c e p s Journal Enhancing Mathematical Creativity Through a 21st- Century-Based Learning Model: A Mixed Methods Study Alimuddin Tampa* 1 , Ja’faruddin 2 , Fauziyyah Alimuddin 2 and Andi Mulawakkan Firdaus 3 • The present study investigates the effectiveness of the 21st-Century- Based Mathematics Learning Model in enhancing junior high school students’ mathematical creativity on a large scale. A quasi-experimental design with a mixed-methods approach was employed, involving 369 students from 12 schools located in urban, suburban and rural areas of South Sulawesi, Indonesia. The primary instrument was an open-ended mathematical creativity test designed to measure students’ fluency and flexibility in problem solving. Quantitative data were collected through pretests and posttests and analysed using descriptive statistics, paired- samples t-tests, Kruskal–Wallis tests and effect size calculations (Cohen’s d and N-gain). Qualitative data were obtained through classroom ob - servations and analysed using thematic analysis to explore patterns of originality in students’ creative thinking processes. The results indicate significant improvements in fluency and flexibility (Cohen’s d = 3.64; N-gain = 0.50), with significant differences based on school location, where urban students demonstrated higher performance. The qualitative findings reveal emerging patterns of originality in students’ questioning behaviours and mathematical communication. The 21st-Century-Based Mathematics Learning Model integrates structured learning syntax, a collaborative so - cial system and reflective dialogue, all of which support 21st-century com - petencies. The present study provides empirical evidence of the model’s ef - fectiveness across diverse contexts and offers methodological implications for the development of context-sensitive instructional innovations. Keywords: mathematical creativity, 21st-Century-Based Mathematics Learning Model, 21st-century learning framework, geographical variation, mixed methods research 1 *Corresponding Author. Faculty of Mathematics and Natural Sciences, Makassar State University, Indonesia; alimuddin@unm.ac.id. 2 Faculty of Mathematics and Natural Sciences, Makassar State University, Indonesia. 3 Faculty of Teacher Training and Education, Universitas Muhammadiyah Makassar, Indonesia. DOI: https://doi.org/10.26529/cepsj.2268 Received: 10 August 2025, Accepted: 30 March 2026, Published on-line as Recently Accepted Paper: June 2026 enhancing mathematical creativity through a 21st-century-based learning ... 2 Krepitev matematične ustvarjalnosti s pomočjo modela za učenje, zasnovanega za 21. stoletje: študija z mešanimi metodami Alimuddin Tampa, Ja’faruddin, Fauziyyah Alimuddin in Andi Mulawakkan Firdaus • Ta študija preučuje učinkovitost modela za učenje matematike, zasno- Ta študija preučuje učinkovitost modela za učenje matematike, zasno - vanega za 21. stoletje, pri krepitvi matematične ustvarjalnosti učencev zadnjega triletja osnovne šole v širšem obsegu. Uporabili smo kvazieks - perimentalni načrt s pristopom mešanih metod, v katerega je bilo vklju - čenih 369 učencev iz 12 šol v mestnih, predmestnih in v podeželskih območjih južnega dela Sulavezija v Indoneziji. Glavni instrument je bil odprt test matematične ustvarjalnosti, zasnovan za merjenje spretnosti in prilagodljivosti učencev pri reševanju problemov. Kvantitativni po - datki so bili zbrani s predtestiranjem in potestiranjem ter analizirani z uporabo deskriptivne statistike, t-testov za pare vzorcev, Kruskal-Walli - sovih testov in izračunov velikosti učinka (Cohenov d in N-gain). Kva - litativni podatki so bili pridobljeni z opazovanjem v razredih in ana - lizirani s pomočjo tematske analize, da bi raziskali vzorce izvirnosti v procesih kreativnega mišljenja učencev. Izsledki kažejo na znatno izbolj - šanje spretnosti in prilagodljivosti (Cohenov d = 3,64; N-gain = 0,50), pri čemer so bile ugotovljene znatne razlike glede na lokacijo šole, saj so se učenci iz mestnih območij bolje odrezali. Kvalitativne ugotovitve razkrivajo nastajajoče vzorce izvirnosti v vprašujočem vedenju učencev in matematični komunikaciji. Model za učenje matematike, zasnovan za 21. stoletje, združuje strukturirano učenje, sistem sodelovanja in so - cialne interakcije ter refleksivni dialog, kar vse podpira kompetence 21. stoletja. Ta študija prinaša empirične dokaze o učinkovitosti modela v različnih kontekstih in ponuja metodološke implikacije za razvoj inova - tivnih pedagoških pristopov, prilagojenih posameznim kontekstom. Ključne besede: matematična ustvarjalnost, model za učenje matematike, zasnovan za 21. stoletje, okvir učenja za 21. stoletje, zemljepisne razlike, raziskava z mešanimi metodami c e p s Journal 3 Introduction Amid global competition and rapid technological advancement, teacher education reform emphasises the need to align pedagogy with digital trans - formation and global challenges (Perla et al., 2024). The demand for higher- order thinking skills, particularly mathematical creativity, continues to increase (OECD, 2018; Beswick & Fraser, 2019). Mathematical creativity is understood as the ability to generate novel ideas and solve complex problems in flexible and innovative ways, a competence that is crucial for junior high school students when addressing real-world problems (Al Moray, 2024; Cevikbas et al., 2022). Numerous studies indicate that students’ mathematical creativity in open-ended tasks remains low. Students tend to produce a single solution, rely heavily on routine procedures, and demonstrate limited fluency and flexibility in their strategies (Nurkaeti et al., 2020; Tanudjaya & Doorman, 2020). These findings are consistent with the preliminary analysis of the present study, which shows that in a sample of 63 students, 97.3% produced only one or no solution, while only 2.7% were able to generate two correct solutions. One major contributing factor is the dominance of one-way instruc - tional approaches (Hu & Wang, 2024; Bicer et al., 2024). Although problem- based learning, discovery learning and project-based learning have been widely adopted, creativity is often positioned as an implicit outcome rather than an explicit instructional objective. Students are not systematically supported to transform problems into multiple representations, compare alternative strate - gies or revise ideas through critical feedback. As a result, the development of mathematical creativity remains limited and sporadic (Milin Šipuš et al., 2021; Marwa et al., 2024; Mariani et al., 2025). This condition underscores the need for instructional interventions that explicitly and systematically foster math - ematical creativity. In response to this need, the authors have developed and validated the 21st-Century-Based Mathematics Learning (21st-BML) model. Mathematical Creativity Mathematical creativity refers to the ability to generate diverse and adaptive solutions by integrating convergent and divergent thinking (Kwon et al., 2025). Schoevers et al. (2019) distinguish professional creativity in research contexts from school-level creativity, which emphasises students’ exploration of problem-solving processes. Within the Guilford–Torrance framework, math - ematical creativity comprises three indicators: fluency, flexibility and original - ity (Guilford, 1967; Torrance, 1974; Bingol & Ozyaprak, 2025). These indicators enhancing mathematical creativity through a 21st-century-based learning ... 4 show different measurement characteristics (Leikin & Sriraman, 2017): fluency and flexibility can be operationalised through the number of valid solutions and the diversity of strategies, whereas originality is contextual because it de - pends on the perceived uncommonness of solutions (Schoevers et al., 2019). In school settings, an idea may be uncommon for students even if it is not mathematically original, which makes originality difficult to assess objectively through written tests alone. Accordingly, prior studies often separate approaches, using quantitative analyses for fluency and flexibility, while qualitative analyses is used for origi - nality based on observations, student artifacts and verbal interactions. Math - ematical creativity is also viewed as a process evident in students’ engagement in problem solving, problem posing and mathematical communication across representations (Silver, 1997; Sfard, 2008; Sitorus & Masrayati, 2016), shaped by task design, social interaction and learning autonomy (Leikin, 2020; Wessels, 2014). Therefore, analysing patterns of creative activity is essential for under - standing how mathematical creativity develops. The 21st-BML Model The 21st-BML model was developed through an R&D study based on the Plomp development model and Nieveen’s evaluation criteria (Plomp, 2013; Nieveen, 2010). Prior work indicates that the model meets validity and practi - cality criteria and can enhance students’ mathematical creativity (Tampa et al., 2024). However, the development phase relied on a small-scale trial with one class (32 students) in a relatively homogeneous setting, so the model’s empirical base and conceptual structure require further strengthening. Accordingly, large-scale implementation across diverse contexts (e.g., school location, status and accreditation) is needed to provide more robust evi - dence. Conceptually, the 21st-BML model comprises five interrelated compo - nents that function as an integrated learning system, with their relationships illustrated in Figure 1. c e p s Journal 5 Figure 1 Coherence Among Components of the 21st-BML Model Learning Framework of the 21st-BML Model Based on Figure 1, the learning framework of the 21st-BML model (T able 1) summarises the core functions of each learning syntax and is operationalised into instructional materials (lesson plans, student worksheets, learning media, enrichment materials and assessment instruments) as the model’s concrete im - plementation in classroom practice (Tampa et al., 2024). Table 1 Learning Framework of the 21st-BML Model Learning Syntax Description of the Learning Syntax (Functions and General Activities) 1. Object Translation Students are guided to translate contextual problems into mathematical representations through the process of enactive metaphorising, which bridges concrete experiences with abstract ideas. This process is supported by embodied metaphors, enabling the cognitive transition from iconic to symbolic thinking. 2. Decomposition Students are directed to organise and connect mathematical concepts through visual mapping and analysis of relationships between elements, thereby forming a coherent structure of understanding. enhancing mathematical creativity through a 21st-century-based learning ... 6 Learning Syntax Description of the Learning Syntax (Functions and General Activities) 3. Cognitive Construction Students are facilitated in building cognitive structures developed through the preceding learning syntax. This is achieved through exploration, elaboration and synthesis of ideas, conducted individually or collaboratively, in completing problem-based student worksheets featuring contextual open- ended tasks. 4. Rotational Peer Review Students are engaged in exchanging work outcomes and conducting peer evaluations to develop mathematical communication and critical thinking, and to broaden their perspectives on problem solving. 5. Cognitive Advancement Students extend and integrate prior cognitive structures through open- ended, independent, time-bound explorations based on real-life situations from their environment. Activities involve observation, analysis and creating mathematical models in symbolic, visual, verbal or digital forms, documented with progress reports, online communication and periodic feedback for reflective strategy adjustments. The Role of the Learning Syntax of the 21st-BML Model in Fostering Mathematical Creativity Each syntax in the 21st-BML model is designed to support specific di - mensions of mathematical creativity. Object Translation supports the transi - tion from contextual experience to abstraction through enactive and embodied metaphors, strengthening fluency and flexibility (Bernadez & Montero, 2025; Bicer, 2021) and aligning with evidence on the role of problem posing and problem solving in fostering flexibility and originality (Papadopoulos et al., 2022). Decomposition develops relational mapping and conceptual organisa - tion, promoting fluency, flexibility and originality (Hamid et al., 2024). Cogni - tive Construction builds these structures through open-ended, context-based worksheet tasks, supporting strategic synthesis and reinforcing fluency and originality (Kholid et al., 2024; Arifin et al., 2021), consistent with structured learning management principles (Kónya & Kovács, 2021). Rotational Peer Re - view expands perspectives through peer evaluation and argumentation, en - hancing flexibility and originality (Y ayuk et al., 2020). Cognitive Advancement extends prior cognitive structures through independent, context-based explo - ration supported by documentation and feedback, fostering flexibility, fluency and collective creativity (Jonsson et al., 2022; Meier et al., 2023; Ban Hassan Majeed et al., 2023). Geographical Factors in the Implementation of the 21st-BML Model School geographical context shapes mathematics learning conditions through differences in resource access, learning culture and institutional c e p s Journal 7 support, thereby creating unequal learning opportunities for implementing in - novative models (Ma & Liu, 2022; Niu et al., 2025; Zhang et al., 2025). Interna - tional evidence shows consistent achievement patterns by school location, with urban schools typically outperforming rural schools and suburban schools tending to fall between these contexts, reflecting variations in educational structures and instructional management (Mohammadpour & Yon, 2024; De Nadai et al., 2020). In Indonesia, similar disparities have been reported, particularly related to technological readiness and teacher quality (Susanta et al., 2025; Baharuddin & Burhan, 2025). More broadly, technological readiness, teacher quality and institutional support vary geographically and may influence the effectiveness of student-centred and collaborative learning models (Li, 2025). Therefore, the present study positions geographical context as a key factor for understanding variation in the implementation and outcomes of the 21st-BML model and for formulating hypotheses regarding location-based differences in effectiveness. The present study examines the large-scale effectiveness of the 21st-BML model across diverse contexts in order to strengthen its empirical foundation and conceptual structure. It also explores patterns of students’ creative activi - ties during implementation to clarify how the model supports mathematical creativity. Accordingly, the study addresses the following research questions: 1. How effective is the 21st-BML model in enhancing lower secondary stu - dents’ mathematical creativity? 2. Do students’ creative gains vary significantly by school location (urban, suburban, rural)? 3. What patterns of creative activity emerge during implementation? Research Hypotheses H₁: The 21st-BML model significantly enhances students’ mathematical creativity. H₂: Improvements in students’ creativity differ significantly across geo - graphic school types. Method Participants The study involved 369 eighth-grade students from 12 junior high schools in South Sulawesi Province, Indonesia. The sample was selected using stratified cluster random sampling based on school location (urban, suburban, enhancing mathematical creativity through a 21st-century-based learning ... 8 rural), school status (public/private) and accreditation level (A/B). Table 2 pre - sents the participant profile Table 2 Profile of Quantitative Research Participants Geographical Location School Code School Type Accreditation Status Number of Participants M F Urban KM-1 Public A 36 15 21 KM-2 Public B 34 13 21 KM-3 Private A 30 11 19 KM-4 Private B 29 9 20 Suburban PK-1 Public A 32 11 21 PK-2 Public A 32 13 19 PK-3 Public B 30 13 17 PK-4 Private B 31 12 19 Rural LK-1 Public A 32 15 17 LK-2 Public A 30 12 18 LK-3 Public B 26 9 17 LK-4 Public B 27 9 18 Total 369 142 227 Instruments The primary instrument was a mathematical creativity test (MCT) con - sisting of three open-ended items on systems of linear equations in two vari - ables (SLETV). The MCT measured fluency (generating multiple valid solu - tions) and flexibility (using diverse problem-solving strategies). Scoring used an analytic rubric (0–3 for each indicator). Fluency was scored by the number of distinct correct solutions (3 = ≥5; 2 = 3–4; 1 = 1–2; 0 = none), and flexibility by the diversity of the strategies used (e.g., graphical representation, tables, substi - tution, algebraic manipulation) (3 = ≥3 strategies; 2 = 2; 1 = 1; 0 = unclear). The total score range was 0–18. The MCT was piloted with 40 students and analysed using the Rasch Model; all items showed acceptable fit ( Infit/Outfit MNSQ). The rubric and scoring examples are provided in the Appendices. An observation sheet documented creativity-related classroom activities during the implementation of the 21st-BML model, focusing on interactional indicators that are difficult to capture from written documents (spontaneous questions, responses to peers’ questions, verbal explanations, argumentation/ c e p s Journal 9 rebuttals and collaboration). Originality was identified from observed interac - tional episodes, especially uncommon questions or arguments relative to the dominant classroom discourse. Two raters independently coded the observa - tion records (87% agreement). The observation sheet is included in the Ap - pendices. The written test rubric was used to score fluency and flexibility only. Research Design The study employed a mixed-methods approach to examine mathemati - cal creativity through the implementation of the 21st-BML model (Creswell & Plano Clark, 2018; Tashakkori & Teddlie, 2010). The eight-week study com - prised a pretest, six instructional sessions and a posttest using the same in - struments. The intervention used six expert-validated 21st-BML instructional packages aligned with the 2013 Curriculum ( M = 4.2/5.0), and participating teachers attended a two-day microteaching workshop to support implementa - tion consistency. Quantitative analysis included Rasch-based instrument checking (eRm package), descriptive statistics, paired-samples t-tests, N-gain calculations, school-location comparisons using the Kruskal–Wallis H test with Mann– Whitney U post hoc tests, and effect size estimation (Cohen’s d). Qualitative data comprised group artifacts from 67 collaborative groups (four to five stu - dents per group) and observation episodes collected across 72 instructional sessions (six per school). Qualitative data were analysed using a six-phase the - matic analysis (Braun & Clarke, 2019), with trustworthiness supported through source triangulation and an audit trail. For further academic purposes, the complete set of instructional docu - ments is available from the corresponding author upon request. Results Quantitative Findings on Students’ Mathematical Creativity Table 3 summarises the descriptive results, showing that the 21st-BML model improved the participating students’ mathematical creativity across regions. At pretest, all of the students (100%; n = 369) were in the very low category (0 ≤ X ≤ 3; M = 1.36, SD = 0.92). At posttest, none of the students re - mained in this category and the mean increased to 9.60 ( SD = 2.50), indicating substantial gains in fluency and flexibility (Table 3). Regional patterns showed the greatest improvement in urban schools, followed by suburban and rural schools (Table 3). enhancing mathematical creativity through a 21st-century-based learning ... 10 Further analyses confirmed these gains. The overall N-gain was 0.50 (moderate), with regional values of 0.60 (urban), 0.50 (suburban) and 0.40 (rural) (Table 4). The paired-samples t-test indicated a statistically significant improvement with a very large effect size (Cohen’s d = 3.64; p < 0.001) (Table 4). Posttest scores differed significantly by school location (Kruskal–Wallis H = 129.014, p < 0.001), and post hoc Mann–Whitney U tests showed a consistent gradient (urban > suburban > rural) (Table 5). These differences suggest that although the model was effective across contexts, the magnitude of its impact varied with contextual readiness and support. Table 3. Comparison of Pretest–Posttest Mathematical Creativity Scores Region N Pretest Mean SD Category Posttest Mean SD Category Distribution (%) Δ Mean Overall 369 1.36 0.92 Very Low (100%) 9.60 2.50 Low–Moderate: 42 (11.4%); Moderate: 193 (52.3%); High: 134 (36.3%) +8.24 Urban 129 1.53 0.95 Very Low (100%) 11.50 1.90 Low–Moderate: 7 (5.4%); Moderate: 38 (29.5%); High: 84 (65.1%) +9.97 Sub urban 125 1.39 0.91 Very Low (100%) 9.30 2.20 Low–Moderate: 18 (14.4%); Moderate: 79 (63.2%); High: 28 (22.4%) +7.91 Rural 115 1.26 0.89 Very Low (100%) 7.90 2.10 Low–Moderate: 31 (27.0%); Moderate: 69 (60.0%); High: 15 (13.0%) +6.64 Note. Very Low (0 ≤ X ≤ 3), Low–Moderate (3 < X ≤ 6), Moderate (6 < X ≤ 10), High (10 < X ≤ 14). Further analyses confirmed these gains (Tables 4 and 5). The overall N- gain was 0.50 (moderate), with regional values of 0.60 (urban), 0.50 (suburban) and 0.40 (rural) (Table 4). The paired-samples t-test showed a statistically sig - nificant improvement with a very large effect size (Cohen’s d = 3.64; p < 0.001) (Table 4). Posttest scores differed significantly by school location (Kruskal– Wallis H = 129.014, p < 0.001), and post hoc Mann–Whitney U tests indicated a consistent gradient (urban > suburban > rural) (T able 5). These findings suggest that although the 21st-BML model was effective across contexts, the magnitude of its impact varied with contextual readiness and support. c e p s Journal 11 Table 4 Effectiveness of the 21st-BML Model: N-Gain and Statistical Significance Region N-Gain Category Mean Difference t(df) Cohen’s d Overall 0.50 Moderate -8.271 -69.83 (368)*** -3.64 Urban 0.60 Moderate → High Suburban 0.50 Moderate Rural 0.40 Low–Moderate Note. *** p < 0.001 Table 5 Regional Differences in Posttest Creativity Scores Statistical Test df / Comparison H / U Value p-value Kruskal–Wallis 2 129.014 < 0.001 Mann–Whitney U Urban vs. Suburban 12,530.0 < 0.001 Urban vs. Rural 13,136.5 < 0.001 Suburban vs. Rural 9,902.5 < 0.001 These regional differences not only indicate statistical variations in creativity outcomes but also reflect differences in contextual readiness, such as access to resources, teacher competence and institutional support. Patterns of Students’ Creative Activities During the Implementation of the 21st-BML Model In order to support the representativeness of the qualitative findings, all of the qualitative data were coded across the full dataset. Table 6 summarises the distribution of creativity indicators across group artifacts and classroom interaction episodes. The examples presented in the following sections were se - lected to represent dominant patterns identified across the dataset rather than isolated cases. enhancing mathematical creativity through a 21st-century-based learning ... 12 Table 6 Distribution of Mathematical Creativity Indicators Across the Entire Qualitative Dataset Activity Category Unit of Analysis Total Units Analysed Fluency Flexibility Originality Representative Examples Discussed Task-Based Mathematical Problem Solving Group artifacts (student worksheets) 67 groups 378 artifacts 352 artifacts (93%)* 306 artifacts (81%)* 174 artifacts (46%)* T-1, T-2, T-3 Student Questioning Observed questioning episodes 487 episodes 158 (32%)** 213 (44%)** 291 (60%)** P-1 to P-5 Mathematical Communication Observed explanation episodes 513 episodes 174 (34%)** 220 (43%)** 282 (55%)** J-1 to J-4 Notes. (1) Percentages indicate the proportion of groups exhibiting the creativity indicator in their artifacts. A single group may demonstrate multiple indicators. (2) Percentages indicate the proportion of activity episodes coded according to the dominant creativity indicator. An episode may involve multiple indicators but is classified based on the most salient feature. (3) Codes T, P and J refer to activity types and sequences of occurrence rather than to individual students or group identities. Qualitative analysis indicated three main categories of creative activity: task-based mathematical problem solving reflected in collaborative products, students’ spontaneous questioning during classroom interactions, and math - ematical communication at individual and group levels. Creativity in Task-Based Mathematical Problem Solving The students’ responses to three representative tasks ( T-1 to T-3) showed fluency and flexibility in mathematical problem solving and also indicated emerging originality in how the students constructed and presented strategies. This suggests that 21st-BML tasks support solution and strategy diversity while enabling more unique and contextualised approaches to problem solving. c e p s Journal 13 Data Evidence Task T-1: Examples and Non-Examples of SLETV Figure 2 Student Work Sample for Task T-1 on Examples and Non-Examples of SLETV Data Evidence Task T-2: Inconsistent System of Equations Figure 3 Student Work Sample for Task T-2. Inconsistent System of Equations enhancing mathematical creativity through a 21st-century-based learning ... 14 Data Evidence Task T-3: Mind Mapping SLETV Equations Figure 4 Student Work Sample for Task T-3. Mind Mapping SLETV Equations Fluency was evident across tasks. In T-1, the students produced five valid variations of systems of equations (A1–A5) using integer, fractional and deci - mal coefficients. In T-2, they generated multiple parameter sets that yielded in - consistent systems through different constructions. In T-3, fluency appeared in the production of varied mind maps that organised concepts related to SLETV . Flexibility was also consistent. In T-1, the students used different con - struction approaches, including both linear and non-linear forms, to generate diverse systems. In T-2, they applied three strategies: gradient-based analysis, algebraic manipulation of constant ratios and visual verification using GeoGe - bra. In T-3, flexibility was reflected in linking algebraic and geometric ideas through different mind-map structures (e.g., radial–hierarchical and modular layouts). Originality appeared when the students introduced uncommon repre - sentations or strategy sequences beyond those modelled in instruction. In T-1, this was visible in alternative notations and visual layouts. In T-2, the students chose and combined analytical, algebraic and visual strategies in self-directed sequences. In T-3, originality emerged in autonomous mind-map organisation, including colour coding, relational structures among concepts and distinctive presentation formats. c e p s Journal 15 Creativity in Questioning and Mathematical Communication The students’ creativity in questioning and communicating mathemati - cal ideas was examined through nine representative expressions: five creative questions ( P-1 to P-5) and four explanatory responses ( J-1 to J-4 ). These expres - sions illustrate how the students formulated questions and explanations that reflected fluency, flexibility and originality in classroom interaction. Table 7 Student Expressions of Creativity in Questioning and Communication Code Student Quote Triggering Instructional Phase P-1 What if we replace the variables in a system of equations with geometric shapes? For example:  + Δ = 10 and 2  − Δ = 5. Can we calculate the values of  and Δ? Cognitive Construction P-2 If the graphs of two equations are parallel, why don’t they have a solution? They’re both straight and go in the same direction. Peer Review Rotation P-3 Do all systems of equations have to be solved with calculations? Can we solve them with pictures or stories instead? Cognitive Construction P-4 What if the equations change every second, like in a game or a sensor? Can we still find a fixed solution? And can we make a system where the solution is a fraction, not a whole number? Cognitive Advancement P-5 If a store sells orange and apple juice at different prices, how can a system of equations help the owner determine profit? Peer Review Rotation J-1 Those two equations are like two roads. If they meet at one point, that means there’s one solution that fits both, but if they never meet, then there’s no solution. Peer Review Rotation J-2 I drew two lines using different colours. If they intersect, I put a circle at that point, because that’s the solution. Peer Review Rotation J-3 Suppose 3 donuts and 2 teas cost 15 thousand, and 2 donuts and 4 teas cost 16 thousand. We can find the price of one donut and one tea from that. Cognitive Construction J-4 Guessing numbers in a game is like solving a system of equations. You need numbers that fit both conditions. If they don’t, it’s not correct yet. And when I changed 2x + y = 8 into y = 8 – 2x, it felt easier, like rewriting a sentence to make it clearer. Cognitive Advancement Note. Code Description: P = Creative Questioning; J = Creative Communication in Mathematics Based on Table 7, fluency in questioning was reflected in generating multiple distinct, relevant and mathematically valid questions, as shown in P-2 and P-5, which connected SLETV to parallel-line properties and everyday eco - nomic contexts. enhancing mathematical creativity through a 21st-century-based learning ... 16 Flexibility in questioning was indicated by shifts across representation - al modes and solution approaches. In P-1 and P-3, the students moved from symbolic forms to visual or narrative representations (e.g., replacing variables with geometric symbols or proposing story-based approaches). Flexibility also appeared in P-4, where the students linked SLETV to dynamic technological contexts, shifting from static to dynamic perspectives. Originality in questioning was evident when the students proposed un - common ideas not exemplified in instruction, particularly in P-1 (geometric symbols replacing algebraic variables) and P-4 (temporally changing systems), while remaining mathematically coherent. Similar patterns appeared in mathematical communication. Fluency was shown in coherent contextualised explanations (e.g., J-3 modelling a real- world situation into a system). Communicative flexibility emerged through shifts from symbolic to coloured visual representations and through linking algebraic transformations with linguistic explanation ( J-2 and J-4). Original - ity appeared when the students used uncommon explanatory frames such as analogies and game contexts to explain solution concepts ( J-1 and J-4 ). Overall, these findings suggest that the Cognitive Construction, Peer Review Rotation and Cognitive Advancement syntaxes provide cognitive and social space for students to generate diverse ideas, shift representations and propose unique questions and explanations through autonomy and explora - tory opportunities. Integration of Quantitative and Qualitative Evidence on Creativity Integrating quantitative and qualitative results strengthens the interpre - tation of the 21st-BML model’s effectiveness. Quantitative findings show sub - stantial gains in fluency and flexibility, while qualitative evidence explains how these gains were supported through task engagement, questioning and math - ematical communication during implementation. The convergence of evidence indicates that structured task design sup - ports solution diversity and strategy variation, whereas interaction and learn - ing autonomy enable uncommon questions, arguments and explanations to emerge within classroom discourse. Thus, the 21st-BML model enhances both measurable creativity outcomes and the learning processes through which mathematical creativity develops. c e p s Journal 17 Discussion Effectiveness of the 21st-BML Model in Enhancing Mathematical Creativity The findings support Hypothesis 1, indicating that the 21st-BML model enhances students’ mathematical creativity, particularly fluency and flexibility. This aligns with evidence that innovative approaches such as problem-based learning, flipped classroom, realistic mathematics education, STEM-based learning and open-ended tasks can foster mathematical creativity and espe - cially flexibility and originality (Winasis et al., 2025; Mariani et al., 2025; Khalil et al., 2023; Desi et al., 2025). The added value of the 21st-BML model lies in integrating these influences through five interrelated learning syntaxes. Object Translation and Decomposition support multiple representations and rela - tional organisation of ideas, strengthening fluency and flexibility (Bicer, 2021; Sitorus & Masrayati, 2016). Cognitive Construction and Peer Review Rotation expand idea exploration, synthesis and reflective exchange, stimulating ques - tioning and mathematical communication (Yaniawati et al., 2020). Cognitive Advancement complements this process through autonomous, context-based exploration supported by feedback and technology integration (Nurkaeti et al., 2020; Desi et al., 2025). Differences in Creativity Outcomes Based on Geographical Location The findings support Hypothesis 2, showing that the magnitude of the gains varies by school location. These differences can be interpreted through contextual conditions: urban schools generally have stronger technological readiness, more consistent teacher professional development and more estab - lished collaborative learning cultures, thus enabling more optimal enactment of the 21st-BML syntaxes. In suburban and rural contexts, constraints in ac - cess and training continuity, as well as the students’ readiness for discussion- based and open-ended learning, may limit implementation, although gradual development across later sessions suggests adaptive potential. This interpreta - tion is consistent with studies linking location-based differences to learning resources and interaction opportunities (Triana et al., 2024; Nadjet et al., 2024; Kriewaldt et al., 2023) and with evidence that technology-supported collabo - ration can strengthen creativity where readiness is sufficient (Cisnaulin et al., 2025). Therefore, implementation should be context-sensitive, with stronger enhancing mathematical creativity through a 21st-century-based learning ... 18 scaffolding, phased digital integration and more structured collaborative rou - tines in resource-constrained settings. Patterns of Students’ Creative Activities During the Implementation of the 21st-BML Model The qualitative findings indicate that creative activity during implemen - tation manifests in task-based problem solving, questioning and mathematical communication, highlighting creativity as a process not fully captured through written tests alone. Task-based work primarily revealed fluency and flexibility through diverse solutions and strategy variation supported by Object Trans - lation and Decomposition. Interactional episodes more strongly highlighted originality when the students modified problems, introduced alternative sym - bols or used new analogies and metaphors, supported by Cognitive Construc - tion and Peer Review Rotation, while Cognitive Advancement enabled the students to integrate knowledge into new contexts through autonomous explo - ration. Overall, these patterns suggest complementary roles of the five syntaxes and support evidence that integrative, contextual, collaborative and digitally supported learning can enhance students’ creativity (Dilekçi & Karatay, 2023; Rahman et al., 2025; Ritter et al., 2020). Interpreting the Convergence of Quantitative and Qualitative Evidence on Mathematical Creativity The convergence of evidence suggests that gains reflect changes in stu - dents’ engagement in mathematical thinking activities. Quantitative improve - ments in fluency and flexibility provide outcome indicators, while qualitative evidence clarifies the processes through which students explore meaning, con - nect representations and develop more personalised mathematical communi - cation. This supports the view that mathematical creativity develops through the interaction of structured instructional design and learning autonomy, me - diated by social negotiation of meaning. The theoretical contribution of the 21st-BML model lies in articulating a coherent syntactic framework that con - nects cognitive transitions, collaborative amplification and autonomous explo - ration within a unified learning trajectory. c e p s Journal 19 Conclusion The present study shows that integrating structured learning syntaxes in the 21st-BML model enhances junior high school students’ mathematical crea - tivity, and especially improves fluency and flexibility. The model was effective across geographical contexts, although gains were strongest in urban schools and weaker in suburban and rural schools, likely reflecting differences in tech - nological readiness, collaborative learning culture and teacher professional development intensity. The qualitative findings further indicate that creativity develops not only in test performance but also through learning processes, par - ticularly in task-based work, questioning and mathematical communication, where originality emerges in interactional episodes. Practically, the 21st-BML model provides an adaptive framework for strengthening collaborative learning, integrating technology strategically and adjusting instructional routines to local conditions. In resource-constrained contexts, implementation may require stronger scaffolding, phased digital in - tegration and sustained professional support in order to maintain students’ en - gagement in creative mathematical processes. The present study does have some limitations. The one-group pretest– posttest design limits causal inference. The intervention duration and single educational level restrict conclusions about long-term effects across develop - mental stages. The geographical focus on one province limits broader national representation. The absence of a comparative design constrains claims about the model’s unique contribution relative to other innovative approaches. Fi - nally, quantitative measurement focused on fluency and flexibility, while origi - nality and other outcomes beyond mathematical creativity (e.g., mathematical dispositions) were not systematically examined. Guidelines for Further Research Future studies should: 1. examine the long-term implementation of the 21st-BML model across educational levels to assess sustainability and developmental consistency; 2. test effectiveness across diverse regions of Indonesia to strengthen rep - resentativeness and generalisability; 3. explore adaptation beyond mathematics (e.g., science, technology, so - cial sciences) to evaluate cross-domain creativity potential; 4. compare the 21st-BML model with other approaches (e.g., problem- based learning, flipped classroom, realistic mathematics education) to enhancing mathematical creativity through a 21st-century-based learning ... 20 clarify unique contributions. 5. investigate additional outcomes beyond mathematical creativity, includ - ing instructional and nurturant effects. Ethical Statement This study received ethical approval from the Institute for Research and Community Service (LP2M), Universitas Negeri Makassar, South Sulawesi, Indonesia. All of the participants provided written informed consent after re - ceiving a clear explanation of the study’s objectives, procedures and confidenti - ality protocols. The entire data collection process was conducted in strict com - pliance with research ethics standards, with all personal identifiers removed to safeguard participant privacy. Data Availability Statement The data generated in the current study are not publicly available due to ethical restrictions. However, anonymised data may be made available from the corresponding author upon reasonable request. Disclosure Statement The authors have no conflict of interest to declare. When preparing the final revision of this article, the authors used Chat - GPT (OpenAI), version GPT-5, on 13 March 2026, with the following prompts: (1) condense the manuscript without changing the meaning, results or conclu - sions; (2) remove repetition; and (3) keep all in-text citations consistent. The authors subsequently reviewed and edited the output as necessary and accept full responsibility for the content and integrity of the publication. References Al Moray, N. A. (2024). The integration of 21st-century skills in grade eight mathematics curriculum. Journal of Curriculum and Teaching, 13 (2), Article 271. https://doi.org/10.5430/jct.v13n2p271 Arifin, S., Zulkardi, Z., Putri, R. I. I., & Hartono, Y . (2021). On creativity through mathematization in solving non-routine problems. Journal on Mathematics Education, 12 (2), 313–330. https://doi.org/10.22342/jme.12.2.13885.313-330 Baharuddin, B., & Burhan, B. (2025). Urban and rural teacher perspectives on Indonesian educa - tional reform: Challenges and policy implications. Cogent Education, 12(1), Article 2497142. c e p s Journal 21 https://doi.org/10.1080/2331186X.2025.2497142 Ban Hassan Majeed, A., Hassan, A. K., & Hammadi, S. S. (2023). The effect of cognitive modeling in mathematics achievement and creative intelligence for high school students. International Journal of Emerging Technologies in Learning, 18(9), 203–215. https://doi.org/10.3991/ijet.v18i09.39413 Bernadez, E., & Montero, J. (2025). Enhancing student performance in mathematics through concrete-representational-abstract (CRA) approach. International Journal of Science and Research Archive, 14(3), 887–895. https://doi.org/10.30574/ijsra.2025.14.3.0887 Beswick, K., & Fraser, S. (2019). Developing mathematics teachers’ 21st century competence for teaching in STEM contexts. ZDM Mathematics Education, 51(6), 955–965. https://doi.org/10.1007/s11858-019-01084-2 Bicer, A. (2021). Multiple representations and mathematical creativity. Thinking Skills and Creativity, 41, Article 100960. https://doi.org/10.1016/j.tsc.2021.100960 Bicer, A., Aleksani, H., Butler, C., Jackson, T., Smith, T. D., & Bostick, M. (2024). Mathematical crea - tivity in upper elementary school mathematics curricula. Thinking Skills and Creativity, 51, Article 101462. https://doi.org/10.1016/j.tsc.2024.101462 Bingol, B., & Ozyaprak, M. (2025). Enhancing higher education: Differentiating the curriculum and instruction to mathematical creativity and motivation. The Journal of Creative Behavior, 59 (2), Article e70000. https://doi.org/10.1002/jocb.70000 Bond, T. G., & Fox, C. M. (2020). Applying the Rasch model: Fundamental measurement in the human sciences (4th ed.). Routledge. Braun, V ., & Clarke, V . (2019). Reflecting on reflexive thematic analysis. Qualitative Research in Sport, Exercise and Health, 11(4), 589–597. https://doi.org/10.1080/2159676X.2019.1628806 Cevikbas, M., Kaiser, G., & Schukajlow, S. (2022). A systematic literature review of the current dis - cussion on mathematical modelling competencies: State-of-the-art developments in conceptualizing, measuring, and assessing. Educational Studies in Mathematics, 109(2), 205–236. https://doi.org/10.1007/s10649-021-10104-6 Cisnaulin, D., Ghosh, M., & Putra, A. (2025). The effect of the creative problem solving learning model assisted by Google Sites on creative thinking ability in geography learning. Future Space: Stud- ies in Geo-Education, 2(1), 41–56. https://doi.org/10.69877/fssge.v2i1.56 Creswell, J. W ., & Plano Clark, V . L. (2018). Designing and conducting mixed methods research (3rd ed.). SAGE Publications. De Nadai, M., Xu, Y ., Letouzé, E., González, M. C., & Lepri, B. (2020). Socio-economic, built envi- Socio-economic, built envi - ronment, and mobility conditions associated with crime: A study of multiple cities. Scientific Reports, 10(1), Article 13871. https://doi.org/10.1038/s41598-020-70808-2 Desi, D., Putra, H., & Hendriana, H. (2025). Enhancing mathematical understanding ability, creative thinking skills, and self-confidence through an open-ended approach on junior high school students. Journal of Innovative Mathematics Learning, 8 (1), 91–99. https://doi.org/10.22460/jiml.v8i1.24788 Dilekçi, A., & Karatay, H. (2023). The effects of the 21st century skills curriculum on the development of students’ creative thinking skills. Thinking Skills and Creativity, 47, Article 101229. enhancing mathematical creativity through a 21st-century-based learning ... 22 https://doi.org/10.1016/j.tsc.2022.101229 Guilford, J. P . (1967). The nature of human intelligence. McGraw-Hill. Hamid, E., Isnarto, I., & Kharisudin, I. (2024). Creative thinking ability in mathematics problem solving in terms of self-regulated learning in learning creative problem solving model contextual ap - proach. Journal of Mathematics Education , 9(1), 69–82. https://doi.org/10.31327/jme.v9i1.2158 Hu, L., & Wang, H. (2024). Unplugged activities in the elementary school mathematics classroom: The effects on students’ computational thinking and mathematical creativity. Thinking Skills and Creativity, 52, Article 101653. https://doi.org/10.1016/j.tsc.2024.101653 Jonsson, B., Mossegård, J., Lithner, J., & Karlsson Wirebring, L. (2022). Creative mathematical rea - soning: Does need for cognition matter? Frontiers in Psychology, 12, Article 797807. https://doi.org/10.3389/fpsyg.2021.797807 Khalil, I., Charitas, R., & Prahmana, I. (2023). Mathematics learning orientation: Mathematical crea - tive thinking ability or creative disposition? Journal on Mathematics Education, 15 (1), 253–276. https://doi.org/10.22342/jme.v15i1.pp253-276 Kholid, M., Mahmudah, M., Ishartono, N., Putra, F., & Forthmann, B. (2024). Classification of students’ creative thinking for non-routine mathematical problems. Cogent Education, 11(1), Article 2394738. https://doi.org/10.1080/2331186X.2024.2394738 Kónya, E., & Kovács, Z. (2021). Management of problem solving in a classroom context. Center for Educational Policy Studies Journal, 11 (2), 115–132. https://doi.org/10.26529/cepsj.895 Kriewaldt, J., Robertson, L., & Ziebell, N. (2023). Creating the conditions for geographic conceptual development in post-primary students through collaborative guided inquiry. Education Sciences, 13(11), Article 1098. https://doi.org/10.3390/educsci13111098 Kwon, H., & Lee, Y . (2025). A meta-analysis of STEM project-based learning on creativity. STEM Education, 5(2), 275–290. https://doi.org/10.3934/steme.2025014 Leikin, R., & Lev, M. (2020). Mathematical creativity in generally gifted and mathematically excelling adolescents: What makes the difference? ZDM Mathematics Education, 52(1), 183–195. https://doi.org/10.1007/s11858-019-01114-z Leikin, R., & Sriraman, B. (2017). Creativity and giftedness: Interdisciplinary perspectives from math- ematics and beyond. Springer International Publishing. https://doi.org/10.1007/978-3-319-38840-3 Li, M. (2025). Exploring the digital divide in primary education: A comparative study of urban and rural mathematics teachers’ TPACK and attitudes towards technology integration in post-pandemic China. Education and Information Technologies, 30(3), 1913–1945. https://doi.org/10.1007/s10639-024-12890-x Li, Q., & Kim, S. (2025). Meta-analysis of PMRI studies related to mathematical creativity. Frontiers in Psychology, 15, Article 1400328. https://doi.org/10.3389/fpsyg.2024.1400328 Linacre, J. M. (2017). A user’s guide to Winsteps: Rasch-model computer programs. Winsteps.com. Ma, J., & Liu, S. (2022). Effects of culture on the balance between mathematics achievement and sub - jective wellbeing. Frontiers in Psychology, 13, Article 894774. https://doi.org/10.3389/fpsyg.2022.894774 Mariani, D., Mustaji, M., & Dewi, U. (2025). Literature study: The effect of the problem-based c e p s Journal 23 learning model assisted by the flipped classroom on mathematical creative thinking ability. Jurnal Pendidikan Indonesia, 6(1), 75–90. https://doi.org/10.59141/japendi.v6i1.6668 Marwa, M., Nurfaisal, N., Muliardi, M., Awal, R., Putri, A. Y ., Irawan, H., Kadir, R. A., & Ramli, F. (2024). Assessment of students’ 4C skills in research article writing projects for publication. E3S Web of Conferences, 593, Article 04003. https://doi.org/10.1051/e3sconf/202459304003 Meier, M. A., Gross, F., Vogel, S. E., & Grabner, R. H. (2023). Mathematical expertise: The role of domain-specific knowledge for memory and creativity. Scientific Reports, 13(1), Article 12500. https://doi.org/10.1038/s41598-023-39309-w Milin Šipuš, Ž., Bašić, M., Doorman, M., Špalj, E., & Antoliš, S. (2021). MERIA – Conflict lines: Ex - perience with two innovative teaching materials. Center for Educational Policy Studies Journal, 11 (1), 45–64. https://doi.org/10.26529/cepsj.987 Mohammadpour, E., & Y on, H. (2024). Mathematics achievement at rural and urban secondary schools: A trends analysis. Mathematics Education Research Journal . Advance online publication. https://doi.org/10.1007/s13394-024-00511-2 Nadjet, H., Chahrazed, K., Khenioui, A., & Salah, A. (2024). Mind maps: How to improve critical learning and creative thinking using geographic maps. Studies in Engineering and Exact Sciences, 5(2), Article e11838. https://doi.org/10.54021/seesv5n2-699 Nieveen, N. (2010). Formative evaluation in educational design research. In T. Plomp, & N. Nieveen (Eds.), An introduction to educational design research (pp. 89–102). SLO. Niu, J., Xu, H., & Yu, J. (2025). Identifying multilevel factors on student mathematics performance for Singapore, Korea, Finland, and Denmark in PISA 2022: Considering individualistic versus collectiv - istic cultures. Humanities and Social Sciences Communications, 12, Article 151. https://doi.org/10.1057/s41599-025-04466-y Nurkaeti, N., Turmudi, Karso, Pratiwi, V ., Aryanto, S., & Gumala, Y . (2020). Enhancement of mathematical creative thinking ability through an open-ended approach based on metacognitive strategies. Journal of Physics: Conference Series , 1521(3), Article 032030. https://doi.org/10.1088/1742-6596/1521/3/032030 OECD. (2018). The future of education and skills: Education 2030. OECD Education Policy Perspectives, No. 98. OECD Publishing. https://doi.org/10.1787/54ac7020-en Papadopoulos, I., Patsiala, N., Baumanns, L., & Rott, B. (2021). Multiple approaches to problem posing: Theoretical considerations regarding its definition, conceptualisation, and implementation. Center for Educational Policy Studies Journal, 11 (3), 49–66. https://doi.org/10.26529/cepsj.878 Perla, L., Vinci, V ., & Agrati, L. S. (2024). The Italian way to the Europeanisation of teacher educa - tion: An analysis of reforms and the ongoing experience of digital transformation. Center for Educa- tional Policy Studies Journal, 14 (2), 67–84. https://doi.org/10.26529/cepsj.1714 Plomp, T. (2013). Educational design research: An introduction. In T. Plomp, & N. Nieveen (Eds.), Educational design research (Vol. 1, pp. 11–50). SLO. Rahman, Z., Muttaqin, A., & Putri, R. (2025). The effect of the STEM-PJBL model on students’ crea - tive thinking skills. Indonesian Science Education Research, 7(1), 1–6. enhancing mathematical creativity through a 21st-century-based learning ... 24 https://doi.org/10.24114/iser.v7i1.68034 Ritter, S., Gu, X., Crijns, M., & Biekens, P . (2020). Fostering students’ creative thinking skills by means of a one-year creativity training program. PLoS ONE, 15(3), Article e0229773. https://doi.org/10.1371/journal.pone.0229773 Schoevers, E. M., Kroesbergen, E. H., & Kattou, M. (2019). Mathematical creativity: A combination of domain‐general creative and domain‐specific mathematical skills. The Journal of Creative Behav - ior, 54(2), 242–252. https://doi.org/10.1002/jocb.361 Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press. https://doi.org/10.1017/CBO9780511499944 Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing. ZDM Mathematics Education, 29(3), 75–80. https://doi.org/10.1007/s11858-997-0003-x Sitorus, J., & Masrayati, M. (2016). Students’ creative thinking process stages: Implementation of realistic mathematics education. Thinking Skills and Creativity, 22, 111–120. https://doi.org/10.1016/j.tsc.2016.09.007 Sureeyatanapas, P ., Panitanarak, U., Kraisriwattana, J., Sarootyanapat, P ., & O’Connell, D. (2024). The analysis of marking reliability through the approach of gauge repeatability and reproducibility (GR&R) study: A case of English-speaking test. Language Testing in Asia, 14(1), Article 1. https://doi.org/10.1186/s40468-023-00271-z Susanta, A., Susanto, E., Rusnilawati, Sumardi, H., & Ali, S. R. B. (2025). Literacy skills through the use of digital STEAM-inquiry learning modules: A comparative study of urban and rural elemen - tary schools in Indonesia. Eurasia Journal of Mathematics, Science and Technology Education, 21 (4), Article em2615. https://doi.org/10.29333/ejmste/16170 Tampa, A., Ikram, M., & Firdaus, A. M. (2024). Development of a 21st century based mathematics learning model: A validity study on the investigation-construction stages. Journal of Mathematics, Science and Technology Education, 1(1), 31–55. https://doi.org/10.12973/jmste.1.1.31 Tanudjaya, C. P ., & Doorman, M. (2020). Examining higher order thinking in Indonesian lower secondary mathematics classrooms. Journal on Mathematics Education , 11(2), 277–300. https://doi.org/10.22342/jme.11.2.11000.277-300 Tashakkori, A., & Teddlie, C. (2010). Mixed methodology: Combining qualitative and quantitative approaches. SAGE Publications. Torrance, E. P . (1974). Torrance tests of creative thinking: Norms-technical manual. Scholastic Testing Service. Triana, S., Utaya, S., & Soekamto, H. (2024). Geographical creative thinking ability (GCTA) of stu - dents in the diamond panning Cempaka area, Banjarbaru. TEM Journal, 13 (3), 2350–2364. https://doi.org/10.18421/tem133-62 Wang, H., Zhang, S., Li, X., & Gu, B. (2024). The embodied effect in the comprehension of Chinese action-verb metaphors. Journal of Psycholinguistic Research, 53 (4), Article 54. https://doi.org/10.1007/s10936-024-10094-5 c e p s Journal 25 Wessels, H. (2014). Levels of mathematical creativity in model-eliciting activities. Journal of Math - ematical Modelling and Application, 1(9), 22–40. Winasis, A., Rejeki, A., & Cahyono, A. (2025). Challenge-based learning assisted by virtual reality STEAM trails at a cultural heritage museum to promote mathematical creative thinking skills. Jurnal Elemen, 11(3), 741–756. https://doi.org/10.29408/jel.v11i3.29778 Y aniawati, P ., Kariadinata, R., Sari, N., Pramiarsih, E., & Mariani, M. (2020). Integration of e-learning for mathematics on resource-based learning: Increasing mathematical creative thinking and self- confidence. International Journal of Emerging Technologies in Learning (iJET), 15 (6), 60–78. https://doi.org/10.3991/ijet.v15i06.11915 Y ayuk, E., Purwanto, P ., As’ari, A., & Subanji, S. (2020). Primary school students’ creative thinking skills in mathematics problem solving. European Journal of Educational Research , 9(3), 1281–1295. https://doi.org/10.12973/eu-jer.9.3.1281 Zhang, Y ., Y ang, W ., Li, B., Liu, M., Wang, S., Chen, L., & Rodriguez, M. (2025). Convergence of primary education development in urban and rural China: Empirical analysis of historical trends and future projections. Humanities and Social Sciences Communications, 12, Article 475. https://doi.org/10.1057/s41599-025-04752-9 enhancing mathematical creativity through a 21st-century-based learning ... 26 Appendices The appendices only provide representative excerpts of the instruments, learning materials and media. The complete documents can be requested from the corresponding author. Appendix A. Research Instruments A.1 Mathematical Creativity Test (Sample Item) Question 1 – Specific Solution Given the system of linear equations below: ax + by = c 3x - 2y = 11 If ( x, y) = (3, -1) is a solution to the system and a, b, c are real numbers, a,b, determine as many pairs ( a, b, c) as possible that satisfy the conditions. Explain the reasoning and method you used to obtain the solutions. You may apply different approaches such as substitution, tables, graphs or coefficient manipulation. A.2 Mathematical Creativity Assessment Rubric Assessed Dimensions This rubric evaluates students’ mathematical creativity based on two quantitative indicators, as defined operationally in the Instruments section. 1. Fluency The number of distinct and valid solutions or systems of equations gen - erated by the student in accordance with the problem requirements. 2. Flexibility The variety of distinct and valid mathematical approaches, strategies or representations employed by the student in solving the problem. Scoring Scheme per Item A. Fluency Score Description 3 Produces ≥ 5 distinct and correct solutions/systems 2 Produces 3–4 distinct and correct solutions/systems 1 Produces 1–2 correct solutions/systems 0 Produces no correct solutions/systems c e p s Journal 27 B. Flexibility Score Description 3 Uses ≥ 3 different strategies or representations (e.g., graphs, tables, substitution, algebraic manipulation) 2 Uses 2 different strategies or representations 1 Uses only 1 strategy or representation 0 Shows no clear or logical strategy Total Score – Maximum score per item: 6 (Fluency 3 + Flexibility 3) – Number of items: 3 – Maximum total score per student: 18 – Minimum score: 0 Mathematical Creativity Score Criteria Score Range Category 15–18 Very High 11–14 High 7–10 Moderate 4–6 Low 0–3 Very Low Illustration of Student Test Result Assessment Problem Given a system of linear equations in two variables (SLETV): If ( x, y) = (3, −1) is a solution and a, b ≠ 0, determine as many ordered triples ( a,b,c) as possible, and explain the method used. Sample Student Response Method 1: Table (Algebraic Exploration) By substituting (3, −1) into ax + by = c, we obtain c = 3a − b. The student selects several values of and , then computes the corre - sponding value of , resulting in the following ordered triples: (1,1,2), (2,1,5), (3,4,5), (1,−2,5), (4,2,10), (5,5,10) enhancing mathematical creativity through a 21st-century-based learning ... 28 Method 2: Slope (Geometric Approach) The equation is rewritten as . By selecting several different slopes and ensuring that the line passes through the point (3, −1), the following ordered triples are obtained: (2,2,4), (6,3,15), (3,6,3), (6,8,10) Scoring Aspect Score Rationale Fluency 3 Generates ≥ 5 distinct and valid solution pairs Flexibility 2 Uses two different approaches: a table (algebraic) and a slope- based (geometric) approach Total 5/6 Near-maximum score A.3 Observation Sheet (Excerpt) Indicators of students’ creative activity during the implementation of the 21st- BML model included: • use of multiple mathematical representations (symbolic, graphical, ta - bular, verbal); • spontaneous student questioning; • verbal reasoning in problem-solving discussions. Observation Sheet 21st-BML Syntax Observation Focus Observation Notes Object Translation • Ma thema tical r epr esen ta tions cr ea t ed b y s tuden ts when translating from concrete objects. • Ques tions posed b y s tuden ts. • S tuden t e xplana tions or me thods of c ommunica ting ideas. Decomposition • Mind maps created by students. • Mathematical concept components included in the mind maps. • Questions posed by students. • Student explanations or methods of communicating ideas. Cognitive Construction • Student outputs in solving open-ended problems (based on strategy variations). • Student outputs in solving open-ended problems (based on solution variations). • Questions posed by students. • Student explanations or methods of communicating ideas. c e p s Journal 29 Appendix B. Sample Lesson Plan (Excerpt of Meeting 1 – Object Translation Syntax) A. Learning Activities Object Translation  Stepwise Animated Video Presentation • T he t eacher pr esen ts V ideo 1 , sho wing Mrs Sari shopping at Toko Buah Segar, buying 5 kg of apples and 4 kg of oranges for a total of Rp 42.000. • T he t eacher c on tinues with V ideo 2, wher e Mrs Sari returns the next day and buys 3 kg of apples and 2 kg of oranges for Rp 23.000,- • S tuden ts ar e ins truc t ed t o observ e both videos car efully while r ec or ding k e y information on a structured observation sheet.  Initial Group Discussion • T he t eacher f acilita t es a gr oup discus sion t o help s tuden ts e x tr ac t in f orma tion from both video scenarios. • S tuden ts ar e guided t o iden tif y variables and constants involved in the real-life context.  Mathematical Representation Transformation • Each gr oup tr ans f orms the in f orma tion fr om V ideo 1 in t o the fir s t linear equa- tion. • T hen, the y tr ans f orm the in f orma tion fr om V ideo 2 in t o the sec ond linear equation. • T he t eacher pr o vides adaptiv e scaff olding using guiding ques tions such as: 0 What is unknown in this situation? 0 How can you express the price of apples and oranges using variables? 0 What is the relationship between the quantity of fruit and the total cost?  Concept Formulation: System of Linear Equations in Two Variables • T he t eacher guides s tuden ts t o f ormula t e the definition of a s y s t em of linear equa tions in t w o v ariables ( SLET V) based on the equa tions the y deriv ed. • S tuden ts e xpr es s the c onc ept in t w o f orma ts: 0 Verbal definition: A system of linear equations in two variables is… 0 General mathematical form: ax + by = c and px + qy = r, where a, b, p, q ≠ 0  Dialogic Interaction and Conceptual Reinforcement • T he t eacher pr o vides opportunities f or s tuden ts t o ask ques tions about c on- cepts they do not yet understand. • A triadic in t er ac tion s tr a t egy is f acilita t ed: 0 Teacher ↔ Student: Clarification through probing questions 0 Student ↔ Student: Peer teaching and inter-group discussion 0 Student ↔ Learning Resources: Use of worksheets and digital references  Scaffolding-Based Learning Support • T he t eacher c onduc ts guided pr ac tic e t o help s tuden ts r each their z one of proximal development. • Scaff olding is tailor ed t o each gr oup’ s c ompr ehension le v el: 0 Basic Level: Direct assistance with variable identification 0 Intermediate Level: Strategic hints for forming equations 0 Advanced Level: Reflective questions for deepening understanding  Documentation and Transition • S tuden ts documen t their gr oup w ork in their individual digital portf olios, including: • T r ans f orma tion of video sc enarios in t o linear equa tions • Definition of SLET V ( v erbal and ma thema tical) • R eflec tion on wha t the y learned during the ac tivit y enhancing mathematical creativity through a 21st-century-based learning ... 30 B.2 Sample Student Worksheet (Nasi Kuning Context) Mas Joko sells nasi kuning (yellow rice) at his food stall called NASI KUNING MACCINI. The menu and price list are presented in the image below. Step 3: Mathematical Modelling (Symbolic Representation) Tasks to Complete: 1. Discuss the results of Step 2 within your group’s WhatsApp chat. 2. Identify two unknowns in the situation – these will serve as variables x and y. 3. Determine relevant constraints, such as price limits, quantity limits and so on. 4. Formulate a context-based problem that reflects the observed situation. 5. Construct a mathematical model based on the given context (i.e., a sys - tem of linear equations in two variables). Appendix C. Sample Digital Media c e p s Journal 31 Biographical note Alimuddin Tampa, PhD, is a professor of mathematics education whose area of expertise is mathematics learning oriented toward 21st-century skills, particularly mathematical creativity. He has been a lecturer in the Depart - ment of Mathematics, Faculty of Mathematics and Natural Sciences, Universi - tas Negeri Makassar, Indonesia, since 1988. He earned his doctorate in mathe - matics education from Universitas Negeri Surabaya, Indonesia, with a research focus on mathematical creativity. His research interests include mathematics education, 21st-century skills in mathematics learning, mathematical creativity, and the development of higher-order thinking skills. He has authored ISBN- registered books, registered intellectual property works, and scholarly articles indexed in Scopus and nationally accredited Indonesian journals. He can be contacted at alimuddin@unm.ac.id. Ja’faruddin, PhD, is an Associate Professor of Applied Mathematics whose scholarly expertise lies at the intersection of mathematics and culture, with particular emphasis on ethnomathematics. Since 2010, he has served as a faculty member in the Department of Mathematics, Faculty of Mathemat - ics and Natural Sciences, Universitas Negeri Makassar, Indonesia. He obtained his doctoral degree in Applied Mathematics from Tunghai University, Taiwan, where his research focused on the integration of mathematical structures and cultural contexts. His research interests include knot theory, projective geom - etry, mathematics and culture, ethnomathematics, and mathematics education. He has authored several ISBN-registered books, holds registered intellectual property rights, and has published scholarly articles in Scopus-indexed jour - nals and nationally accredited Indonesian journals. He can be contacted at jafaruddin@unm.ac.id. Fauziyyah Alimuddin, M.Pd., is an assistant professor in the field of mathematics education in the Department of Mathematics, Faculty of Math - ematics and Natural Sciences, Universitas Negeri Makassar, Indonesia. Her re - search interests include mathematics education, mathematics learning, and the development of learning resources in mathematics. She has authored several scientific articles and contributed to intellectual property works. She can be contacted at fauziyyah.alimuddin@unm.ac.id. enhancing mathematical creativity through a 21st-century-based learning ... 32 Andi Mulawakkan Firdaus, PhD, is an Associate Professor of Mathematics Education with expertise in foundational mathematics learn - ing, particularly number patterns. Since 2014, he has served as a lecturer in the Mathematics Education Program, Faculty of Teacher Training and Educa - tion, Universitas Muhammadiyah Makassar, Indonesia. He earned his doctoral degree in Mathematics Education from Universitas Negeri Surabaya, with a dissertation focusing on number pattern learning based on APOS theory. His research interests include mathematics education, number patterns, APOS theory, mathematical critical thinking, and the development of higher-order thinking skills. He has authored ISBN-registered books, produced intellectual property-protected works, and published scholarly articles in Scopus-indexed journals and nationally accredited journals. He can be contacted at andi.mula - wakkan@unismuh.ac.id.