Collaborative Preferences for Learning Mathematics
- The paper demonstrates the development and validation of the CPLM scale, showing a robust single-factor structure with strong model fit and longitudinal invariance.
- CPLM is a context-specific measure that quantifies the continuum from individual to collaborative learning in mathematics, applicable to scenarios like sense-making, assessments, and problem-solving.
- Empirical evidence reveals stable semester-long scores, no significant overall gender differences, and systematic variations based on tutorial engagement.
Collaborative Preferences for Learning Mathematics (CPLM) denotes students’ preferences for collaborative versus individual learning in mathematics. The construct was introduced to address the absence of a mathematics-specific quantitative instrument for capturing how students prefer to learn across contexts such as sense-making, exposure to novel concepts, preparation for high-stakes assessments, and low-stakes problem solving. Current research treats CPLM as a measurable and longitudinally comparable construct: the five-item CPLM scale exhibits a single-factor structure, good model fit, and strong invariance over time, while undergraduate studies report stable semester-long scores, no significant overall gender difference, and systematic variation by tutorial engagement (Kim et al., 17 Aug 2025, Kim et al., 20 Aug 2025, Kim et al., 19 Aug 2025).
1. Construct definition and conceptual scope
CPLM was developed to measure student preferences for collaboration specifically within mathematics, rather than generic attitudes toward group work. The underlying rationale is that collaboration within mathematics has been established as being effective in providing students with crucial opportunities to develop critical thinking, effective communication, and teamwork skills, and that shared learning experiences may help students gain deeper insights into mathematical concepts and approach challenges from multiple perspectives (Kim et al., 17 Aug 2025).
The construct is explicitly framed as a preference continuum rather than a binary disposition. In the validated instrument, respondents indicate preference from $0$ for individual learning to $100$ for collaborative learning. The resulting scores are therefore interpretable as degree of preference rather than categorical affiliation. This design also permits comparison across distinct mathematical situations, including learning new ideas, preparing for examinations, and engaging in homework or practice exercises (Kim et al., 17 Aug 2025).
A notable feature of the construct is that it is context-specific to mathematics. The scale was motivated by the claim that quantitative tools capturing collaborative preferences in mathematics were previously underdeveloped, despite the relevance of such preferences to group dynamics, instructional planning, and student engagement. This disciplinary specificity distinguishes CPLM from broader measures of collaborative learning attitude (Kim et al., 17 Aug 2025).
2. Scale design and psychometric validation
The CPLM scale is a 5-item instrument administered with sliders ranging from $0$ (“Individually”) to $100$ (“Collaboratively”). Respondents are prompted: “Consider yourself learning mathematics. By moving the slider, state the extent to which you prefer to do it from 0 = Individually to 100 = Collaboratively for each of the items:” (Kim et al., 17 Aug 2025).
| Item code | Prompt |
|---|---|
| CPLM_1 | What is the most effective way for you to learn mathematics? |
| CPLM_2 | What is the best way for you to make sense of mathematics? |
| CPLM_3 | What is the most effective way for you to study for high-stakes maths assessments (e.g., exams)? |
| CPLM_4 | In what social setting do you prefer to be exposed to novel concepts? |
| CPLM_5 | In what social setting do you prefer to engage in problem-solving in low-stakes assessment (e.g., homework, practice exercises)? |
Psychometric validation proceeded in two samples. In Sample 1, recruited via Prolific from undergraduate students in the UK/USA in STEM or commerce majors, exploratory factor analysis was conducted on after cleaning. The analysis used principal axis factoring, with appropriateness supported by and Bartlett’s test of sphericity . The result was a single-factor structure with eigenvalue , explaining of variance; item loadings ranged from to $100$0, communalities from $100$1 to $100$2, and Cronbach’s alpha was $100$3 (Kim et al., 17 Aug 2025).
In Sample 2, collected in a second-year mathematics service course at a large New Zealand university, confirmatory factor analysis on $100$4 produced good model fit: $100$5, $100$6, $100$7, $100$8, and $100$9. Internal consistency was higher in this sample, with Cronbach’s alpha $0$0; convergent validity indices were $0$1 and $0$2 (Kim et al., 17 Aug 2025).
Longitudinal comparability was examined using multi-group confirmatory factor analysis over three time points with $0$3. Strong invariance was achieved across configural, metric, and scalar models, with $0$4 for those comparisons. Strict invariance was not achieved, as the residual model yielded $0$5, but the study notes that strong invariance is sufficient for many research purposes involving mean-level comparisons over time (Kim et al., 17 Aug 2025).
3. Semester-long evidence on gender and temporal stability
A longitudinal undergraduate study examined whether gender influenced CPLM over a 12-week semester in a second-year undergraduate service mathematics course at a large New Zealand university. Weekly one-hour tutorials encouraged, but did not require, peer collaboration. Of 294 enrolled students, 201 completed all three surveys; after excluding three “declined to answer” gender responses, the analytic sample was $0$6. CPLM was measured at T1 (Week 1), T2 (post-midsemester), and T3 (final week), with the mean of the five items used as the composite score at each timepoint (Kim et al., 20 Aug 2025).
| Timepoint | Male mean CPLM | Female mean CPLM |
|---|---|---|
| T1 | $0$7 | $0$8 |
| T2 | $0$9 | $100$0 |
| T3 | $100$1 | $100$2 |
The statistical analysis used a two-way mixed ANOVA with gender as the between-subjects factor and time as the within-subjects factor. Mauchly’s test indicated violation of sphericity, $100$3, so the Greenhouse-Geisser correction with $100$4 was applied. The interaction between gender and time was not significant, $100$5; the main effect of time was also not significant, $100$6; and the main effect of gender was not significant, $100$7 (Kim et al., 20 Aug 2025).
These results support two claims specific to that context. First, CPLM scores did not change significantly across the semester. Second, male and female students showed comparable preferences overall. Although male students had numerically higher mean CPLM scores at each timepoint, the effect sizes were very small and the differences were not statistically significant. The reported 95% confidence intervals for means across gender and time were between $100$8 and $100$9, which the study interpreted as indicating a balanced preference rather than an extreme preference for either collaboration or independence (Kim et al., 20 Aug 2025).
This evidence directly challenges a common assumption that gender differences in mathematical collaboration preferences are stable and readily detectable. In this study, no significant gender difference emerged, despite prior literature cited by the authors as suggesting gendered patterns in collaboration. The paper therefore argues for further exploration of contextual factors rather than reliance on generalized gender stereotypes (Kim et al., 20 Aug 2025).
4. Tutorial engagement as a contextual correlate
A second undergraduate study examined tutorial activity engagement as a correlate of CPLM in a tertiary mathematics context. The sample comprised 201 students in a second-year mathematics course at the University of Auckland who completed surveys at the start, middle, and end of the semester. Tutorial behavior was measured with a single Likert-scale item ranging from “always individually” to “always with others,” then recoded into three groups: Individual Learners (0), Mixed Learners (1), and Collaborative Learners (2) (Kim et al., 19 Aug 2025).
| Group | T1 mean CPLM | T3 mean CPLM |
|---|---|---|
| Individual Learners | 43.68 | 38.36 |
| Mixed Learners | 46.08 | 51.56 |
| Collaborative Learners | 53.10 | 56.03 |
The analysis again used a two-way mixed ANOVA, with tutorial engagement group as the between-subjects factor and time as the within-subjects factor. Sphericity was checked and the Greenhouse-Geisser correction was applied with 3. The main effect of tutorial engagement was statistically significant, 4, indicating that collaborative learners consistently reported stronger preferences for collaborative learning than individual learners. The main effect of time was not significant, 5. The interaction effect was not statistically significant, though it approached significance, 6 (Kim et al., 19 Aug 2025).
The principal interpretation is that differences in CPLM between engagement groups persist rather than clearly diverge or converge during a single semester. The paper states that the absence of an interaction effect suggests CPLM differences remain stable, and that familiar modes of tutorial engagement may reinforce existing collaboration preferences. It further relates this interpretation to the mere-exposure effect: repeated engagement with a particular tutorial mode may strengthen the corresponding preference pattern (Kim et al., 19 Aug 2025).
A second misconception is therefore not supported by the available evidence: repeated participation in tutorials does not automatically imply rapid convergence toward a common collaborative preference. In the reported data, tutorial engagement differentiated students, but semester-long trajectories remained comparatively stable (Kim et al., 19 Aug 2025).
5. Instructional antecedents and related pedagogical mechanisms
Earlier work in collaborative learning and mathematics instruction provides mechanisms that are plausibly relevant to CPLM, even when the specific CPLM scale was not used. One line of work emphasized meta-cognitive reflection on collaborative activity. In the course “Learning Management,” designed for first-year undergraduates to “learn collaborative learning,” students in the experimental condition used KBDeX to analyze their own discourse. The study reported a transformation in beliefs from “just experiences of the participation of the collaborative learning” to “active contribution for collaborative knowledge creation,” and the course “succeeded in changing the students’ preferences about collaborative learning from negative to positive as well.” Quantitatively, preference for collaborative learning increased from 7 to 8 on a 1–5 scale, 9; average RubKB scores on ideal group activity improved from 0 to 1, 2 (Matsuzawa et al., 2013).
A second line of work in calculus framed formative peer review as a collaborative mathematical practice. A sequence of collaborative writing assignments in Calculus I and II distinguished writing-to-learn from writing-in-the-disciplines and used structured peer review, Google Docs/Drive, and web-based LaTeX editors. The reported outcomes were improved engagement, conceptual understanding, and mathematical exposition, with peer review structured around agendas, questions, summative feedback, and revision plans (Eaton et al., 2014).
A third line of work focused on collaborative tagging and annotation in e-learning mathematics. A Moodle-integrated framework encoded mathematical content using presentation MathML with RDFa annotations and allowed students to highlight and annotate specific parts of mathematical content. Tag clouds were used as an agreement measure across students, and the most agreed-upon tags informed the bottom ontology. This system was intended to add meaning to e-learning contents, create relationships between contents, and facilitate search (Doush et al., 2012).
Taken together, these studies suggest several mechanisms by which collaborative preferences in mathematics may be formed or reinforced: structured reflection on discourse, peer explanation and revision, and shared semantic organization of mathematical artifacts. That interpretation extends beyond the direct CPLM studies, but it is consistent with the pedagogical processes explicitly reported in those adjacent literatures.
6. AI-mediated extensions, limitations, and research directions
Recent AI-mediated work does not operationalize CPLM with the five-item undergraduate scale, but it examines closely related preferences in collaborative mathematical interaction. In a participatory design study with 24 middle school students, AI peers for collaborative problem solving were preferred when they were mathematically competent yet explicitly deferential, provided progressive scaffolds such as hints and checks under clear student control, and used a tone of friendly expertise rather than exaggerated personas. Students strongly preferred scaffold-first support, with “give hints, not the full answer” and error detection among the most valued features, while direct answer-giving was treated as a “dealbreaker” by many participants (Lyu et al., 25 Jan 2026).
A complementary strand concerns simulated learners for teacher education. In fraction-comparison dialogues, three approaches—Fine-tuning, Multi-agent, and Direct Preference Optimization (DPO)—were evaluated for their ability to simulate authentic student reasoning, language, uncertainty, and willingness to elaborate. All approaches improved cognitive and linguistic authenticity over few-shot prompts, and DPO achieved the highest measured authenticity at 3 language and 4 cognition, while generating diverse reasoning that spanned brief, hesitant, uncertain, and elaborated response styles (Cao et al., 6 Apr 2026). This suggests a possible future link between measured CPLM and the modeling of collaborative discourse styles, though that connection remains to be established empirically.
The direct CPLM literature also states clear limitations. The longitudinal gender study was conducted in a single course at one university, within a service mathematics population, so generalizability is limited. The analytic sample of 5 may lack power to detect small effects, raising the possibility of Type II error. The study did not assess prior collaboration experiences, academic achievement, or cultural background; the data were self-reported; and there were no qualitative or observational data to explain why preferences were held or how they manifested in classroom experience (Kim et al., 20 Aug 2025).
The resulting research agenda is correspondingly specific. Future work is described as needing larger and more diverse samples, integration of qualitative and observational data, and investigation of additional psychosocial, cultural, and situational correlates of CPLM. For practice, the current evidence supports a restrained conclusion: educators should not assume universal or gendered collaborative preferences in undergraduate mathematics, and contextual adaptation is preferable to blanket approaches based on stereotypes (Kim et al., 20 Aug 2025, Kim et al., 19 Aug 2025).