Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The submitted article addresses a timely and pedagogically relevant topic—mental calculation. The authors designed and conducted the research carefully on a representative sample and supported it with appropriate statistical analyses, including the use of the Bonferroni post hoc test, which significantly contributes to the reliability of the results and reduces the risk of Type I error.

Despite the overall methodological quality, I find some of the conclusions less convincing, for the following reasons:

1. While I agree with the authors on the importance of encouraging students to find and use their own calculation strategies, I do not believe that simply acquiring a greater number of strategies automatically leads to greater mathematical success. Mental calculation places a considerable burden on working memory, which has limited capacity. More complex strategies often involve an increased number of intermediate steps, which raises the likelihood of computational errors—particularly among students who have not yet fully automatized basic operations.
2. The study does not take into account the variable “students’ general success in mathematics”, which could represent a crucial moderating factor. It is possible that students with weaker mathematical skills may choose appropriate strategies but are unable to carry them out mentally due to limited working memory or insufficient procedural fluency. Even the authors themselves acknowledge that some participants, despite the instructions, resorted to using written algorithms or mentally visualizing them—possibly a sign of this developmental asymmetry. Without controlling for this factor, it is difficult to determine whether higher success in mental calculation is truly related to the number of strategies used, or rather to overall mathematical ability.

Textual weaknesses and recommendations for improvement:

1. A) Unclear and inconsistent use of key terms

The manuscript lacks clear definitions of fundamental terms—particularly strategy, algorithm, and specific procedure. As a result, several parts of the text, especially in the introduction, appear ambiguous or even contradictory. Two examples illustrate the issue:

“Standard algorithms privilege knowledge of basic facts and implementation of taught procedures whereas mental calculations focus on the structure of number operations and their relationships” (Rathgeb-Schnierer & Green, 2019).

The authors define mental calculation as performing calculations “in the head.” But what if a student mentally applies a standard algorithm—does it still qualify as mental calculation? Is an algorithm a strategy or not? Without precise definitions, the interpretation is unclear.

Another example:

“After learning standard algorithms, students often abandon previously taught number-based strategies, even when those are more efficient or appropriate for specific problems. Sequentially teaching strategies by example appears to hinder the development of flexibility.”

The authors first claim that students abandon strategies, then assert that they persist in using the first-learned procedures—this is logically contradictory. Furthermore, the suggestion that standard algorithms are not based on numerical properties is debatable. Is replacing an informal strategy with a more efficient algorithm inherently negative?

These inconsistencies suggest that the authors use key concepts too loosely and inconsistently, making it difficult to interpret the argument and understand what is actually being claimed. This also affects the clarity of the results presentation, which comes across as unstable and difficult to follow.

 

1. B) Suggested structural revision of the introductio

For better clarity and logical structure, I recommend dividing the Introduction section into two separate parts:

1. Introduction – presenting the motivation and the research objectives,
2. Theoretical Background or Review of Literature – where key concepts are clearly defined and a concise overview of the current state of research in the field is provided.

Such a division would help readers better distinguish between the motivation, theoretical framing, and research goals of the study.

 

1. C) Separate data analysis from discussion

Currently, the Results and Discussion sections are merged, which reduces clarity and weakens the interpretative depth. I recommend clearly separating:

• a Results section, in which only the statistical findings (e.g., descriptive data, test results) are presented,
• and a Discussion section, in which the authors interpret the findings, highlight the study’s contribution, and relate the results to existing theory and literature.

The discussion should go beyond merely repeating the results—it should offer critical reflection, acknowledge limitations, and discuss implications for educational practice.

Author Response

I am sending my report in a word document.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The study focuses on the issue of diagnosing mental calculation skills in school mathematics education. Mental calculations promote creativity and are crucial for cultivating mathematical thinking. However, in many educational systems, mathematics teaching focuses primarily on algorithmic procedures pupils are expected to learn (e.g., the algorithm for written addition and subtraction). Although algorithmic procedures lead to the correct result, they do not promote creativity in pupils and students when solving mathematical problems. The authors of the text point out the interesting fact that the diverzity of mental calculations does not change with the increasing age of pupils/students. It is therefore important to pay attention to the development of mental calculation already in primary mathematics education. I consider the detailed taxonomy of Mental Calculation Strategies (Table 1 – Table 3, lines 169-172) to be very useful.

The research was based on the evaluation of the results of a set of didactic tests completed by 233 respondents aged from 4th grade of primary school to the last year of univerzity education (7 age groups of respondents). The authors' goal was to find new relationships/correlations between the phenomena under study. The data obtained was systematically analyzed, and the authors reached some conclusions, which they present in the article. I believe that the results obtained are an authentic reflection of the everyday experiences of mathematics teachers. 

Given the small sample of respondents in the experiment in the context of a large age range, it is not possible to generalize the conclusions of the research. However, the experiment can be an inspiring source for further research in this area. The findings provide ideas for further research in this area.

The authors cite an extensive set of publications that are current and directly related to the issue at hand.

Structure and comments on the text:

Ad 1) Introduction

In the introductory part of the text, the issue addressed is anchored in detail in the context of previous research and available literary sources. The references cited are relevant to the issue addressed and the research.

Ad 2) Materials and Methods

Comments on the text:

• The text does not specify the criteria used to select the research respondents (line 224).
• The text does not specify the focus of the selected secondary schools and universities. The focus may play a significant role in the results of the research (line 224).
• The text does not specify the time allowed for solving the didactic text (lines 230-269).
• The text does not provide examples of test tasks. It is not clear whether these were only numerical tasks (in the sense of the tasks listed in Table 1 (line 169)) or whether the test also included, for example, verbal tasks or tasks leading to logical thinking processes. I recommend adding this information.
• The text does not specify how the Mental Calculation Success (MCS) value was calculated. Neither the procedure nor the numerical rating scale is specified. I recommend clarifying this.

 

 

Ad 3) Results and Discussion

• The text does not specify the variable Number of Mental Calculation Strategies (NSTR) (line 305). Does this refer to different strategies within a single task or the number of different strategies within the entire set of tasks solved?
• For Tables 6 and 8, Figure 2, I would recommend adding a legend.
• I would recommend standardizing the headings for Table 4 (Girls x Boys), Table 10 (M x F), Figure 2 (m x z), and Table 11 (F).

 

Ad 4) Conclusion and Implication

It is beneficial that the research confirmed the school's experience with the issue of low mental problem-solving levels by pupils/students in mathematics. The research also confirmed that the causes must be sought in primary mathematics education, where unnecessary emphasis is placed on the acquisition of algorithmic procedures instead of mental problem-solving of arithmetic tasks. This is a complicated area that provides scope for further similar professional research.

Author Response

I am sending my report in a word document.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Dear Authors.

The manuscript entitled "The Use of Different Strategies and Their Impact on Success in Mental Calculation" presents an interesting and well-structured study on the relationship between the number of mental calculation strategies employed by students and their success in this matter. In general terms, drawing on a sample of 233 Croatian students across a wide age range, the authors combine quantitative analysis with qualitative interviews to explore how strategy diversity correlates with performance, and how this varies by age and gender. The study is grounded in a robust conceptual framework and represents a contribution to the discourse on flexibility, number sense, and relational thinking in mathematics education. However, the manuscript presents some aspects to be improved, which are explained in greater detail in the following paragraphs.

My review of this manuscript is divided into three sections: first, I highlight the strengths of the study; second, I comment the weaknesses and make some suggestions for improvement; and third, I present my conclusions.

Strengths

The first aspect to highlight of this manuscript is its thorough theoretical foundation in the literature on mental calculation, flexibility, and relational thinking. The authors present, in a concise way, the key contributions from both cognitive psychology and mathematics education, providing a nuanced understanding of how strategy diversity supports mathematical reasoning.

The second aspect to highlight of this manuscript is its well-articulated methodological design, with a clear rationale for the use of both a mental calculation test and follow-up interviews with the students. The decision to avoid multiple-choice formats in favour of open-ended interviews is pedagogically sound, allowing for a more authentic capture of the strategic thinking of the participating students.

The third aspect to highlight of this manuscript is its statistical rigour, since the authors utilised the appropriate tools (e.g., Pearson's correlation, ANOVA, Bonferroni post hoc tests) that reinforce the credibility of the findings. In addition, the authors interpret the results in a way that these are clearly linked back to the study questions and hypotheses. Based on it, the authors point out their reflections on the implications for classroom practice.

Weaknesses and Areas for Improvement

Although this study is situated in the Croatian context, it would be interesting if the authors could expand their background by encompassing a broader international perspective. For example, by comparing Croatian mental calculation practices with other educational systems (inside and/or outside Europe). This could enrich the discussion and highlight the strengths and weaknesses of different educational contexts.

In addition, the authors point out that students often abandon mental strategies in favour of written algorithms due to curricular pressures. The authors could explore this point further with a more critical analysis of how national curricula and assessment methods shape pedagogical priorities and, consequently, student behaviour.

Although interviews were conducted with the participating students, the manuscript does not include direct excepts from their responses. It would be interesting to include some anonymised quotes, as they would provide richer insight into the students' reasoning and make the findings more tangible for readers.

Conclusions

This manuscript makes a valuable contribution to the field of mathematics education by providing evidence about the link between strategies diversity and mental calculation success. I consider it a theoretically grounded, methodologically sound, and pedagogically relevant study. Nevertheless, while generally clear, the manuscript would benefit from minor linguistic polishing. Some sentences are overly long or contain awkward phrasing (e.g., "students slowly deviate from 'school' methods…" in line 425). A careful copy-edit would enhance the readability of the manuscript. 

Author Response

I am sending my report in a word document.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

Thank you for the opportunity to review your manuscript. Your study addresses a timely and relevant topic in mathematics education. However, there are a number of important areas that require clarification, revision, or enhancement to improve the quality and scholarly rigour of the work.

Title and Abstract

The title is concise and accurately reflects the focus of the study, which is the use of different strategies in mental calculation and their impact on success. The abstract presents the aim, methodology, key findings, and recommendations. However, it omits specific details on the study’s design (e.g., whether it’s correlational, descriptive, or mixed-methods) and lacks information on the statistical methods used in data analysis. It is recommended that the authors briefly specify the research design and mention the key statistical techniques employed (e.g., correlation analysis, ANOVA) within the abstract to improve clarity.

Introduction

The introduction provides a comprehensive background on mental calculation, emphasising its importance in educational contexts and everyday life. The authors review relevant literature on mental calculation strategies, flexibility, and relational thinking, effectively building a case for the study. However, while the discussion touches upon theoretical concepts such as cognitive flexibility and relational thinking, there is no explicit section labelled as "Theoretical Framework." Although the cited theories serve as an implicit framework, the authors should formally define this section and clearly articulate how the theoretical concepts underpin their hypotheses and analysis approach. This will strengthen the link between theory and empirical investigation.

Purpose of the Study and Hypotheses

The study's aim is clearly stated, and the hypotheses are well-articulated, focusing on the correlation between strategy use and success in mental calculations, as well as age and gender differences. This section is well-written and aligns with the issues highlighted in the introduction. However, a brief mention of the study design (correlational, comparative, or cross-sectional) within this section would provide additional clarity for readers.

Materials and Methods

Study Design

The manuscript does not explicitly state the design of the study. However, from the description of data collection and analysis, it is evident that the study follows a quantitative, cross-sectional, correlational-comparative design. This should be clearly articulated in the methods section to provide readers with a clear understanding of the study’s structure.

Measuring Instruments

The authors developed a Mental Calculation Test inspired by McIntosh et al. (1995), containing 20 tasks across four arithmetic operations. The instrument was refined through a pilot study involving 31 participants. However, it is unclear whether the instrument was adopted, adapted, or newly developed—this distinction should be made explicit.

Furthermore, the manuscript does not report any psychometric validation (e.g., content validity through expert reviews or construct validity via factor analysis) of the final instrument. The lack of formal validation is a significant limitation that needs to be acknowledged. While the pilot study helped adjust task length and structure, no data is provided regarding the reliability or validity of the instrument in measuring the intended constructs.

Data Collection Procedures

The data collection process is well-documented. The authors conducted tests using oral instructions to minimise written computation, followed by individual interviews to explore the strategies used. This approach is suitable for capturing qualitative nuances of strategy use. However, no information is provided on inter-rater reliability or coder agreement in categorising participant responses, which raises concerns about subjective bias during data coding.

Results Section

The results are presented in a logical order, directly addressing the study's hypotheses. Tables and figures are used effectively to summarise descriptive and inferential statistics. The correlation between strategy use and calculation success is clearly demonstrated, and findings regarding age and gender differences are thoroughly analysed. However, the interpretation of some findings lacks caution. For example, stating that a hypothesis is "partially confirmed" despite non-significant p-values should be avoided. Non-significant findings should be acknowledged as such, with possible explanations explored in the discussion.

Discussion Section

The discussion effectively links findings to prior research, particularly in terms of flexibility, strategy use, and gender differences. However, the absence of a formally defined theoretical framework earlier in the paper weakens the connection between theory and empirical results. Additionally, the discussion does not address limitations related to the instrument’s lack of validation, potential interviewer bias, or the cross-sectional nature of the study, which limits causal inferences.

Recommendation: The discussion should explicitly address these methodological limitations and suggest directions for future research.

Conclusion and Implications

The conclusion is well-drafted, summarising the practical implications for mathematics education. It rightly emphasises the importance of teaching a variety of mental calculation strategies to foster flexibility and confidence among students. However, the authors should add a paragraph discussing the study’s limitations.

APA Referencing Style Compliance

The manuscript does not consistently follow APA 7th edition guidelines:

• In-text citations sometimes use inconsistent punctuation.
• The reference list lacks uniform formatting (e.g., italics for journal titles, volume numbers, missing DOIs).
• Some references list incomplete author information (e.g., missing initials or inconsistent name formatting).

Recommendation: A thorough reference audit is needed to ensure full compliance with APA 7th edition standards.

Author Response

I am sending my report in a word document.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Comments and Suggestions for Authors

All concerns raised have been addressed.