Visual Multiplication Through Stick Intersections: Enhancing South African Elementary Learners’ Mathematical Understanding

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presents an innovative and contextually significant contribution to mathematics education research. It addresses a pressing educational problem in South Africa and provides compelling evidence for the potential of a visual teaching approach. To meet the standards of reputable international journals, revisions should focus on (1) sharpening the positioning of the study within global research debates, (2) strengthening methodological rigor and addressing issues of generalizability, and (3) elaborating on the discussion of scalability and international implications. Detailed suggestions are provided below.

The title, “Visual Multiplication Through Stick Intersections: Enhancing Intermediate Phase Learners’ Understanding”, is clear and descriptive. It highlights the innovative teaching method and the learner group under study. However, it could be made more concise by omitting less essential words (e.g., “through stick intersections” could be revised to “using stick intersections”). Adding a geographical marker such as “in South Africa” may further enhance relevance for an international readership.

The introduction establishes a strong rationale by situating the problem within South Africa’s mathematics achievement gap, referencing TIMSS results and contextual challenges such as language diversity and resource constraints. To strengthen this section:

• Explicitly link the global discourse on visual and embodied pedagogy to the specific South African challenges.
• Provide a sharper articulation of the research gap, explaining why stick intersections were selected and how this approach differs from established array- or lattice-based multiplication methods.
• Incorporate more recent international studies (2020–2024) on visual, embodied, and multimodal learning in mathematics to better situate the contribution in global scholarship.

The mixed-methods design is appropriate and clearly articulated, combining pre-/post-tests, classroom observations, interviews, and teacher reflections. Strengths include the triangulation of data sources and the dual focus on conceptual understanding and learner engagement. Nevertheless, several aspects require clarification or expansion:

• Sample size and generalizability: The study is limited to 45 learners in one school. This limitation should be more explicitly acknowledged, and claims carefully bounded.
• Test validity and reliability: Provide details on how the pre- and post-tests were developed, piloted, or adapted from existing validated instruments.
• Ethics: Outline procedures for obtaining consent from learners, parents, and the school, and indicate whether ethical clearance was secured.
• Qualitative analysis: Offer more detail on the coding process for thematic analysis, including coder training, inter-rater checks, or steps taken to ensure trustworthiness.

The results are clearly presented, demonstrating both statistical and practical significance. Tables and effect sizes strengthen the quantitative analysis, and the improvements among lower-performing learners are especially compelling. Suggested refinements include:

• Adding measures of variability (e.g., standard deviations) to tables for more complete statistical reporting.
• Enriching the qualitative findings with illustrative excerpts from learner interviews and teacher reflections.
• Strengthening the integration of quantitative and qualitative strands to show how the two data sources complement and reinforce each other.

The discussion situates the findings within frameworks of embodied cognition, place value understanding, and cultural relevance, offering valuable insights for multilingual South African classrooms. Nonetheless, this section could be enhanced by:

• Positioning the novelty of the stick intersection method more critically against other visual multiplication strategies (e.g., lattice methods, dot arrays, digital visualizations).
• Expanding the cultural argument with reference to broader ethnomathematics literature, including comparative perspectives beyond African contexts.
• Addressing scalability in greater depth, including practical considerations such as large class sizes, teacher professional development, and curriculum integration.
• Expanding directions for future research, particularly the potential for digital or technological adaptation of the method.

The conclusion is concise and highlights the significance of the findings, including a compelling learner quotation. To enhance its impact:

• Restate the core empirical contribution (improved conceptual understanding of multiplication) and methodological contribution (use of a culturally relevant, visual teaching approach).
• Explicitly address implications for curriculum reform, teacher education, and professional development.
• Emphasize the international relevance of the approach, particularly for other developing or multilingual contexts.
• Reframe the conclusion to avoid overly advocacy-driven language, grounding claims more firmly in the evidence presented.

The reference list demonstrates broad engagement with both foundational and context-specific sources. However, revisions are needed to ensure compliance with APA 7th edition:

• Correct inconsistencies in formatting (capitalization, italics, sentence case, DOIs).
• Ensure that all in-text citations appear in the reference list, and vice versa, avoiding duplication or omission.
• Update entries listed as “advance online publication” with full bibliographic details where available.
• Incorporate more recent international studies (2020–2024) on visual pedagogy, digital mathematics education, and embodied cognition to strengthen global positioning.
• Balance South African scholarship with comparative international studies to enhance the manuscript’s appeal for a global readership.

In summary, the manuscript is innovative and contextually significant, but revisions are needed to sharpen its positioning in global debates, strengthen methodological rigor and transparency, and expand the discussion of scalability and international implications. Addressing these areas will substantially enhance the manuscript’s scholarly and practical contribution and increase its suitability for publication in a leading international journal.

Comments on the Quality of English Language

The English could be improved to more clearly express the research.

Author Response

Thank you for the opportunity to revise our manuscript based on the valuable feedback from three reviewers. We have carefully considered all comments and have made substantial revisions that significantly strengthen the manuscript's methodological rigor, theoretical positioning, and practical applicability. Below, I provide a detailed point-by-point response to each reviewer's comments.

Comment 1.1: Title Revision for Conciseness and Geographic Clarity
Reviewer's Suggestion: Make title more concise and add geographical marker.
My Response: I agree this enhances international readability. 
Action Taken: Revised title from "Visual Multiplication Through Stick Intersections: Enhancing Elementary Pupils' Understanding" to "Visual Multiplication Using Stick Intersections: Enhancing South African Elementary Pupils' Mathematical Understanding" 
Location: Title page
Comment 1.2: Strengthen Global Research Positioning
Reviewer's Suggestion: Explicitly link global discourse on visual pedagogy to South African challenges.
My Response: This connection strengthens the manuscript's international relevance. 
Action Taken: Added paragraph connecting global visual mathematics research to the specific South African context, emphasizing the research gap in multilingual developing contexts.
Location: Section 1, Introduction, end of paragraph 2
Comment 1.3: Research Gap Articulation
Reviewer's Suggestion: Explain why stick intersections were selected over other visual methods.
My Response: This clarification enhances the study's theoretical foundation. 
Action Taken: Added detailed rationale comparing stick intersections to lattice multiplication and dot arrays, highlighting three key advantages: minimal resources, universal accessibility, and natural scalability. 
Location: Section 1, after research questions
Comment 1.4: Methodological Rigor - Ethics and Validity
Reviewer's Suggestion: Provide details on ethical procedures and instrument validation.
My Response: These details are essential for replication and research integrity. 
Action Taken: Added comprehensive Section 3.2.1 "Ethical Considerations and Instrument Validation" including ethics approval details, consent procedures, content validity establishment, and reliability coefficients. 
Location: New subsection after Section 3.2
Comment 1.5: Qualitative Analysis Detail
Reviewer's Suggestion: Offer more detail on thematic analysis procedures.
My Response: Enhanced methodological transparency is crucial for trustworthiness. 
Action Taken: Expanded Section 3.5 with detailed description of Braun and Clarke's six-phase approach, inter-rater reliability statistics (κ = 0.84), member checking procedures, and triangulation strategies. 
Location: Section 3.5 Data Analysis
Comment 1.6: Statistical Completeness
Reviewer's Suggestion: Add measures of variability to tables.
My Response: Standard deviations provide essential statistical completeness. 
Action Taken: Revised all quantitative tables to include standard deviations alongside means. 
Location: Tables 1, 2, 3, and 5
Comment 1.7: Critical Positioning Against Other Visual Methods
Reviewer's Suggestion: Position stick intersections more critically against other visual strategies.
My Response: This provides balanced perspective rather than advocacy. 
Action Taken: Added detailed comparison with lattice multiplication and digital array visualizations, positioning stick intersections as uniquely suited to specific contexts rather than universally superior. 
Location: Beginning of Section 6 Discussion
Comment 1.8: Scalability Discussion
Reviewer's Suggestion: Address practical implementation considerations.
My Response: Scalability analysis enhances practical applicability. 
Action Taken: Added new Section 6.1 "Scalability and Implementation Considerations" addressing large class sizes, teacher training requirements, curriculum integration, and cost implications. 
Location: New subsection in Discussion
Comment 1.9: Enhanced Conclusion Impact
Reviewer's Suggestion: Restate empirical and methodological contributions more clearly.
My Response: Clearer contribution statements strengthen manuscript impact. 
Action Taken: Revised conclusion opening to explicitly state both empirical findings (d=1.31 effect size) and methodological contributions (resource-minimal intervention validation). 
Location: Section 7 Conclusion, paragraph 1
Comment 1.10: International Implications
Reviewer's Suggestion: Emphasize international relevance for other developing contexts.
My Response: This broadens the manuscript's global appeal and applicability. 
Action Taken: Added paragraph detailing relevance for other countries with rural populations, limited resources, or linguistic diversity. 
Location: Section 7 Conclusion, before final paragraph
Comment 1.11: Recent Literature Integration
Reviewer's Suggestion: Incorporate more recent international studies (2020-2024).
My Response: We acknowledge this suggestion but respectfully maintain focus on foundational theoretical frameworks. The current literature base provides strong theoretical grounding through seminal works (Lakoff & Núñez, 2000; Arcavi, 2003; Vergnaud, 1983) that remain highly relevant. Adding recent references solely for currency without clear theoretical enhancement would dilute the focused argument. Our contribution lies in applying established visual learning theories to a novel method in an under-researched context.

Reviewer 2 Report

Comments and Suggestions for Authors

Comments

I found the article very interesting and thought-provoking to read. The challenge of finding materials and/or models which make mathematical concepts meaningful to students is a perennial one. In the context of South Africa, I can only imagine the difficulties faced by teachers who are teaching in the English language which is not the first language of the students, in large classes, and with few educational materials.

The idea of using crossed sticks has appeal to me: it would be easy to physically collect and model single-digit multiplication problems. It would also be easy to draw representations of the sticks on paper or onto the ground.  I am keen to try that myself with younger children (5 to 6-year-olds) in my home country Australia. Where the visual representation provided in the article falls down in my view is when 22 X 4 is split into tens and ones with no diagrammatic differentiation between the sticks that are ones and the sticks that are tens. My other misgiving is the reliance on counting by ones to find solutions. I would prefer a model that encouraged mental calculation and flexibility. It seemed to me that that stick intersections were constructed to try to make sense of the standard written algorithm for multiplication.

However, I was convinced by the testing that was presented to show the students had learned to calculate multiplication problems. It might have been more convincing if the tests were included as an Appendix to the article. I would also advise that the Retention percentages column of the data were deleted. They are quite meaningless and the retention can be easily seen from the raw scores. It was good to see that the students had adopted the ideas introduced by the stick diagrams. I would be interested to read an article that reports your study about how you encourage students to transition from counting to calculate products to mentally manipulating numbers to calculate products.

I have written detailed comments on the article for your consideration. As you will see, I would have liked to see more detail in the Method section. As it is written at present, the experiment could not be replicated.

Take care not to over-claim in your findings section. I liked the use of comments by students which enlivened the article. My suggestion is to read what they say carefully for unintended messages. When a student says he/she no longer needs to memorise numbers, while I see what they mean, and I see what you intend, but I also want students to have knowledge of numbers.

Detailed Comments

Check Line.180-184. Is this your opinion? I thought that this was the matter you were studying. If it is not your view it needs a reference here please.

You wrote well about the limitations of aspects of your study. However, in Line 343-348 These two developments are most promising. What might be done to foster these approaches?

While you have included some excellent writers in the field. However, three of the 27 references cited were written in the last 10 years and the most recent citation has little to do with the specifics of your study being about secondary school. I understand that seminal writers should be cited but is there no recent research on teaching and learning early multiplication?

The weaknesses in this article in my view are the suitability of the experimental methodology, and sufficient details given to replicate the testing procedures.

Author Response

Thank you for the opportunity to revise our manuscript based on the valuable feedback from three reviewers. We have carefully considered all comments and have made substantial revisions that significantly strengthen the manuscript's methodological rigor, theoretical positioning, and practical applicability. Below, I provide a detailed point-by-point response to each reviewer's comments.

RESPONSE TO REVIEWER 2
Comment 2.1: Place Value Visual Differentiation Concern
Reviewer's Suggestion: Address lack of visual differentiation between tens and ones sticks.
My Response: We acknowledge this limitation while noting its pedagogical value. 
Action Taken: Added explanation in Section 4.3.1 showing how physical separation of place values makes concepts explicit, and acknowledged this as an area for future refinement in limitations section. 
Location: Section 4.3.1 and Section 5.3
Comment 2.2: Methodology Detail for Replication
Reviewer's Suggestion: Provide sufficient detail for replication.
My Response: Replication detail is essential for research utility. 
Action Taken: Significantly expanded Section 3.4 with week-by-week lesson structure, specific timing (45 minutes), materials used, and detailed implementation procedures. 
Location: Section 3.4 Instructional Intervention
Comment 2.3: Assessment Instrument Details
Reviewer's Suggestion: Include test details or as appendix.
My Response: Test specifications enhance replicability. 
Action Taken: Added new Section 3.3.1 detailing exact problem types, distribution, scoring methods, and administration procedures. 
Location: New subsection after Section 3.3
Comment 2.4: Table 4 Retention Percentages
Reviewer's Suggestion: Delete retention percentages as meaningless.
My Response: I agree this column is redundant. 
Action Taken: Removed "Retention Rate (%)" column from Table 4, retaining only raw scores that clearly show retention patterns. 
Location: Table 4 revision
Comment 2.5: Student Quote Interpretation
Reviewer's Suggestion: Carefully analyze student quote about not memorizing.
My Response: Important clarification to prevent misinterpretation. 
Action Taken: Added explanatory paragraph clarifying that the visual method supports rather than replaces number sense development and meaningful memorization. 
Location: Section 5.2.3, after student quotes

Reviewer 3 Report

Comments and Suggestions for Authors

This manuscript is well-written and well-organized. The details provided in the explanations and figures makes replicating the intervention of stick intersections seamless.  The strength of your research comes from the literature review in which you connect the importance of visual representations to multilingual students and the cultural backgrounds of the students. 

In the literature review, section 2.3, line 108, a specific example is needed on how teachers are able to use stick intersections to connect with the students' backgrounds and prior experiences. 

Your literature review names that stick intersections are a visual representation, however on line 10, 44, and 354 it is stated that stick intersections are also a concrete model. Did the students create models of the stick intersections with materials that were not on paper? If not, then remove the word concrete when describing stick intersections. 

In the literature review, lines 87-89, a brief connection was made to the array model, however, similarities and differences were never stated between the two models. I also wonder if a limitation should be added related to the array model since this model is also used to solve fractions, decimals, and polynomials and it might create misconceptions for the students later in their learning. 

This research seems like a multimethod study not a mixed method study. Several types of data were collected, however, there was not any indication of how the data was mixed, what type of mixed method study it was, or any references to the methods used.  If the data was mixed during either the design, data collection, or data analysis please include those details along with the specific type of mixed methods was implemented. 

Thank you for sharing your research to support students with their learning of mathematics.

Author Response

Thank you for the opportunity to revise our manuscript based on the valuable feedback from three reviewers. We have carefully considered all comments and have made substantial revisions that significantly strengthen the manuscript's methodological rigor, theoretical positioning, and practical applicability. Below, I provide a detailed point-by-point response to each reviewer's comments.

RESPONSE TO REVIEWER 3
Comment 3.1: Cultural Connection Examples
Reviewer's Suggestion: Provide specific examples of cultural background connections.
Our Response: Concrete examples strengthen the cultural relevance argument. 
Action Taken: Added detailed paragraph with specific examples from Sepedi culture, traditional counting practices, and connections to indigenous artistic traditions. 
Location: Section 2.3 Visual Learning in the South African Context
Comment 3.2: Concrete vs Visual Terminology
Reviewer's Suggestion: Remove "concrete" when describing paper-based method.
Our Response: Accurate terminology is important for clarity. 
Action Taken: Replaced "concrete" with "visual" or "visual-tactile" throughout the manuscript where referring to the drawn stick method. 
Location: Lines 10, 44, and 354 (multiple locations)
Comment 3.3: Array Model Comparison
Reviewer's Suggestion: State similarities and differences with array model and address potential limitations.
My Response: This comparison provides important theoretical context. 
Action Taken: Added detailed comparison in Section 4.1 highlighting advantages and acknowledging transition needs for advanced concepts. Also added limitation regarding future mathematical concepts. 
Location: Section 4.1 and Section 5.3
Comment 3.4: Mixed vs Multimethod Study Design
Reviewer's Suggestion: Clarify methodology type and data integration.
My Response: While I acknowledge the terminology distinction, my study employed triangulation across multiple data sources with integrated analysis as described in Section 3.5. The label "mixed-methods" accurately reflects our combination of quantitative assessments with qualitative observations and interviews, analyzed both separately and in integration. We believe changing this terminology would not enhance the manuscript's contribution or clarity.

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Accept in present form

Author Response

Reviewers comment: Accept in present form

Response: Decision accepted by all the authors

Reviewer 3 Report

Comments and Suggestions for Authors

Thank you for explaining your rationale for your mixed methods research study and the phase of when the data was integrated. Please add this detail to Section 3.5 because this is not clear as it is currently written. Understanding the phase the data was integrated and what data was mixed along with the priority, purpose, and sequence of the methods will help the readers understand the research design and methodology related to your research. See APA 7th edition pages 107. 

Author Response

COMMENT1: Thank you for explaining your rationale for your mixed methods research study and the phase of when the data was integrated. Please add this detail to Section 3.5 because this is not clear as it is currently written. Understanding the phase the data was integrated and what data was mixed along with the priority, purpose, and sequence of the methods will help the readers understand the research design and methodology related to your research. See APA 7th edition pages 107. 

RESPONSE: 

This study employed an explanatory sequential mixed methods design (Creswell & Plano Clark, 2018), with quantitative data collection and analysis prioritized and conducted first, followed by qualitative data collection and analysis to explain and expand upon the quantitative findings. The integration of quantitative and qualitative data occurred during the interpretation phase, after both strands had been independently analyzed. Specifically, pre- and post-test quantitative data were analyzed first to determine the statistical significance and magnitude of performance improvements. Subsequently, qualitative data from classroom observations, student interviews, and work samples were collected and analyzed to provide explanatory insight into how and why these improvements occurred—addressing questions such as what specific aspects of the method enhanced understanding, how students engaged with the visual approach, and what their subjective experiences were.

The purpose of mixing methods was complementarity and expansion: quantitative measures assessed the extent of learning gains, while qualitative data illuminated the mechanisms, processes, and student perspectives underlying these gains. During the final interpretation stage, quantitative results (Tables 1-5) were integrated with qualitative themes (Sections 5.2.1-5.2.3) through a process of comparison and connection, whereby statistical patterns were explained through corresponding qualitative evidence. For example, the quantitative finding of significant improvement among low-performing students (Table 5) was enriched by qualitative observations of increased engagement and confidence among previously passive learners. This integration approach allowed for a more comprehensive understanding of both the effectiveness and the experiential dimensions of the stick intersection method than either strand alone could provide.
NOTE: All the revisions are highlighted on pages 5-6. Relatedly, the new citation has also been referenced, see page 17.