Abstract
Background: The transition from arithmetic to algebra remains a persistent challenge in early mathematics education. While structural awareness is widely recognised as foundational to this shift, the developmental mechanism through which learners come to operate on mathematical structure remains insufficiently theorised.
Aim: This article advances the Structural Extension Spine as a theoretically derived model explaining how recursive numerical activity is progressively reorganised into structurally generalised reasoning, thereby accounting for the emergence of early algebra prior to formal symbolism.
Setting: Six Grade 2 teachers in public schools participated in a series of three workshops foregrounding the use of structure in the teaching of patterns.
Methods: Adopting a theory-generative interpretive orientation, the study emerged through disciplined analytic engagement with classroom practice within a Foundation Phase professional learning initiative in South Africa. Document analysis of curriculum materials, workshop activity and classroom observations functioned as theoretically generative sites revealing persistent tensions between recursive participation and structurally oriented reasoning.
Results: The analysis renders visible a developmental reorganisation in which mathematical activity shifts from succession to relation to generality.
Conclusion: The resulting six-phase Structural Extension Spine models the progressive availability of structure as an object of thought, culminating in reasoning capable of sustaining algebraic generalisation.
Contribution: By theorising how learners come to operate on invariant relationships, the Structural Extension Spine offers a principled developmental account of algebraic emergence and repositions structure as the primary object of mathematical reasoning.
Keywords: structural extension; number patterns; sequencing; early algebra; relational understanding; structural reasoning; generalisation; covariance; teacher pedagogy.
Introduction
The transition from arithmetic to algebra remains one of the most enduring challenges in mathematics education. While research consistently highlights the importance of structural awareness in supporting this shift, the developmental mechanism through which learners begin to operate on mathematical structure remains insufficiently theorised. This article advances the position that the origins of algebraic thinking lie not in the adoption of symbolic notation but in a cognitive reorientation towards invariant relationships that reorganise mathematical activity.
From this perspective, early algebra is best understood as an emergent property of structurally oriented reasoning. Symbols stabilise relationships that learners have already begun to coordinate; they do not themselves produce algebraic thought. The decisive shift is therefore structural rather than symbolic: when learners operate on relationships that remain invariant across instances, mathematical activity becomes relationally organised rather than procedurally driven.
Accordingly, this article introduces the Structural Extension Spine (SES) as a theoretically grounded developmental model explaining how recursive numerical engagement is progressively reorganised into structurally generalised reasoning. By modelling how learners come to operate on systemic structure, the framework positions early algebra as the cognitive outcome of structural generalisation rather than the product of symbolic instruction.
Conceptual framework
Algebraic reasoning as structural reorganisation
The emergence of algebraic reasoning is conceptualised here not as a symbolic transition but as a structural reorganisation in how learners engage with mathematical relationships. Rather than locating algebra in the adoption of formal notation, this article proceeds from the premise that algebraic thinking originates in learners’ developing capacity to discern, coordinate and operate on invariant relationships across numerical instances.
The SES is advanced as a theoretical lens for modelling this developmental reorganisation. Structural extension refers to the process through which recursive numerical engagement is transformed into relational reasoning grounded in systemic invariance. In modelling qualitative shifts in the use of structure – from perceiving regularity to coordinating relationships that sustain generalisation – the framework moves beyond accounts centred on procedural progression. Structure is thus repositioned from an implicit feature of activity to an explicit object of reasoning, establishing the cognitive conditions for the emergence of early algebra.
Viewed in this way, the arithmetic–algebra transition is developmental rather than curricular. What reorganises is not merely the symbolic register of mathematics but the organisation of attention itself: learners increasingly orient towards relationships that remain stable amid variation. The SES, therefore, provides an analytic architecture for interpreting how mathematical activity is reorganised from recursive participation towards operating on systemic structure.
Structuralist foundations of mathematical activity
A structuralist orientation provides the theoretical grounding for this position. Freudenthal (1986) characterised mathematics as the activity of structuring reality, positioning patterns as epistemic tools through which underlying relationships are discerned and articulated. Progress in mathematics, therefore, depends less on computational fluency than on a reorganisation of attention towards relational structure.
Extensive research reinforces the centrality of structure, pattern and generalisation in supporting this shift, cautioning against instructional approaches that privilege procedural execution detached from conceptual networks (Du Plessis 2018, 2025; Mason, Stephens & Watson 2009; Mulligan et al. 2008; Mulligan et al. 2012, 2013). Structure, in this sense, constitutes the very object of mathematical activity (Mason et al. 2009). Mathematical development thus entails a movement from attending to successive calculations towards discerning general properties instantiated across cases – a transition resonant with Sfard’s (1991) shift from operational to structural conceptions of mathematical ideas.
Empirical work in early mathematics positions awareness of pattern and structure as a foundational developmental pathway rather than an enrichment trajectory (Mulligan & Mitchelmore 2009). Structural thinking emerges as learners recognise regularities, abstract relational features, and progressively coordinate these into integrated systems of reasoning (Mulligan et al. 2004). Such coordination supports movement beyond recursive continuation towards reasoning capable of sustaining generalisation – the cognitive condition from which algebraic thinking becomes possible (Blanton & Kaput 2011; Radford 2014).
This literature does not dismiss procedural knowledge; rather, it challenges pedagogies in which procedures become detached from the relational structures that confer mathematical meaning. Conceptual and procedural knowledge develop most productively when mutually reinforcing, supporting explanation, transfer, and adaptive expertise (National Research Council 2001; Rittle-Johnson & Alibali 1999). When arithmetic is experienced primarily as rule execution, learners may generate correct answers without apprehending the invariants that sustain generalisation.
Disrupting counting-based participation, therefore, requires more than increasing task difficulty; it necessitates a pedagogical reorientation that foregrounds relationships, invariance and covariation as objects of attention. Instruction that presses learners to articulate general rules, reason functionally and justify structural shortcuts reorganises mathematical activity from surface continuation towards relational organisation (Warren & Cooper 2008). Within such environments, structure becomes not merely noticed but operated upon – signalling a qualitative transformation in mathematical engagement.
Collectively, structuralist scholarship converges on a central claim: algebraic reasoning emerges when learners begin to treat structure as an object of thought. The SES is advanced precisely to model this developmental reorganisation.
Cognitive–semiotic mediation of structural awareness
While structuralist perspectives clarify the object of mathematical attention, a cognitive–semiotic lens explains the mechanisms through which such attention becomes possible. Radford (2003, 2014) argues that mathematical reasoning emerges through the intertwined activity of cognition and semiotics, wherein relationships are simultaneously constructed and expressed through language, gesture, symbols and diagrams. Semiotic activity stabilises relationships across representational forms, rendering them available for reflection and deliberate manipulation.
Working structurally, therefore, entails more than identifying regularities; it involves learning to signify relationships in ways that support generalisation. As learners increasingly externalise relationships through inscriptions, diagrams and linguistic formulations, structure becomes progressively objectified – available for intentional reasoning rather than tacit recognition. Semiotic mediation thus functions as a developmental mechanism through which structural awareness is consolidated and extended.
Relational understanding and relational reasoning
Skemp’s (1976) distinction between instrumental understanding – knowing the rules – and relational understanding – knowing why they work – remains foundational for interpreting this reorganisation. Relational understanding provides the connected conceptual network necessary for explanation, transfer, and justification, enabling mathematical activity to move beyond execution towards sense-making.
Relational reasoning, by contrast, is a cognitive act involving the discernment and coordination of relationships to organise mathematical information into generalised forms (Kaput 2008; English 2016). In pattern contexts, such reasoning becomes visible when learners explain growth, predict distant terms or extend rules beyond immediate cases (Carpenter, Franke & Levi 2003). Crucially, relational reasoning depends upon prior relational understanding; without a coherent conceptual network, reasoning risks remaining procedural.
When instruction deliberately cultivates relational understanding, it creates conditions under which relational reasoning can emerge. Structural extension may therefore be understood as the developmental convergence of these processes – a qualitative shift in which learners begin to operate on relationships rather than successive values, reorganising mathematical activity around invariant structure.
Theoretical synthesis and analytic implications
Taken together, structuralist, cognitive–semiotic and relational perspectives point towards the necessity of a developmental account capable of explaining how structure becomes progressively available for reasoning. Structuralism clarifies the object of attention, semiotics explains its stabilisation, and relational theory accounts for the conceptual coherence required to operate upon it.
The SES arises from this synthesis as a theoretical articulation of the developmental reorganisation through which learners come to operate on mathematical structure. Rather than introducing a novel progression, the framework renders explicit a trajectory already implied within these traditions – from recursive engagement with numerical instances towards structurally generalised reasoning.
Accordingly, the framework shifts analytic focus from task completion to the evolving use of structure as the primary indicator of mathematical development. Algebraic reasoning is thus conceptualised not as the consequence of symbolic instruction, but as the outcome of a developmental shift in which structure becomes an object of thought.
Table 1 summarises the analytic distinction between relational understanding as a conceptual condition and relational reasoning as its enactment in mathematical activity.
| TABLE 1: Key differences between relational understanding and reasoning compared over four dimensions. |
The SES is therefore advanced as a theoretical contribution to mathematics education. It explains how learners come to operate on mathematical structure, thereby providing a developmental account of the emergence of early algebra.
Research methods and design
This study adopted a theory-generative interpretive orientation in which conceptual development arose through disciplined analytic engagement with classroom practice. Rather than reporting teacher outcomes, the inquiry interrogated the structural affordances of patterning tasks and the pedagogical conditions under which structurally oriented reasoning may become available. The SES is therefore advanced as an analytic construct emerging from the productive interplay between theory and practice.
The research was situated within a professional learning initiative involving six Foundation Phase teachers in the Ekurhuleni district in South Africa. Document analysis focused on classroom resources – including materials from Department of Basic Education and teacher-selected texts – to examine the forms of patterning constituted for instruction and the extent to which these privileged recursive activity or supported engagement with invariant relationships capable of sustaining generalisation. Workshops introduced mathematical structure as an organising principle for reasoning about patterns, with classroom observations attending to subsequent shifts in pedagogical selection and task enactment.
Although these engagements informed the analytic process, teacher data are not presented here; rather, they functioned as theoretically generative sites revealing persistent tensions between counting-based participation and structurally oriented reasoning. Analysis attended to relational features, cognitive demand, structural use, growth regularity and task design. The recurring disjunction between latent structure and enacted practice rendered visible the need for a developmental account of learners’ movement from surface regularities towards systemic structure, from which the SES emerged as an empirically grounded contribution to theory building.
The formulation of the structural extension spine
The SES (see Figure 1) was formulated to account for the developmental reorganisation through which learners’ mathematical activity becomes structurally oriented. It responds to a longstanding theoretical problem in mathematics education: how to explain the transition from recursive numerical participation to reasoning grounded in invariant relationships without reducing this shift to symbolic acquisition or curricular sequencing.
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FIGURE 1: The Structural Extension Spine phases and interrelated analytical dimensions. |
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Structural extension is conceptualised here as a mode of mathematical engagement in which learners increasingly recognise, coordinate and operate on both surface and systemic structures, orienting their activity towards the formation of generalised mathematical objects (Du Plessis 2025). Converging lines of research position such structural reasoning as developmental rather than incidental, emerging as learners move from perceptual encounters with regularity towards integrated forms of relational coordination (Mulligan & Mitchelmore 2009). Structural awareness is therefore not an ancillary outcome of learning but a constitutive condition for mathematical advancement, warranting explicit theoretical articulation (Lenz 2022).
Within this perspective, structural extension denotes engagement with patterns in ways that expose, stabilise, and elaborate their relational organisation. It involves identifying invariance, coordinating co-varying quantities and expressing these relations in forms that hold beyond particular instances. When such coordination becomes possible, mathematical activity is reorganised: operations, representations and outcomes are no longer loosely connected but form a coherent relational system capable of sustaining generalisation (Gared 2023; Mulligan et al. 2008). Structural extension thus marks a qualitative transformation in the nature of mathematical engagement – structure shifts from being tacitly perceived to becoming available for deliberate manipulation.
The formulation of the SES is situated within, yet analytically extends, established accounts of structural development. Mulligan and colleagues’ Awareness of Mathematical Pattern and Structure (AMPS) framework, for example, describes learners’ increasing structural sophistication (Mulligan et al. 2013; Mulligan & Mitchelmore 2009). The present model builds on this insight by theorising the reorganisation of mathematical activity itself. Where AMPS characterises degrees of structural awareness, the SES explains how structure becomes progressively constituted as an object of reasoning.
This theoretical positioning resonates with semiotic accounts of algebraic generalisation, which describe learners’ movement from noticing regularities to articulating generality through coordinated linguistic, gestural, and inscriptional activity (Radford 2006, 2010), as well as with relational perspectives emphasising the conceptual networks that support explanation and justification (Hiebert & Carpenter 1992). Emerging scholarship further indicates that learning environments can enhance structural–mathematical thinking (Semenets et al. 2024), reinforcing the legitimacy of modelling such development at a theoretical level.
Taken together, these traditions converge on the need for an analytic account capable of explaining how structure becomes progressively available for reasoning. The SES arises from this convergence not as an instructional proposal, but as a theoretically derived model that renders explicit a developmental trajectory already implicit across structuralist, semiotic and relational scholarship – a trajectory through which recursive engagement is reorganised into structurally generalised reasoning.
Importantly, the Spine should not be understood as prescribing stages of learning nor as classifying learners. Rather, it provides an analytic architecture for interpreting transformations in the organisation of mathematical activity. In this sense, the model aligns with a theory-generative orientation: it emerged through analytically productive encounters between established theory and classroom practice and is advanced here as a conceptual lens for explaining the conditions under which algebraic reasoning becomes possible.
Analytic architecture of the structural extension spine
The SES is structured through four interrelated analytic dimensions – relational attributes, intended mathematical work, cognitive activity and use of structure – which together function as indicators of developmental reorganisation.
The phases comprise:
- Phase 0: Perceptual recognition grounded in counting.
- Phase 1: Recognition and extension within constrained numerical ranges.
- Phase 2: Recognition and extension across expanded numerical ranges.
- Phase 3: Coordination of surface features with systemic structure.
- Phase 4: Calculation through coordinated structural relations.
- Phase 5: Structural reasoning oriented towards generalised mathematical objects.
Developmental reorganisation across the structural extension spine
The SES is articulated through six analytically distinguishable phases that render visible transformations in the organisation of mathematical activity. These phases should not be read as learner stages or classificatory categories; rather, they function as theoretical distinctions that explain how structure becomes progressively available for reasoning as attention shifts from successive calculation towards invariant relationships.
Across the Spine, what reorganises is not task complexity but the object of mathematical attention. Mathematical activity becomes increasingly stabilised around relational structure rather than recursive production, enabling a developmental movement from participation oriented towards succession to reasoning organised around generality.
Phases 0–2 describe forms of participation in which structure supports action without yet becoming an object of reasoning. Activity remains temporally organised – proceeding stepwise from term to term – and is governed primarily by recursive production.
Phase 0: Perceptual participation through counting
Engagement is grounded in perceptual recognition supported by skip-counting routines. Sequences typically provide consecutive terms that invite recall-based continuation, and responses are evaluated primarily in terms of correctness. Structure remains implicit: although present in the sequence, it is not taken up as an object of attention. Activity concludes once missing terms are produced, signalling completion rather than conceptual advancement. Mathematical participation is therefore oriented towards succession rather than interpretation.
Phase 1: Recursive extension through surface regularity
The first analytic shift occurs as learners explicitly identify the repeating difference and apply it recursively to extend the sequence. Structure is now accessed but treated procedurally; the growth relation functions as a rule for continuation rather than as a relationship to be examined. Mathematical activity remains focused on production, and structural extension is neither required nor strongly afforded. Structure is thus used but not yet conceptualised.
Phase 2: Stabilised recursion across numerical range
Recursive reasoning becomes stabilised under expanded numerical conditions, reinforcing reliance on the growth relationship while further reducing perceptual strategies. Learners demonstrate procedural fluency with successive differences, yet activity remains oriented towards term generation rather than relational interpretation. Structure is operationalised but not interrogated, and mathematical activity continues to unfold sequentially rather than structurally. Table 2 shows the typical activity that can be classified Procedural Participation at the three phases.
| TABLE 2: Sample phase 0, 1 and 2 growth pattern activities. |
Taken together, these phases describe a domain of procedural participation in which surface structure enables action but does not yet reorganise reasoning.
Phase 3: Coordinating surface and systemic structure
Phase 3, the pivot point, marks a transitional reorganisation in which learners begin to coordinate surface features with emerging systemic structure. A consecutive pair of terms affords calculation of the growth quantity, which can then be used to evaluate the consistency of non-consecutive terms. Mathematical activity increasingly involves verification rather than mere continuation.
Although checking remains more strongly afforded than generative reasoning, relationships begin to orient activity. Learners encounter the possibility that sequences are governed not by successive terms but by invariant structures. Phase 3 therefore functions as a conceptual bridge: structure shifts from being operational to becoming interpretable. Table 3 illustrates phase 3 activities.
| TABLE 3: Sample phase 3 growth pattern activities. |
Phase 4: Operating on coordinated structural relations
At Phase 4, mathematical activity is reorganised around explicit engagement with structural relationships. The absence of consecutive terms disrupts recursive strategies and compels learners to coordinate surface and systemic features to determine growth and traverse numerical gaps efficiently.
Reasoning becomes increasingly motivated by relationships rather than succession. While recursive trials remain possible, they are rendered mathematically inefficient relative to structurally grounded approaches. Attention is directed towards invariance, and calculation is guided by relational coherence. Table 4 illustrates phase 4 activities, and Table 5 illustrates phase 4 solution strategies that access relational structural features. The shift enacted here is decisive: structure is no longer merely applied – it is actively operated upon.
| TABLE 4: Sample phase 4 growth pattern activities. |
| TABLE 5: Possible solution strategies for phase 4 activities, using the structural attributes embedded in the sequence. |
Phase 5: Structural generalisation and the formation of mathematical objects
Phase 5 marks the point at which structure becomes fully constituted as an object of reasoning. Activity moves beyond completing sequences towards constructing and operating on a generalised mathematical form.
Surface and systemic relations are coordinated covariantly, linking growth, starting value and term position within a unified relational system capable of generating any term. Importantly, symbolic notation is not a prerequisite; generality may be expressed linguistically, diagrammatically or algebraically.
Any activity situated in earlier phases may function at Phase 5 when extended towards generalisation. Mathematical activity is thus reorganised around the general rather than the particular.
Representational tools such as diagrams, spatial configurations and input–output tables often support this work by rendering invariance visible and tracking covariation. Tasks operating across the grain (Watson 2000) enable reasoning beyond the given sequence, positioning generality as the organising principle of activity.
Phase 5 therefore represents structural reasoning in its fullest sense: learners operate with structure rather than on sequences.
Table 6 and Table 7 illustrate activities that work with structure to work beyond the perceptual and generalise ideas.
| TABLE 6: Sample phase 5 activities that introduces the functional notion of covariance. |
| TABLE 7: Foregrounding the structural invariants – Starting point and growth. |
The generalised object can be described in words: (starting number + growth number x the term number).
Conceptual shifts across the phases
Phases 0–5 articulate a developmental reorganisation from instrumental engagement with number sequences towards relational and structurally oriented reasoning. Phase 0 aligns with Skemp’s (1976) notion of instrumental understanding, as sequences are completed through recall of skip-counting procedures with correctness prioritised over explanation. Phases 1 and 2 remain predominantly instrumental: surface growth regularities are identified and applied recursively, with distinctions arising from numerical range rather than relational complexity. Phase 3 marks a transitional shift in which a consecutive pair affords calculation of the growth quantity and its use in checking consistency across the sequence, signalling emergent attention to structure without yet operating on it generatively. Phase 4 represents a substantive move towards relational understanding, as surface and systemic structures are coordinated to calculate growth and traverse non-consecutive terms, reorganising activity from verification towards structurally motivated reasoning. Phase 5 reflects fully developed structural awareness (Mulligan & Mitchelmore 2009), where growth, position and starting value are coordinated covariantly to form a generalised object capable of generating any term. At this point, mathematical activity operates with structure rather than on sequences, establishing the cognitive conditions for early algebraic reasoning through structural extension.
Developmental coherence across the spine
The phases describe a developmental reorganisation in which mathematical activity shifts from succession to relation to generality. What differentiates the phases is not difficulty, numerical range or task format, but the evolving availability of structure as an object of thought. The SES thus provides an analytic account of how learners come to operate on invariant relationships – offering a theoretical explanation for the emergence of algebraic reasoning prior to formal symbolism.
These shifts trace a theoretically coherent movement: instrumental participation then operational structure progressing to relational interpretation then structural operation and finally objectified generality.
Conclusion
This article advances the SES as a theoretical model for explaining how mathematical activity is reorganised from recursive participation towards operating on invariant structure. Rather than positioning algebra as the consequence of symbolic instruction, the framework conceptualises algebraic reasoning as emerging through a developmental shift in which structure becomes the primary object of thought.
By rendering analytically visible the progressive availability of structure for reasoning, the Spine contributes to longstanding theoretical efforts to account for the arithmetic–algebra transition. It offers an interpretive architecture for examining the conditions under which generalisation becomes possible and provides a language for describing transformations in the organisation of mathematical activity.
Importantly, the framework is not intended as a prescriptive progression nor as a classificatory scheme for learners. Its significance lies in its explanatory power: it makes explicit a developmental trajectory already implicit across structuralist, semiotic, and relational traditions.
Future research may examine how instructional environments mediate learners’ movement across forms of structural engagement and how pedagogical practices might support the reorganisation of attention towards invariant relationships. In this way, the SES opens new theoretical and empirical pathways for investigating the emergence of early algebra.
Acknowledgements
This article is based on research originally conducted as part of Jacques D. du Plessis’ doctoral thesis titled ‘Number Pattern: Developing a sense of structure with primary school teachers’, submitted to the Faculty of Humanities, School of Education, University of the Witwatersrand, Johannesburg, South Africa in 2017. The thesis was supervised by Professor Karin Brodie. The supervisor wase not involved in the preparation of this article and was not listed as co-author. Portions of the data, analysis, and discussion have been revised, updated, and adapted for publication as a journal article. The original thesis is publicly available at: http://hdl.handle.net/10539/23252. The author affirms that this article complies with ethical standards for secondary publication, and appropriate acknowledgement has been made of the original work.
The author would like to thank Karin Brodie for their guidance and supervision during the original research conducted as part of the doctoral thesis, which served as the basis for this article. The authors acknowledge that Karin Brodie is not listed as a co-author of this article and confirm that the supervisor(s) had no objection to this arrangement.
Competing interests
The author, Jacques D. du Plessis, declares that no financial or personal relationships inappropriately influenced the writing of this article.
CRediT authorship contribution
Jacques D. du Plessis: Conceptualisation, Methodology, Formal analysis, Investigation, Writing – original draft, Visualisation, Project administration, Software, Data curation, Resources, Writing – review & editing and funding acquisition. The author confirms that this work is entirely their own, has reviewed the article, approved the final version for submission and publication, and takes full responsibility for the integrity of its findings.
Ethical considerations
Ethical clearance to conduct this study was obtained from the University of the Witwatersrand, Ethics Committee in Education of the Faculty of Humanities (No. 2011ECE153).
Funding information
This work forms part of a research study supported in part by the National Research Foundation of South Africa (Unique Grant no: 98246).
Data availability
The data that support the findings of this study are not openly available because of reasons of sensitivity and are available from the corresponding author, Jacques D. du Plessis, upon reasonable request.
Disclaimer
The views and opinions expressed in this article are those of the authors and are the product of professional research. They do not necessarily reflect the official policy or position of any affiliated institution, funder, agency or that of the publisher. The authors are responsible for this article’s results, findings and content.
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