Exploring Polygonal Number Sieves through Computational Triangulation
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsI noticed "...in shown in the bottom part ..." line 287
It's good to show the need to view our results critically.
I like explanations concerning deviations in the sieves.
Author Response
R1: I noticed "...in shown in the bottom part ..." line 287
A: Corrected. Please see line 319
R1: It's good to show the need to view our results critically.
A: Thank you.
R1: I like explanations concerning deviations in the sieves.
A: Thank you.
Reviewer 2 Report
Comments and Suggestions for AuthorsI would like to acknowledge that the topic of this manuscript, exploring polygonal number sieves through computational triangulation, is highly interesting. The paper present how digital computations in elementary number theory can support educational activities with polygonal numbers. Two methods underpin these activities: computational triangulation and the TITE (technology-immune/technology-enabled) framework. This framework represents a synergy of computational thinking, resulting digital computations, and the epistemic interpretation of those outcomes. The paper is relevant and important as it demonstrate the study of the mathematical phenomenon based on new experimental approach to mathematics, that builds on previous approaches to mathematics, yet adds something new - extensive use of computer tools to deal with mathematical problems.
However, an issue arises in the last sentence of the paper (lines 449-452), where the instrumental method is briefly mentioned without sufficient elaboration. This mention may be unclear to readers unfamiliar with the method and requires further explanation for better comprehension.
Author Response
R2: I would like to acknowledge that the topic of this manuscript, exploring polygonal number sieves through computational triangulation, is highly interesting. The paper present how digital computations in elementary number theory can support educational activities with polygonal numbers. Two methods underpin these activities: computational triangulation and the TITE (technology-immune/technology-enabled) framework. This framework represents a synergy of computational thinking, resulting digital computations, and the epistemic interpretation of those outcomes. The paper is relevant and important as it demonstrate the study of the mathematical phenomenon based on new experimental approach to mathematics, that builds on previous approaches to mathematics, yet adds something new - extensive use of computer tools to deal with mathematical problems.
Thank you.
R2: However, an issue arises in the last sentence of the paper (lines 449-452), where the instrumental method is briefly mentioned without sufficient elaboration. This mention may be unclear to readers unfamiliar with the method and requires further explanation for better comprehension.
A: Please see new text, lines 134-144.
Reviewer 3 Report
Comments and Suggestions for AuthorsThe author has done a lovely job in showcasing how technology may be used to promote exploration within secondary mathematics. This article is firmly grounded in his rich and robust research tradition, methods, and perspectives. If offers exellent insight into mathematical pedagogy as well as best practice use of technological tools.
I am grateful for the opportunity of reviewing this work. There was much here that I found personally interesting and enlightening.
This was an interesting review as several of the prior review prompts seemed centered around a more research centered paper. This particular paper illustrates a pedagogical approach in developing technology enabled skills in secondary teachers. As such, it provided a rich illustration of pedagogy where technology was useful in exploring number sieves - a non-traditional topic via a non-traditional approach.
This means that a research centered set of review questions does not adequately address the focus of the paper. This was why I indicated the NOT APPLICABLE option on those areas of the review and did not address them.
The main topic of pedagogical development (I am purposely using the terms topic - not question - and pedagogical development rather than research) and lies in how one can use technological resources available in the schools (in this case Wolfram ALPHA, spreadsheets, MAPLE, and the Online Encyclopedia of Integer Sequences), to explore the topic of polygonal number sieves. This is a non-traditional approach to an admittedly non-traditional mathematical topic and one which I feel the readers would find intensely interesting. It is definitely a fresh exemplar of a well researched technology-immune/technology-enabled (TITE) pedagogy.
This very well written description and rich walk through of process provides a fresh description to how such tools might be used in a TITE teaching environment. It serves as both an introduction to this approach as well as tying into the extensive research and development into the TITE framework.
This ties the paper into a much larger set of demonstrations and descriptions. Given the uniqueness of the problems explored in this article, while supporting prior writings surroundings TITE approaches, stands on its own and expands upon approaches and processes.
This paper illustrates a very non-traditional approach to an admittedly non-traditional mathematical topic. Many ideas in mathematics education make use of a similar set of problem spaces. This new approach toward number sieves is truly unique and extremely interesting. I found myself revisiting the OEIS and exploring a few ideas after reading the article.
This is clearly a stand-alone article. However, it provides a powerful bridge to other writings surrounding the TITE framework. To have such a rich demonstration presented by one of the leading thinkers in TITE pedagogy makes this a strong contribution to the field.
Author Response
R3: The author has done a lovely job in showcasing how technology may be used to promote exploration within secondary mathematics. This article is firmly grounded in his rich and robust research tradition, methods, and perspectives. If offers exellent insight into mathematical pedagogy as well as best practice use of technological tools.
I am grateful for the opportunity of reviewing this work. There was much here that I found personally interesting and enlightening.
This was an interesting review as several of the prior review prompts seemed centered around a more research centered paper. This particular paper illustrates a pedagogical approach in developing technology enabled skills in secondary teachers. As such, it provided a rich illustration of pedagogy where technology was useful in exploring number sieves - a non-traditional topic via a non-traditional approach.
This means that a research centered set of review questions does not adequately address the focus of the paper. This was why I indicated the NOT APPLICABLE option on those areas of the review and did not address them.
The main topic of pedagogical development (I am purposely using the terms topic - not question - and pedagogical development rather than research) and lies in how one can use technological resources available in the schools (in this case Wolfram ALPHA, spreadsheets, MAPLE, and the Online Encyclopedia of Integer Sequences), to explore the topic of polygonal number sieves. This is a non-traditional approach to an admittedly non-traditional mathematical topic and one which I feel the readers would find intensely interesting. It is definitely a fresh exemplar of a well researched technology-immune/technology-enabled (TITE) pedagogy.
This very well written description and rich walk through of process provides a fresh description to how such tools might be used in a TITE teaching environment. It serves as both an introduction to this approach as well as tying into the extensive research and development into the TITE framework.
This ties the paper into a much larger set of demonstrations and descriptions. Given the uniqueness of the problems explored in this article, while supporting prior writings surroundings TITE approaches, stands on its own and expands upon approaches and processes.
This paper illustrates a very non-traditional approach to an admittedly non-traditional mathematical topic. Many ideas in mathematics education make use of a similar set of problem spaces. This new approach toward number sieves is truly unique and extremely interesting. I found myself revisiting the OEIS and exploring a few ideas after reading the article.
This is clearly a stand-alone article. However, it provides a powerful bridge to other writings surrounding the TITE framework. To have such a rich demonstration presented by one of the leading thinkers in TITE pedagogy makes this a strong contribution to the field.
A: Thank you.
Reviewer 4 Report
Comments and Suggestions for AuthorsThe article is highly informative and beneficial for readers seeking a comprehensive understanding of triangular number sieves. It provides valuable insights into the subject matter and is particularly inspiring for the mathematical community, especially students studying mathematics.
However, to enhance the reader's comprehension and improve the article's utility, I would like to suggest incorporating two additional figures in Section 3 - "Triangular number sieves: the first approach" (line 147). These figures could illustrate the input box of Wolfram Alpha for both "Finding a closed-form formula for the triangular number sieve of order zero" and "Finding a closed-form formula for the triangular number sieve of order one." Integrating these visual aids would significantly assist readers in navigating through the text and grasping the concepts more efficiently.
Furthermore, while Figure 9 - "Triangular number sieves of orders 0, 1, 2, ..., 10 horizontally displayed" (line 261) provides a broad overview, it might be more effective to exclude it. Instead, emphasizing the vertically displayed figures, which convey similar information, would maintain consistency throughout the article and align with the presentation style of the visuals.
In conclusion, despite these suggestions for potential improvements, I highly recommend publishing the article. Its inspirational content and relevance to the broader mathematical audience, especially students, make it a valuable addition to the academic discourse in mathematics.
Author Response
R4: The article is highly informative and beneficial for readers seeking a comprehensive understanding of triangular number sieves. It provides valuable insights into the subject matter and is particularly inspiring for the mathematical community, especially students studying mathematics.
A: Thank you.
R4: However, to enhance the reader's comprehension and improve the article's utility, I would like to suggest incorporating two additional figures in Section 3 - "Triangular number sieves: the first approach" (line 147). These figures could illustrate the input box of Wolfram Alpha for both "Finding a closed-form formula for the triangular number sieve of order zero" and "Finding a closed-form formula for the triangular number sieve of order one." Integrating these visual aids would significantly assist readers in navigating through the text and grasping the concepts more efficiently.
A: Please see new text and new Figs 1 and 2, lines 167-192.
R4: Furthermore, while Figure 9 - "Triangular number sieves of orders 0, 1, 2, ..., 10 horizontally displayed" (line 261) provides a broad overview, it might be more effective to exclude it. Instead, emphasizing the vertically displayed figures, which convey similar information, would maintain consistency throughout the article and align with the presentation style of the visuals.
A: Figure 9 has been left out.
R4: In conclusion, despite these suggestions for potential improvements, I highly recommend publishing the article. Its inspirational content and relevance to the broader mathematical audience, especially students, make it a valuable addition to the academic discourse in mathematics.
A: Thank you.
Reviewer 5 Report
Comments and Suggestions for AuthorsThe paper presents an exploration of subsequences of polygonal numbers with various sides, derived through a step-by-step elimination of terms from the original sequences. This process is akin to the development of the classic sieve of Eratosthenes, which involves the elimination of multiples of primes. The paper is positioned in the context of elementary number theory, targeting technology-enhanced secondary mathematics education. Tools such as spreadsheets, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences are utilized. The paper introduces general formulas for subsequences of polygonal numbers, termed as polygonal number sieves of order k, incorporating base-two exponential functions of k. Various problem-solving approaches, both technology-immune and technology-enabled, are explored in deriving these and other sieves. The accuracy of computations and mathematical reasoning is substantiated through computational triangulation, employing multiple digital tools. The paper also briefly references relevant historical aspects of mathematics.
Strengths:
1. The topic and content are intriguing, presenting a novel approach to exploring polygonal numbers.
2. The methodology, involving computational triangulation and the use of multiple digital tools, adds depth and reliability to the findings.
3. The integration of technology in exploring elementary number theory concepts is well-aligned with modern educational needs.
Weaknesses:
1. The language requires refinement by a native speaker for clarity and precision.
2. The novelty and contribution of the article are not clearly articulated. It lacks a distinct statement of its advancements over previous works.
3. The objectives of the paper, both primary and secondary, are vague and need to be more explicitly defined.
4. The paper should clarify whether its focus is predominantly didactic or research-oriented.
The article "Exploring Polygonal Number Sieves through Computational Triangulation" presents a valuable and innovative approach to understanding polygonal numbers using modern computational tools. However, it requires minor revisions to enhance clarity, particularly in defining its novel contributions and objectives. The potential impact of the work in the field of mathematics education is notable, provided these adjustments are made.
Comments on the Quality of English Languagenone
Author Response
R5: The paper presents an exploration of subsequences of polygonal numbers with various sides, derived through a step-by-step elimination of terms from the original sequences. This process is akin to the development of the classic sieve of Eratosthenes, which involves the elimination of multiples of primes. The paper is positioned in the context of elementary number theory, targeting technology-enhanced secondary mathematics education. Tools such as spreadsheets, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences are utilized. The paper introduces general formulas for subsequences of polygonal numbers, termed as polygonal number sieves of order k, incorporating base-two exponential functions of k. Various problem-solving approaches, both technology-immune and technology-enabled, are explored in deriving these and other sieves. The accuracy of computations and mathematical reasoning is substantiated through computational triangulation, employing multiple digital tools. The paper also briefly references relevant historical aspects of mathematics.
A: Thank you.
R5: Strengths:
- The topic and content are intriguing, presenting a novel approach to exploring polygonal numbers.
- The methodology, involving computational triangulation and the use of multiple digital tools, adds depth and reliability to the findings.
- The integration of technology in exploring elementary number theory concepts is well-aligned with modern educational needs.
A: Thank you.
R5: Weaknesses:
- The language requires refinement by a native speaker for clarity and precision.
A: With due respect, may I share with you my experience with seeking help of a native speaker. It was more than 30 years ago. My paper (hard copy) was given to a native speaker to edit. The paper had the word equation written many times (as you can imagine). The edited version had each equation changed to equasion. Apparently, the native speaker knew how to speak the word equation but did not know how to write it. Since then, I have been very reluctant seeking help of a native speaker. So, this week (after completing the second pass of page proofs for my 13th book), I proofread the paper by myself and hope that it displays clarity and precision.
- The novelty and contribution of the article are not clearly articulated. It lacks a distinct statement of its advancements over previous works.
A: Please see new text, lines 470-472, including two new references.
- The objectives of the paper, both primary and secondary, are vague and need to be more explicitly defined.
A: Please see new text, lines 90-96, 441-442.
- The paper should clarify whether its focus is predominantly didactic or research-oriented.
A: Please see new text, line 20, lines 88-90.
R5: The article "Exploring Polygonal Number Sieves through Computational Triangulation" presents a valuable and innovative approach to understanding polygonal numbers using modern computational tools. However, it requires minor revisions to enhance clarity, particularly in defining its novel contributions and objectives. The potential impact of the work in the field of mathematics education is notable, provided these adjustments are made.
A: Please see my answers above.
Reviewer 6 Report
Comments and Suggestions for AuthorsThis manuscript was written to explore subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences using special rules. General formulas for subsequences of polygonal numbers have been developed using different problem-solving approaches. The accuracy of computations and mathematical reasoning was confirmed through the technique of computational triangulation enabled by using more than one digital tool.
The manuscript was well written and organized. The idea was clearly presented with suitable historical background. The types of materials have been used seem to be appropriate. The methods and the techniques have been used seem to be correct. The conclusion summarized the outcomes of the manuscript in a good way.
However, the punctuation marks need to be used throughout. For example, in the abstract: General formulas for subsequences of polygonal numbers, referred to in the paper as polygonal number sieves, of order k that include base-two exponential functions of k have been developed.
Comments on the Quality of English Language
The punctuation marks need to be used throughout. For example, in the abstract: General formulas for subsequences of polygonal numbers, referred to in the paper as polygonal number sieves, of order k that include base-two exponential functions of k have been developed.
Author Response
R6: This manuscript was written to explore subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences using special rules. General formulas for subsequences of polygonal numbers have been developed using different problem-solving approaches. The accuracy of computations and mathematical reasoning was confirmed through the technique of computational triangulation enabled by using more than one digital tool. The manuscript was well written and organized. The idea was clearly presented with suitable historical background. The types of materials have been used seem to be appropriate. The methods and the techniques have been used seem to be correct. The conclusion summarized the outcomes of the manuscript in a good way.
A: Thank you.
However, the punctuation marks need to be used throughout. For example, in the abstract: General formulas for subsequences of polygonal numbers, referred to in the paper as polygonal number sieves of order k that include base-two exponential functions of k have been developed.
A: I marked commas added to the abstract to address your comment. Other places in the paper were proofread with attention to commas in mind.
