Stepping Stones: Adopting a Fading Programme Design to Promote Teachers’ Use of Metacognitive Strategies for Mathematical Problem Solving

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The study described in the paper"Stepping Stones: Adopting a Fading Programme Design to Promote Teachers' Use of Metacognitive Strategies for Mathematical Problem-Solving" aims at the professional development of primary school teachers to promote metacognitive strategies in problem solving of primary school children. The aim of the paper is to find out to what extent the metacognitive strategies introduced by the teachers changed in the course of the intervention and how they perceived the intervention. Qualitative and quantitative data were collected for this purpose. The study contributes to evaluating the effectiveness of teacher training programmes and is therefore of great interest to the scientific community.

The weaknesses of the paper lie in the fact that the content of the intervention, the metacognitive strategies, is not sufficiently considered: To what extent the teachers really continued to promote metacognitive strategies, what this looked like, whether the type was meaningful, is not sufficiently presented. However, in order to be able to evaluate this, the concept of metacognition should also be presented in more detail in the theoretical part of the paper. In particular, with reference to this paper, I miss the fact that for independent application of metacognition the students must be familiar with the application and have experience with it (as justification for the duration of the intervention), that the use of metacognitive strategies only makes sense for tasks that have a certain level of difficulty, that teachers must therefore (initially) adaptively take over the monitoring for their students (e.g. concept of meta-meta-cognition according to Stillman). In addition, it should be emphasised that it is necessary for learners' skills acquisition that fading also takes place here, i.e. that teachers do not permanently take over monitoring and ask their students to use meta-cognition.

The second question is about the meaningfulness of quantitative data (and calculated mean values) with n=7 people.

If I have understood the paper correctly, this is a sub-study of a larger project. In this context, perhaps the content of the intervention, the metacognition, is not so relevant for this paper, but rather the question of the extent to which the teachers accepted and processed the content for themselves. The three reconstructed areas are of particular importance in this case.

On the linguistic level, I would like to point out that in some places very long, complex sentences were used, which makes reading difficult for non-native speakers. The bibliography should also be checked again for consistency.

Author Response

Many thanks for your feedback on our submission. We have used this to inform our revisions. All of our changes are highlighted in yellow in the text. An overview of our responses and amendments is provided below.

 

Comment:

The weaknesses of the paper lie in the fact that the content of the intervention, the metacognitive strategies, is not sufficiently considered: To what extent the teachers really continued to promote metacognitive strategies, what this looked like, whether the type was meaningful, is not sufficiently presented. However, in order to be able to evaluate this, the concept of metacognition should also be presented in more detail in the theoretical part of the paper. In particular, with reference to this paper, I miss the fact that for independent application of metacognition the students must be familiar with the application and have experience with it (as justification for the duration of the intervention), that the use of metacognitive strategies only makes sense for tasks that have a certain level of difficulty, that teachers must therefore (initially) adaptively take over the monitoring for their students (e.g. concept of meta-meta-cognition according to Stillman). In addition, it should be emphasised that it is necessary for learners' skills acquisition that fading also takes place here, i.e. that teachers do not permanently take over monitoring and ask their students to use meta-cognition.

 

Response:

Thank you for this comment. We have added further detail to clarify the nature of the Stepping Stones programme where appropriate throughout our submission. This is highlighted in yellow in the text. We have also added 3 further figures (Figure 1-3) to provide further information about the Stepping Stones programme and the professional development offer we used to support participating teachers. We also appreciate the comment about meta-metacognition. We have reviewed this literature, and have also considered wider sources, such as Wall & Hall’s (2016) work on the role of teachers as metacognitive role models. We have added discussion of this to the literature review and as appropriate throughout the description of our study and our results.

 

Comment:

The second question is about the meaningfulness of quantitative data (and calculated mean values) with n=7 people. If I have understood the paper correctly, this is a sub-study of a larger project. In this context, perhaps the content of the intervention, the metacognition, is not so relevant for this paper, but rather the question of the extent to which the teachers accepted and processed the content for themselves. The three reconstructed areas are of particular importance in this case.

 

Response:

Thank-you for the comment. We wanted to clarify that this is not a sub-study from a larger project - it is a pilot study of a faded programme delivered in primary schools. On sample size, we think the first part of the comment refers to the ‘mean ranks’ as presented in Table 2. We think it is important to clarify that the ‘mean rank’ statistics in Table 2 are used as ‘descriptive measures’ to present the relative prioritisation of each checklist item based on their observed implementation frequencies. We believe that these data are meaningful as they provide detailed ‘descriptive’ insights into implementation fidelity and, crucially, demonstrate that certain associations were strong enough to be statistically detected even with a small participant base. However, we do caveat these findings in the limitations section noting the small sample size and need for complete data sets and use of parametric tests that would inevitably have more statistical power and opportunity to establish cause and effect’ where possible.

 

Comment:

On the linguistic level, I would like to point out that in some places very long, complex sentences were used, which makes reading difficult for non-native speakers. The bibliography should also be checked again for consistency.

 

Response:

Thank you for this prompt. We have reviewed the document to shorten sentences where possible. We have also reviewed references for consistent presentation. We have highlighted places where this has been done within the document.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Authors,

This study offers valuable insights into the Stepping Stones professional development (PD) program, which was designed to support teachers in scaffolding metacognitive strategies specifically for mathematical problem solving. I appreciate the focus on both teacher learning and instructional practice, particularly within the context of promoting strategic thinking in mathematics classrooms. However, I have several critical points and considerations regarding the study’s methodology, interpretation of results, and the clarity of program implementation, which I outline below.

Qualitative Data Analysis Concerns
In the qualitative analysis phase, the authors employed 18 codes grouped under three main categories: professional development design, feelings of increased ownership, and the value of collaboration. However, it remains unclear whether these codes emerged inductively through open coding or were derived deductively from a pre-established framework informed by existing literature. This methodological ambiguity raises questions about the coding approach and its alignment with grounded theory or theory-driven analysis.

Furthermore, there appears to be an inconsistency between the categories derived during coding and the thematic findings reported in the results section. The three reported themes—such as building knowledge and confidence—primarily reflect teachers’ conceptualizations of planning and enacting metacognitive strategies, which do not directly correspond to the original coding categories. This discrepancy calls for clarification: Were the reported themes the result of a secondary level of axial or selective coding? Or do they reflect an interpretive synthesis rather than a direct extension of initial codes?

On the Use of a Convergent Mixed-Methods Design
Although the study claims to adopt a convergent mixed-methods design, its execution raises methodological concerns. Both qualitative and quantitative datasets are presented, and their sources and uses are transparent. However, the study falls short in demonstrating how these two strands were meaningfully compared or integrated to generate comprehensive interpretations. It is difficult to trace where the qualitative and quantitative data were systematically compared or merged to yield integrative insights—two core criteria for validating a convergent design.

Instead, the role of quantitative data seems to be limited to offering a snapshot of the presence or absence of targeted metacognitive strategies—particularly through frequency tables such as Table 3. Items that yielded non-significant findings are not addressed in the discussion, undermining the interpretive balance. This raises a critical question: Does the mere inclusion of both qualitative and quantitative data in evaluating a PD initiative suffice to categorize the study as a convergent mixed-methods design? A more rigorous integration process would be required to substantiate such a claim, particularly when the qualitative strand appears to dominate the interpretive narrative.

Results and Interpretive Depth: A Critical Commentary

The findings suggest that the Stepping Stones PD program positively influenced teachers’ development of knowledge, confidence, and motivation in teaching metacognitive strategies for mathematical problem solving. While such outcomes are indeed promising, the mechanisms by which the program brought about these gains remain insufficiently articulated. Specifically, it is unclear which particular features of the PD design—such as instructional routines, reflective tasks, or feedback mechanisms—directly contributed to these improvements. The results appear to rely heavily on teacher self-reports and survey responses, yet a more explicit and evidence-based account of how the PD fostered change would strengthen the study’s claims of impact.

Lack of Detail on Instructional Design and Strategy Implementation

The program materials, including pupil-facing slides and teacher scripts, are referenced as central tools; however, the actual content and pedagogical structure of these resources are not adequately described. For instance, were teachers provided with written scripts that included metacognitive prompts or problem-solving tasks? The mention of scripting tasks in the early weeks of the program suggests a scaffolded approach to classroom implementation, possibly intended to support the internalization of metacognitive teaching strategies. Yet, without concrete examples or excerpts from these scripts, it is difficult to ascertain how such tools contributed to teachers’ instructional repertoires.

Similarly, references to worked examples and visual representations imply a structured instructional model, but the absence of detail regarding their content, cognitive demand, or integration into the teaching sequences leaves the reader unconvinced of their instructional power. If the goal is to demonstrate the efficacy of Stepping Stones in developing metacognitive strategy instruction, a clearer exposition of these materials is essential.

Collaboration as a Mechanism: Underdeveloped Link to Metacognition

While the discussion rightly underscores the importance of collaboration in PD contexts, its specific role in fostering metacognitive strategy development is not well theorized. Were collaborative structures—such as peer feedback, joint lesson planning, or group reflection—explicitly leveraged to promote metacognitive awareness and teaching practices? Could it be argued that teachers co-constructed metacognitive strategies in groups? Such questions remain unanswered, leaving a conceptual gap between the collaborative processes described and the cognitive outcomes desired.

Call for Stronger Programme-Level Evidence

Overall, the assertion that Stepping Stones is an effective PD program is primarily supported through teacher perspectives and self-reported gains. While valuable, these forms of evidence require triangulation with detailed program content and concrete examples of instructional change. For instance, claims such as “worked examples are effective means of scaffolding learning” would be more convincing if accompanied by illustrative excerpts or classroom enactments showing how these examples were used to model metacognitive strategies.

To make a compelling case for the program’s impact, the study needs to articulate not only that teachers improved, but how the specific design features of the PD program supported that improvement—particularly in relation to the teaching of metacognitive strategies.

 

 

 

Author Response

Many thanks for your feedback on our submission. We have used this to inform our revisions. All of our changes are highlighted in yellow in the text. An overview of our responses and amendments is provided below.

 

Comment: Qualitative Data Analysis Concerns
In the qualitative analysis phase, the authors employed 18 codes grouped under three main categories: professional development design, feelings of increased ownership, and the value of collaboration. However, it remains unclear whether these codes emerged inductively through open coding or were derived deductively from a pre-established framework informed by existing literature. This methodological ambiguity raises questions about the coding approach and its alignment with grounded theory or theory-driven analysis.

 

Response:

We appreciate the careful attention to our analytical approach. We have clarified in the revised version of 2.6 Qualitative Data Analysis that we adopted a reflexive thematic analysis (RTA) framework as outlined by Braun and Clarke (2022), which explicitly allows for flexibility between inductive and deductive modes of analysis. RTA is distinct from grounded theory and does not require a strict commitment to either induction or deduction. Instead, our coding reflected a reflexive engagement with the data, shaped by both our insider and outsider researcher positions, subjectivity and theoretical understandings. We have clarified this approach and its rationale more explicitly in the revised section, and have also added Figure 6 to represent this analysis process – from our 18 initial codes, through to 4 candidate themes, and then the final 3 themes presented in this study.

 

Comment:

Furthermore, there appears to be an inconsistency between the categories derived during coding and the thematic findings reported in the results section. The three reported themes—such as building knowledge and confidence—primarily reflect teachers’ conceptualizations of planning and enacting metacognitive strategies, which do not directly correspond to the original coding categories. This discrepancy calls for clarification: Were the reported themes the result of a secondary level of axial or selective coding? Or do they reflect an interpretive synthesis rather than a direct extension of initial codes?

 

Response:

Thank you for this comment. We have revised section 2.6 and included Figure 6 to more clearly explain the progression from initial coding to candidate themes and then final themes. Rather than following a grounded theory-style axial or selective coding process, our development of themes was through the six phases of reflexive thematic analysis (Braun and Clarke, 2022).

 

The shift from initial codes to the final themes was not intended as a direct one-to-one categorisation but as a synthesis developed collaboratively by the research team. This synthesis allowed us to bring together patterns of meaning that spoke more broadly to the participants’ experiences and the research aims. We have made this process more visible in both our description of the phases of analysis and in the new Figure 6, which notes the initial codes through to final themes.

 

We hope these clarifications demonstrates how this is in fact a characteristic of reflexive thematic analysis, where themes are constructed, but rather explore patterns of meaning interpreted by the researchers.

 

Comment: On the Use of a Convergent Mixed-Methods Design
Although the study claims to adopt a convergent mixed-methods design, its execution raises methodological concerns. Both qualitative and quantitative datasets are presented, and their sources and uses are transparent. However, the study falls short in demonstrating how these two strands were meaningfully compared or integrated to generate comprehensive interpretations. It is difficult to trace where the qualitative and quantitative data were systematically compared or merged to yield integrative insights—two core criteria for validating a convergent design.

 

Response:

Thank you for the comment. We agree there is a lack of detail concerning the mix-method convergent design employed in the study. As such, we’ve added a Figure to illustrate the process of data collection, analysis, integration and interpretation. We have also added a detailed explanation outlining each step from data collection to interpretation to increase clarity for the reader.

 

Comment:

Instead, the role of quantitative data seems to be limited to offering a snapshot of the presence or absence of targeted metacognitive strategies—particularly through frequency tables such as Table 3. Items that yielded non-significant findings are not addressed in the discussion, undermining the interpretive balance. This raises a critical question: Does the mere inclusion of both qualitative and quantitative data in evaluating a PD initiative suffice to categorize the study as a convergent mixed-methods design? A more rigorous integration process would be required to substantiate such a claim, particularly when the qualitative strand appears to dominate the interpretive narrative (I think this is covered in my response above).

 

Response:
Thank-you for your comment. While Table 3 provides frequency counts of teacher-generated materials and the degree of their inclusion within Stepping Stones, it also presents Goodman and Kruskal's Gamma (Γ) values and associated p-values. These inferential statistics allowed us to move beyond ‘snapshots’, indicating the strength and direction of association between changes in teacher-generated materials and the progression of the programme's fading scaffolding. Our quantitative analysis provides a measure of how specific aspects of teacher planning changed during the programme. We appreciate the importance of exploring non-significant results to add an interpretive balance to our results and discussion more generally. As such, we’ve included a new paragraph in the results section and further interpretations are explored in the discussion section.

 

Results and Interpretive Depth: A Critical Commentary

The findings suggest that the Stepping Stones PD program positively influenced teachers’ development of knowledge, confidence, and motivation in teaching metacognitive strategies for mathematical problem solving. While such outcomes are indeed promising, the mechanisms by which the program brought about these gains remain insufficiently articulated. Specifically, it is unclear which particular features of the PD design—such as instructional routines, reflective tasks, or feedback mechanisms—directly contributed to these improvements. The results appear to rely heavily on teacher self-reports and survey responses, yet a more explicit and evidence-based account of how the PD fostered change would strengthen the study’s claims of impact.

 

Response:

Thank you for this comment. We have reviewed our submission to provide greater detail regarding the mechanisms by which the programme positively influenced teachers’ knowledge, confidence and motivation.

 

We have also made further amendments to clarify the data that supports our claims. Whilst our study does use teacher self-reported data, this is triangulated with 2 sets of observational data collected by the research team: 1) checklist of implemented programme features, completed by 2 members of the research team after observing participating teachers’ delivery of a programme session; and 2) checklist of programme features, completed from a review of participant-generated materials in the final 4  (fading) weeks of the programme. These checklists were also completed by 2 members of the project team. 12 of these checklists were completed in total. We have added additional references to the role of the research team in collecting this data where appropriate throughout our submission to try to clarify this to the audience.

 

Comment: Lack of Detail on Instructional Design and Strategy Implementation

The program materials, including pupil-facing slides and teacher scripts, are referenced as central tools; however, the actual content and pedagogical structure of these resources are not adequately described. For instance, were teachers provided with written scripts that included metacognitive prompts or problem-solving tasks? The mention of scripting tasks in the early weeks of the program suggests a scaffolded approach to classroom implementation, possibly intended to support the internalization of metacognitive teaching strategies. Yet, without concrete examples or excerpts from these scripts, it is difficult to ascertain how such tools contributed to teachers’ instructional repertoires.

 

Similarly, references to worked examples and visual representations imply a structured instructional model, but the absence of detail regarding their content, cognitive demand, or integration into the teaching sequences leaves the reader unconvinced of their instructional power. If the goal is to demonstrate the efficacy of Stepping Stones in developing metacognitive strategy instruction, a clearer exposition of these materials is essential.

 

Response:

Thank you for your comment. We have significantly extended the ‘Present Study’ section of our submission to include greater detail regarding the instructional design of the programme. We have also included example programme materials to complement this description in Figures 1-3. All changes have been highlighted in yellow.

 

Comment: Collaboration as a Mechanism: Underdeveloped Link to Metacognition

While the discussion rightly underscores the importance of collaboration in PD contexts, its specific role in fostering metacognitive strategy development is not well theorized. Were collaborative structures—such as peer feedback, joint lesson planning, or group reflection—explicitly leveraged to promote metacognitive awareness and teaching practices? Could it be argued that teachers co-constructed metacognitive strategies in groups? Such questions remain unanswered, leaving a conceptual gap between the collaborative processes described and the cognitive outcomes desired.

 

Response:

Thank you for this comment. We have added further detail regarding the collaborative processes used within the Stepping Stones model between lines 290 and 304. This section is also highlighted in yellow.

 

Comment: Call for Stronger Programme-Level Evidence

Overall, the assertion that Stepping Stones is an effective PD program is primarily supported through teacher perspectives and self-reported gains. While valuable, these forms of evidence require triangulation with detailed program content and concrete examples of instructional change. For instance, claims such as “worked examples are effective means of scaffolding learning” would be more convincing if accompanied by illustrative excerpts or classroom enactments showing how these examples were used to model metacognitive strategies. To make a compelling case for the program’s impact, the study needs to articulate not only that teachers improved, but how the specific design features of the PD program supported that improvement—particularly in relation to the teaching of metacognitive strategies.

 

Response:

The mixed methods design of our study provides opportunities to triangulate teachers’ perspectives and self-reported gains with review of their implementation of programme features, during both an observation of programme delivery in participating teachers’ schools, and through review of participant-generated programme materials during weeks 7-10. We have added greater detail of the nature of this additional data and its purpose of triangulation in our method. We have also added detail throughout the findings and discussion section to clarify that we are drawing upon multiple, complementary data sets to inform our assertions.

In addition to this detail about the nature of our data, we have also added increased information about the design features of both the Stepping Stones programme and its accompanying PD offer. We hope that this serves to make our argument more convincing to the reader.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Many thanks for the careful revision. My previous comments have been incorporated satisfactorily.

Reviewer 2 Report

Comments and Suggestions for Authors

Following the recommended revisions, the authors have thoroughly addressed previously unclear or insufficiently explained aspects of the study and implemented the necessary improvements. These revisions have clearly enhanced the overall quality of the manuscript. The study is expected to make a valuable contribution to the field. 

With kind regards,