Jurnal Elemen, 11.
, 898-911.
October 2025 https:/doi.
org/10.
29408/jel.
Secondary studentsAo informal strategies in solving enumeration problems prior to formal combinatorics instruction Aan Putra 1.
Zulkardi 1 *.
Ratu Ilma Indra Putri 1.
Laswadi 2 Mathematics Education Department.
Universitas Sriwijaya.
South Sumatra.
Indonesia Mathematics Education Department.
State Islamic Institute of Kerinci.
Jambi.
Indonesia Correspondence: zulkardi@unsri.
A The Author.
2025 Abstract Although enumeration problems are fundamental in combinatorics, little is known about how students intuitively approach such enumeration problems before receiving formal instruction.
This exploratory qualitative study investigated the initial strategies employed by twelve-grade students in solving enumeration problems prior to formal instruction on enumeration rules.
Fifteen students from a public senior high school in Kerinci.
Indonesia, who had not yet learned combinatorics in the curriculum, participated in this study.
Data were collected through studentsAo written responses to three combinatorial problems presented in different real-life contexts and further explored through semi-structured interviews.
Only responses demonstrating coherent and interpretable strategy were analyzed.
The findings reveal three dominant strategies: listing all possible arrangements, generalizing patterns, and applying the multiplication principle.
These findings indicate that students possess intuitive approaches that can serve as a foundation for formal combinatorial reasoning.
The study aligns with the Realistic Mathematics Education (RME) perspective, emphasizing the importance of guided reinvention and contextual mathematization, and proposes implications for designing learning trajectories that build on studentsAo informal reasoning in secondary mathematics education.
Keywords: combinatorial reasoning.
counting principles.
student initial strategies How to cite: Putra.
Zulkardi.
Putri.
, & Laswadi.
Secondary studentsAo informal strategies in solving enumeration problems prior to formal combinatorics instruction.
Jurnal Elemen, 11.
, 898-911.
https://doi.
org/10.
29408/jel.
Received: 22 June 2025 | Revised: 18 August 2025 Accepted: 15 October 2025 | Published: 5 November 2025 Jurnal Elemen is licensed under a Creative Commons Attribution-ShareAlike 4.
0 International License.
Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi Introduction The ability to solve problems involving selection and arrangement is increasingly vital in todayAos complex and data-driven world.
Whether consciously or not, individuals are often required to make choices from a set of possibilitiesAisuch as planning travel routes, organizing schedules, or selecting items within constraintsAiwhich inherently involve combinatorial reasoning and numeracy.
Globally, such skills are recognized as core components of mathematical literacy (OECD, 2019, 2.
, yet reports indicate that many students struggle with tasks requiring these forms of thinking (Mullis et al.
, 2.
In Indonesia, the 2022 PISA results revealed that the majority of 15-year-old students perform below minimum proficiency in mathematics, particularly in tasks that require creative problem solving and application of combinatorial reasoning in everyday contexts (Kusmaryono & Kusumaningsih, 2.
These findings raise serious concerns about the preparedness of students to role effectively in quantitative environments, both in academic and real-world settings.
Researchers and educators worldwide have responded to this issue by emphasizing the need to develop both combinatorial thinking and numeracy as integral parts of mathematics education (Geiger et al.
, 2015.
Lockwood, 2.
Combinatorial thinking refers to the ability to systematically explore and quantify possibilities in a given problem context, such as through enumeration, arrangement, and selection (English, 1991.
Lockwood, 2013.
Salavatinejad et al.
Numeracy, by contrast, entails the capacity to interpret, evaluate, and act on quantitative information to make reasoned decisions (Geiger et al.
, 2015.
OECD, 2.
Both are essential not only for academic success but also for informed participation in everyday life, such as budgeting, planning, and risk assessment.
However, previous research has largely focused studentsAo post-instructional performance, often evaluating how well students apply permutation or combination formulas after formal As Matitaputty et al.
that focus on identifying student mistakes conceptually and procedurally in permutations and combinations.
While such studies offer insights into procedural knowledge and conceptual errors, they overlook how students initially make sense of counting problems using their informal mathematical understanding.
The study has compared the strategies of students with and without instruction combinatorics instruction.
For example.
Lamanna et al.
, but this study did not identify in detail the strategies of students who had not received combinatorics instruction.
These studies leaving a research gap in how students approach enumeration problems intuitively prior to instruction.
Addressing this gap is crucial for two reasons.
First, identifying studentsAo initial strategies provides a window into their intuitive and informal thinking, revealing the cognitive resources they bring to bear when confronting unfamiliar problems.
Understanding studentsAo initial strategies in solving enumeration problems is essential for supporting the development of relational rather than merely instrumental understanding of combinatorial reasoning (Skemp.
Second, it offers a powerful basis for designing instructional trajectories that begin with studentsAo existing knowledge, consistent with the principles of Realistic Mathematics Education (RME).
RME, developed in the Netherlands and adapted in Indonesia .
an den HeuvelPanhuizen & Drijvers, 2014.
Zulkardi et al.
, 2.
, emphasizes learning mathematics is seen Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
as a process of guided reinvention, where students reconstruct formal mathematical concepts from their informal experiences through mathematizationAithe progressive organization and formalization of problem situations (Solomon et al.
, 2.
By understanding studentsAo spontaneous approaches to enumeration problems, educators can design tasks that bridge informal reasoning with formal counting principles in a cognitively authentic procedure.
Exploring such intuitive yet sometimes incomplete strategies aligns with the notion of productive failure (Kapur, 2.
, where studentsAo initial struggles to make sense of problems can serve as a valuable foundation for subsequent conceptual learning.
combinatorics, this means that students can move from intuitive enumeration .
, listing all possibilitie.
toward systematic organization .
, tree diagram.
and eventually toward abstract symbolic reasoning, such as the multiplication principle (English, 1991.
Lockwood.
This study investigates how Indonesian twelfth-grade students solve enumeration problems before receiving any formal instruction in counting principles.
The problems were embedded in everyday-related contexts that allowed for multiple solution strategies.
The aim is to identify and categorize the initial strategies students use and to discuss their implications for designing RME-based instructional sequences.
By doing so, the study contributes to a deeper understanding of studentsAo informal reasoning in enumeration rules, and supports the development of mathematics learning that is both contextually relevant and cognitively Methods This study employed a qualitative exploratory design to investigate studentsAo initial strategies in solving enumeration problems before receiving any formal instruction.
Qualitative research is particularly suited for uncovering the meaning behind students' responses and exploring their thought processes (Creswell & Poth, 2.
The study focused on identifying and classifying of naturally occurring strategies as a foundation for future instructional design based on Realistic Mathematics Education (RME) principles.
The research was conducted before participantsAitwelfth-grade studentsAireceived any formal instruction in counting principles .
ermutations, combinations, or the multiplication principle.
in their mathematics curriculum.
This timing was crucial to ensure that students relied solely on their prior mathematical knowledge and informal reasoning.
Participants consisted of 15 students .
ged 18Ae19 years, 7 males and 8 female.
from a public senior high school Jambi.
Indonesia, who voluntarily agreed to take part in the study.
The selection was based on purposive sampling, a common practice in qualitative research when the goal is to explore specific cognitive phenomena in depth (Patton, 2.
Data saturation was achieved when no new strategies emerged after analyzing all participantAos response, indicating that the diversity of strategies had stabilized across students.
To elicit students' initial strategies, the researcher administered a pretest consisting of three enumeration problems presented in different real-life contexts but requiring equivalent mathematical reasoning.
Each student was asked to choose and solve one problem to reduce Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi cognitive load and allow deeper reasoning within a single context.
However, this approach also limited cross-problem comparisons, which is acknowledged as a methodological limitation of the study.
The problems are as follows:
Problem 1.
In an exam, the teacher gives 4 questions.
To ensure that students do not cheat on each other, students are not allowed to answer the questions in the same order as other students.
How many different orders are there for answering the 4 questions? Problem 2.
During Eid al-Fitr, you plan to visit the homes of Mr.
Beni.
Mrs.
Dalius.
Mrs.
Emi, and Mrs.
Fitri.
How many different routes are there to visit the 4 teachers' homes? Problem 3.
Suppose you want to create a PIN for your cellphone screen lock consisting of 4 digits consisting of the numbers 0, 1, 2, and 3.
How many different PINs can you create? Problems 1 and 2 represent permutation tasks without repetition .
! = .
, whereas Problem 3 is intentionally open to dual interpretationAistudents may treat it as either with or without repetition.
This ambiguity was deliberately retained as a methodological feature to reveal how studentsAo reason about combinatorial conditions when the context does not explicitly specify constraints.
Data analysis was conducted using content analysis, studentsAo written responses were collected and reviewed, focusing on identifying distinct and meaningful solution strategies.
Only responses that reflected distinct strategiesAiwhether correct or partially correctAiwere analyzed in depth, while responses exhibited misconceptions or relied on external aids were excluded from the classification.
Following the test, semi-structured interviews were conducted with selected students to clarify their reasoning, confirm their thinking process, and avoid misinterpretation of their written work.
Interviews are a well-established method for capturing students' cognitive strategies in mathematics education (Goldin, 2.
Triangulation between written responses and interview data ensured credibility and internal validity of the findings so that research results were more valid, accurate, and reduced potential bias.
Ethical procedures were followed throughout the study: informed consent was obtained from all participants and initials were used to ensure anonymity.
Results The written test was administered under a specific rule: students seated next to each other were not allowed to solve the same problem.
This rule aimed to minimize academic dishonesty and encourage individual reasoning.
Figure 1 presents an illustration of the students' seating arrangement and their respective problem choices .
RD .
indicates that student RD attempted problem number .
Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
Figure 1.
Ilustration of studentsAo seating arrangement and their problem choices Problem 1: Question order arrangement Only two students.
AL and FZ, attempted Problem 1.
AL manually attempted to list all possible orders of answering four questions using trial-and-error.
AL managed to produce only six sequences, such as 1234, 4312, 2413, 1324, 2134, and 3142.
However, the responses lacked structure and did not suggest a systematic method of enumeration.
AuMy answer is still lacking but I hesitate to make all the arrangements because it is too Ay Ae AL FZ, in other hand, proposed a distinctive strategy by starting with six unique arrangements starting with Question 1.
Then generalizing the number of possible arrangements that would result if the sequence began with other questions.
AuI suspect that if the sequence .
f working on the question.
starts with another number .
it will produce 6 sequences as well.
Ay Ae FZ Although FZ could not fully justify why each starting point would lead to exactly six arrangements .
ee Figure .
, their strategy demonstrated an emerging awareness of recursive or multiplicative reasoning, which could serve as an intuitive basis for introducing tree diagrams or the multiplication principle in future instruction.
Translation:
(Arrangement.
Because it consists of 4 digits and every first digit like 1 can make 6 arrangements.
So, the number of different arrangement that can be made are 6 x 4 = 24.
Figure 2.
FZAos attempt to generalize the total number of orders from different starting Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi Problem 2: Visiting teacherAos houses Five studentsAiRT.
NZ.
IQ.
DZ, and ADAiattempted this problem.
RT and IQ independently applied the multiplication principle by noting the decreasing number of choices at each stage of the trip.
IQ, for instance, reasoned that the first house could be chosen in 4 ways, the second in 3, and so on .
ee Figure .
However, during interviews, neither RT nor IQ could justify why multiplication was the appropriate operation to combine these choices, indicating partial understanding of the principle.
AuIf you add 4, 3, 2, and 1, the result is only 10.
So I tried using multiplication.
Ay Ae IQ Translation:
(Give.
4 options, chosen 1, left 3 options (Lef.
3 options, chosen 1, left 2 options (Lef.
2 options, chosen 1, left 1 options (Lef.
1 4 x 3 x 2 x 1 = 24 routes or 4! ABCD BCD CD D Figure 3.
IQAos use of logical step-by-step reasoning before multiplication AD used a listing strategy similar to that of FZ in Problem 1.
AD started by assuming the trip began at Mr.
BeniAos house, enumerated six different routes from that point, and then inferred a total of 24 routes by multiplying by 4 potential starting points .
ee Figure .
Interviews revealed that AD had already listed all the possibilities on a separate sheet before making the AuAfter I tried starting from Mrs.
Dalius, the result was also 6.
I guess this applies to others too.
Ay Ae AD Translation:
24 routes Because each route that begin from one teacher house have 6 .
For example, here the routes that begin from Mr.
BeniAos house.
(Arrangement.
It means 6 x 4 = 24 .
NB:
6 = the number of routes from each teacher house that visited first 4 = the number of teachers .
that will be Figure 4.
ADAos strategy: listing from one starting point, then multiplying by the number of starting points Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
NZ presented only four routes, mistakenly assuming that different starting points alone constituted different routes.
This reflects a misconception, failing to recognize that multiple permutations can originate from the same starting point .
ee Figure .
AuI think there are only 4 routes because I started from Mr.
Beni.
Mrs.
Dalius.
Mrs.
Emi, and Mrs.
Fitri.
Ay Ae NZ Translation:
There are 4 differnet routes.
A Mr.
Beni - Mrs.
Dalius - Mrs.
Emi - Mrs.
Fitri A Mrs.
Dalius - Mrs.
Emi - Mrs.
Fitri - Mr.
Beni A Mrs.
Emi - Mrs.
Fitri - Mr.
Beni - Mrs.
Dalius A Mrs.
Fitri - Mr.
Beni - Mrs.
Dalius - Mrs.
Emi Figure 5.
NZAos misinterpretation of route variation based solely on starting points DZ showed a more fundamental misunderstanding by interpreting AurouteAy as a single segment of travel between two houses rather than a full sequence.
For example.
DZ considered traveling from Mr.
BeniAos to Mrs.
DaliusAos house as a complete route.
This error indicates a failure to grasp the problemAos structure.
Problem 3: Creating a 4-digit PIN This problem was chosen by most students: OK.
RZ.
AG.
BG.
IT.
RD.
HS, and PR.
OK.
RZ,
AG.
IT, and PR used enumeration by listing possible PINs.
Among them, only RZ successfully listed 24 unique combinations by fixing the starting digit and varying subsequent digits systematically .
ee Figure .
Translation:
Each digit have 6 arrangements, so 6 6 6 6 = 24.
(Arrangement.
So.
I can make 24 different PIN, by different digits.
Figure 6.
RZAos systematic enumeration of unique pins based on starting digit Interestingly.
RZ used repeated addition rather than multiplication, suggesting a completely manual enumeration process.
On the other hand.
OK and AG produced repeated combinations due to a lack of systematic listing, as shown in Figure 7.
AuI didn't check the same order.
What matters is the total is 6 y 4.
Ae OK Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi AuI believe it is true because sequences starting from different numbers have the same Ay Ae AG Figure 7.
Overlapping PINs in OKAos .
and AGAos enumerations .
In interviews.
OK revealed that the listing assumed no repeated digits, indicating partial awareness of repetition rules.
However, both IT and PR failed to list all possible combinations, producing only 22 and 6 combinations, respectively, some of which involved repeated digits .
, 1122, 0.
RD applied the multiplication principle by multiplying 4 y 4 y 4 y 4 = 256, assuming each digit could be filled with any of the four available numbers .
Ae.
While RD was confident in this approach, interviews revealed that RD could not justify the use of multiplication .
ee Figure .
AuThese numbers can be repeated, so the number of choices remains the same .
t each Just multiply them to get the largest number.
Ay Ae RD Translation:
That is I can make.
That consist of 4 PIN.
So, 4 x 4 x 4 x 4 = 256 So, the number of PINs that can be made are Figure 8.
RDAos use of the multiplication rule without supporting explanation HS used the strategy of listing combinations starting with one digit .
, similar to FZ and AD, then multiplied by 4 to arrive at 24 PINs.
Lastly.
BG gave two answers: 256 .
ith repetition allowe.
and 24 .
o repetitio.
, but admitted during interviews that the answers were generated using ChatGPT, and not through personal reasoning.
Consequently.
BGAos response was excluded from strategy classification for reasons of methodological integrity.
Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
Variation of studentsAo strategies Apart from DZ and BG, students' solutions were classified into three general strategies: .
listing all unique arrangements, .
generalizing patterns of arrangement, and .
using the multiplication principle.
The distribution of these strategies is summarized in Table 1.
Table 1.
Summary of studentsAo strategies in solving enumeration problems Frequency Strategy Students Percentage Description .
= .
Listing Unique OK.
RZ.
Manually listing all Arrangements AG.
IT.
NZ, possible outcomes AL.
Generalizing FZ.
AD.
Identifying structure or Patterns pattern before generalizing result Using RT.
IQ.
Directly applying Multiplication multiplication without Principle full justification No Clear DZ.
Misunderstanding or Strategy or relying on external Misguided As summarized in Table 1, seven students .
%) relied on listing unique arrangements, three .
%) generalized structural patterns, and three .
%) directly applied the multiplication The remaining two students .
%) either misunderstood the problem (DZ) or relied on external tools (BG).
These results indicate that while most students were able to engage in combinatorial reasoning at an intuitive level, their strategies reflected varying degrees of formalizationAifrom exhaustive manual listing to emerging multiplicative reasoning.
Discussion The findings of this study reveal that students employ various initial strategies in solving enumeration problems, reflecting diverse levels of understanding related to permutations and counting principles.
The three main strategies identified were: .
listing all possible arrangements, .
generalizing patterns of arrangement, and .
applying the multiplication These findings suggest that studentsAo intuitive approaches reflect early forms of relational understanding, as opposed to purely instrumental reasoning (Skemp, 1.
Recognizing such informal reasoning is essential for designing learning trajectories that connect intuitive thinking with formal combinatorial principles.
The development of these strategies aligns with the theoretical framework of Realistic Mathematics Education (RME), where understanding progresses from contextual to formal reasoning through guided reinvention (Solomon et al.
, 2.
This classification was guided by previous research on studentsAo combinatorial reasoning (Lockwood, 2013.
Salavatinejad et al.
, 2.
Lockwood .
proposed a model of studentsAo combinatorial thinking that illustrates how learners progress from constructing explicit lists of Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi possible outcomes to identifying structural patterns and, eventually, applying formal counting principles such as the multiplication rule.
This developmental trajectory reflects a gradual shift from informal to formal reasoning.
Similarly.
Salavatinejad et al.
found that students initially rely on intuitive strategies such as listing and drawing visual representations to organize possibilities, with only a few able to generalize patterns or correctly use the multiplication Together, these studies provide a strong theoretical basis for the three strategy categories identified in this researchAilisting unique arrangements, generalizing patterns of arrangement, and using the multiplication principleAias they represent distinct yet connected stages in studentsAo evolving understanding of enumeration problems.
The first strategy, listing all possible arrangements, was employed by the majority of This strategy is aligned with Lamanna et al.
that unveil that the majority of students without combinatorics instruction use listing, mostly systematic, to help the enumeration process.
The listing strategy corresponds to the situational level of RME, where reasoning remains embedded in the problem context .
an den Heuvel-Panhuizen, 2.
This strategy represents an intuitive and concrete approach to problem-solving and is characteristic of the early stage of mathematization in the framework of Realistic Mathematics Education (RME) (Freudenthal, 1.
Lockwood et al.
argue that partial lists of the set of outcomes which created by students led to significant improvements in solving problems, implying that systematic listing worthwhile for students as they learn to count.
The second strategyAigeneralizing patterns of arrangementAiwas demonstrated by students who attempted to make a tree diagrams however not complete but they can extrapolate them to solve whole problems.
This approach marks a transition toward horizontal mathematization, where students begin to connect contextual problem situations to more structured mathematical representations .
an den Heuvel-Panhuizen, 2.
For example, students like FZ and AD showed an intuitive grasp of repeated structural patterns, even though they were not able to articulate them formally.
This indicates that students were beginning to construct informal models of mathematical structure, which can serve as a foundation for formal instruction on permutations and tree diagrams (Litwiller & Bright, 2.
This symbolic representation like tree diagrams enables students to perceive how elements can be combined in a systematic manner and help them develop their combinatorial reasoning (Borba et al.
The third strategy, applying the multiplication principle, reflects a shift toward vertical mathematizationAimoving from informal representations to formal mathematical procedures (Gravemeijer, 2.
However, it is important to note that most students using this strategy did not demonstrate a deep conceptual understanding of why multiplication was appropriate in the given context.
Although the use of the multiplication principle suggests potential for formal reasoning, pedagogical support is needed to guide students toward a full conceptual grasp of the method.
The research shows that it is not always natural for students to use the multiplication rule in solving combinatorics problems because it involves sequential steps that tend to be complicated for beginners (Lockwood & Purdy, 2.
In the specific case of the PIN problem, studentsAo understanding became more complex due to the involvement of digit repetition.
Some students limited their responses to non907 Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
repetitive arrangements, while others conflated repetition and non-repetition scenarios.
This finding underscores the need for explicit instruction on the distinction between permutations with and without repetition, as recommended by Lockwood .
in her study on combinatorial representations.
Two students.
DZ and BG, displayed responses that did not align with any of the three major strategies.
DZ misunderstood the concept of "route" as a complete tour rather than a sequence of destinations, while BG provided a correct answer but admitted to relying on AI assistance (ChatGPT).
The latter case illustrates a growing issue in mathematics education related to academic integrity and the responsible use of digital tools.
While technology can be a valuable tool in fostering mathematical thinking, it can also hinder cognitive engagement if used as a substitute for reasoning rather than a support for it (Zbiek et al.
, 2.
A particularly meaningful feature of this study is its cultural context.
The enumeration problemsAiespecially the Eid al-Fitr visiting scenarioAiare deeply familiar to Indonesian Such culturally resonant contexts align with the RME emphasis on realistic .
xperientially meaningfu.
an den Heuvel-Panhuizen & Drijvers, 2.
The use of tourism and holiday activities connects mathematics to students lived experiences, enabling authentic engagement and supporting horizontal mathematization.
This underscores that realistic contexts need not be Western or generic.
they can and should be locally and culturally grounded to enhance relevance and motivation.
These findings have significant implications for instructional design informed by RME First, students need learning experiences that stimulate natural mathematization processes from contexts that are meaningful to themAisuch as the tourism scenarios used in this Second, teachers play a crucial role in encouraging open exploration of student strategies prior to introducing formal procedures.
Third, studentsAo initial strategies should serve as starting points for developing a bottom-up learning trajectory, in accordance with the principle of guided reinvention in RME (Eerde, 2.
The findings carry several implications for mathematics instruction, particularly in the domain of counting and permutations.
The problems in this study provided meaningful contexts that facilitated student engagement and elicited natural strategies.
Teachers should integrate culturally relevant and experientially rich contexts that allow students to mathematize real-life situations .
an den Heuvel-Panhuizen & Drijvers, 2.
Teachers should create space for students to share and discuss their own problem-solving strategies before introducing formal This aligns with the RME principle of guided reinvention, allowing students to construct mathematical understanding from their own thinking.
Solomon et al.
emphasized that guided reinvention requires environments that encourage exploration before Although some studentsAo strategies appeared incomplete or incorrect, these attempts can be seen as instances of productive failure (Kapur, 2.
, where the struggle to construct solutions independently lays the groundwork for deeper conceptual understanding in subsequent instruction.
Instruction should focus on developing studentsAo reasoning about when and why multiplicative structures apply, using tools such as tree diagrams and systematic listing as transitional models (English, 1991.
Lockwood, 2.
Teachers also play a crucial role in Aan Putra.
Zulkardi.
Ratu Ilma Indra Putri.
Laswadi orchestrating whole-class discussions, connecting informal and formal strategies, and bridging intuitive ideas with canonical mathematical representations (Stein et al.
, 2.
While digital tools like AI can support learning, their integration should aim to enhanceAinot replaceAi student thinking.
Educators must carefully monitor how students use such tools to ensure productive engagement (Zbiek et al.
, 2.
Conclusion This study demonstrates that even without formal instruction, students naturally use a range of strategiesAilisting all possible arrangements, generalizing patterns, and attempting to apply the multiplication principleAireflecting varying levels of informal combinatorial reasoning.
While some students relied heavily on trial-and-error or incomplete listing strategies, others began to show signs of structural understanding and pattern generalization.
Only a small number attempted formal mathematical reasoning.
These findings underscore the importance of recognizing and leveraging studentsAo intuitive thinking as a foundation for meaningful learning trajectories in combinatorics.
Teachers should view studentsAo informal approaches not as misconceptions, but as seeds of mathematical understanding that can be nurtured through guided reinvention.
Future research should extend this work in several directions.
First, design experiments could be conducted to test learning trajectories that build on studentsAo initial strategies and progressively develop conceptual understanding of counting principles.
Second, cross-cultural studies could explore whether similar intuitive patterns emerge in different sociocultural settings, thereby enriching the global understanding of combinatorial reasoning.
Third, the integration of digital toolsAisuch as dynamic tree diagram applications or AI-based visualization aidsAishould be examined for their potential to support studentsAo exploration and deepen their relational understanding of combinatorial structures.
Conflicts of Interest The authors declare no conflict of interest regarding the publication of this manuscript.
Funding Statement This work received no specific grant from any public, commercial, or not-for-profit funding Author Contributions Aan Putra: Conceptualization, formal analysis, investigation, writing - original draft, and Zulkardi: Conceptualization, methodology, validation, and writing - review & editing, supervision.
Ratu Ilma Indra Putri: Methodology, validation, and supervision.
Laswadi: Writing - original draft, validation, and supervision.
Secondary studentsAo informal strategies in solving enumeration problems prior to formal .
References