-achieving individuals who attribute their success to external factors rather than to their own ability — a pattern that can manifest as early as elementary school.
Several intervention approaches have demonstrated effectiveness in addressing these gaps, including goal-free problem-posing tasks, structured equivalence instruction, attributional feedback and value affirmation, and approaches that give students genuine voice in their learning (Davenport et al., 2022; Messiou, 2025; Parker-Holliman & Marcel-Herbert, 2024; Kroesch et al., 2024). Given this background, the current study sought to (1) measure mathematical beliefs and problem-solving ability among fifth graders, (2) examine the relationship between the two, (3) identify the proportion of students falling into belief-performance gap categories, and (4) explore through interviews the factors that drive each type of gap. Making sense of word problems requires far more than computational skill alone, underscoring the need for an integrated approach that considers both cognitive and affective factors together (Verschaffel et al., 2000).
This study employed an explanatory sequential mixed-methods design (Creswell & Plano Clark, 2018), consisting of a quantitative phase followed by a qualitative phase. The quantitative phase measured the relationship between mathematical beliefs and problem-solving ability and identified students in the gap categories. The qualitative phase explored the factors underlying these gaps through in-depth interviews with purposively selected students.
Participants in the quantitative phase were 50 fifth-grade students from three public elementary schools in Pinrang Regency, South Sulawesi, Indonesia, selected using proportional random sampling to ensure balanced representation across schools. Complete participant characteristics are presented in Table 1.
Note: full table available in the underlying dataset; percentages rounded to the nearest whole number.
For the qualitative phase, eight students were selected purposively based on the following criteria: classification in either the HBLP or LBHP category, willingness to be interviewed with parental consent, and the ability to communicate fluently.
Mathematical Beliefs Questionnaire. The questionnaire was developed based on the theoretical framework of Op’t Eynde, De Corte and Verschaffel (2002); the blueprint is presented in Table 2. Responses used a five-point Likert scale ranging from strongly disagree to strongly agree. The instrument was pilot-tested on 25 students from a different school, yielding a Cronbach’s alpha of 0.886 (very high reliability); after item validity testing, 37 of 44 items were retained for analysis.
Problem-Solving Ability Test. The test was developed based on Schoenfeld’s (1985) framework and consisted of four essay questions with a maximum total score of 100 ( Table 3). Reliability was estimated with KR-20 (0.78, reliable category); all items were declared valid, and content validity was confirmed by two mathematics-education experts (average Content Validity Index = 0.89).
Interview Guide. The semi-structured interview guide consisted of ten core questions covering students’ affective responses toward mathematics, problem-solving strategies, planning habits, answer-checking habits, strategies for difficult questions, sources of self-confidence, external support, attributions for success, and experiences of confidence-performance mismatch in both directions.
Data collection proceeded in three stages. The preparation stage included obtaining school permission, sending consent forms to parents, and pilot-testing instruments with 25 students from a different school. In the quantitative phase, students completed the beliefs questionnaire (35 minutes) and the problem-solving test (50 minutes) on separate days at each school; instructions were read aloud to accommodate differences in reading ability. In the qualitative phase, conducted after quantitative analysis, eight students meeting HBLP/LBHP criteria across the three schools were interviewed individually (20–30 minutes each) in a separate room, with audio recorded on two devices and transcribed verbatim.
Quantitative data were analyzed in SPSS version 26. Descriptive statistics (means, medians, standard deviations, minimum/maximum) were computed, followed by a Kolmogorov–Smirnov normality test to determine whether parametric statistics were appropriate. A Pearson correlation examined the relationship between mathematical beliefs and problem-solving ability. Belief-performance gaps were identified using the median-split method, producing four groups: HBHP (at/above median on both), LBLP (below median on both), HBLP (high beliefs, low performance), and LBHP (low beliefs, high performance).
Qualitative data were analyzed using the six-phase thematic analysis framework of Braun and Clarke (2006): familiarization, initial coding, searching for themes, reviewing themes, defining/naming themes, and reporting. Credibility was strengthened through source triangulation (cross-checking student statements against teacher notes and test scores) and member checking (sharing summary interpretations with participants for confirmation).
Descriptive statistics for both variables are presented in Table 4. The mean mathematical-beliefs score was 76.04% (SD = 7.12), and the mean problem-solving score was 65.78% (SD = 10.84) — high and moderate categories respectively. The larger standard deviation for problem solving indicates substantially greater between-student variation in that domain.
Kolmogorov–Smirnov tests indicated normal distributions for both mathematical beliefs (D(50) = 0.089, p = 0.200) and problem solving (D(50) = 0.094, p = 0.200), both p > 0.05, supporting the use of parametric correlation.
Pearson correlation ( Table 5) yielded r = 0.52, p < 0.01, 95% CI [0.28, 0.70] — a significant, moderate positive relationship. The coefficient of determination (r2 = 0.27) indicates that mathematical beliefs accounted for 27% of the variance in problem-solving ability, leaving 73% attributable to other, unmeasured factors.
Using median cutoffs (mathematical beliefs = 75.50; problem solving = 65.00), students were distributed into four categories ( Table 6): HBHP, n = 20 (40%); LBLP, n = 13 (26%); HBLP, n = 8 (16%); LBHP, n = 6 (12%); three students fell exactly on the median and were not classified. In total, 14 students (28% of participants) were classified into gap categories. The larger proportion of HBLP relative to LBHP indicates that overconfidence cases were somewhat more common than underconfidence-with-compensation cases in this sample.
Interview analysis of four HBLP students produced three themes ( Table 7). Ineffective metacognitive control was reflected in a lack of readiness to plan solution steps, inability to monitor understanding during the process, and negligence in re-evaluating answers; representative students reported working directly on problems without rough notes because they felt confident their answers were correct. Hasty working habits were marked by carelessness in reading problems and procedural calculation errors. Situational anxiety emerged during examinations, with several students reporting they could solve problems well at home but became anxious during timed tests, particularly when comparing their pace to classmates.
Interview analysis of four LBHP students produced three themes ( Table 8). Strong external support included after-school tutoring and intensive parental guidance. Compensatory persistence was reflected in students continuing to try despite uncertainty and checking answers repeatedly. Attribution of success to external factors was evident in students crediting ease of problems, luck, or others’ help rather than their own ability — consistent with the impostor phenomenon (Clance & Imes, 1978).
Table 9 presents a systematic comparison of the two groups. The most pronounced difference was in metacognitive control, where HBLP students showed marked weaknesses and LBHP students showed relative strengths; the groups also displayed opposite profiles on perseverance and anxiety.
This study examined how mathematical beliefs relate to problem-solving ability among fifth graders. A moderate positive correlation emerged (r = 0.52, p < 0.01), but with a coefficient of determination of only 0.27, indicating that beliefs alone are an incomplete predictor — consistent with Op’t Eynde et al.’s (2002) argument that beliefs matter without being the whole story. The interview data help explain the remaining variance. For HBLP students, three mechanisms recurred: weak metacognitive control (consistent with Schoenfeld’s, 1985, account of belief systems shaping planning, monitoring, and evaluation), hasty working habits, and situational anxiety, which is consistent with Ashcraft’s (2002) account of math anxiety depleting working-memory resources needed for the task.
For LBHP students, three compensatory mechanisms recurred: external support (consistent with Vygotsky’s, 1978, zone of proximal development), compensatory persistence (consistent with Duckworth et al.’s, 2007, construct of grit), and external attribution of success, a pattern consistent with the impostor phenomenon (Clance & Imes, 1978), which prior work on discrimination and mental health among minority students suggests can co-occur with broader wellbeing difficulties (Cokley et al., 2017); whether this extends specifically to perfectionism and mood in elementary-age children, as suggested in the manuscript’s earlier draft, could not be independently confirmed and is not asserted here as an established finding.
The moderate correlation found here is broadly consistent with findings among Chinese elementary students (Chen, Van Dooren & Verschaffel, 2013) and Chilean students (Saadati et al., 2023), suggesting the pattern is not confined to a single educational system. The proportion of students in gap categories (16% HBLP, 12% LBHP) was somewhat lower than the 20–24% reported among secondary students (Muis et al., 2015), which may reflect developmental differences in the stability of belief-performance patterns among younger children. The HBLP pattern resembles the Dunning–Kruger effect (Kruger & Dunning, 1999), a resemblance strengthened by evidence that this effect is observable among elementary children across multiple European countries (Yang Hansen et al., 2024).
An aspect that received comparatively less attention in prior elementary-mathematics literature is the role of attributional patterns in explaining the LBHP profile specifically; this study’s qualitative evidence — planning breakdowns, monitoring breakdowns, and checking breakdowns for HBLP, and external attribution for LBHP — offers a more granular account of the mechanisms that Muis et al. propose mediate the beliefs-performance relationship.
Theoretically, these findings (a) extend the Dunning–Kruger effect to an elementary-mathematics context, (b) sharpen Schoenfeld’s (1985) model by specifying exactly where metacognitive control breaks down (planning, monitoring, checking), (c) document the impostor phenomenon among elementary-age students in a mathematics-specific context, and (d) support Vygotsky’s (1978) sociocultural account by showing that external scaffolding can offset low self-belief. These are interpretive syntheses grounded in a moderate correlation and a small qualitative sample; they should be read as plausible mechanisms consistent with the data, not as causally established pathways, given the cross-sectional quantitative design and the non-random, purposive qualitative subsample.
Practically, three implications follow, offered as testable recommendations rather than established prescriptions: (1) early-semester screening of belief-performance profiles using brief questionnaires paired with short problem-solving tasks; (2) for HBLP students, explicit metacognitive-control instruction (e.g., think-aloud, self-checking routines) combined with anxiety-regulation strategies; and (3) for LBHP students, attributional retraining that reframes effort-based explanations of success (Perry et al., 2014).
This study has several limitations. First, the sample was small (N = 50) and drawn from three schools in a single district, limiting generalizability; studies with larger samples (ideally >200) are needed. Second, the cross-sectional design precludes causal inference regarding the proposed mediating role of metacognitive control; longitudinal designs are required to test directionality. Third, the median-split method is sample-dependent — group membership may not replicate in a differently distributed sample; latent profile analysis offers a more stable alternative. Fourth, only eight students were interviewed, which is acceptable for an exploratory qualitative component but limits thematic saturation and transferability. Fifth, mathematical beliefs were measured via self-report, which is susceptible to social desirability bias, particularly among young children. Sixth, the problem-solving test covered only four items across a limited topic range (integers, GCF, LCM, decimals); findings may not generalize to other mathematical content domains.
Future work should pursue larger samples (>200) with latent profile analysis rather than median split, longitudinal designs to clarify causal direction, broader and repeated qualitative sampling to reach saturation, experimental/RCT designs testing metacognitive-control and attributional-retraining interventions directly, and systematic cross-cultural comparison of gap-category proportions across educational systems and age ranges.
This study found that fifth-grade students in this sample held, on average, high mathematical beliefs but only moderate problem-solving ability, with considerable between-student variation. Mathematical beliefs and problem-solving ability were significantly and positively correlated (r = 0.52, p < 0.01), with beliefs accounting for 27% of the variance in performance. More than one-quarter of participants (28%) fell into a belief-performance gap category, with HBLP (16%) more prevalent than LBHP (12%). Qualitative findings indicate that HBLP is primarily explained by deficits in metacognitive control (planning, monitoring, evaluating), compounded by hasty working habits and situational anxiety, whereas LBHP is explained by external support, compensatory persistence, and attribution of success to external factors consistent with the impostor phenomenon. These findings indicate that mathematical beliefs are a significant but partial predictor of problem-solving success and that intervention should be differentiated by gap profile: metacognitive-control training for HBLP students and attributional retraining for LBHP students. Given the cross-sectional design and modest sample size, these conclusions should be treated as hypothesis-generating rather than confirmatory, pending replication with larger, longitudinal, and more diverse samples.
This study involved elementary school students and was conducted with formal research permission from the host institutions, as this article reports part of a broader doctoral dissertation project. Data collection was authorized by official permission letters issued by the participating schools: UPT SD Negeri 266 Pinrang (Surat Keterangan No. 421.2/72/UPTSDN266/IX/2025, issued by the school principal, Hj. Rusni, S.Pd., M.M., on 23 September 2025) and UPT SD Negeri 175 Pinrang, Kec. Duampanua (Surat Keterangan No. 421.2/117/UPT SDN 175/IX/2025, issued by the school principal on 24 September 2025), representing the participating school clusters involved in data collection. Because participants were recruited through their schools rather than through the authors’ home institution (Universitas Pendidikan Indonesia), ethical oversight for data collection was provided by the respective host schools under the authority of the Pinrang Regency Education and Culture Office (Dinas Pendidikan dan Kebudayaan Kabupaten Pinrang), consistent with standard practice for school-based research in this context; the authors are not affiliated with the participating schools. Written informed consent was obtained from parents/guardians of all participating students prior to data collection, and verbal assent was obtained from students themselves. Participants were informed of their right to withdraw at any time without consequence. All data were anonymized prior to analysis.
Repository name Zenodo: Underlying data for “Exploring the Gap Between Mathematical Beliefs and Problem Solving in Elementary School: A Mixed Methods Study.” https://doi.org/10.5281/zenodo.21438570 (Irianto et al., 2026).
This project contains the following underlying data: (1) mathematical beliefs questionnaire responses (N = 50, anonymized, .xlsx); (2) problem-solving test scores (N = 50, anonymized, .xlsx); (3) interview transcripts for 8 selected students (anonymized, .docx/.pdf, with any identifying details redacted).
Data are available under the terms of the Creative Commons Attribution 4.0 International licence (CC-BY 4.0).
This mixed-methods study is reported with reference to the Good Reporting of a Mixed Methods Study (GRAMMS) guideline.
The authors thank the participating schools, teachers, students, and parents/guardians in Pinrang Regency for their cooperation during data collection, and the Indonesian Education Scholarship (Beasiswa Pendidikan Indonesia, BPI), administered by the Center for Higher Education Funding and Assessment (Pusat Pembiayaan dan Pendanaan Pendidikan Tinggi, PPAPT), Ministry of Higher Education, Science, and Technology of the Republic of Indonesia, and by the Indonesian Endowment Fund for Education (Lembaga Pengelola Dana Pendidikan, LPDP).
This research was supported by the Indonesian Education Scholarship (Beasiswa Pendidikan Indonesia, BPI), administered by the Center for Higher Education Funding and Assessment (Pusat Pembiayaan dan Pendanaan Pendidikan Tinggi, PPAPT), Ministry of Higher Education, Science, and Technology of the Republic of Indonesia, and by the Indonesian Endowment Fund for Education (Lembaga Pengelola Dana Pendidikan, LPDP).
The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
© 2026 Irianto A et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Current Reviewer Status: ?
Key to Reviewer Statuses VIEW HIDE
ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested
Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.
Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions
Version 1
VERSION 1
PUBLISHED 11 Aug 2026
Reviewer Report 17 Aug 2026
Torang Siregar, UIN Syekh Ali Hasan Ahmad Addary Padangsidimpuan, Padangsidimpuan, North Sumatra, Indonesia
Approved with Reservations
VIEWS 0
Is the work clearly and accurately presented and does it cite the current literature?
Partly
Is the study design appropriate and is the work technically sound?
Partly
Are sufficient details of methods and analysis provided to allow replication by others?
Partly
If applicable, is the statistical analysis and its interpretation appropriate?
Partly
Are all the source data underlying the results available to ensure full reproducibility?
Yes
Are the conclusions drawn adequately supported by the results?
Partly
Competing Interests: No competing interests were disclosed.
Reviewer Expertise: Mathematics Education, Mathematics Education Research, STEM Education, STEAM Education, Science Education, Educational Technology, Artificial Intelligence in Education, GeoGebra and Dynamic Mathematics Software, Ethnomathematics, Curriculum and Instruction, Educational Research and Development (R&D), Mathematical Literacy, Mathematical Problem Solving, Higher-Order Thinking Skills (HOTS), Project-Based Learning (PjBL), Realistic Mathematics Education (RME), Learning Analytics, Digital Learning, and Innovative Mathematics Learning.
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