From Theory to Practice, and Back: Student Evidence Testing ZPD, APOS, CLT, and Constructivism in Mathematical Thinking Workshops

Abstract

University mathematics-support programs rarely test their theoretical foundations against student evidence, particularly in the Global South. This study addresses that gap by analyzing how students’ experiences in Mathematical Thinking Workshops (MTWs) at a South African university confirm, nuance, or challenge assumptions the Zone of Proximal Development (ZPD), Action–Process–Object–Schema (APOS) theory, Cognitive Load Theory (CLT), and constructivism. We conducted a qualitative secondary analysis of six focus-group interviews (n = 17), using abductive reflexive thematic analysis and an Assumption–Indicator–Evidence matrix that linked design rationales to student narratives. Student accounts strongly supported ZPD, with facilitation and peer norms fostering psychological safety and risk-taking, while also showing that equitable participation required explicit role-rotation routines. APOS-informed task sequencing enabled coordination across representations but operated recursively, with students calling for planned revisiting sessions to consolidate difficult ideas. CLT claims were affirmed where venue conditions and timing inflated extraneous load, highlighting the need for short debriefs and load-aware logistics. Constructivist activity fostered belonging, confidence, and more social views of mathematics but generated uncertainty when tasks ended without brief closure. We conclude by proposing context-aware refinements to these frameworks and outlining a replicable routine for testing educational theory through student evidence.

1. Introduction

1.1. Context and Problem

In many educational systems, the transition into first-year university mathematics is abrupt: students are expected to pivot from largely procedural, example-driven school mathematics to a discipline organized around definition, abstraction, generalization, and proof (Gueudet & Bosch, 2017; Rach & Ufer, 2020). In South Africa, this challenge is amplified by uneven prior preparation, multilingual classrooms, and material constraints (Jojo, 2019; Durand-Guerrier et al., 2015; Sedibe, 2011). Over the past three decades, universities have introduced supplemental instruction programs, peer-assisted study sessions, mathematics support centers, foundation and extended curriculum offerings, inquiry-oriented and workshop-style mathematics interventions to bridge these mismatches between school preparation and university expectations (Arendale, 1993; Beckley et al., 2015; Mouton & Rewitzky, 2024; Rasmussen et al., 2006).

Yet, despite the proliferation of all these offerings, many of these interventions are reported in a practice-first, outcomes-orientated way, and only a minority make their theoretical foundations explicit (Boughey, 2010; Dawson et al., 2014). Even fewer interrogate whether the constructs they invoke, for example, scaffolding, collaborative knowledge-building, or social constructivism, are actually visible in students’ own accounts of participation and learning, and this is especially true in African and South African contexts where student-voice studies of support programs remain comparatively scarce (Clarence, 2016; Leibowitz et al., 2009; Mac an Bhaird et al., 2020).

1.2. Aim and Contribution

This paper addresses that gap by completing the cycle from theory to practice to theory for a set of Mathematical Thinking Workshops (MTWs) implemented at a research-intensive South African university (Mokhithi et al., 2025). Our central aim is to ask whether the four frameworks that guided the design, the Zone of Proximal Development (ZPD) (Vygotsky, 1978), Action-Process-Object-Schema (APOS) theory (Asiala et al., 1996), Cognitive Load Theory (CLT) (Sweller, 1988), and constructivism (von Glasersfeld, 1995), can be seen in what students say about participation and learning, and where student evidence in fact pushes back against the theory. The contribution is twofold: empirically, we offer evidence-backed refinements to widely used frameworks in an under-represented context; methodologically, we demonstrate an assumption-to-indicator approach to theory-testing in program evaluation that can be used across different contexts.

1.3. Theoretical Orientation of the MTWs

The MTWs were intentionally designed using a composite of theoretical frameworks. However, the four frameworks did not exhaustively determine every design choice; we also drew on practitioner experience, accumulated tuition in first-year mathematics courses, and contextual judgement from working with this and similar cohorts. From ZPD (Vygotsky, 1978; Wood et al., 1976), the design foregrounded structured peer collaboration, calibrated facilitation, and planned scaffolding, which gradually faded as students developed greater independence. From APOS (Asiala et al., 1996), it sequenced tasks to support movement from actions on objects to internalized processes, reified objects, and interconnected schemas. From CLT (Sweller, 1988), it aimed to manage intrinsic load, reduce extraneous load, and create conditions for germane load through worked examples, prompts, and consolidation. From constructivism (von Glasersfeld, 1995), the design emphasized interactive tasks, social meaning-making, and reflection. Our earlier paper from the same project (Mokhithi et al., 2025) documented students’ affective and identity-related gains, but it did not center the evaluation of these theoretical claims.

1.4. Analytical Approach and Expectations

Guided by design-based research perspectives, we treat theory as most useful when its claims are operationalized into assumptions that generate observable consequences in specific settings (Cobb et al., 2003; McKenney & Reeves, 2019; Sandoval, 2014). We therefore undertook a secondary analysis of six focus-group interviews with workshop participants (n = 17). We used an abductive, reflexive thematic analysis (Braun & Clarke, 2019, 2023; Timmermans & Tavory, 2022; Thompson, 2022) that moved between (a) a deductive codebook derived from the four frameworks, cast as an assumption–indicator structure to make claims testable, and (b) inductive openness to disconfirming cases and to previously under-articulated mechanisms that the frameworks background or omit.

We expected to find evidence of ZPD-type expansion of participation and of APOS-style shifts from action to process, and we anticipated disconfirming cases where CLT assumptions were challenged by venue and resource constraints. Findings informed context-aware refinements to the workshop design, especially around equitable participation and cognitive-load management in multilingual, resource-constrained settings.

1.5. Research Questions

The study is guided by the following research questions:

• To what extent do students’ accounts of participation and learning in the MTWs confirm or challenge core assumptions from ZPD, APOS, CLT, and constructivism?
• Where do the frameworks require extension or qualification to accommodate contextual features such as multilingual interaction, uneven prior preparation, and environmental constraints?
• What design implications follow for subsequent iterations of the workshops?

2. Materials and Methods

2.1. Study Design and Intervention Context

This study is a secondary qualitative analysis of semi-structured focus-group interviews generated within a larger, convergent-parallel mixed-methods case study of Mathematical Thinking Workshops (MTWs). The parent project collected quantitative and qualitative data in the same academic year to examine the effects of MTWs on learning and engagement; the present paper analyzes only the qualitative strand to test theoretical claims against student accounts.

As described in Section 1.3, four theoretical frameworks provided the primary design logic, but decisions were also shaped by the team’s accumulated practitioner experience and tuition in first-year mathematics, as well as the practical constraints of rooms, time, and cohort size. A detailed account of how ZPD, APOS, CLT, and constructivism informed the workshop design is provided in our earlier work (Mokhithi et al., 2025). For readers’ convenience, an abridged comparative summary of the theoretical framework in relation to the MTWs, with constructivism providing the overarching integrative frame, is included in Appendix A (

Table A1. Comparative summary of ZPD, Cognitive Load Theory, and APOS Theory in relation to Mathematical Thinking Workshops (adopted from Mokhithi et al., 2025).

Category	ZPD	CLT	APOS Theory
Primary Focus	Socially mediated learning through support	Mental efficiency during learning	Cognitive development in mathematical thinking
Schema Acquisition	Occurs via social interaction and scaffolding/mediation within the ZPD	Optimized by managing intrinsic, extraneous, and germane load	Emerges through progression from actions to processes to schemas
Role of the Learner	Active participant in a social context	Cognitive processor whose capacity must be managed	The learner progresses through internal mental constructions
Role of the Facilitator/Tutor	Provides scaffolding/mediation to bridge gaps in understanding	Designs tasks that avoid overload and promote schema construction	Designs activities that trigger transitions through APOS phases
Nature of Support	External and temporary; tailored to learner(s) readiness	Instructional design-driven; reduces unnecessary demands	Internalized through mathematical activity and reflection
Relevance to Workshops	Supports collaborative and peer-facilitated discussions	Informs task design to manage cognitive demand	Frames cognitive growth in mathematical understanding
Tensions/ Challenges in applying the theory	Identifying the learner(s)’s current zone and ensuring the support provided is neither overwhelming nor under-stimulating the learner(s).	Striking the right balance between reducing extraneous load and maintaining sufficient task complexity to promote meaningful learning.	Moving from theoretical genetic decomposition to practical task design requires deep pedagogical content knowledge and iterative refinement.

).

2.2. Participants, Recruitment, and Data Collection

The MTWs supported students enrolled in a large, high-risk introductory mathematics course aimed at science and actuarial science students. Workshop attendees were invited via email to participate in focus groups. Six focus-group interviews were conducted across the two semesters of the 2023 academic year as part of the parent mixed-methods evaluation of the MTWs. Groups comprised 2–6 participants, and the focus group interviews ran for 45–60 min. An external psychologist, experienced in qualitative education research, conducted all interviews to reduce social-desirability pressures linked to lecturer–student power relations. Sessions were audio-recorded with consent and conducted in English. Recordings were transcribed verbatim and checked against the audio by the first author.

The interview guide probed (a) salient workshop moments, (b) perceived benefits and challenges, (c) changes in mathematical engagement, and (d) views on facilitation and peer interaction, domains chosen to surface talk relevant to ZPD, APOS, CLT and constructivism.

2.3. Data Analysis, Trustworthiness, and Ethics

As outlined in Section 1.4, we adopted an abductive, reflexive thematic approach to the six focus-group transcripts. Coding was led by the first author, a mathematics education researcher who co-designed and facilitated the MTWs, bringing insider knowledge of the program. The second and fourth authors, who were not involved in day-to-day workshop activities, acted as independent reviewers by reviewing code reports, questioning interpretations, and helping to surface assumptions we might otherwise have taken for granted. We treat these differing positions (insider/outsider to the workshops) as both a resource and a potential source of bias, and we reflected them in our analytic memos.

Operationally, we began with a deductive codebook that translated constructs from ZPD, APOS theory, CLT, and constructivism into testable Assumption–Indicator statements and then coded transcripts line by line on NVivo 15. During coding we remained inductively open to disconfirming cases and to previously under-articulated mechanisms that the frameworks background or omit.

To make the logic of theory-testing transparent, we built an Assumption–Indicator–Evidence matrix (

Table 1. Analytic Framework for Theory Testing.

Framework	Core Theoretical Assumption	Observable Indicators in Student Narratives	Typical Forms of Evidence (Anticipated)
ZPD Cobb and Yackel (1996); Vygotsky (1978); Wood et al. (1976).	Calibrated mediation and temporary scaffolds enable performance beyond independent capability; equitable participation and safety are prerequisites.	Mentions of scaffolding, guidance, or peer explanation; talk about support gradually fading; references to inclusive or uneven participation.	Descriptions of collaborative problem-solving, supportive facilitation, or risk-taking within a non-judgmental group climate.
APOS Theory Asiala et al. (1996); Cottrill et al. (1996); Dubinsky and McDonald (2001).	Sequenced tasks promote movement from actions to internalized processes, reified objects, and integrated schemas.	Language signaling coordination across representations, explanations of “why it works”, or calls for structured consolidation.	Student reflections distinguishing procedural from conceptual understanding; requests to revisit or consolidate difficult ideas.
CLT Sweller (1988); Sweller et al. (2019); van Merriënboer and Sweller (2005).	Managing intrinsic and extraneous load frees working memory for schema construction; physical and temporal environments affect load.	Attributions of fatigue or confusion to crowding, time pressure, or unclear instructions; mentions of relief after breaks or debriefs.	References to environmental comfort, pacing, or clarity improving understanding.
Constructivism Cobb and Yackel (1996); von Glasersfeld (1995); Sfard (1998).	Social activity and reflection foster meaning-making, belonging, and identity when coupled with clear shared standards.	Narratives of collaboration, recognition, or community; self-reports of confidence or expressive growth; uncertainty when closure is missing.	Students describing mathematics as more social, reflective, or personally meaningful learning.

) and populated it with extracts that confirmed, extended, or challenged each assumption.

Consistent with reflexive thematic analysis, we did not seek intercoder reliability or treat coding as a procedure for generating consensus scores (Braun & Clarke, 2019, 2023). Instead, we prioritized depth and reflexivity by engaging in repeated analytic dialogues, sharing memos, and revisiting extracts when disagreements or alternative readings emerged; these practices functioned as our primary strategies for enhancing qualitative rigor.

Because two authors were involved in designing/implementing the MTWs, we treated insider knowledge as both a resource and a risk and addressed this through the use of an external interviewer and explicit reflexive commentary.

Ethical approval for the parent project was granted by the Faculty of Science Research Ethics Committee at the University of Cape Town, under approval number FSREC 089-2022. All participants gave informed consent to audio-recording and to the use of de-identified transcripts in future research on teaching and learning, and pseudonyms are used in reporting.

3. Results: Assumption-to-Evidence Analyses

We present the results framework by framework. For each, we restate the assumption we set out to test, identify the indicators we expected to see in student narratives, show what students actually said, and then offer a judgement with a concrete return to design. Across all four lenses, students largely recognized the design intentions, but they also named boundary conditions, equitable talk, time for consolidation, and the quality of the workshops venue that must be met if the frameworks are to hold in the specific context.

3.1. ZPD: Mediation, Safety, and Designed Equity

Assumption. If workshops provide calibrated, temporary support in a non-judgmental climate, students should report doing things they could not yet do alone; in mixed-ability, multilingual groups, they should also report that participation was deliberately distributed, not left to chance.

Evidence. Students were very clear that the space was non-judgmental. Jane (FG1) captured this sense of safety: “The workshops for me are so helpful because you are not being judged at all … everyone is so involved in helping one another.” That safety is exactly what ZPD says must precede risk. They also noticed deliberate participation routines, as Andile (FG3) put it, “They did this thing of giving us different whiteboard markers and saying they want to see different colors on the board so that they see that everyone’s engaging and everyone writes their piece of work there.” Mbali (FG6) described being told to have a timer so each group could work at its own pace and “it’s OK” if the group did not finish, an example of contingent mediation. Andiswa (FG6) contrasted peer talk with lecturer talk, emphasizing a shared way of speaking: “Sometimes it’s better to learn from your peers than from lecturers, because they actually communicate in the same language and have similar experiences, so it’s easier to understand what they’re saying.” From a ZPD perspective, this suggests that mediation worked best when it was delivered by peers who could tune their explanations to students’ everyday ways of talking and thinking, rather than only more formal lecturer discourse.

At the same time, students were frank about the equity boundary: “Some people won’t really participate … somebody would just tell you, ‘I’m just tired’, and that could be a bit frustrating” (Siphesihle, FG3). Other described shy peer who “just keep quiet even though they are confused” unless a routine such as “giving them a marker” forced them on the board (Andile, FG3), and moments when faster peers “don’t value your opinion” and leave you “feeling stupid” (Musa, FG1). Without the color markers, the rotating roles, or the facilitator’s prompting, collaboration slipped back into dominance and silence. That is students pushing back on a naïve reading of ZPD that assumes collaboration automatically distributes mediation.

Finally, a few students were anxious about losing the scaffold altogether. As Jane (FG1) put it, “I was panicking because I don’t know how I’m going to do it without this support … maybe we can have a group chat … so that in second year we can do it by ourselves.” This is a call for support that tapers off gradually rather than stopping suddenly.

Judgement. In this first-year, mixed-ability mathematics support setting, student accounts support ZPD’s core claim that calibrated mediation in a safe space enables students to do more than they could alone. However, they also challenge the assumption that such mediation will distribute itself equitably once a supportive climate is established. Mediation, questioning, and peer help worked because participation was visibly structured through markers, roles, timers, and affective openings, not because students were naturally inclined to share opportunities equally.

Return to design. Treat equity scaffolds as part of ZPD, not as optional add-ons. Keep rotating roles (scribe/timekeeper/reporter), visible board contributions, and short think-time plus prompted uptake so quieter students get mediated too. Plan a gradual fade-out into peer-led circles in second year, reusing the same routines.

3.2. APOS: Movement from Action to Process to Object and Schema

Assumption. If tasks are sequenced in an APOS-informed way, students should talk less about cramming procedures and more about approaches, coordination, and wanting to finish the idea; they should also notice when that final consolidation does not happen.

Evidence. Students’ descriptions matched the assumption quite closely. Karabo (FG6) said, “It’s not even about getting the answers right. It’s more about the process of learning how to get to that answer or the approach,” which is almost a textbook statement of the action to process shift APOS is after, valuing how you did it over whether you got it. Palesa (FG1) added that the workshops “bring us to the more useful systematic way of approaching problems … not just taking out that procedure you have crammed in your brain,” again, signaling movement away from memorized actions toward interiorized processes. Together, these accounts show that students experienced the workshops as pressing for explanation, method, and structure rather than for final answers.

Students also noticed when the APOS trajectory was interrupted. Kea (FG2) reported that “sometimes we work on a hard question, we struggle for a long time, and then time runs out and we never see a solution, so I’m left wondering whether my approach was actually right.” This suggests that a process has formed but has not yet been encapsulated into a stable object that can be carried forward. Ayanda (FG2) immediately proposed the remedy: “In the next workshop we could come back to those questions, not necessarily give us the answer, but explain how we should approach them.” This is an APOS-friendly request; they are not asking for final answers, but for the teacher to show APOS operationalization by sketching a genetic decomposition (Trigueros, 2022).

There were also concrete moments of interiorization supported by materials. In the mathematical optimization activity, Hector (FG4) explained that getting to handle “little 3D shapes” helped him “look at the shape, understand it and identify the height” before doing the algebra, which is exactly the type of support that helps a performed action become a thinkable process and then a reified object.

Judgement. APOS assumptions are supported, but as spiral rather than linear. Students recognized the intended progression from doing to explaining to coordinating, but they experienced it as incomplete whenever there was no planned return to a difficult task.

Return to design. Keep APOS-led sequencing but include a 5–8 min carry-over segment at the start of every session to revisit one unfinished or conceptually heavy task and close it at the level of approach (strategy, structure, representation) rather than at the level of answer.

3.3. CLT: Cognitive Load and the Learning Environment

Assumption. If the workshops reduce extraneous load and pace intrinsic load, students should link attention and understanding to clarity, pacing, and comfort; when load is not well managed because the room is hot, crowded, or distracting, or because time runs out, they should say so.

Evidence. Students were very explicit that the physical environment sometimes made it hard to learn, even when the pedagogy itself was working. Palesa (FG1) described the venue as “very humid and cluttered and hot in … most of the time I just want to get out of the place to get a breath of fresh air,” directly connecting thermal discomfort to loss of focus. Jane (FG1) added, “Sometimes when you come in there, it’s so hot and then I start to feel sleepy and I can’t really focus well.” Both are descriptions of extraneous load: effort is diverted to staying awake and physically comfortable rather than to mathematics. Busisiwe (FG4) made a similar plea for “a bigger venue,” noting that “there’s not really air conditioning in there.”

Layout and visibility also mattered. Musa (FG1) highlighted how the seating arrangement interfered with attention:

I could say maybe it’s the way we are seated, with [the facilitator] in front and just talking to everyone. Sometimes it’s not easy to pay attention to him fully because you’re not facing him because of the way the tables are placed … you feel a little bit left out … Maybe set the tables in a certain order before we even start the session so that everyone has a feel of what’s happening.

Andile (FG3) similarly complained that “when a person just passes in front of the projector … I just get distracted,” suggesting that the venue “should just [be] book[ed] for only us and then close it so that it can be more private … people are talking here and there.” In CLT terms, blocked sightlines, background talk, and shared use of the room are all forms of extraneous load that are unrelated to the task.

Time also functioned as load. As Kea (FG2) noted before, “Time runs out and we don’t finish it … I’m left wondering if my approach was even correct,” while Ayanda (FG2) suggested that in the next workshop “we could look at those questions … not necessarily give us the answer, but explain the way we should approach the question.” Unfinished problems leave a cognitive residue that occupies working memory; a short, structured debrief at the next session is a CLT-friendly way to convert that residue into consolidated understanding. Students also linked scheduling to load: Busisiwe (FG4) and others asked for “a bigger venue and maybe not on a Friday afternoon,” and Andiswa (FG6) described wellness sessions at “such inconvenient times” during heavy assessment weeks as “two hours when I could have done something better,” intensifying the sense of competing demands.

Finally, students noticed and valued explicit regulation shifts that supported germane load. Jane (FG1) said she liked starting with “the whole breathing exercise,” and Musa (FG1) explained that after a long day it helped you “calm yourself down before you do something.” Andile (FG3) similarly described being told to “close your eyes and take a deep breath” as helping you “be concentrated … even though you have [distractions], just because you had that moment of calming down.” Karabo (FG6) framed the brief meditation as “resetting your mind and just relaxing into the workshop … allowing yourself to be in the space,” so that “you’re telling yourself … I’m going to engage in this session.”

Judgement. In this first-year mathematics support setting, student accounts strongly support CLT’s claim that extraneous demands can derail learning; however, they challenge narrow interpretations that locate extraneous load primarily within task or material design. Students made clear, causal links from heat, cramped or noisy venues, awkward seating, and rushed or poorly timed sessions to loss of focus and incomplete learning, and they spontaneously proposed CLT-consistent fixes (cooler, private rooms; re-arranged tables; short debriefs; better timing). Once the venue was booked for exclusive use and layouts were adjusted, the main environmental sources of extraneous load were addressed, confirming that the problem lay in the learning context rather than in the workshop pedagogy.

Return to design. Treat the venue, seating, noise, and timeslot as instructional variables, not neutral background. Where possible, secure exclusive use of a well-ventilated venue and pre-arrange tables so all students can see and hear the facilitator without sacrificing workspace. When this is not possible, shorten work segments and build in micro-breaks. Script 5–8 min start-of-session debriefs to clear temporal load from unfinished items and keep brief breathing/reflection exercises so that the working memory students do have is spent on schema-building rather than on fighting the room.

3.4. Constructivism: Social Meaning-Making, Recognition, and Closure

Assumption. If students learn mathematics through social activity, explanation, and reflection in a humanized space, they should talk about the workshops as communal, about doing mathematics and doing life, and about feeling more confident to explain. They may, however, still want a small signal of what counts at the end.

Evidence. Naledi (FG4) memorably called the workshops “math therapy”, describing them as a space where “you’re actually doing [mathematics]” but also learning “to be comfortable with being uncomfortable and knowing that it is okay not to get it right the first time” and still staying in the struggle. Hector (FG4) recalled a session where they shared about people they were grateful for and reflected on their first semester, saying that in moments like that “we forget that it’s even a [mathematics] workshop … it goes really deeper than that.” Aphiwe (FG5) captured the same blend of academic and personal work: “We have these sentences we like to say … we’re doing [mathematics], doing life. We speak about life, and sometimes we even bring everyday life things into the problems.” Taken together, these accounts point to constructivist identity work: academic and personal sense-making braided together in a humanized mathematics space.

Students also noticed and valued explicit attention to mathematical communication. Ayanda (FG2) explained that the workshops “teach the students how to really understand what they’ve been doing and how to voice it out and be able to explain it to someone else,” adding that “if you can explain it to someone else, then it means you really understand it.” The same student linked this directly to assessment, noting that in their test “there are bonus marks for good mathematical writing … so even my writing is important now,” and that the workshop focus on clear, structured explanations had helped them “make sure that I am writing properly, showing all the steps where there’s a need for an explanation.”

Several participants contrasted this designed talk with more individualized tutorials. Hector (FG4) described tutorials as a space where “no one seems to show any interest there … you can work by yourself. Nobody cares if you dialogue or not,” whereas in the workshops “they give us three minutes, you have to do it right now with the group,” which made dialogue happen and made “the group more cohesive.” This is constructivist design in which participation is actively structured.

Where the boundary appeared was when a task ended without closure. Kea (FG2) said they “just want to see if my approach is good or what,” because sometimes “we work on a hard question, we struggle for a long time, and then time runs out and we never see a solution.” This is not resistance to open, social problem-solving; it is a call for a brief signal that names what would count as an acceptable solution or approach. It also shows the tension in our design, because the workshops were intentionally set up not to hand out solutions; we wanted the emphasis to stay on reasoning and approach rather than on getting the right answer.

Judgement. In this first-year mathematics support setting, constructivist intentions are strongly affirmed: students describe the workshops as communal, humanizing spaces in which they “do [mathematics]” and “do life,” learn to explain their thinking, and rework their relationship with mathematics. However, their accounts also highlight a need for openness paired with brief signals of what counts—rich, relational, exploratory work followed by a short indication of an acceptable solution or approach.

Return to design. Keep the humanized, reflection-rich format; student accounts indicate that it changed both how they feel about mathematics and how they communicate it. But close at least one hard task per session with a two–three sentence canonical approach or mini-rubric (for example, key features of a good explanation, diagram, or proof). That small signal helps less experienced students locate their own contributions against expected mathematical standards without undermining the exploratory, social character of the workshops.

3.5. Cross-Framework Synthesis

Students’ narratives validate the composite MTW design while sharpening where each framework needs context-aware guardrails: (a) ZPD requires designed equity in participation, not only safety; (b) APOS progressions should be spiralized with scheduled returns; (c) CLT must extend to the venue and times at which we teach; and (d) constructivist openness should close with brief, explicit criteria for what counts. These refinements are not departures from the underlying theories but practice-tested adjustments for first-year mathematics in this South African university setting.

Drawing on the framework-specific analyses above, we summarize our theoretical contribution as four named propositions that refine canonical accounts of ZPD, APOS, CLT, and constructivism for this setting:

Proposition 1 (ZPD with Designed Equity).

Canonical ZPD formulations stress calibrated, temporary scaffolding in a safe social climate. In mixed-ability, multilingual first-year groups, psychological safety is necessary but not sufficient; equitable access to mediation requires designed participation structures (for example, rotating scribe/timekeeper/reporter roles, “everyone writes” visible contributions, timed think-time). Without these routines, help concentrates with dominant speakers and the ZPD is unevenly available.

Proposition 2 (Spiralized APOS for Transitional Workshops).

Canonically, APOS emphasizes progression from actions to processes, objects, and schemas. In first-year transitional workshops, our results suggest that this progression is better understood as spiral rather than linear, requiring planned return visits to difficult tasks so that emerging processes can be encapsulated into stable objects and integrated into schemas.

Proposition 3 (Contextualized CLT for Support Spaces).

CLT typically foregrounds the management of intrinsic and extraneous load through task design. Our results show that, in workshop-style support, extraneous load is also strongly shaped by room conditions, layout, and timeslot; temperature, noise, blocked sightlines, and scheduling can negate otherwise well-designed tasks unless they are treated as instructional variables in their own right.

Proposition 4 (Constructivism with Light, Explicit Closure).

Constructivist accounts emphasize social activity, explanation, and reflection in meaning-making. In this context, students welcomed open, relational problem-solving but also asked for brief signals of “what counts”. Constructivist workshop designs therefore need light, explicit closure on at least some hard tasks (for example, a short canonical approach or mini-rubric), so that less experienced students can locate their contributions against expected mathematical standards.

These propositions, together with their empirical grounding and design implications, are summarized in

Table 2. Student Evidence and Design Implications by Theoretical Framework.

Framework	Observed Indicators and Student Evidence	Judgement 1	Design Refinement/ Implication
ZPD	Students described the MTWs as non-judgmental and collaborative, enabling risk-taking and peer help. Some also noted dominance in groups without structured roles.	Supported, with equity qualification.	Embed explicit role rotation and participation routines to ensure equitable access to scaffolding.
APOS Theory	Learners contrasted “cramming steps” with deeper understanding and coordination across representations and requested brief revisits for consolidation.	Supported, with recursive nuance.	Institutionalize short revisit segments to reinforce encapsulation and schema building.
CLT	Students cited heat, crowding, layout, and time pressure as barriers to focus, and appreciated short debriefs, clear layouts, and regulation features (e.g., breathing exercises).	Strongly supported.	Manage extraneous load through venue selection, seating, pacing, and 5–8 min debriefs on key tasks.
Constructivism	Learners reported community, belonging (“it feels like a family”), and confidence gains, but some felt lost without signals of what would count as a good solution.	Affirmed, with a need for clearer closure.	End sessions with a concise “what counts” summary (e.g., features of a good explanation/diagram/proof) to anchor understanding.

¹ Judgements indicate whether student narratives confirmed, nuanced, or challenged each framework’s core assumptions.

.

4. Discussion

In this section we interpret the results in the light of the four frameworks that helped guide the design of the MTWs: the ZPD, APOS theory, CLT, and constructivism, and in relation to the working assumption that theory, when made testable, should be visible in student voices (Design-Based Research Collective, 2003; Cobb et al., 2003). The findings show that the core claims of these frameworks did travel into the South African first-year mathematics context, and they did so under conditions that students themselves named. In this sense, the results do not contradict the frameworks; they refine them.

4.1. Where the Frameworks Held, Where Context Bent Them, and Where Students Pushed Back

Student narratives confirmed that ZPD’s central idea was realized. Vygotsky’s (1978) formulation and the scaffolding refinement by Wood et al. (1976) both emphasize that learners can function at a higher level when assistance is calibrated, temporary, and gradually withdrawn. Participants described the MTWs as spaces in which they could attempt work just beyond what they would risk in tutorials or alone, precisely because the atmosphere was non-judgmental and help was available. They spoke about peers explaining to them, tutors circulating and asking probing questions, and structured roles that made them take a turn at the board-practices that align with sociocultural views of classroom norms and participation (Cobb & Yackel, 1996). These are all recognizably ZPD practices. Students linked this support to emotional safety, calling the workshop “math therapy” and saying that it made them comfortable with not getting something right the first time. In this context, cognitive mediation and relational scaffolding were fused, which means that the ZPD was not only a cognitive zone but also a social and affective one.

The APOS component of the design was likewise recognizable to students. They noticed that the workshops were not simply about cramming steps but about being pushed to explain why something worked, to coordinate different representations, and to carry a line of reasoning through to a clear, well-structured conclusion. Some students could recall particular sessions where concrete materials or visual prompts helped them move from doing a procedure to actually thinking of the procedure as an object, consistent with work on encapsulation and reification (Sfard, 1991). However, they also made it clear that this progression did not feel like a straight staircase. When time ran out on a difficult task, they were left unsure whether their approach was right, and several proposed that the next workshop should begin with a short return to that item so that the approach could be made explicit. That is, the APOS trajectory was experienced as spiral and incomplete unless there was a planned opportunity to consolidate.

CLT’s claim that extraneous demands can derail learning was very strongly corroborated. Students did not blame the pedagogy when their attention wandered; they blamed the venue and the timing. They described the venue as hot, crowded, humid, and sleep-inducing. They linked poor ventilation and Friday-afternoon slots to an inability to focus. These accounts align with research in educational psychology showing that physical environmental discomfort, including heat, crowding, and poor ventilation, can significantly increase extraneous cognitive load and detract from attention and learning (Choi et al., 2014). Recent studies similarly report that elevated classroom temperatures and inadequate ventilation depress attention and learning performance, with measurable declines in cognitive task performance as thermal load rises and air quality deteriorates (Kirkil, 2025; Vasilakopoulou & Santamouris, 2025). Students also pointed to time pressure: when tasks could not be finished, they carried cognitive residue into the next session. In their talk, then, cognitive load was not only a property of task design or worksheet layout; it was also a property of air, space, noise, and scheduling.

Constructivist intentions were also visible. Students reported that the MTWs made mathematics feel social, that they could “do [mathematics] and do life” in the same space, that they met people from different degree programs, and that they regained or deepened an affection for mathematics. This aligns with constructivist emphases on social meaning-making and integration of participation, identity, and learning (von Glasersfeld, 1995; Cobb & Yackel, 1996; Sfard, 1998). Students noticed that clear mathematical writing was recognized, and this motivated them to explain better. The only voiced friction appeared when an open, exploratory task ended without a small closing note; in those moments, some students said they did not know whether their approach counted.

What these four strands have in common is that the theory held, but it held conditionally. ZPD worked if participation was designed to be inclusive, not only safe. APOS worked if there was time to come back to difficult tasks and close them at the level of approach. CLT worked if we treated room conditions and timing as part of the instructional design, not as neutral background. Constructivism worked if open, relational problem-solving was paired with brief, explicit signals of what counts as a good solution or explanation. In each case, the context of multilingual, unevenly prepared students working in sometimes physically uncomfortable spaces bent the framework in ways that needed to be documented so that future designers and researchers can see how context shapes theoretical application.

4.2. Refined Theoretical Propositions

On the basis of the evidence gathered and reported, each framework can be restated in a way that is true to its intent but attentive to local conditions.

The version of ZPD that emerged from the student accounts is best described as equity-refined ZPD. In mixed-ability, multilingual groups, access to mediation does not automatically follow from the presence of a friendly or supportive atmosphere. As critical readings of Vygotskian pedagogy have noted, collaborative settings can still distribute participation unevenly unless equity is deliberately attended to (Daniels, 2008). This concern has also been documented in recent a recent intervention study showing that equitable participation in classroom dialogue requires deliberate design rather than reliance on collaboration alone (Sedova et al., 2025). This also aligns with recent reviews of equity-oriented groupwork approaches, including Complex Instruction, which emphasize that participation structures and role design are necessary to prevent status-based participation patterns in collaborative learning environments (Villa & Sedlacek, 2025). In the present study, students themselves reported group members who dominated and others who did not speak. They also reported how much it helped when facilitators insisted on rotating roles, when everyone had to write in a different color, or when there was a visible expectation that each person contributed. In this setting, then, participation structures and relational scaffolding are not extras; they are part of the enabling conditions of the ZPD.

APOS can be restated as spiralized APOS. The original genetic decompositions that informed the workshop tasks remained useful; they gave the designers a target sequence from action to process to object to schema. However, the data show that students needed a second pass, and this need was evident in their responses. Requests for “next time, please just show us the approach,” or for “a little more time to finish that hard one,” are requests for encapsulation to be made public and shared. APOS, in this context, is therefore not a single ascent; it is a designed return, consistent with arguments that concept formation often involves revisiting and reorganizing earlier actions (Gray & Tall, 1994).

CLT, likewise, can be extended to include the venue and the timetable. Students were not only grappling with how to focus on the task itself; they were also contending with room conditions and scheduling that continually pulled attention away from the mathematics. It was heat, cluttered furniture, noise, and sessions that clashed with lab deadlines. In other words, extraneous load here was spatial and temporal as much as textual. A CLT-informed workshop in this context must therefore treat venue choice, ventilation, seating, session length, and calendar placement as instructional decisions.

Constructivism can be restated as openness with brief signaling. Students clearly valued the social, humanized, exploratory quality of the workshops, and this quality produced confidence, a sense of recognition, and a wider view of what mathematics could be. But some of them also wanted a small signal at the end of a task that said, “this is what a good version of this looks like.” That signal need not be a full solution; it can be a short description of the approach or of the features of a good explanation. When that signal is present, openness serves less experienced students rather than confusing them.

4.3. Implications for Workshop Design, Facilitation, and Continuity

When these refined propositions are translated back into practice, they point to a workshop model that is not very different in spirit from the one already implemented but that is more explicit, more repeatable, and more resilient to context, in line with calls for design principles that can travel across settings (Gravemeijer & Cobb, 2006).

The learning arc should be designed to include short, predictable return passes. Since students asked for unfinished items to be revisited, each workshop can begin with a five to eight-minute segment in which the previous session’s most difficult problem is reopened, the main approaches are compared, and the underlying structure is named. This is enough to complete the APOS move for those who were almost there and to stabilize the new object inside an emerging schema.

Participation should be designed every time. Rather than assuming that small groups will distribute talk fairly, facilitators should continue to assign rotating roles, to insist on everyone writing on the board or on the sheet, and to give a short thinking period before inviting responses. These routines make the ZPD available to students who would otherwise remain silent. Because students have already experienced these participation structures as helpful, there is a good chance of uptake if they are made non-negotiable. The physical and temporal environment needs to be treated as part of pedagogy. Where it is possible to choose venues, those with ventilation, movable furniture, and enough space to walk around should be prioritized, echoing recommendations from mathematics support center research about welcoming, functional spaces (Croft, 2000; Lawson et al., 2003). Where it is not possible, sessions can be adjusted by shortening intense segments, inserting brief movement or breathing breaks, or devoting that week to consolidation rather than new material. Attention to exam weeks and lab deadlines can also prevent avoidable load.

Closure should be light but consistent. Ending one task per workshop with a two or three-sentence summary or circulating a short note afterward that names the approach rather than giving the answer is enough to give students the ‘what counts’ cue they sought without eroding the exploratory character of the workshop. Finally, students were explicit that they worried about losing this kind of support in subsequent years. For that reason, the workshop routines—roles, return passes, debriefs, and reflection—can be carried over into lighter, student-led problem circles in second year. This provides a designed fade-out rather than an abrupt withdrawal of scaffolding, aligning with recommendations for building sustainable peer-learning cultures (Tinto, 2012).

4.4. Limitations and Future Research

This study worked with six focus-group interviews from a single, research-intensive South African university and analyzed them using an abductive, reflexive thematic approach. The findings are therefore situated and interpretive rather than generalizable; they offer context-rich insight into how these frameworks operated for this particular cohort and setting and are best read as theoretically informed claims about plausibility and transferability rather than as universal effects. Two of the researchers were also involved in designing and facilitating the workshops, which offered contextual sensitivity but also introduced the possibility of confirmation bias, a familiar challenge in practitioner-research and design-based work (Cobb et al., 2003). We mitigated this through the use of an external interviewer and by making our assumption–indicator–evidence logic explicit, yet these steps do not remove all bias. The analysis also privileges student accounts of participation and learning over fine-grained interactional data; it tells us how students made sense of the workshops, not exactly how each group discussion unfolded.

Future work can build on this in at least three ways. First, a longitudinal design could examine whether the participation and debrief routines that worked in the workshops can be sustained when the explicit facilitation is withdrawn, for example, in second-year modules. Second, close analysis of multilingual interaction in these workshops would help to clarify how language resources were used to construct understanding and whether certain students were systematically sidelined. Third, linking specific design choices—such as planned revisits of key tasks, role rotation, or room changes—to item-level performance or retention measures in a mixed-methods or design-based research cycle would strengthen the evidence for causal claims while still recognizing that learning is context-bound.

4.5. Conclusions

This study set out to do something that many support-program studies do not: to return to the theories that justified the design and ask whether they were visible, qualified, or challenged in student voices. By translating ZPD, APOS, CLT, and constructivism into a small set of assumptions, attaching observable indicators to those assumptions, and examining them against six focus-group interviews, we showed that the theories did in fact guide practice and were recognized by students. Access to the ZPD depended on participation being deliberately structured so that many voices, not just a few, mediated learning. APOS progressions required planned opportunities to revisit and complete difficult tasks. CLT needed to extend to the room and the timing. Constructivist openness needed small, explicit closing signals. Articulating these conditions is not a move away from theory; it is a refinement that enables theory to continue guiding design in a meaningful way.

In relation to the research questions, the findings provide clear answers. With respect to Research Question 1, students’ accounts largely confirmed the core assumptions of ZPD, APOS, CLT, and constructivism, but they also challenged the sufficiency of these frameworks when applied without attention to contextual conditions. Regarding Research Question 2, student narratives showed that uneven prior preparation and material conditions such as venue quality and scheduling shaped how each framework was realized in practice, requiring extensions or qualifications to canonical formulations. Finally, in response to Research Question 3, these findings translate into concrete design implications for subsequent workshop iterations, including structured participation routines, spiralized task revisiting, attention to physical and temporal learning environments, and the inclusion of brief, explicit closure in otherwise open, exploratory activities.

The main contribution of this work, therefore, is twofold. First, it offers a transferable way of making theory testable in program evaluation, the assumption–indicator–evidence approach that underpinned the results, in line with calls for “inspectable” theory–practice links in design research (Cobb et al., 2003). Second, we offer to the literature a set of concise but consequential refinements that translate directly into routines which can be built into timetables, facilitator scripts, and venue features for first-year mathematics support in similarly constrained contexts.