The Role of Spatial Reasoning in Growing and Spatial-Repeating Patterns in First and Second Graders’ Structural Development of Mathematics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you for giving me the opportunity to read the manuscript. I believe it is an important work in the field of mathematics education, and I enjoyed reading it. Below are some suggestions and comments that may improve the report.

The aim  of this report is to develop a more coherent understanding of the interrelationships between different forms of pattern (repetitions and growing) and  between patterning, spatial skills and mathematical knowledge. Through a descriptive analysis of estructural levels they establish the Integration of spatial skills with repetitions and the growing patterns.

Its strengths is the micro.level analyses which let knowing children’ strategies evidencing the interrelation between spatial skills and repetitions patterns.

The contribution comes from empirically demonstrating the complexity of children's patterning and spatial concepts and the integration of a range of mathematical skills, which provides teachers with information for designing pattern-related tasks that enable them to connect mathematical skills and concepts.

One of the weakest sections of the article is the methodology.

In section 4.1 Context and participants, it would be interesting to explain the participants' prior knowledge and learning of the patterns. The learning context is relevant, as is the curriculum developed in schools, in order to understand the results in the context of the research, which will surely differ from other countries.

In page 5 of 19, lines 231-233, what was the purpose of observing and recording class sessions? How was this data subsequently handled? If it was not used in this research, this should be stated; otherwise, the type of analysis and how the recordings were used should be explained.

Was Table 1 completed based on the students' responses?  [...] Coding descriptors for the PASA interview were developed to distinguish qualitative        248

differences in the variation of responses according to five increasing levels of structural   249

development of patterning (see XXXX & Mitchelmore, 2009; XXXX et al., 2020a,  XXXX et              250

al., 2020b)

Did the coding follow an inductive or deductive analysis?

Is this a mixed or exclusively qualitative interpretative study? This aspect is relevant for understanding section 5.1 of the results section.

Referring to result section:

In particular, the one referring to descriptive macro-analysis (5.1. Phase 1–Macro-level analysis, p. 7 of 18). The authors simply establish the frequencies of the responses and obtain the percentage. It does not seem strange that there may be a slight increase in the frequency of G2 students moving from the structural level to the advanced level, in relation to G1 students. This aspect should be discussed in detail in the discussion section. On the other hand, a more detailed statistical study is needed to examine the correlation of responses and standard deviations and thus determine the significance of the mean values proposed by the authors in Figure 5.2.

With such a large sample of students, a statistical study, such as latent classes or another, could have confirmed or refuted the hypothesis about the interrelation of the variables and supported the empirical evidence. If possible, and without this being another study, it is suggested that this methodological section be further explored and revised.

Although the figure and table of percentages allow the results to be visualised, they do not provide information relevant to the objective of the article. The question I suggest to the authors is: if we were to dispense with section 5.1 as it currently stands, would this affect the results described in section 5.2? Is section 5.1 dispensable, or does it corroborate the empirical results exemplified in section 5.2? The authors may have conducted a confirmatory statistical study, but its purpose is unclear as it appears in the report.

Other comments and suggestions:

The article is well written, coherent and clear. Overall, I consider it to be a relevant piece of research, although the authors should elaborate further on the contributions of the research and the relevance of the findings, which are not obvious and must be inferred by the reader after several readings of the manuscript.

As the authors clearly show in the introduction, there is a great deal of research on this topic, not only at the early childhood education level, but also at the early primary education level. In addition, the authors have already conducted other specific research on the subject, which gives them the status of experts. However, I would like to focus my attention on the following paragraph:

(p, 4 de 18, lines 177-183):

Explicit interpretation of structural levels of responses supported by video data, in- 177

formed pedagogical ‘pathways’, resulting in the Pattern and Structure Mathematics Aware- 178

ness Program (PASMAP) (XXXX & Mitchelmore, 2018).    Findings gleaned from the PASA   179

interview  and  PASMAP  implementation  across  educational  contexts  confirmed  that,    180

while repetitions (unit of repeat) are fundamental, they limit pedagogical processes for     181

promoting advanced structural development, and for engaging children in growing and    182

spatial patterns.

 

Given that PASMAP was used and implemented in this study and two groups (G1 and G2) were analysed, the authors do not make it clear how this can be overcome. [...] they limit pedagogical processes for            181

promoting advanced structural development, and for engaging children in growing and    182

spatial patterns.

It is necessary to delve deeper into the methodological aspects that have changed with respect to previous research and that therefore allow them to delve deeper into student learning and the relationship with mathematical concepts.

In the discussion section (page 14 of 18, lines 432-433), to enrich the article, it would be desirable for the authors to delve deeper into possible learning sequences and necessary changes in the curriculum that, in view of the results, would support student progression. The comment is quite general.

Author Response

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Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The topic of the study is of practical significance and research value. The paper is generally well-structured and clearly written. However, there are several issues that require further revision and clarification before publication.

1. One of the listed keywords is “early childhood.” Although some definitions extend “early childhood” to include Grade 1 and Grade 2, in the mathematics education literature the term more commonly refers to preschool and kindergarten. Using this keyword may mislead readers into thinking that the study focuses on younger children.

2. The introduction cites a large body of literature on “early childhood mathematics.” However, the paper needs to explain more clearly why Grade 1 and Grade 2 students are included under this category.

3. At lines 156 and 184, please ensure that the MDPI heading format is used correctly.

4. The authors refer to “patterning skills,” “structural awareness,” and “spatial structuring,” but the meanings and distinctions among these core concepts are not sufficiently clarified. I recommend providing clearer definitions of these terms and explaining their specific roles within the context of the study.

5. The Methods section states that the total sample included Kindergarten children (n = 213), but the subsequent analysis appears to use only the Grade 1 and Grade 2 subsample (n = 405). Please provide an explanation.

6. Please ensure that the formatting of Table 1 is consistent.

7. Ethical considerations should be addressed more explicitly.

8. The “Implications” section covers a wide range of areas (teacher education, curriculum, learning trajectories, etc.), but some statements are relatively general. It is recommended that the authors be more specific: What are the direct implications of these findings for classroom instructional design or for mathematics curricula in the early primary grades?

Author Response

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Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

The primary structural revision needed involves clarifying the role of the classroom task in Section 5.3; you should either better integrate this section by reframing the paper's aims and discussion to create a cohesive narrative that connects the interview findings with their classroom application, or remove it to create a more focused article, saving the rich classroom data for a separate publication. Furthermore, the theoretical argument can be significantly sharpened by more explicitly applying the AMPS framework's distinction between its cognitive (knowledge of structure) and meta-cognitive (tendency to seek and analyze patterns) components within the micro-level analysis of the children's reasoning . Finally, a minor but helpful clarification should be made in the results section to explicitly link the finding that 15% of children visualized the spatial pattern as a rotation back to the Advanced structural level data presented in Table 2, thereby strengthening the connection between the qualitative and quantitative findings.

Author Response

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Author Response File: Author Response.docx

Reviewer 4 Report

Comments and Suggestions for Authors

Dear Authors.

The manuscript entitled “Integrating Spatial Reasoning with Repeating and Growing Patterns in the Early Development of Mathematical Knowledge” presents an interesting study based on a descriptive and interpretive analysis of young children’s responses to two mathematical patterning tasks (Growing Arrays and Spatial Patterns) within the framework of Awareness of Mathematical Pattern and Structure (AMPS). This study is grounded in a robust theoretical foundation and contributes meaningfully to the field of early childhood mathematics education by exploring the integration of spatial reasoning with patterning skills.

The manuscript is well-structured, clearly written, and demonstrates a strong command of the relevant literature, where the inclusion of micro-level analyses and classroom-based illustrations enriches the findings and provides valuable pedagogical insights. However, there are areas where the manuscript could be strengthened to enhance its academic rigour and clarity, which are explained in greater detail in the following paragraphs.

Regarding the title and abstract, the title is informative and accurately reflects the content of the manuscript, while the abstract succinctly summarises the study’s aims, methods, and key findings. However, it could benefit from a clearer articulation of the research questions and a brief mention of the implications of the study for pedagogy. Therefore, I suggest considering explicitly stating the two research questions in the abstract and briefly noting the pedagogical relevance of the findings of the study.

Regarding the introduction, it provides a comprehensive overview of the literature on early patterning and spatial reasoning, with an important breadth of references. However, it could be convenient to streamline the literature review to focus more sharply on the gap this study addresses.

Regarding the theoretical framework, the discussion of AMPS and spatial structuring is conceptually rich and well-integrated. The definition of spatial structuring (Batista, 1999) is appropriate, but the manuscript could benefit from a clearer explanation of how this concept operationally informs the coding scheme. Therefore, I recommend elaborating briefly on how spatial structuring was observed or measured by the authors in the children’s responses.

Regarding the methodology, the sample size and demographic details are clearly presented, where the use of a sub-sample for micro-analysis is justified. The coding scheme is well-defined and supported by inter-coder reliability measures. Although the description of the PASMAP implementation is informative but lacks detail regarding the instructional design and teacher training, so I recommend providing a bit more of information on the nature of the PASMAP lessons and how they were aligned with the AMPS framework.

Regarding the results, the macro-level analysis is clearly presented with appropriate tables and figures, and the micro-level analyses are a strength of the manuscript, offering rich qualitative insights into children’s reasoning. Furthermore, the integration of interview excerpts and visual representations is effective. Just as a minor suggestion, consider summarising key themes emerging from the micro-analyses to enhance coherence and facilitate comparison across AMPS levels.

Regarding the discussion, it effectively addresses the research questions and situates the findings within the broader literature. The interpretation of children’s dynamic visualisation and transformation skills is insightful. Furthermore, the manuscript acknowledges limitations appropriately. However, it would be convenient – in the conclusions section – to strengthen the discussion by more explicitly linking findings to implications for curriculum design and teacher education.

Finally, I highlight that this manuscript presents a valuable contribution to the field of early mathematics education. With minor revisions addressing clarity, completeness of references, and elaboration of pedagogical implications, it would be suitable for publication in Education Sciences. Therefore, the decision made on this manuscript is to ACCEPT AFTER MINOR REVISION.

Author Response

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Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Thank you to the authors for their efforts to improve their article. The authors have addressed most of the suggestions made in the previous review.
After reading the manuscript, I would just like to point out that I have detected some typographical errors:
at the end of line 86, a full stop is needed;
in line 246, Patterns should be in capital letters to be consistent with the acronym (SP).
In the references section, I found quite a few references where the DOI is missing, for example: reference line 595; lines 625-627; 634-635; 636-637; 641 and so on.
The references in lines: 596-598, which are missing the volume and pages, must also be completed, as must the reference between lines 603-604, which needs pages and DOI.

Author Response

The author thanks the reviewer for their second review of the paper.  The omissions and typographical errors have been corrected and the missing doi details have been added to many references. 

In addition, multiple fine edits and corrections for consistency of terms and sentence structure have been completed.

Several updated references have been added.