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.

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.

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.

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.

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.

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.

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.

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.

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:

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.

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.

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.