Analysis of a Structural Thinking Task Solved by Pre-Service Mathematics Teachers

Abstract

Structural thinking is widely recognized as an important aspect of mathematical thinking, yet the role of structuring representations in supporting generalization has received relatively little attention. This study examined how pre-service mathematics teachers construct mathematical structure when solving a pattern generalization task. Written solutions produced by 62 pre-service mathematics teachers at different stages of their teacher education program were analyzed using qualitative thematic analysis informed by theoretical perspectives on emergent and mathematical structure. Six distinct solution strategies were identified, differing primarily in the structuring representations participants used to reorganize the mathematical situation. The findings indicate that participants differed not only in their solution procedures but also in the mathematical relationships highlighted by their representations. These differences were reflected in the general rules participants formulated and in the mathematical reasoning they used to support their conclusions. The study highlights the importance of structuring representations as a component of structural thinking and suggests that developing pre-service teachers’ ability to construct and interpret mathematically productive representations may support the transition from recognizing local relationships to formulating general mathematical relationships and supporting them through mathematical reasoning.

1. Introduction

In this paper, we analyze solutions to a task focused on elements of structural thinking produced by pre-service mathematics teachers. In the theoretical part, we examine the concept of structure from the perspectives of mathematics and theory of mathematics education, and present examples of mathematical problems that illustrate various aspects of structural thinking. The empirical part of the paper describes a research study aimed at analyzing solution strategies used in a task addressing aspects of structural thinking related to patterns and regularities. The task was completed by 62 pre-service mathematics teachers in 2022 and 2026. Their solutions were subjected to qualitative thematic analysis, through which the most common strategies were identified. Based on this analysis, we also propose pedagogical interventions that may contribute to improving the preparation of future mathematics teachers in the area of structural understanding. In view of the research problem and objectives outlined above, we formulated the following research questions:

RQ1: What solution strategies do pre-service mathematics teachers use when solving a task focused on identifying and generalizing a pattern?

RQ2: To what extent do their solutions demonstrate work with local relationships, and to what extent do they involve the formulation of a generally valid relationship?

RQ3: What types of justification do participants use when formulating their conclusions?

2. Theoretical Background

The concept of mathematical structure permeates modern mathematics and reaches its culmination in the work of Birkhoff and Mac Lane (1958) and in subsequent developments, including the Bourbaki project aimed at codifying and interrelating the structures of known mathematics (Beaulieu, 1990; Mashaal, 2006). The roots of the search for structure can be traced back to Euclid, through Gauss, and further through the rich development of nineteenth-century mathematics, including Peano’s axioms of arithmetic, various axiomatizations of non-Euclidean geometries, as well as the theories of groups, rings, fields, and other algebraic structures (Cohn, 1965). The essence of this modern approach lies in the identification and isolation of properties that serve as the sole foundation for mathematical reasoning, such that all derivations hold for every instance possessing these properties.

Warren (2003), in her conception of mathematical structure in the context of arithmetic, emphasizes attention to both relationships and mathematical properties (i.e., those characteristics that remain invariant within a particular class and/or define that class) in order to support the transition to algebraic thinking. According to the author, mathematical structure relates specifically to relationships between quantities (for example, whether quantities are equal, or whether one quantity is smaller or greater than another); group properties of operations (for example, whether an operation is associative and/or commutative, and whether inverse and identity elements exist); relationships between operations (for example, whether one operation is distributive over another); and relational properties (for example, the transitivity of equality and inequality).

As Gross (1978) points out, one reason for the importance of mathematical structures is that many areas that appear distinct at first glance are, in fact, governed by the same underlying rules. If we assume that we accept only those consequences that necessarily follow from a given set of rules, then whenever two different models satisfy the same rules, every consequence that is held in one model must also be a necessary consequence in the other. This provides a remarkable shortcut in the study of topics that share the same structure as areas that have already been investigated. As an example, the author presents the introduction of an area measure based on three rules:

(1)

The area of a rectangle is equal to the product of the lengths of its sides.

(2)

If one region is contained within another region, then the area of the former cannot exceed the area of the latter.

(3)

If a region is the union of pairwise disjoint parts, then the area of the entire region is equal to the sum of the areas of its individual parts.

In the study of volume, we find that, with appropriate changes in terminology, the same three rules that apply to area can also be applied to volume. In fact, the only modification required in the second and third rules is the replacement of the word “area” with “volume.” Since the rules capture relationships among concepts, preserving these relationships while changing the concepts themselves means that the underlying structure remains unchanged. As a result, many statements about volume can be proved simply by adapting the corresponding proof developed for area.

It is also important to recognize that when the rules differ, the structures differ as well. For example, when studying the length of a curve (perimeter), we must proceed with caution, because in this case it is not possible simply to adopt the same three rules by replacing the word “area” with “length.” Specifically, if we attempted to do so in rule (2), we would obtain the following statement:

(2′) If one region is contained within another region, then the length (perimeter) of the contained region cannot be greater than the length of the region that contains it.

This statement is not always true. In many cases, a region with a smaller area may have a larger perimeter, as illustrated by the following shapes (Figure 1).

The mathematical way of reasoning about structure presented above provides a natural theoretical foundation for educational interpretations of structure in mathematics education, which will be the focus of the following section.

Within the theory of mathematics education, structures and structural understanding are among the frequently studied and discussed topics (Hejný & Litter, 2007). Despite this fact, the concept of structure is often treated within the mathematics education community as if it were an undefined concept; at the same time, it is assumed that there is a general consensus on what this concept means (Kieran, 2018). One of the causes of the difficulties in defining the concept of structure more precisely is the fact that this term is often intertwined with a set of other concepts that are sometimes considered semantically distinct, but at other times are understood as synonyms. This set of concepts includes structure, relationships, generalizing, and properties. Kieran (2018) also points to a certain bias in the mathematics education literature, which more frequently focuses on the semantically narrower concept of generalizing and tends to relegate attention to structure to the background. However, the ambiguity of the concept of structure itself may be one of the reasons for this lack of attention.

According to Kieran (2018), structure emerges through “structuring activity,” which primarily refers to what she describes as the operations of “decomposing and recomposing.” We decompose a mathematical object, such as a number or an expression, into parts in order to reveal its internal structure, and then these parts can be rearranged in different ways, as in the following examples:

989 = 9 × 109 + 8 or 9 × 110 − 1

x² − 5x + 6 = (x − 2)(x − 3)

Such operations create alternative ways of representing particular arithmetic or algebraic objects. This corresponds to Venkat et al. (2019), who, among other things, state that the concept of structure has an architectural character—it refers to a spatial organization created by specific relationships that arrange a particular element or elements into specific configurations with other elements, rather than into random arrangements.

For example, the elements 6, 8, and 48 can be “structured” into the arrangement 6 × 8 = 48, but not into the arrangement 6 × 48 = 8. Structure is therefore tied to a mathematically appropriate relationship that underlies the spatial architecture, which provides a useful clarification:

structure ⇔ mathematical relationship between elements

This focus on relationships between elements is also related to determining the degree of awareness of structures, or organization into a mathematical relationship, as proposed by Mulligan and Mitchelmore (2009) in their analysis of students’ responses across a wide range of tasks. Mulligan et al. (2009) and Tapia Yagual et al. (2026) state that awareness of structure is important for mathematical representation, symbolization, abstraction, generalizing, and proving; however, they do not specify in greater detail what exactly constitutes awareness of structure. While their formulation suggests abstraction as an outcome of structural awareness, Warren (2005), by contrast, points out that abstracting patterns is the starting point of structural knowledge. Across these descriptions, local “organizing” is present, which is also central to Freudenthal’s (1991) understanding of “structuring” as “emphasizing form,” within which phenomena are organized either according to internal relationships or according to their relationships with other phenomena.

A common feature of all these definitions of the concept of structure is the perception or arrangement of elements, symbols, or images into a particular configuration that highlights mathematical relationships. This leads to an understanding of structure as something based on noticing or creating local relationships that are present within a particular case, although they may also apply across a broader class of cases. The authors in Venkat et al. (2019) refer to such structures as emergent structures.

In their description of mathematical structure, Mason et al. (2009) emphasize the need to be aware not only of local relationships but also of general properties. They define properties as internal relationships within the class of mathematical objects under consideration. The authors therefore understand mathematical structure as the identification of general properties that are realized in specific situations such as relationships between elements. These elements may be mathematical objects, such as numbers and triangles, sets, relations on sets, and even relations between relations. In this conception, structure is described as something that becomes visible through relationships between elements. Recognizing a relationship between two or more objects does not in itself constitute structural or relational thinking. Instead, the identification of general properties through the analysis of a specific realization of a relationship is required. For example, in the specific case 6 × 4 = 4 × 6, a student recognizes the general validity of the commutative law for multiplication.

Freudenthal (1983) places exceptionally high demands on the concept of mathematical structure, using it to denote the entire network of both fundamental and derived properties that can be associated with an initial relationship. For example, this involves recognizing that the relationships 2 × 3 = 6 and 3 = 6 ÷ 2 are equivalent based on the inverse nature of the operations of multiplication and division for natural numbers. According to Freudenthal (1983), structural understanding is therefore manifested in a student’s understanding of how general properties are expressed in specific cases.

Stehlíková (2004) understands mathematical structure as a general concept that encompasses both individual pieces of knowledge and their organization (for example, in terms of accessibility: some pieces of knowledge are always immediately available, e.g., 5 × 6 = 30, while others require a certain effort to recall, e.g., the axioms of rings). The main characteristic of structure is its capacity for development. The process of structuring mathematical knowledge leads to the emergence of new individual pieces of knowledge and connections, as well as to qualitative changes in existing connections.

This leads to a refinement of the concept of structure: a distinction between arranging elements into a local or emergent relationship (thereby creating an emergent structure) and arranging elements into a relationship that is more general and is understood as valid across a broader class of examples, thereby creating a mathematical structure. Thus, emergent structure involves analyzing, creating, and perceiving local relationships. Mathematical structure involves analyzing, creating, and perceiving general relationships. The relationship between these concepts is illustrated in Figure 2.

Based on the synthesis of the theoretical approaches presented above, we understand structural thinking as a process in which the problem solver creates or selects a structuring representation of a mathematical situation that highlights its essential properties, uses it to identify patterns and relationships, reorganizes mathematical objects or their representations, recognizes general properties that extend beyond the specific case, and formulates a general rule on their basis. We do not consider the individual components of this process to be strictly sequential phases, but rather interconnected aspects of structural thinking that may manifest themselves in different sequences and with varying intensity when solving a specific task. In this article, we use the term structuring representation of a mathematical situation to refer to a representation (visual, symbolic, algebraic, graphical, or other) that highlights essential mathematical relationships and creates conditions for identifying general properties and formulating a general rule. The elements of structural thinking are summarized in the following

Table 1. The elements of structural thinking.

Aspect of Structural Thinking	Characteristic	Theoretical Basis	Manifestation in Our Research
Structuring representation of the mathematical situation	Creation or selection of a representation that highlights essential mathematical properties and relationships (visualization, algebraic transformation, graph, table, diagram, etc.)	structuring activity (Kieran, 2018); organizing elements (Venkat et al., 2019); organizing phenomena (Freudenthal, 1983)	Visualization of the solid as a single rectangle, a square and a rectangle, staircases, and cross-sections
Recognition or creation of a pattern	Identification of local relationships mediated by the selected representation	emergent structure (Venkat et al., 2019; Mulligan et al., 2009)	Identifying the way the number of cubes increases
Reorganization of mathematical objects or representations	A change in the representation or arrangement of objects leading to a new perspective on the situation	decomposition/recomposition (Kieran, 2018)	Rearrangement of the solid into a rectangle or another configuration
Identification of general properties	Recognition of properties extending beyond the specific case	(Mason et al., 2009; Freudenthal, 1983)	Awareness of the general relationship between the dimensions and the number of cubes
Formulation of a general rule	Expression of a general relationship using algebraic notation	(Warren, 2005; Mason et al., 2009)	Derivation of a formula for the number of cubes for height n

.

In mathematics education, the concept of structure is not clarified primarily through formal definitions, but through the analysis of situations in which a qualitative change occurs in the way learners understand and use mathematics. The following examples, inspired by Mason et al. (2009), illustrate different aspects of structural thinking.

Example 1: Patterns and Regularities. One of the fundamental contexts in which structure emerges is that of repeating patterns and sequences, e.g., the sequence of letters A, ABB, ABBBB, ABBBBBB, … Another example is a task involving determining the number of cubes in a construction as a function of its height (Figure 3).

Students are often able to identify that “something is repeating” or that “the pattern changes according to some rule,” but this ability alone does not yet imply structural understanding. In many cases, it involves recognizing a local relationship between adjacent elements without being aware of the general principle that generates the pattern. Such situations are well captured by the concept of an emergent structure: the elements are arranged in a way that makes sense locally, but the student does not yet perceive them as a manifestation of a general property or rule. The transition to mathematical structure occurs only when attention shifts from observing individual cases to identifying and formulating a general rule, for example, expressing the relationship for the n-th term of a sequence. This example shows that structure does not emerge suddenly, but gradually, as part of the development of understanding. First, an emergent structure appears, which is subsequently transformed through generalization into an explicit mathematical structure (as illustrated in Figure 2).

Example 2: Structures in Arithmetic and Number Theory. A classic example of structural thinking in arithmetic is provided by strategies based on transformations of expressions that do not change their validity, for example, expressions of the form 47–38 = 49–40. Some students solve these tasks without explicit calculation, arguing that if the same value is added to both numbers, the difference remains unchanged. In this case, it is not a matter of recognizing a specific relationship between two numbers, but of applying a general property: the preservation of the value of the difference when both terms are changed by the same amount. This property holds for all numbers, not only for the specific example. It is precisely this awareness of generality that distinguishes structural thinking from procedural calculation.

Example 3: Missing Numbers in Equations. Tasks involving missing numbers in equations (Stephens & Wang, 2008) make it possible to distinguish different levels of structural understanding in a nuanced way (Figure 4). Some students solve these tasks using local strategies, while others treat the equation as an object and use its properties independently of the specific values. This shows that structural thinking is not a binary category (has/does not have), but a continuum along which the student gradually moves from working with specific relationships to the conscious use of general properties.

Example 4: Structure as the Basis of Proof. Another example of structural thinking is the justification of the sum of the exterior angles of a triangle, with the procedure subsequently being generalized to any polygon. Instead of empirically measuring the angles of different triangles, attention shifts to an invariant property of planar motion: the total angle of rotation when traversing a closed polygonal line (Abelson & diSessa, 1981). When drawing any triangle in “turtle” graphics, after returning to the starting point and turning to its original orientation, the turtle has performed rotations through angles whose measures add up to the sum of the exterior angles of the triangle. Since the turtle is oriented in the same direction as at the beginning, this sum is exactly 360 degrees. This reasoning can subsequently be generalized to any polygon.

Example 5: Completing the Square. The transformation of a quadratic trinomial, known as completing the square, is often understood by students as a sequence of algebraic steps. However, the general process of completing the square itself illustrates an underlying structure that makes it possible to develop the important realization that the graphs of all quadratic functions are parabolas, because their graphs are obtained through transformations of the graph of the function $y=x^2$.

The examples presented illustrate that structure does not exist as a fixed object but rather emerges gradually through mathematical activity, engagement with a variety of tasks, and reflection on their solutions. Developing an understanding of structure represents a qualitative shift in mathematical thinking. Achieving structural understanding requires the ability to generalize a mathematical situation rather than relying solely on locally valid relationships (Example 1). An understanding of structure enables students to develop conceptual understanding instead of merely mastering procedures (Example 2). It also allows them to solve problems by drawing on general properties and relationships rather than on isolated cases (Example 3). Furthermore, recognizing mathematical structures can serve as a basis for justification and proof while supporting the formulation of generalizations (Example 4). Finally, structural awareness enables students to identify invariant properties across a class of mathematical objects (Example 5).

3. Materials and Methods

In this study, we analyze prospective mathematics teachers’ solutions to a selected task (Example 1). The research was conducted at Comenius University Bratislava, Slovakia, within a teacher education program for prospective secondary school mathematics teachers. This is a five-year study program consisting of a three-year bachelor’s program and a two-year master’s program. To qualify as a mathematics teacher in Slovak schools, students must complete both the bachelor’s and master’s program. The task was completed by 13 fifth-year (final year of the master program) and 28 first-year (first year of the bachelor program) prospective mathematics teachers in 2022, and by 21 fifth-year prospective mathematics teachers in 2026. The total research sample therefore consisted of 62 prospective mathematics teachers. A purposive sampling strategy was employed to enable comparisons of strategies across the years 2022 and 2026. Specifically, the first-year students in 2022 and the fifth-year students in 2026 represent the same cohort; the difference in sample size is due to the fact that some students did not successfully progress to the fifth year of the study program. Although the first-year students surveyed in 2022 and the fifth-year students surveyed in 2026 originated from the same initial cohort, the responses were collected anonymously. It was therefore not possible to identify which students participated at both measurement points or to match their responses at the individual level. Consequently, the study does not constitute a longitudinal analysis of individual development. The comparisons are interpreted as cohort-level descriptive patterns observed in the participating groups at the respective measurement points. The design also allows for a comparison of the solution strategies of fifth-year students in 2022 and 2026. Consequently, the study enables an overall comparison of solution strategies used by students at the beginning and at the end of the teacher education program, as well as a comparison of final-year prospective mathematics teachers across the 2022 and 2026 cohorts. However, because of participant attrition and the lack of individual-level matching, these comparisons cannot be used to explicitly trace development within the study program and should be interpreted only as cohort-level descriptive patterns. The overall research design is illustrated in Figure 5.

The task itself (Example 1) formed part of a test designed to investigate functional and structural thinking in mathematics. The anonymized responses were subjected to an in-depth thematic qualitative analysis (Creswell, 2012) using the MAXQDA Plus 2022 (Release 22.8.0) software. The analysis combined inductive and deductive coding and comprised three complementary coding dimensions corresponding to the three research questions. Accordingly, although the Results section is organized by research question, each subsection presents the findings obtained through the corresponding coding dimension and its associated categories.

To address RQ1, an inductive open-coding procedure was used because the categories of solution strategies were not determined in advance. This process resulted in six categories of solution strategies: four visually based strategies (V1–V4) and two algebraic strategies based on arithmetic sequences (A1–A2). Because a participant could use more than one distinct strategy within the same response, multiple strategy codes could be assigned to a single solution. The operational definitions and inclusion criteria for these categories are provided in Appendix A.1.

The coding related to RQ2 was theory-informed and deductive. It was based on the distinction between identifying local relationships and formulating a generally valid mathematical relationship, as reflected in the theoretical framework of emergent and mathematical structure. Each response was classified according to whether the participant identified a correct local relationship and whether they formulated a correct general relationship. This produced the categories C0–C4: task not solved; incorrect local relationship without a general relationship; correct local relationship without a general relationship; correct local relationship with an incorrect general relationship; and correct local relationship with a correct general relationship. The detailed operational definitions and inclusion criteria are also presented in Appendix A.1.

The analysis addressing RQ3 was also deductive. The types of justification were coded using the analytical framework developed by Sevinç et al. (2022), as summarized in

Table 2. Ways of Reasoning (source: Sevinç et al., 2022).

Different Ways of Reasoning	Brief Description
Appeal to authority	No explanation or reasoning, e.g., Euclid, a textbook, etc., says it is so.
Simple (1-step) deductive reasoning	A single deduction from one or more premises.
Mathematising	The explanation/justification of transformation/decontextualization of a word problem/a problem defined in the real world to a strictly mathematical form.
Reasoning by analogy	Involves making a conjecture based on similarities between two cases, one well known (the source) and another usually less well understood (the target).
Reasoning with empirical arguments/specific cases (Making claims and generalizing; Justification of claim)	Reasoning begins with specific cases and produces a generalization from these cases; Testing claims using evidence from examples (sometimes just one example) of direct measurements of quantities, substitutions of specific numbers in algebraic expressions, and so forth.
Developing conclusions/justifying/refuting through deductive reasoning (Generic example; Counterexample; Systematic enumeration; Other)	Conclusions are derived from known information (premises) based on formal logic rules, where conclusions are necessarily derived from the given information and there is no need to validate them by experiments.
Other	e.g., abductive reasoning—the search for a general rule from which a specific case would follow.

. The categories from this framework were applied to the participants’ written reasoning, with subcategories used to distinguish generic-example reasoning, reasoning based on previously established knowledge, and empirical reasoning based on specific cases. Within the deductive reasoning category, we used the label “Recognizing known information” to identify justifications based on previously established mathematical knowledge. Responses containing no justification were recorded separately as “No reasoning”.

Both researchers jointly examined and coded the participants’ responses. Initial differences in interpretation were discussed by referring to the previous responses and comparing them with the definitions and boundaries of the emerging categories. Cases that did not clearly fit an existing category were discussed jointly by the two authors. The discussion focused on the structural features of the solution, its correspondence with the emerging operational definitions, and its similarity to previously coded responses. Codes and categories were assigned only after both researchers had reached consensus. The identified solution strategies were subsequently interpreted in relation to the theoretical framework of structural thinking.

4. Results

The results are presented according to the individual research questions.

4.1. RQ1: What Solution Strategies Do Pre-Service Mathematics Teachers Use When Solving a Task Focused on Identifying and Generalizing a Pattern?

The analysis of the participants’ solutions revealed six main solution strategies. Four of these were based on visual representations (V1–V4), whereas the remaining two were purely algebraic and relied on recognizing the sequence as an arithmetic progression with a known common difference (A1, A2). These strategies are described below.

Strategy V1: “Square and Rectangle”. Within this strategy, participants decomposed the solid into two shapes whose front views have dimensions $n\times n$ and $n\times (n-1)$, yielding the expression $n·n+n\left(n-1\right)$ as illustrated in Figure 6.

Strategy V2: “Tower and Four Staircases”. In this strategy, participants expressed the total number of cubes as the sum of the cubes forming the central tower and the four surrounding “staircases”, yielding the expression $n+4(1+2+\dots +\left(n-1\right))$ (Figure 7).

Strategy V3: “Single Rectangle”. Participants reorganized the solid into a single shape whose front view has dimensions $n\times \left(2n-1\right)$ (Figure 8).

Strategy V4: “Number of Cubes as the Sum of Parallel Slices”. In this strategy, participants determined the total number of cubes by decomposing the solid into parallel slices and summing the numbers of cubes in the individual slices, for example: $1+\left(4+1\right)+\left(2·4+1\right)+\dots +\left(\left(n-1\right)·4+1\right)$ or $4+2·4+3·4+\dots +\left(n-1\right)·4+n$ (Figure 9).

Strategy A1: “Number of Cubes as the Sum of an Arithmetic Sequence with a Known Common Difference”. In this strategy, participants determined the total number of cubes by expressing it as the sum of the first n terms of an arithmetic sequence, for example: $a_1=1, d=4, S_n={\displaystyle \frac{n\left(a_1+a_n\right)}{2}}={\displaystyle \frac{n·(1+\left(1+4·\left(n-1\right)\right))}{2}}$. Less frequently, they used the equivalent arithmetic sequence with $a_1=4\left(n-1\right)+1, d=-4, S_n={\displaystyle \frac{n\left(a_1+a_n\right)}{2}}={\displaystyle \frac{n·(\left(4·\left(n-1\right)+1\right)+1)}{2}}$.

Strategy A2: “Number of Cubes as the Sum of an Arithmetic Sequence Defined Recursively”. In this infrequently observed strategy, one participant expressed the total number of cubes as the sum of a recursively defined sequence: $S_n=S_{n-1}+5+4\left(n-2\right)$, with the initial condition $S_1=1$, but did not develop this expression any further.

Table 3. Frequencies of the Identified Solution Strategies.

	2022, (Year 5)	2022, (Year 1)	2026, (Year 5)	∑
Strategy V1 Square and Rectangle	0	0	2	2
Strategy V2 Tower and Four Staircases	8	19	14	41
Strategy V3 Single Rectangle	0	0	1	1
Strategy V4 Number of Cubes as the Sum of Parallel Slices	0	2	5	7
Strategy A1 Number of Cubes as the Sum of an Arithmetic Sequence with a Known Common Difference	4	5	0	9
Strategy A2 Number of Cubes as the Sum of a Recursively Defined Arithmetic Sequence	1	0	0	1
Not solved	0	2	0	2
∑	13	28	22	63

presents the frequencies of the identified solution strategies within the research sample. Since one participant in the fifth-year 2026 cohort employed two different strategies, this participant is represented twice in the table.

The frequencies presented in Table 3 indicate that strategies V2, A1, and V4 were used most frequently, whereas the remaining strategies occurred only sporadically. An interesting pattern emerged in the results of the fifth-year students assessed in 2026. Unlike the fifth-year cohort assessed in 2022, none of the students used strategies A1 or A2; instead, all of them relied on visual solution strategies. In contrast, strategy A1 was relatively common among the fifth-year students assessed in 2022. Comparing the students who were in their first year in 2022 with the same cohort in their fifth year in 2026 reveals an increase in the diversity of visual solution strategies. Whereas in 2022, these students relied predominantly on strategy V2, with only occasional use of strategy V4, by 2026 they employed all four visual strategies (V1–V4). At the same time, none of them used a solution strategy based on an arithmetic sequence. One possible, although tentative, interpretation of this difference is that it may be associated with the emphasis placed during the teacher education program on moving flexibly between multiple mathematical representations, including visual representations. Examples include the use of figurate numbers, visualizations of algebraic identities and trigonometric relationships, and visual representations of probability problems. However, the study design does not allow this association to be confirmed.

4.2. RQ2: To What Extent Do Their Solutions Demonstrate Work with Local Relationships, and to What Extent Do They Involve the Formulation of a Generally Valid Relationship?

To address RQ2, participants’ solutions were classified according to the correctness of the local relationship they identified and the general relationship they formulated. Five categories were defined: (C0) task not solved; (C1) incorrect local relationship and no formulation of a general relationship; (C2) correct local relationship and no formulation of a general relationship; (C3) correct local relationship and incorrect formulation of a general relationship; (C4) correct local relationship and correct formulation of a general relationship. No participants exhibited either an incorrect local relationship accompanied by an attempt to formulate a general relationship or a correct general relationship derived from an incorrect local relationship. Consequently, these categories are not reported. The frequencies of the remaining categories are presented in

Table 4. Frequencies of categories C0–C4.

	2022, (Year 5)	2022, (Year 1)	2026, (Year 5)	∑
(C0) Task not solved	0	2	0	2
(C1) Incorrect local relationship; no formulation of a general relationship	0	0	0	0
(C2) Correct local relationship; no formulation of a general relationship	0	2	0	2
(C3) Correct local relationship; incorrect formulation of a general relationship	0	3	4	7
(C4) Correct local relationship; correct formulation of a general relationship	13	21	17	51
∑	13	28	21	62

.

Only two participants did not attempt to solve the task. Most participants were able to formulate a relationship describing the number of cubes for the specific case of a solid of height 12, representing reasoning at the local level. Two participants correctly identified the local relationship but did not attempt to formulate a general relationship. Seven participants identified the correct local relationship but formulated an incorrect general relationship, whereas 51 participants correctly formulated both the local and the general relationship. Comparing the two fifth-year cohorts reveals an interesting difference. Whereas none of the fifth-year students assessed in 2022 formulated an incorrect general relationship, four students in the 2026 cohort did so. Of the 13 fifth-year students in 2022 who correctly formulated the general relationship, 12 expressed it as a function of the variable $n$, for example, $n+4{\displaystyle \frac{n(n-1)}{2}},$ whereas one student expressed it using the recurrence relation $S_n=S_{n-1}+5+4\left(n-2\right).$ Of the 17 fifth-year students in 2026 who correctly formulated the general relationship, 15 expressed it as a function of the variable $n$, while two expressed it in the summative form $n+4\left(1+2+\dots +\left(n-1\right)\right)$.

A particularly interesting pattern emerged for the students who were in their first year in 2022. Within this cohort, two participants did not attempt the task, two correctly identified the local relationship but did not attempt to formulate a general relationship, and three identified the correct local relationship but formulated an incorrect general relationship. Twenty-one participants correctly identified both the local and the general relationship; however, only 10 of them were able to express the general relationship as a function of the variable n, representing the height of the solid. When the same cohort was assessed again in their fifth year in 2026, the results were more favorable. Only four participants formulated an incorrect general relationship, while 15 correctly expressed the general relationship as a function of the variable n. These results indicate an improvement in the cohort’s ability to generalize and to express the resulting relationship as a function over the course of the teacher education program. One possible, although tentative, explanation is the emphasis placed during the study program on working with multiple mathematical representations, particularly visual representations. However, the observed difference may also reflect participant attrition, including the possibility that lower-performing students were less likely to remain in the cohort. The present study design does not allow these possible explanations to be distinguished.

4.3. RQ3. What Types of Justification Do Prospective Mathematics Teachers Use When Formulating Their Conclusions?

To classify the types of justification used by the participants, we adopted the analytical framework developed by Sevinç et al. (2022). The framework was originally proposed for analyzing the argumentation of prospective mathematics teachers and has also been used to analyze argumentation tasks in mathematics textbooks (Işıksal Bostan et al., 2025). The categories used in the present study are summarized in Table 2 in Materials and Methods section of our paper.

The frequencies of the different types of justification are presented in

Table 5. Frequencies of the Different Ways of Reasoning.

	2022, (Year 5)	2022, (Year 1)	2026, (Year 5)	∑
Developing conclusions through deductive reasoning—Generic example	9	22	20	51
Developing conclusions through deductive reasoning (Recognizing known information—arithmetic sequence)	3	3	0	6
Developing conclusions through deductive reasoning (Recognizing known information—recursive definition of a sequence)	1	0	0	1
Reasoning with empirical arguments/specific cases—Making claims and generalizing	0	1	0	1
No reasoning	0	2	1	3
∑	13	28	21	62

. Across all three cohorts, the dominant type of justification was Developing Conclusions through Deductive Reasoning—Generic Example, which was used by 51 of the 62 participants. The second most common type was Developing Conclusions through Deductive Reasoning—Recognizing Known Information, in which participants justified the general relationship by drawing on previously established mathematical knowledge, most commonly the properties of, or the formula for, the sum of an arithmetic sequence. The remaining types of justification occurred only sporadically. Empirical justification (Reasoning with Empirical Arguments/Specific Cases—Making Claims and Generalizing) was identified in only one solution, while three participants did not provide any justification for their solution.

The predominance of deductive forms of justification suggests that participants generally did not seek to validate the derived relationship by testing additional specific cases. Instead, they attempted to justify its general validity.

Considered together with the findings for RQ1 and RQ2, these results reveal a close relationship between the structuring representation of the mathematical situation, the formulation of a general relationship, and the way in which that relationship was justified. By structuring representation, we refer to a representation that highlights the essential mathematical relationships and provides a basis for formulating a general rule. Representations that made the underlying mathematical structure more explicit—for example, reorganizing the solid into a single rectangle or decomposing it into two generally describable parts—were more frequently associated with generic example argumentation (Balacheff, 1988). In these cases, participants did not treat the representation as an isolated example but rather as a representative of an entire class of objects, using it to justify the general validity of the derived relationship. In contrast, representations that primarily highlighted local regularities—for example, the staircases or the parallel slices of the solid—facilitated the identification of the growth pattern but more often required additional mathematical knowledge when formulating and justifying the general relationship.

These findings suggest that justification should not be viewed as a separate stage of the solution process but rather as a natural continuation of structuring and generalization. In RQ1, participants constructed structuring representations of the mathematical situation; in RQ2, they used these representations to formulate a general relationship; in RQ3, they justified its general validity. The findings further suggest that different structuring representations differ not only in the mathematical relationships they make accessible but also in the extent to which they support the formulation and subsequent justification of general relationships. These findings should, however, be interpreted in light of the nature of the task. Its strong spatial and visual character may have naturally favored generic example argumentation. Consequently, the proposed interpretation should be examined further using a broader range of tasks involving different mathematical content and different types of structuring representations.

5. Discussion

The findings related to RQ1–RQ3 can be interpreted as interconnected stages of the process of structural thinking. Participants first constructed a structuring representation of the mathematical situation, through which they organized its elements into mathematically meaningful relationships. Based on this representation, they subsequently formulated a general mathematical relationship and finally justified its validity. These findings therefore suggest that structuring, generalization, and justification should not be viewed as three independent activities but rather as mutually dependent components of a single process of mathematical thinking.

The identified solution strategies can be interpreted as different manifestations of structuring activity in the sense described by Kieran (2018). Although all participants worked on the same task, they differed in the structuring representations they constructed, or selected, as the basis for generalization. Consequently, the identified strategies should be understood primarily as alternative representations of the mathematical situation rather than as alternative solution procedures. These representations highlighted different mathematically significant relationships and consequently provided different conditions for the formulation of a general relationship.

All visual strategies (V1–V4) were based on reorganizing the original configuration, but they differed in the representations that resulted from this reorganization. Strategies V1 and V3 transformed the solid into representations (“square and rectangle” and “single rectangle”, respectively) from which the general relationship could be formulated directly. In contrast, strategies V2 and V4 represented the situation in terms of “staircases” or parallel slices of the solid. Although these representations revealed the pattern of growth, formulating a general relationship required additional mathematical knowledge, such as the properties of an arithmetic sequence or the formula for the sum of its terms.

This interpretation is consistent with the conception of structure proposed by Venkat et al. (2019), who describe structure as the organization of elements established through the relationships between them, rather than as the elements themselves or their particular form. Our findings further suggest that not all representations provide equal support for the process of generalization. Some enable an almost immediate transition to a general relationship, whereas others establish only a local organization of the situation that requires further mathematical processing before a general relationship can be formulated.

From the perspective of the theory of emergent and mathematical structure (Venkat et al., 2019), all identified solution strategies can be interpreted as manifestations of constructing emergent structure. Whether participants reorganized the geometric configuration or recognized a numerical regularity in the form of an arithmetic sequence, they first identified local relationships that enabled them to organize the situation mathematically. The resulting representations, however, differed in the extent to which they made the underlying mathematical structure, that is, the general property of the situation, accessible. Representations V1 and V3 enabled a relatively direct transition to the formulation of a general mathematical relationship, whereas strategies V2, V4, and A1 required further mathematical processing based on previously established mathematical knowledge.

Our findings therefore do not suggest that some strategies involve the construction of emergent structure whereas others involve the construction of a mathematical structure. Rather, they indicate that different representations differ in their potential for generalization because they make different mathematical relationships explicit and therefore provide different levels of support for the transition from emergent to mathematical structure.

This interpretation also extends the theoretical distinction between emergent and mathematical structure proposed by Venkat et al. (2019). Our results show that not all representations of emergent structure are equally conducive to generalization. Some representations make the general property of the mathematical situation almost immediately accessible, whereas others remain at the level of local relationships and require further mathematical processing. These findings suggest that an important aspect of structural thinking is not only the ability to construct a structuring representation of a mathematical situation but also the ability to construct or select a representation with a high potential for generalization.

The findings related to RQ3 further suggest that a similar relationship exists between the way the mathematical situation is structured and the form of justification. The dominant form of justification was the generic example, which occurred most frequently in solutions based on representations that enabled a direct transition to a general mathematical relationship. By contrast, representations that primarily highlighted local regularities, such as staircases or parallel slices of the solid, enabled participants to recognize the pattern of growth but did not, by themselves, provide sufficient support for formulating a general relationship. Consequently, participants more often relied on previously established mathematical knowledge, particularly the properties of arithmetic sequences, when justifying their generalizations. In these cases, the justification was based not directly on the representation itself but on its subsequent mathematical interpretation.

The study demonstrates that justification should not be viewed as a separate stage of problem solving but rather as a natural continuation of the processes of structuring and generalization. The structuring representation appears to play a mediating role by linking the formulation of a general mathematical relationship with its subsequent justification. The relationships among structuring representations, generalization, and justification are summarized in

Table 6. Relationships among Structuring Representations, Generalization, and Justification.

Type of Structuring Representation	Representation Highlights	Transition to a General Relationship	Basis of Justification	Potential for Generalization	Interpretation
Representation revealing the general structure directly (V1, V3)	Reorganization of the situation directly exposes the underlying mathematical relationship	Direct	Developing conclusions through deductive reasoning—Generic example	High	The representation provides direct support for both the formulation and the justification of the general relationship
Representation revealing local regularities (V2, V4)	Visualizes growth or repetition	Indirect; requires further mathematical processing.	Developing conclusions through deductive reasoning—Generic example	Medium	The representation reveals a pattern, but a general relationship emerges only after additional mathematical interpretation.
Algebraic representation (A1)	Based on an already established mathematical structure (arithmetic sequence)	Direct	Developing conclusions through deductive reasoning—Recognizing known information (arithmetic sequence)	High	Generalization is supported by activating known mathematical relationships.
Recursive representation (A2)	Describes the situation through a recursive growth rule	Indirect	Developing conclusions through deductive reasoning—Recognizing known information (recursive definition of a sequence)	Low	The recursive representation captures the growth process but does not itself provide an explicit general relationship

.

A further observation emerged from the comparison of the 2022 and 2026 cohorts. In the more recent cohort, solution strategies based on constructing alternative representations of the mathematical situation occurred more frequently, whereas strategies relying on the direct use of established mathematical structures became less common. Given the size of the research sample, however, this finding should not be interpreted as evidence of a developmental trend. Rather, it suggests the hypothesis that, with increasing mathematical experience, prospective mathematics teachers may shift from recognizing familiar mathematical structures towards constructing their own structuring representations. Testing this hypothesis will require further longitudinal research.

The findings should also be interpreted in light of the characteristics of the task. Its pronounced spatial and visual nature may have naturally encouraged both the construction of visual representations and the use of generic example justifications. Future research should therefore examine whether similar relationships among structuring representations, generalization, and justification can also be observed in tasks from other areas of mathematics.

6. Conclusions

The aim of this study was to examine how prospective mathematics teachers progress from identifying local relationships to formulating a general mathematical relationship when solving a task designed to elicit structural thinking.

The results show that structuring representations differ not only in the mathematical relationships they make explicit but also in the extent to which they support the transition from locally identified regularities to the formulation and justification of a general mathematical relationship. In this respect, the study extends existing theoretical perspectives on structuring activity and the relationship between emergent and mathematical structure by highlighting the differing potential of structuring representations to support generalization. The findings further suggest that more advanced structural thinking may be reflected not only in the ability to construct a representation that reveals the underlying structure of a mathematical situation but also in the ability to construct or select representations that provide more effective support for generalization and mathematical justification.

From an educational perspective, the results indicate that the development of structural thinking cannot be reduced to recognizing regularities or identifying patterns. Equally important is providing opportunities for learners to construct, compare, and critically evaluate structuring representations of mathematical situations. Consequently, teachers’ attention should not be directed solely towards the correctness of the final solution but also towards how students represented the situation, which mathematical relationships their chosen representations made explicit, and how these representations influenced their subsequent processes of generalization and justification. These findings should also be taken into account in the preparation of prospective mathematics teachers.

The study is limited by the size of the research sample and the specific nature of the task employed. Another limitation of the study is that the coding was conducted jointly and did not include an independent coding phase or the calculation of an inter-rater agreement statistic. Although all codes and categories were established through consensus between both researchers, this approach does not provide a quantitative measure of coding reliability and may have limited the identification of alternative interpretations. Another limitation linked to the research design is that the first-year participants surveyed in 2022 and the fifth-year participants surveyed in 2026 originated from the same initial cohort, but their anonymized responses could not be matched across the two measurement points. The study therefore cannot be considered longitudinal, and individual developmental trajectories could not be examined. Moreover, the number of participants decreased from 28 to 21. It is not known whether the students who participated in 2026 differed systematically from those who did not, creating a potential attrition bias. Accordingly, the reported differences should be interpreted as cohort-level descriptive patterns rather than evidence of development resulting from the teacher education program. The relatively small and context-specific sample also precludes causal conclusions or generalization to the wider population of Slovak prospective mathematics teachers. Future research should therefore examine whether the identified relationships can be confirmed using a larger sample of prospective mathematics teachers and a wider range of mathematical tasks. A particularly promising direction for future research is to investigate in greater detail and using a more rigorous coding methodology, how different types of structuring representations influence the process of generalization, the choice of justification strategies, and the development of structural thinking.