Fostering Collateral Creativity in School Mathematics: Paying Attention to Students’ Emerging Ideas in the Age of Technology (Mathematics Education in the Digital Era, 23) 3031406389, 9783031406386

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Mathematics Education in the Digital Era

Sergei Abramovich Viktor Freiman

Fostering Collateral Creativity in School Mathematics Paying Attention to Students’ Emerging Ideas in the Age of Technology

Mathematics Education in the Digital Era Volume 23

Series Editors Dragana Martinovic, University of Windsor, Windsor, ON, Canada Viktor Freiman, Faculté des sciences de l’éducation, Université de Moncton, Moncton, NB, Canada Editorial Board Marcelo Borba, State University of São Paulo, São Paulo, Brazil Rosa Maria Bottino, CNR – Istituto Tecnologie Didattiche, Genova, Italy Paul Drijvers, Utrecht University, Utrecht, The Netherlands Celia Hoyles, University of London, London, UK Zekeriya Karadag, Giresun Üniversitesi, Giresun, Türkiye Stephen Lerman, London South Bank University, London, UK Richard Lesh, Indiana University, Bloomington, USA Allen Leung, Hong Kong Baptist University, Kowloon Tong, Hong Kong Tom Lowrie, University of Canberra, Bruce, Australia John Mason, The Open University, Buckinghamshire, UK Sergey Pozdnyakov, Saint Petersburg Electrotechnical University, Saint Petersburg, Russia Ornella Robutti, Dipartimento di Matematica, Università di Torino, Torino, Italy Anna Sfard, University of Haifa, Haifa, Israel Bharath Sriraman, University of Montana, Missoula, USA Eleonora Faggiano, University of Bari Aldo Moro, Bari, Italy

The Mathematics Education in the Digital Era (MEDE) series explores ways in which digital technologies support mathematics teaching and the learning of Net Gen’ers, paying attention also to educational debates. Each volume will address one specific issue in mathematics education (e.g., visual mathematics and cyber-learning; inclusive and community based e-learning; teaching in the digital era), in an attempt to explore fundamental assumptions about teaching and learning mathematics in the presence of digital technologies. This series aims to attract diverse readers including researchers in mathematics education, mathematicians, cognitive scientists and computer scientists, graduate students in education, policy-makers, educational software developers, administrators and teacher-practitioners. Among other things, the highquality scientific work published in this series will address questions related to the suitability of pedagogies and digital technologies for new generations of mathematics students. The series will also provide readers with deeper insight into how innovative teaching and assessment practices emerge, make their way into the classroom, and shape the learning of young students who have grown up with technology. The series will also look at how to bridge theory and practice to enhance the different learning styles of today’s students and turn their motivation and natural interest in technology into an additional support for meaningful mathematics learning. The series provides the opportunity for the dissemination of findings that address the effects of digital technologies on learning outcomes and their integration into effective teaching practices; the potential of mathematics educational software for the transformation of instruction and curricula and the power of the e-learning of mathematics, as inclusive and community-based, yet personalized and hands-on. Submit your proposal: Please contact the Series Editors, Dragana Martinovic (dragana@ uwindsor.ca) and Viktor Freiman ([email protected]) as well as the Publishing Editor, Marianna Georgouli ([email protected]). For a detailed list of Volumes published in this Series, see Mathematics Education in the Digital Era | Book series home (springer.com)

Sergei Abramovich · Viktor Freiman

Fostering Collateral Creativity in School Mathematics Paying Attention to Students’ Emerging Ideas in the Age of Technology

Sergei Abramovich School of Education and Professional Studies State University of New York at Potsdam Potsdam, NY, USA

Viktor Freiman Faculté des sciences de l’éducation Université de Moncton Moncton, NB, Canada

ISSN 2211-8136 ISSN 2211-8144 (electronic) Mathematics Education in the Digital Era ISBN 978-3-031-40638-6 ISBN 978-3-031-40639-3 (eBook) https://doi.org/10.1007/978-3-031-40639-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

A story of young Gauss supposedly having ingenious insight into the task of adding numbers from 1 to 100 made its way in the literature while drawing attention to opportunities for fostering mathematical creativity by young learners of the twentyfirst century. This book, in the series Mathematics Education in the Digital Era, is written to motivate creative mathematical thinking among students who are not necessary considered mathematically advanced. The book reflects the authors’ experience teaching mathematics to teacher candidates of Canada and the United States and supervising a number of field-based activities by the candidates. The aim of the book is to demonstrate how the appropriate use of technology, both physical and digital, in the teaching of mathematics creates conditions for the emergence of what may be called collateral creativity, a notion similar to the notion of “collateral learning” (Dewey, 1933, p. 49). Just as collateral learning does not result from the immediate goal of the traditional curriculum, collateral creativity does not result from the immediate goal of traditional problem solving. Rather, when problem solving is organized as the instrumental act, something that “enhances and immensely extends the possibilities of behavior by making the results of the work of geniuses available to everyone” (Vygotsky, 1930), mathematical creativity emerges as a collateral outcome of using technology. Furthermore, collateral creativity is an educative outcome of one’s learning experience with pedagogy that motivates students to ask questions about computer-generated or tactile-derived information and assists them in finding answers to their own or the teacher’s questions. Therefore, teachers’ role in promoting collateral creativity is more difficult than in promoting collateral learning—the former phenomenon is sustainable only when a teacher is open to address in some way students’ creative questions. In fact, the emerging collateral creativity of a student may not survive if challenging questions are not addressed at least in some way. The book intends to provide guidance to teachers for fostering collateral creativity in their classrooms.

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The following three examples from different classrooms will illustrate the notion of collateral creativity. To begin, consider a case of a learning environment which did not include any activities expecting students to be creative. An elementary teacher candidate worked with the beginning second graders who, mathematically speaking, were asked to represent the number 5 as a sum of two (whole) numbers. When it was discovered that the students were unable to comprehend the possibility of multiple additive decompositions, finger rings were bought in a dollar store and the students were asked to find all ways to put five rings on two fingers. This intervention with unusual concrete materials congenial to the students’ unconscious futuristic delight allowed them to be successful in representing 5 in six ways as it was needed for the context. But then a child asked: “What about five rings and three fingers, how many ways are there?” This question is an example of collateral creativity on the part of the second grader when the context within which young children worked, while being pretty mundane, had potential to stimulate their thinking outside the box. Put another way, whereas concrete thinking was found to be a barrier in the way of grasping abstraction by young children, concrete activity was used to overcome this barrier and, as a result, quite unexpectedly, an inspirational question was asked. This example also demonstrates that collateral creativity in order to bear fruit requires a knowledgeable teacher who can answer (or at least think how to answer) an unexpected question motivated by the learning environment. The second example of collateral creativity comes from a classroom of elementary teacher candidates who were comparing fractions in context using a spreadsheet designed to display the values of fractions as bar graphs. The context was as follows. There are two boxes of markers with 13 markers in the Dry Erase box and 18 markers in the Glassboard box. If there are 5 green markers in the former box and 7 green markers in the latter box, for which box are the chances higher to pick up without looking a green marker? The teacher candidates compared the fractions 5/13 and 7/18 and concluded that chances for a green marker are higher for the Glassboard box. To continue comparison of fractions, it was suggested to remove a green marker from each box and compare the chances again. The candidates compared the fractions 4/12 and 6/17 to find out that chances for a green marker are still higher for the Glassboard box. It was interesting to see that by taking again a green marker from each box the chances were higher for the Glassboard box. At that point, one of the teacher candidates asked a question: Are there boxes for which chances are reversed after removing one green marker from each box? The class tried many other pairs of fractions and could not find a pair satisfying the question asked. The question turned out to be very motivational demonstrating collateral creativity. Indeed, whereas such pairs of fractions do exist, it is not easy to find them. Once again, it is a learning environment, within which comparison of fractions (chances) was purely visual, that motivated a teacher candidate to ask a (creative) question the answer to which is not immediately available.

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The third example comes from a classroom of future secondary mathematics teachers who were using a spreadsheet to explore the behavior of integer number sequences through modeling the equation Fn+1 = a Fn + bFn−1 , n = 1, 2, 3, . . . , F0 = F1 = 1. It is known that when a = b = 1, the equation generates √ 1+ 5 Fibonacci numbers and the ratios Fn+1 /Fn converge to the number 2 ≈ 1.61803 called the Golden Ratio. Students tried different values of a and b to find a number to which the ratios FFn+1 converge and to express this number, called the generalized n Golden Ratio, through a and b. All of a sudden, a teacher candidate reported that when a = 3 and b = – 9 the ratios do not converge to a number but, instead, form a 3-cycle 1, – 6, 4.5, 1, – 6, 4.5, 1, – 6, 4.5, ... . That is, during a pretty routine use of a spreadsheet as a tool for modeling Fibonacci numbers and their parametric modifications, the following question was raised: Do there exist other values of parameters a and b for which the ratios Fn+1 /Fn form 3-cycles and if so, how can one find them? This question is an example of collateral creativity made possible by the ease of spreadsheet modeling. The answer to this question is not easy, and it can be made even more challenging if the query is extended to the existence of cycles of higher lengths and to finding relations between a and b that give birth to such cycles. That is, the ease of spreadsheet modeling of difference equations enabled the discovery that the generalized Golden Ratio may not only be a number but also a string of numbers. Once again, the discovery of a 3-cycle was accidental and could have been seen as a computational glitch had it not for a teacher to be able to recognize the significance of this “glitch”. As the above three examples demonstrate, collateral creativity is a psychological phenomenon inspired by teachers’ mastery to navigate between questions asked and answers provided by all the actors of a particular educational milieu. Furthermore, collateral creativity develops in a learning environment where teachers are capable of asking questions about a seemingly closed-ended situation and answer questions when a student, perhaps accidentally, turns a closed-ended situation into an open-ended one. An important aspect of the emergence of collateral creativity is that all three learning situations considered above were technology-enhanced. It is technology-generated information that inspires both students and teachers to ask questions and, to a certain extent, facilitates answering questions, and serves as the major encouragement of collateral creativity. The book consists of eight chapters and Appendix. The content of each chapter is introduced by a seemingly routine problem which when explored with technology, both physical and digital, reveals the problem’s hidden mathematical richness and conceptual limitless in terms of questions that can be asked by both teachers and students encouraging and displaying collateral creativity. In Chap. 1 titled Theoretical Foundation and Examples of Collateral Creativity, educational and psychological theories explaining the mechanism of collateral creativity in mathematics are presented. It is shown that all the theories converge to the idea of collateral creativity as an outcome of teaching mathematics with technology. Different classroom examples of collateral creativity demonstrated by K-12 teacher candidates and their future students and stimulated by the use of technology through lens of the instrumental act

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and affordances of tools are discussed. The role of teachers in supporting the emergence of collateral creativity of their students is emphasized. Overall, the chapter provides an introduction to other chapters of the book. Chapter 2 titled From Additive Decompositions of Integers to Probability Experiments deals with decomposition of integers into two and three summands with restrictions is presented in the context of distributing legs among artificial creatures using trial and error. A spreadsheet enabling the variation of data is used as an instrument conducive to interpreting and questioning the results of numeric modeling. Different questions with and without obvious answers are considered as examples of emerging collateral creativity within the instrumental act. It is shown how questions of the types “How many?”, “Why?”, “What if?”, and “Why not?” appear naturally as one plays with the spreadsheet and reflects on the change in numeric data. An inquiry into additive decomposition of integers is presented as one of the origins of probability theory. Different probability problems associated with rolling dice are considered. Their connection to the problems with artificial creatures is demonstrated. Experimental and theoretical probabilities of the corresponding events have been calculated and compared. Collateral creativity is shown as a means of enhancing credibility of independent experiments. In Chap. 3 titled From Number Sieves to Difference Equations, a number sieve is defined as a subsequence of natural numbers developed according to a certain rule. Basic examples of number sieves are presented. A spreadsheet is used as a modeling tool for generating different number sieves. These include even and odd numbers and polygonal numbers with different subsequences of those numbers. Different activities conducive to the emergence of collateral creativity and motivated by manipulative representations of number sieves are discussed. Several shape reconstruction problems in connection to productive thinking are interpreted in terms of collateral creativity. Chapter 4 titled Explorations with the Sums of Digits begins with the question: What can you say about year 2021? Using this question, a discussion with students can lead to interesting investigations and discoveries about the properties of the number itself and the sum of its digits. In this context, the authors show how using Wolfram Alpha’s function in finding the number of permutations of letters in the word can help to learn about the years with different sums of digits. Going deeper in the topic, namely, studying differences between two consecutive years with the sum of digits 5 while manipulating with base-ten blocks, provides a possibility of further generalizations about calculating the century number to which a year belongs. Finally, collaterally creative questions that arise from these investigations will lead the reader toward ordered partitions of a number into sums of two positive integers. The use of a spreadsheet in this case supports modeling of the mathematical results and their further interpretation. In Chap. 5 titled Collateral Creativity and Prime Numbers, different informal approaches to deciding the primality of integers are introduced through the lens of developing collateral creativity using the year 2021 as an example. Several divisibility tests and their connection to collateral creativity are discussed. The use of Wolfram Alpha is demonstrated. The notion of an integrated spreadsheet is applied

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to the context of divisibility when exploring formulas that generate primes sequentially. Euler’s formula that generates 40 primes sequentially is explored using a spreadsheet. Reasoning with twin primes based on empirical induction is demonstrated in the context of collateral creativity. In conclusion, Fermat’s primes are used to demonstrate a classic example of collateral creativity leading to an erroneous conjecture. In Chap. 6 titled From Square Tiles to Algebraic Inequalities, the idea about visual comparison of integers is applied to the visual comparison of fractions in the form of bar graphs constructed by a spreadsheet. The context of multicolored M&M candies is used. The activities reflect on the work of teacher candidates with third-grade students. Monotonic behavior of fractions through the sequential decrease of the numerator and denominator is discussed in the context of M&M mathematics and elementary teacher education. Opportunities for collateral creativity of the teacher candidates and their students alike are stipulated. For instance, a question about fractions posed by a teacher candidate as a sign of collateral creativity motivates an inquiry into algebraic inequalities. Different algorithms providing an answer to a creative question are presented. Inequalities between fractions as counterexamples to empirically motivated conjectures are discussed. In Chap. 7 titled Collateral Creativity and Exploring Unsolved Problems, the authors argue that it is rare for the schoolchildren to get a taste on an unsolved mathematical problem. Among those explored in this chapter are the palindromic number conjecture and the 4, 2, 1 sequence (Collatz conjecture). Both problems are open to a variety of questioning and investigation accessible to even elementary school students. For example, based on a simple rule of conducting a sequence by dividing a number by two if it is even or multiplying by 3 and adding 1 if it is odd, one could discover the pattern 4, 2, 1 per se, or look into the length of the sequence before reaching the repeating pattern; also, one can explore its amplitude. The use of a spreadsheet can support both investigation of the pattern and its graphical representation leading to collaterally creative questions. Chapter 8 titled Egyptian Fractions: From Pragmatic Uses of Technology to Epistemic Development and Collateral Creativity deals with the context of exploration of fair division of pizzas that leads to interesting discussions and explorations. This chapter begins with information about the appearance of Egyptian fractions including their presence in the Rhind Papyrus roll, Ahmes as the first mathematician and the explanation of André Weil about mathematical status of Egyptian fractions. Then, the Greedy algorithm which was apparently first used by Fibonacci in converting a common fraction into an Egyptian fraction is discussed. Comparison of the number of pizza pieces provided by the Greedy algorithm and other algorithms is supported by Wolfram Alpha as a friendly tool of converting common fractions into Egyptian fractions. Another method, called semi-fair division of pizzas, is described, and its comparison to the Egyptian fraction division is demonstrated using a spreadsheet leading to interesting collaterally creative questions and further investigations.

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Finally, Appendix provides answers to several questions posed throughout the chapters of the book. It includes spreadsheet programming details and description of dynamic geometry environments used to construct electronic fraction circles. Potsdam, USA Moncton, Canada

Sergei Abramovich Viktor Freiman

Acknowledgements The authors would like to acknowledge that their work on this book has been greatly supported at Springer by Natalie Rieborn—Editor Education who helped us at initiation stages of the project, Helen Van der Stelt—Assistant Editor, for providing feedback. Also, thanks to Marianna Georgouli—Publishing Editor, and Arumugam Deivasigamani—Project Coordinator, for accompanying us during the book preparation and at final stages. A special gratitude is extended to two unanimous reviewers of the book proposal and of the entire manuscript whose support, recommendations, and approval were important to ensure the quality of the manuscript.

References Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the education process. Heath. Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. (Online materials). Available at: https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm.

Contents

1 Theoretical Foundation and Examples of Collateral Creativity . . . . . 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Theories Associated with Collateral Creativity . . . . . . . . . . . . . . . . . . 1.3 Collateral Creativity and the Instrumental Act . . . . . . . . . . . . . . . . . . 1.4 Three More Examples of Collateral Creativity . . . . . . . . . . . . . . . . . . 1.4.1 A Second Grade Example of Collateral Creativity . . . . . . . . 1.4.2 A Fourth Grade Example of Collateral Creativity . . . . . . . . . 1.4.3 Collateral Creativity in a Classroom of Secondary Mathematics Teacher Candidates . . . . . . . . . . . . . . . . . . . . . . . 1.5 Collateral Creativity as Problem Posing in the Zone of Proximal Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Forthcoming Examples of Collateral Creativity Included in the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 From Additive Decompositions of Integers to Probability Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Artificial Creatures as a Context Inspiring Collateral Creativity . . . . 2.3 Iterative Nature of Questions and Investigations Supported by the Instrumental Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Joint Use of Tactile and Digital Tools Within the Instrumental Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Tactile Activities as a Window to the Basic Ideas of Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Historical Account Connecting Decomposition of Integers to Challenges of Gambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 5 7 7 9 10 12 14 16 19 19 20 26 26 29 30 32

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3 From Number Sieves to Difference Equations . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 On the Notion of a Number Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Theoretical Value of Practical Outcome of the Instrumental Act . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 On the Equivalence of Two Approaches to Even and Odd Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Developing New Sieves from Even and Odd Numbers . . . . . . . . . . . 3.6 Polygonal Number Sieves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Connecting Arithmetic to Geometry Explains Mathematical Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Polygonal Numbers and Collateral Creativity . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Explorations with the Sums of Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 About the Sums of Digits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Years with the Difference Nine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Calculating the Century Number to Which a Year Belongs . . . . . . . 4.5 Finding the Number of Years with the Given Sum of Digits Throughout Centuries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Partitioning n into Ordered Sums of Two Positive Integers . . . . . . . . 4.7 Interpreting the Results of Spreadsheet Modeling . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Collateral Creativity and Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . 5.1 ‘Low-Level’ Questions Require ‘High-Level’ Thinking . . . . . . . . . . 5.2 Twin Primes Explorations Motivated by Activities with the Number 2021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Students’ Confusion as a Teaching Moment and a Source of Collateral Creativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Different Definitions of a Prime Number . . . . . . . . . . . . . . . . . . . . . . . 5.5 Tests of Divisibility and Collateral Creativity . . . . . . . . . . . . . . . . . . . 5.6 Historically Significant Contributions to the Theory of Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 The Sieve of Eratosthenes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Is There a Formula for Prime Numbers? . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 From Square Tiles to Algebraic Inequalities . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Comparing Fractions Using Parts-Within-Whole Scheme . . . . . . . . . 6.3 Collateral Creativity: Calls for Generalization . . . . . . . . . . . . . . . . . . 6.4 Collaterally Creative Question Leads to the Discovery of “Jumping Fractions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.5 Algebraic Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Seeking New Algorithms for the Development of “Jumping Fractions” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Collateral Creativity and Exploring Unsolved Problems . . . . . . . . . . . . 93 7.1 Exploring Palindromic Number Conjecture in the Middle Grades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 7.2 The 196-Problem as a Possible Counterexample to Palindromic Number Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.3 Formulation of Collatz Conjecture and Its History . . . . . . . . . . . . . . . 96 7.4 Introducing the Rule: Initial Steps in Conjecturing . . . . . . . . . . . . . . 98 7.5 Deepening Investigation: Some Possible Paths Using a Spreadsheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 7.6 Fibonacci Numbers Emerge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.7 More on the Role of a Teacher in Supporting Problem Posing by Students . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 8 Egyptian Fractions: From Pragmatic Uses of Technology to Epistemic Development and Collateral Creativity . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Brief History of Egyptian Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Egyptian Fractions as a Context for Conceptualizations of Fractional Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Greedy Algorithm and Some Practice in Proving Fractional Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Pizza as a Context for Introducing Egyptian Fractions . . . . . . . . . . . 8.6 Semi-fair Division of Pizzas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 The Joint Use of Wolfram Alpha and a Spreadsheet . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Chapter 1

Theoretical Foundation and Examples of Collateral Creativity

1.1 Introduction This book discusses the concept of collateral creativity in school mathematics introduced by the authors in (Abramovich & Freiman, 2022). The difference between creativity and collateral creativity in mathematics can be described as follows. Creativity, displayed by a student, often leads to a challenging mathematical problem being solved, demonstrating originality, fluency mastery, and insight. Collateral creativity, displayed by a student, leads to an interesting (and potentially engaging) mathematical problem being posed (see Sect. 1.5). In the modern-day classroom, the manifestations of creativity and collateral creativity can be sparked by students’ use of technology, both tactile and digital. Furthermore, an important distinction between creativity and collateral creativity is that the former does not require a teacher in the process of problem solving. At the same time, the latter does require a competent teacher in order to recognize that a student, in fact, has posed a problem that needs at least to be discussed, before it could be solved. To explain this idea, consider two examples. Example 1.1. An elementary teacher candidate was working on the task of using multicolored counters to transform a 4 × 4 square into an isosceles triangle in which each row has two more counters than the previous row. The goal of the task was to recognize the famous connection between square numbers and the sums of consecutive odd numbers starting from one. That is, to re-discover the relationship 4×4 = 1+3+5+7. Tactile experimentation with counters led the candidate to offer an equilateral triangle in which each row has one more counter than the previous row and the construction of which requires one fewer counter than the equilateral triangle. Numerically, the candidate came across the relationship 4×4−1 = 1+2+3+4+5. This relationship was unexpected as part of the original task, yet it was collateral to the use of concrete materials and to the task itself. Furthermore, the candidate’s work did demonstrate collateral creativity for it opened a window for a professor to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_1

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1 Theoretical Foundation and Examples of Collateral Creativity

recognize a creative idea: the student accidentally discovered that there are square numbers one greater than triangular numbers (e.g., 16 = 15 + 1, where 16 is a square number and 15 is a triangular number). This discovery can be seen as posing a problem: Are there other pairs of square and triangular numbers that differ by one? This problem is discussed in detail in Chap. 3, Sect. 3.8. Example 1.2. A pair of elementary teacher candidates worked with their professor on the development of a spreadsheet-based learning environment in the form of bar graphs depicting the values of fractions to be used by young children not familiar with proper fractions as numeric representation of chances (likelihood) of something to happen. For example, the children were supposed to compare chances of selecting 4 red M&Ms out of 7 total with that of selecting 5 red M&Ms out of 8 total by recognizing that in the former case the bar (representing 4/7) is smaller than the bar (representing 5/8) in the latter case. The next step was to recognize a similar relationship between 3 chances out of 6 and 4 chances out of 7, also in the context of selecting red M&Ms. That is, the reduction by one of each numerator and denominator in the above two common fractions yielded another pair of fractions satisfying the same inequality relationship between the chances. Quite unexpectedly, one of the teacher candidates asked the professor: do there exist pairs of fractions representing chances so that the reduction by one of numerators and denominators reverses the chances? This question was collaterally creative as it was sparked by using technology in developing a learning environment in which fractions were used as tools in computing applications. While the answer to this question was not immediately available to a mathematics professor, the teacher candidate, by being collaterally creative, has posed an interesting problem. This problem is discussed in detail in Chap. 6, Sect. 6.4. Three more examples of collateral creativity will be discussed in Sect. 1.4 of this chapter as illustrations of theories introduced in the next two sections.

1.2 Theories Associated with Collateral Creativity When is experience educative? An answer to this question was given by John Dewey, one of the most influential reformers of education in North America, who argued that experience is educative only when it leads to one’s intellectual growth (Dewey, 1938). This argument requires certain educational conditions in order to structure one’s experience. One of such conditions is the pedagogy of reflective inquiry— a problem-solving method which, according to Dewey (1933), integrates knowing and doing. That is, knowledge stems from experience and develops in the course of appropriately designed educational activities which foster reflection. Often, especially in mathematics, reflection includes reorganization and reconstruction of the previous experience which then becomes a basis for the development of new knowledge. This kind of reflection was referred by Piaget (1973) as reflective abstraction and was directly connected to mathematics seeing the subject matter as “a model of

1.2 Theories Associated with Collateral Creativity

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creativity” (Piaget, 1981, p. 227). In the age of technology, a search for a didactic approach to the development of one’s intellectual growth in mathematics through the appropriate use of their diverse learning experience is particularly important in the context of teaching all students to be creative. Including every student in the realm of creativity is not a pretentious statement; rather, it is the main focus of the book. The modern-day learning of mathematics requires the development in students the mind of a mathematician who uses their experience for deepening mathematical knowledge. As Kline (1985) put it, “A farmer who seeks the rectangle of maximum area with given perimeter might, after finding the answer to his question, turn to gardening, but a mathematician who obtains such a neat result would not stop there” (p. 133). Nowadays, the development of such inquisitive attitude towards learning mathematics requires from a teacher to be “familiar with various software programs and technology platforms … to build computational models of mathematical objects, and to perform mathematical experiments” (Conference Board of the Mathematical Sciences, 2012, p. 57), capable of providing students with qualified assistance which includes skills in asking and answering questions prompting further investigation. These skills, augmented by familiarity with technological tools, are important for the success of mathematics education because they create conditions for the emergence of what may be called collateral creativity, a concept motivated by Dewey’s (1933) notion of “collateral learning” (p. 49). In the spirit of Dewey, collateral creativity may be considered as an unintended but favorable and often accidental educative outcome of one’s technology-enhanced learning experience when teaching is organized along the lines of pedagogy that motivates students to ask questions and assists them in finding answers to their own or the teacher’s prompts using technology. In this book, technology includes both physical (e.g., Example 1.1) and digital (e.g., Example 1.2) tools commonly available in the context of mathematics education. Both authors recall multiple teaching–learning situations from their practice when even a routine mathematical task can suddenly be turned into an open-ended problem which prompts further investigations leading to more advanced concepts and structures than those intended to be considered to deal with the original task. Thus, one of the goals of the book is to share ways of nurturing in-service and pre-service mathematics teachers’ awareness of hidden mathematical complexity of seemingly routine tasks. Realization of this complexity, experience in expanding the boundaries of the traditional curriculum as well as willingness to encourage and recognize students’ authorship of such expansions, as well as knowledge of technological tools to provide a grade-appropriate mediation of that kind of didactical evolution of teaching and learning are the major conditions for collateral creativity to take place. The research domain of mathematical creativity has been analyzed and significantly refined at the end of the twentieth century by bringing to the fore the importance of open-ended problems the investigation of which can be beneficial for all students (Hashimoto, 1997; Pehkonen, 1997). These ideas were supported by Sriraman (2009) who underscored the importance of stimulating creativity in students while using problems with an underlying mathematical structure allowing for “a prolonged period of engagement and independence to work on such problems” (p. 26). Furthermore,

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Leikin (2009a, 2009b) stressed the significance of tasks allowing for multiple solutions, including more than one way to prove things, as a milieu for mathematical creativity and advocated for the inclusion of such tasks in mathematics content and methods courses for teacher candidates enabling their awareness of the need to use such tasks as instruments of developing students’ creativity. This awareness makes teachers of mathematics “change their plans and follow unexpected learning trajectories initiated by the students” (Leikin & Dinur, 2007, p. 328). When research on mathematical creativity is integrated into practice, students are expected to be creative and their success in solving mathematical problems is the primary outcome of what happens in the classroom rather than the secondary one. But achieving the self-recognition of creative thinking as an indicator of one’s success in the learning of mathematics may take different routes which are often unexpected both for teachers and their students (Abramovich, 2021). Just as Dewey’s (1938) notion of collateral learning is an educational phenomenon, which does not result from the immediate goal of the traditional curriculum and “may be and often is much more important than the spelling lesson or lesson in geography or history … [celebrating students’] desire to go on learning” (p. 49), collateral creativity celebrates students’ primary interest in asking questions impelling their creative thinking as a consequence of an ensuing discourse. This is where mathematics teachers’ awareness of the importance of such a discourse for students’ learning to occur and their mathematical self-confidence to be developed plays out profoundly. Technology, both physical and digital, can significantly extend the boundaries of students’ productive thinking (Wertheimer, 1959) and enhance continuity of student–teacher creative conversation. Using technology as a pedagogical instrument in support of mathematical creativity, including its collateral manifestation, may also be seen as part of a larger theoretical framework known as TPCK—technological pedagogical content knowledge (Angeli & Valanides, 2009; Mishra & Koehler, 2006). The genesis of TPCK can be traced to an observation by Shulman (1986) that “questions about the content of the lessons taught, the questions asked, and the explanations offered” (p. 8, italics in the original) are unjustifiably missing from research on teaching. This observation, with its emphasis on content and recognition of the importance of questions and explanations, gave birth to the notion of PCK—pedagogical content knowledge. Mapping the notion of PCK, extended to include digital contexts, on the creativity research made it possible for the content of mathematical lessons to include asking information-type and explanation-type open-ended questions (Isaacs, 1930). Often, open-ended questions in digital contexts turn into real problems stemming from exploring numeric patterns (Lee & Freiman, 2006; Wilkie & Clarke, 2016; Abramovich, 2020a). Consequently, creative thinking about context which gave birth to a specific pattern is required in order to generalize the results of explorations. At the same time, dealing with limited number of terms of a numeric sequence enables different, sometimes unforeseen generalizations which, due to their unexpectedness, provide evidence of collateral creativity as an unintended outcome of the use of technology. In a large collection of more than 20 chapters related to the development of mathematical creativity in digital contexts (Freiman & Tassell, 2018), open-ended

1.3 Collateral Creativity and the Instrumental Act

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mathematical problems have been described in a variety of pedagogical settings. For example, Manuel (2018) described the use of such problems in virtual communities of learners of mathematics that can be characterized as having multiple correct answers (when open-endedness affords multiple interpretations), more than one correct problem-solving strategy, and applicable to real life. Dickman (2018) talked about online communities in terms of digital spaces as a way of fostering creativity in collaborative, participatory, and everyday settings. Gerson and Yu (2018) argued that the use of dynamic mathematical software allows students to be creative by making choices that the software affords. Abramovich (2018) suggested that technology use by students in exploring advanced open-ended mathematical situations has both pros and cons in the sense that while the appropriate uses of digital tools can foster creativity, in the absence of acute conceptual awareness of the meaning of mathematical content involved the use of the tools can lead students astray. Moreover, a recent survey of a 30-year long experience of integrating digital technology in mathematics teaching and learning in Russia shows, for example, a potential of “interactive and dynamic tools to support in-depth investigations of geometric relationships, and eventually to enhance creative and productive learning” (Pozdniakov & Freiman, 2021, p. 1503). These authors also argue that the use of computer algebra systems enhances “students’ ability to manipulate experimental models, minimizing the role of routine calculations by focusing on more creative and logical aspects of mathematics” (Pozdniakov & Freiman, 2021, p. 1506). Likewise, Freiman et al. (2017) have documented how teachers’ use of an interactive computer environment to supports primary students’ exploration of mathematical structures of word problems enriches whole class discussion in support of the development of algebraic thinking.

1.3 Collateral Creativity and the Instrumental Act Nowadays, using technology makes it possible to significantly enhance the educative aspect of teacher candidates’ realization of the value of such knowledge and experience when students’ creativity emerges as a collateral outcome of the instrumental act, a concept proposed almost a century ago by Vygotsky (1930). In mathematics education, technology, as an artifact turned into an instrument, can be put to work to animate learners’ interest in problem solving and awaken their slumberous mathematical creativity (Trouche, 2003; Yerushalmy, 2009). Within the instrumental act, nowadays enriched by various tools of technology, on the route from finding a problem to be solved to its solution, one is offered a specially developed representational medium which enables multiple demonstrations of conceptual richness and mathematical complexity of a seemingly simple problem (Kaput, 1986; Arzaerello and Robutti, 2010; Carreira et al., 2016; Abramovich, 2017). The essence of the instrumental act was described by Vygotsky (1930) as follows1 : 1

Throughout the book, this work by Vygotsky, published on-line as an English translation from Russian (https://www.marxists.org/russkij/vygotsky/cw/pdf/vol1.pdf), will be cited with no page

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1 Theoretical Foundation and Examples of Collateral Creativity The inclusion of a tool in the behavioral process, first, sets to work a number of new functions connected with the use and control of the given tool; second, abolishes and makes unnecessary a number of natural processes, whose work is [now] done by the tool; third, modifies the course and the various aspects (intensity, duration, order, etc.) of all mental processes included in the instrumental act, replacing some functions with others, i.e., it recreates, reconstructs the whole structure of behavior just like a technical tool recreates the entire system of labor operations. Mental processes, taken as a whole, form a complex structural and functional unity. They are directed toward the solution of a problem posed by the object, and the tool dictates their coordination and course. They form a new whole – the instrumental act.

This chapter intends to give the modern-day meaning to the words used by Vygotsky in the above quote such as control of the given digital tool, abolishing a number of processes done now by the tool, modification of mental processes included in the act, reconstructing the whole structure of behavior, the tool’s coordination of mental processes and their course. For example, using an electronic spreadsheet programmed to generate sequences defined by a simple recursion xn+1 = xn + a, x1 = 1 with the parameter a, one can control the variation of a to generate different arithmetic sequences, abolish unnecessary mental calculations done now by the spreadsheet, modify the original rule through which a sequence develops (e.g., xn+1 = axn ) towards the reconstruction of the course of thinking about the numbers observed. In other words, if the original problem was to find, using paper and pencil, the sum of the first ten natural numbers (recursively defined as xn+1 = xn +1, x1 = 1 in the context of a spreadsheet), making this uncomplicated problem-solving activity the instrumental act structured by the spreadsheet, may lead a user of the tool (e.g., a student) to generate ideas “that the user may or may not develop … [becoming evident after] the developments will be tried out, validated or rejected … following the diversity of the situations and projects the users set for themselves” (Béguin & Rabardel, 2000, p. 186). Put another way, as a result of the instrumental act, a new problem can be posed. In particular, controlling parameter variations and observing the results of what Euler called “quasi-experiments” (Pólya, 1954) allow one to move from recognition to observation (Abramovich, 2022a) and then from observation to generalization (Artigue, 2002). Through this process, one can integrate a spreadsheet with a tool capable of symbolic computations (e.g., Wolfram Alpha—a computational knowledge engine produced by Wolfram Research and available free on-line at https://www.wolframalpha.com/). Such use of the instrumental act, enabling known ideas to be recognized and new ideas to be observed in a numeric form and, whenever possible, to be generalized in a symbolic form, has to be promoted.

numbers included.

1.4 Three More Examples of Collateral Creativity

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1.4 Three More Examples of Collateral Creativity 1.4.1 A Second Grade Example of Collateral Creativity A second grade student was given eight square tiles and asked to construct all possible rectangles out of the tiles (Abramovich, 2019). Expectations for the student were pretty traditional: to construct two rectangles, 1×8 and 2×4, as shown in the left-hand side of Fig. 1.1. However, the student constructed rectangle with a hole as well shown in the right-hand side of Fig. 1.1. This activity was administered by an elementary teacher candidate who praised the student for their creativity. In the Standards for Preparing Teachers of Mathematics (Association of Mathematics Teacher Educators, 2017) one can find a note that sometimes, mathematics teachers when facing students whose thinking is different from what is expected, “may inadvertently seek to remedy those differences rather than seeing them as strength and resources upon which to build” (p. 22). As Vygotsky (1997) put it, “none of the child’s reactions must go to waste” (p. 347). Indeed, the third rectangle constructed by the second grader is not just a rectangle with a hole. It is special not only because it is a square, but because its area, 8 square units, numerically, is half of its perimeter, 16 linear units (if the border of the hole is considered as part of the perimeter). One can check to see that the same relationship between area and perimeter continues for rectangles with a hole constructed from any even number of square tiles. For example, as shown in Fig. 1.2, in the case of 16 square tiles, there are two rectangles and one square with different size holes, each having perimeter 32 linear units. One can use a spreadsheet to demonstrate that for any number of square tiles being a multiple of four greater than eight, more than one rectangle with a hole can be constructed. In general, out of 4n square tiles, (n − 1) rectangles with a hole may be constructed, each having the perimeter equal to 8n linear units. When n = 4, out of 16 square tiles (Fig. 1.2) three rectangles with a hole can be constructed, one of which is a square; each having perimeter 32 linear units. The fact that out of

Fig. 1.1 A rectangle with a hole as a challenge for a teacher to accept

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1 Theoretical Foundation and Examples of Collateral Creativity

Fig. 1.2 Sixteen square tiles afford two rectangles and one square with a hole

4n square tiles one can construct (n − 1) rectangles with a hole having perimeter 8n is confirmed by the spreadsheet shown in Fig. 1.3 in the case n = 4. Note that this computational confirmation may not be accepted as a formal demonstration (proof). The spreadsheet of Fig. 1.3 can be used when posing problems about constructing rectangles, including those with a hole and special relationship between area and perimeter. Remark 1.1. A proof of the relationship between area and perimeter of rectangles with a hole requires one to note that when 4n tiles are arranged to have m and 2n − m + 2 tiles, 3 ≤ m ≤ n + 1, being a pair of adjacent side lengths of n − 1 (= n + 1 − 3 + 1) rectangles of area 4n, the perimeter of each such rectangle is 2[(m + 2n − m + 2) + (m − 2 + 2n − m)] = 8n. The number of holeless rectangles that can be constructed out of 4n tiles depends on the prime factorization of the number n. Remark 1.2. It appears that the teacher candidate chose to give to the second grader eight tiles by accident. He could have given nine tiles to the child—in that case two traditional rectangles can be constructed only. Thus, it is due to serendipity that the

Fig. 1.3 Three rectangles with a hole generated by a spreadsheet

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child’s hidden creativity, being collateral to the number of tiles accidentally selected by the teacher candidate, was revealed through the construction of a rectangle with a hole.

1.4.2 A Fourth Grade Example of Collateral Creativity In a fourth grade classroom students worked on the task of arranging identical squaresized tables allowing for seating four people, one person at each side. The tables were supposed to share a full side with an adjacent table. The fourth graders were asked to put tables in such a way that exactly 12 people could be seated around a large table. They were given square tiles as a substitute for tables. The first configuration found was a 3 × 3 square each side of which seated three people. A fourth grader was using a document camera to project this example on a large screen. Accidentally, one square tile located at the top-left corner of the large square fell off and the configuration shown in Fig. 1.4 was projected to the screen. Whereas most of the students wanted the missing square tile to be put back, someone noticed that even without it, the configuration of eight square tiles (tables) could still seat 12 people. At that point, other students took on this idea noting that actually each of the corner square tiles may be removed allowing for the preservation of 12 seats. Furthermore, students, without any conscious knowledge that “nothing happens in this world in which some reason of maximum or minimum would not come to light” (Euler, cited in (Pólya, 1954, p. 121)), noted that five is the smallest number and nine is the largest number of tables that can seat 12 people. Moreover, students noted that whereas nine tables can be arranged in only one way to have 12 seats, five tables can be arranged in more than one way to have 12 seats. Other unexpected insights have been shared demonstrating collateral creativity already discussed in the context of grade two. One student suggested to remove a central square tile (table) from a 9-tile configuration. A discussion unfolded whether this is a valid solution to the original task. Another student argued that this would add four new seats inside the ‘hole’. Intuitively, students became aware of the existence of interior and exterior perimeter for such kind of a shape. The idea of having a ‘hole’ inspired other students to look for other possible configurations with more tables (square tiles) thus extending the classroom discussion into an unexpected creative direction. When students of such young age are asked to prove (justify) that each of their configuration gives actually 12 seats, counting sides one by one is their common strategy. Yet, it also happened during the activity when one child found a shortcut for counting (saying that for the configuration with five tables forming a row, the total number of seats can be found through multiplying 5 by 2 and then adding two more seats on two ends, thereby, producing the 2×5+2 rule which other students started to use as a more efficient counting strategy. Another extension of this activity came with the modification of the task changing the number of seats to 13. Students empowered by their success with configurations of 12 seats enthusiastically began searching for at least one solution only to be puzzled that such configuration cannot be found. Suddenly, one student said that this is impossible because 13 is an odd number. The

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Fig. 1.4 An arrangement with a missing table

idea of trying with odd versus even numbers inspired other students’ explorations. This example shows how fourth grade students, inspired by an accidental omission of a corner square tile (table) from a traditional three by three arrangement of nine tiles (tables), were collaterally creative in their simple manipulations of square tiles and have arrived at many discoveries leading the investigation further following unpredictable paths. Put another way, the omission of the tile was the primary incident in the described episode and students’ creativity was secondary or collateral to that incident. Just as in the second grade classroom (Sect. 1.4.1), without teacher being prepared to enter into an uncharted territory unfolded through an accidental omission of a square tile on the document camera, many collaterally creative activities initiated by students would not have taken place.

1.4.3 Collateral Creativity in a Classroom of Secondary Mathematics Teacher Candidates Another example (Abramovich, 2022a) comes from a classroom of future secondary mathematics teachers who were using a spreadsheet to explore the behavior of integer number sequences through modeling the equation Fn+1 = a Fn + bFn−1 , n = 1, 2, 3, . . . , F0 = F1 = 1.

(1.1)

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It is known that when a = b = 1, Eq. (1.1) generates Fibonacci numbers 1, 1, 2, √ 1+ 5 ∼ 3, 5, 8, 13, 21, …, and the ratios Fn+1 /Fn converge to the number 2 = 1.61803 called the Golden Ratio. Teacher candidates tried different values of a and b to find converge and to express this number, called the a number to which the ratios FFn+1 n generalized Golden Ratio, through a and b. All of a sudden, a teacher candidate reported that when a = 2 and b = −4 the ratios do not converge to a number but, instead, form a 3-cycle {1, −2, 4, 1, −2, 4, 1, −2, 4, …}. That is, during a pretty routine use of a spreadsheet as a tool for modeling Fibonacci-like numbers and their parametric modifications, the following question was raised: Do there exist other values of parameters a and b for which the ratios Fn+1 /Fn form 3-cycles and if so, how can one find them? This question is an example of collateral creativity made possible by the ease of spreadsheet modeling. The answer to this question is not easy and it can be made even more challenging if the query is extended to the existence of cycles of higher lengths and to finding relations between a and b that give birth to such cycles. That is, the ease of spreadsheet modeling of difference equations enabled the discovery that the generalized Golden Ratio may not only be a number but also a string of numbers. Once again, the discovery of a 3-cycle was accidental and unintended by the curriculum of the course. It could have been seen as a computational glitch had it not for a teacher candidate to be able to recognize the significance of this “glitch” and report it to the course instructor. But despite an accidental nature of the discovery of cycles, teacher candidates were curious whether other pairs of parameters a and b generating 3-cycles as a generalized Golden Ratio exist. To answer this question, one has to divide both sides = G n+1 , and consider the equation of Eq. (1.1) by F n , then set FFn+1 n G n+1 = a +

b , G 1 = 1. Gn

(1.2)

If G4 = G1 = 1, the sequence Gn forms a 3-cycle. In terms of a and b we have, G4 = a +

b b b b =a+ =a+ =a+ = 1. b G3 a + Gb2 a + a+b b a + a+b G1

Solving the equation a +

b b a+ a+b

= 1 can be outsourced to Wolfram Alpha (Fig. 1.5),

yielding b = −a , b = 1 − a. The latter relation yields a trivial cycle as G n = 1, n = 1, 2, 3, . . .. Note that the pair (a, b) = (2, −4), accidentally found by a teacher candidate, satisfies the former relation the graph of which is a parabola (Fig. 1.6). 2

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Fig. 1.5 Using Wolfram Alpha in solving a two-variable equation Fig. 1.6 Parabola as the locus of cycles of period three in (1.2)

1.5 Collateral Creativity as Problem Posing in the Zone of Proximal Development As was mentioned in the introduction, there is an important distinction between creativity and collateral creativity. Typically, the former does not require assistance or even presence of a teacher in the process of problem solving: a student demonstrates creativity by being able to solve a non-routine problem without any support of a teacher. At the same time, collateral creativity, in order to be recognized, does

1.5 Collateral Creativity as Problem Posing in the Zone of Proximal …

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require the presence of a competent teacher who is capable of making transition from knowing mathematics to using this knowledge in seeing in an unexpected question or comment by a student a problem that a student, in fact, has posed, and then share this vision with other students. Collateral creativity of a student as was shown in Sect. 1.4.1—creation of rectangle with a hole without the presence of a “more knowledgeable other” (Vygotsky, 1978) may go unnoticed. However, this and like examples (e.g., Example 1.1) open a window to exploring new mathematical ideas and may be recognized as unintended problem posing by a student. In general, the activity of problem posing has been a useful method of encouraging the advancement of mathematics and mathematical education for a long time. Problem posing goes back to the fifteenth century Italy when the first printed book on arithmetic written by an unknown author included problems aimed at explaining how to solve problems appearing in the context of trade. In the seventeenth century, many non-mathematicians interested in gambling, posed mathematical problems aiming to become more informed gamblers (see Chap. 2, Sect. 2.6). In the keynote address to the 1900 International Congress of Mathematicians, Hilbert (1902) formulated 23 problems from different branches of mathematics, thereby charting the program of mathematical research for the twentieth century. A few years ago, a book on the use of problem posing in mathematics education was published (Singer et al., 2015). In the specific context of mathematics teacher education, the use of technology in problem posing is discussed in (Abramovich, 2019). This kind of unwitting problem poising manifested by a second grader (constructing rectangle with a hole) and by an elementary teacher candidate (Example 1.1) may be considered as problem posing stemming from doing mathematics in the zone of proximal development (Vygotsky, 1987). This zone can be described as a dynamic characteristic of cognition that, in a problem-solving situation, measures the distance between two levels of the one’s development as determined by independent and assisted performances. The construction of a rectangle with a whole by a second grader was done independently. The child was collaterally creative due to the use of square tiles and in order to recognize his performance as posing a problem—finding the relationship between area and perimeter of rectangle with a hole—it requires competent assistance by a teacher. This competence includes teacher’s ability of recognizing the significance of the student’s independent performance, sharing this recognition with the student (who, otherwise, would remain in the zone of proximal development) and having the entire class to explore the problem posed as a result of collateral creativity by one of their peers that occurred within the zone of proximal development. Likewise, a tactile discovery by an elementary teacher candidate of a pair of square and triangular numbers that differ by one (16 and 15) was recognized and described by the candidate’s professor through the number theory lens. For example, the following problem was posed: How can one find other pairs of square and triangular numbers that differ by one? Alternatively, how can one find the sums of consecutive odd and natural numbers, both sums starting from one, that differ by one?

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1.6 Forthcoming Examples of Collateral Creativity Included in the Book Multiple studies conducted in the past three decades have demonstrated a potential of technology to nurture students’ curiosity and to spark their mathematical creativity (e.g., Abramovich, 2014; Freiman & Tassell, 2018; Kaput, 1992; Mishra, 2012; Sullivan, 2017; Yerushalmy, 2009). Based on the contributions to their book, Freiman and Tassel (2018, p. 20) suggested a model which reflects five factors that enhance (what they called dynamic learning conditions) and foster creativity in technology-rich environments through engaging, enriching, enabling, encouraging and empowering all students through appropriate teaching approaches, activities, and assessment. These factors, when taken into consideration from the initial steps in the investigation of open-ended mathematical problems, can lead to higher-order questioning and continuous exploration of more advanced contents, so it becomes accessible and attractive to all learners of mathematics. The cognitive mechanism of collateral creativity that the authors suggest would help the learners to unleash hidden creative potential, can be finetuned from different theoretical perspectives, such as Dewey’s (1933) reflective inquiry, Isaacs’s (1930) dual questioning, Piagetian (1973) reflective abstraction, Vygotskian (1930) instrumental act, and Wertheimer’s (1959) productive thinking (Fig. 1.7). It’s precisely Dewey’s (1938) notion of collateral learning that inspired the authors to suggest the term collateral creativity as an unintended (but favorable) and often accidental outcome of this open-ended technology-enhanced investigative process. This is where, in the authors’ opinion, all different theoretical perspectives would meet while igniting further steps in a continuous problem-solving and problem-posing cycle where novel questions emerge, and investigation takes new turns towards eventually more advanced levels of mathematical knowledge. The investigative tasks to be analyzed in this book deal with a variety of mathematical topics and technological tools. For instance, the Creatures task (Chap. 2) builds on students’ initial insight of additive partitions of a whole number and it could be further investigated using spreadsheets. This initial exploration leads to questioning which makes use of probabilistic reasoning. Another example of a rich learning environment with potential to prompt collateral creativity deals with interpreting different subsequences of natural numbers as sieves developed through the process of either their elimination or selection and demonstrating equivalence of recursive and closed formulas (Chap. 3). This interpretation allows students to use concrete materials (e.g., multicolored counters) to explore the interplay of different number sieves and their geometric representations. In particular, an elementary teacher candidate when using counters for the construction out of a square an isosceles triangle (Example 1.1) demonstrated collateral creativity by offering unexpected geometric constructions which, when tackled by their professor led to spreadsheet modeling aimed at connecting geometry to number theory (Chap. 3). The use of Wolfram Alpha can support investigations around the year-numbers eventually engaging students with exploration of prime numbers while asking new questions leading to deeper

1.6 Forthcoming Examples of Collateral Creativity Included in the Book

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Fig. 1.7 Collateral creativity as a technology-enhanced dynamic process

insights into number theory (Chaps. 4 and 5). The discovery of “jumping fractions” (Chap. 6) motivates the transition from numeric inequalities justified contextually through pizza sharing to formally demonstrated (proved) algebraic inequalities. The latter tools are needed for a systematic way of constructing such fractions to be used in problem posing which, due to their sensitivity to numeric modifications, cannot be revealed through trial-and-error. Spreadsheet-based explorations of unsolved problems including the Palindromic number conjecture and the 3n + 1 problem (Chap. 7) motivate students asking collaterally creative questions without realizing that their questions (in fact, posed problems) are pretty demanding in response. The variety of ways of dealing with the Pizza Sharing task (Chap. 8) shows how Egyptian fractions generated by Wolfram Alpha and the so-called semi-fair division, the enhanced version of which was offered by an aspiring teacher candidate, can inform fair sharing of a circular pizza more efficiently than the straightforward use of the dividend-divisor context for fractions. The process of exploration, facilitated by technology tools in each of the examples included in the book, will begin with a simple task which allows for different solutions and is rich in follow-up questions. Then, affordances of digital and physical tools will provide learners of mathematics with opportunities to conduct more complex multi-step investigations, eventually prompting new questions or even new directions of collateral creativity, something that teachers need to be aware of. The more students recognize that they are capable of being creative in mathematics, collaterally or primarily, the less the subject matter would be disliked, thus making the very enterprise of mathematical education more and more successful.

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References Abramovich, S. (2014). Computational experiment approach to advanced secondary mathematics curriculum. Springer. Abramovich, S. (2017). Diversifying mathematics teaching: Advanced educational content and methods for prospective elementary teachers. World Scientific. Abramovich, S. (2018). Technology and the development of creativity in advanced school mathematics. In V. Freiman, & J. Tassell (Eds.), Creativity and technology in mathematics education (pp. 371–398). Springer. Abramovich, S. (2019). Integrating computers and problem posing in mathematics teacher education. World Scientific. Abramovich, S. (2020). Paying attention to students’ ideas in the digital era. The Teaching of Mathematics, 23(1), 1–16. Abramovich, S. (2021). Using Wolfram Alpha with elementary teacher candidates: From more than one correct answer to more than one correct solution. MDPI Mathematics (Special issue: Research on Teaching and Learning Mathematics in Early Years and Teacher Training), 9(17), 2112. Available at https://doi.org/10.3390/math9172112 Abramovich, S. (2022a). On the interplay of mathematics and education: Advancing computational discovery from recognition to observation. MDPI Mathematics (Special issue: Advances in the Scientific Interplay of Mathematics and Language, Literature, and Education), 10(3), 359. Abramovich, S., & Freiman, V. (2022). Fostering collateral creativity through teaching school mathematics with technology: What do teachers need to know? International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2022.2113465 Angeli, C., & Valanides, N. (2009). Epistemological and methodological issues for the conceptualization, development, and assessment of ICT–TPCK: Advances in technological pedagogical content knowledge (TPCK). Computers and Education, 52(1), 154–168. Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. Arzarello, F., & Robutti, O. (2010). Multimodality in multi-representational environments. ZDM Mathematics Education, 42(7), 715–731. Available at https://doi.org/10.1007/s11858-0100288-z Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. Available online at https://www.amte.net/ Béguin, P., & Rabardel, P. (2000). Designing instrument-mediated activity. Scandinavian Journal of Information Systems, 12(1), 173–190. Carreira, S., Jones, K., Amado, N., Jacinto, H., & Nobre, S. (2016). Youngsters solving mathematical problems with technology: The results and implications of the Problem@Web Project. Springer. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. The Mathematical Association of America. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the education process. Heath. Dewey, J. (1938). Experience and education. MacMillan. Dickman, D. (2018). Creativity in question and answer digital spaces for mathematics education: A case study of the water triangle for proportional reasoning. In V. Freiman, & J. L. Tassell (Eds.), Creativity and technology in mathematics education (pp. 531–557). Springer. Freiman, V., & Tassell, J. L. (Eds.). (2018). Creativity and technology in mathematics education. Springer. Freiman, V., Polotskaia, E., & Savard, A. (2017). Using a computer-based learning task to promote work on mathematical relationships in the context of word problems in early grades. ZDM Mathematics Education, 29(6), 835–849. Available at https://doi.org/10.1007/s11858017-0883-3

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Gerson, H., & Yu, P. W. D. (2018). Can a kite be a triangle? Aesthetics and creative discourse in an interactive geometric environment. In V. Freiman, & J. L. Tassell (Eds.), Creativity and technology in mathematics education (pp. 347–369). Springer. Hashimoto, Y. (1997). The methods of fostering creativity through mathematical problem solving. International Reviews on Mathematical Education, 29(3), 86–87. Hilbert, D. (1902). Mathematical problems (Lecture delivered before the International Congress of Mathematicians at Paris in 1900). Bulletin of American Mathematical Society, 8(10), 437–479. Isaacs, N. (1930). Children’s why questions. In S. Isaacs (Ed.), Intellectual growth in young children (pp. 291–349.). Routledge & Kegan Paul. Kaput, J. J. (1986). Information technology and mathematics: Opening new representational windows. Harvard Graduate School of Education, Educational Technology Center. Available at https://files.eric.ed.gov/fulltext/ED297950.pdf Kaput, J. J. (1992). In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 515–556). MacMillan. Kline, M. (1985). Mathematics for the non-mathematician. Dover. Lee, L., & Freiman, V. (2006). Developing algebraic thinking through pattern exploration. Mathematics Teaching in the Middle School, 11(9), 428–433. Leikin, R. (2009a). In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 129–145). Sense Publishers. Leikin, R. (2009b). In R. Leikin, A. Berman, & B. Koichu (Eds.), Creativity in mathematics and the education of gifted students (pp. 383–409). Sense Publishers. Leikin, R., & Dinur, S. (2007). Teacher flexibility in mathematical discussion. The Journal of Mathematical Behavior, 26(4), 328–347. Manuel, D. (2018). In V. Freiman, & J. L. Tassell (Eds.), Creativity and technology in mathematics education (pp. 531–557). Springer. Mishra, P. (2012). Rethinking technology and creativity in the 21st century: Crayons are the future. Tech Trends, 56(5), 13–16. Mishra, P., & Koehler, M. J. (2006). Technological pedagogical content knowledge: A framework for teacher knowledge. The Teachers College Record, 108(6), 1017–1054. Pehkonen, E. (1997). The state-of-art in mathematical creativity. International Reviews on Mathematical Education, 29(3), 63–66. Piaget, J. (1973). In G. Howson (Ed.), Developments in mathematics education: Proceedings of the Second International Congress on Mathematical Education. Cambridge University Press. Piaget, J. (1981). In J. M. Gallagher, & D. K. Reid (Eds.), The learning theory of Piaget and Inhelder (pp. 221–229). Brooks/Cole. Pólya, G. (1954). Induction and analogy in mathematics (Vol. 1). Princeton University Press. Pozdniakov, S., & Freiman, V. (2021). Technology-supported innovations in mathematics education during the last 30 years: Russian perspective. ZDM Mathematics Education, 53, 1499–1513. Available online at https://doi.org/10.1007/s11858-021-01279-6 Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14. Singer, F. M., Ellerton, N., & Cai, J. (Eds.) (2015). Mathematical problem posing: From research to effective practice. Springer. Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM, 41(1&2), 13–27. Sullivan, F. R. (2017). Creativity, technology, and learning: Theory for classroom practice. Routledge. Trouche, L. (2003). From artifact to instrument: Mathematics teaching mediated by symbolic calculators. Interacting with Computers, 15(6), 783–800. Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. [Online materials]. Available at https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm Vygotsky, L. S. (1978). Mind in society. MIT Press.

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Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.), The collected works of L. S. Vygotsky (Vol. 1, pp. 39–285). Plenum Press. Vygotsky, L. S. (1997). Educational psychology. CRC Press. Wertheimer, M. (1959). Productive thinking. Harper & Row. Wilkie, K. J., & Clarke, D. M. (2016). Developing students’ functional thinking in algebra through different visualizations of a growing pattern’s structure. Mathematics Education Research Journal, 28(2), 223–243. Yerushalmy, M. (2009). In R. Leikin, A. Berman, & B. Koichu (Eds.), Mathematical creativity and the education of gifted students (pp. 101–113). Sense Publishers.

Chapter 2

From Additive Decompositions of Integers to Probability Experiments

2.1 Introduction A potentially rich problem for an elementary classroom deals with decomposition of positive integers into a sum of like numbers (e.g., Common Core State Standards, 2010, p. 11; Ontario Ministry of Education, 2020, p. 114; Department of Basic Education, 2018, p. 29; Takahashi et al., 2004, p. 103). Such decompositions of integers belong to the very foundation of the additive number theory the problems of which have been continuously recommended to be used even for early grades (Hardy, 1929; Vavilov, 2020). However, one of the authors, working with young children, noticed their difficulty to accept the existence of more than one correct answer when asked to provide a symbolic formulation of additive decompositions of integers (Abramovich, 2021). In particular, when second graders were told that in a period of five days, average temperature has increased by one degree and the temperature increase was recorded during at most two days, they thought that there was only one day when temperature went up by five degrees. In other words, they were unable to overcome the instance of a single day and to grasp the spread of five-degree temperature increase over two days. Indeed, students seemed struggling to understand that there are six decompositions of five into the ordered sums of two non-negative integers: 5 = 5 + 0 = 0 + 5 = 4 + 1 = 1 + 4 = 3 + 2 = 2 + 3. They needed a particular guidance by means of a concrete situation representing this type of repartition (e.g., finding all ways to put five finger rings on two fingers) to overcome the complexity of mathematical abstraction. The intent of activities discussed in this chapter is to demonstrate how turning a seemingly simple additive decomposition of an integer into the instrumental act is supported by a specially designed computational tool (in our case, a spreadsheet). This enables students (elementary teacher candidates included) to control the spreadsheet, reduce mental computations making their duration really insignificant, and, by modifying numeric data involved, to dig deeper into the problem by (re)constructing the way of thinking about mathematics involved. Through this © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_2

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process of (re)construction, students begin asking questions the creative nature of which is, in the authors’ view, a collateral outcome of the instrumental act. The notion of dual questioning by Isaacs (1930), mentioned in Chap. 1, distinguishes between two types of questions that students usually ask: information-type questions when one seeks information (e.g., how many ways can the number 10 be decomposed into the sum of two positive integers?) and explanation-type questions when one requests explanation (e.g., why do we say that cookies are identical when finding all ways to put 10 identical cookies on two plates?). While both information-type and explanation-type questions when asked by students might be indicative of collateral creativity, a question of the latter type often results from deeper thinking, especially when it stems from an intelligent reflection on a method through which a question of the former type was answered.

2.2 Artificial Creatures as a Context Inspiring Collateral Creativity Consider the following problem which was introduced to elementary teacher candidates enrolled in a mathematics education course. Problem 2.1. Two creatures—brimp and drimp—have nine legs among them. If brimp has at least two legs and fewer legs than drimp, how many legs does each creature have? Does the problem have more than one correct answer? Does the problem have more than one correct solution? Discussion. To begin note that there is a difference between answer and solution to a numeric problem: the former means direct information to a question asked and the latter means a specific way of providing the information. This problem (with the first question seeking information about the number of legs) can be solved by using a picture-supported trial and error. Teacher candidates were advised to start assigning legs to the creatures keeping in mind that brimp has at least two legs and, consequently, drimp has at least three legs. A pictorial solution presented by a teacher candidate looked like the one shown in Fig. 2.1. The candidate started with assigning legs to brimp because it has fewer legs than drimp and the smallest number of legs the former creature has, two, was given by the condition of the problem. The candidate’s pictorial solution, displaying three correct answers, was supported by the numeric relations 9 = 2 + 7, 9 = 3 + 6, and 9 = 4 + 5. The use of a diagram as a psychological tool is an element of the instrumental act allowing one to think with images. Thinking with images prompts the second solution by writing the equation b + d = 9 (assuming d > b ≥ 2) the solution of which can be outsourced to Wolfram Alpha, another element of the instrumental act often used by one of the authors with elementary teacher candidates (Abramovich, 2021), by entering into the tool’s input box the command “solve over the integers b + d = 9, d > b ≥ 2” (Fig. 2.2). This second solution (provided by Wolfram Alpha) confirms three correct answers to the problem.

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Fig. 2.1 Pictorial (trial and error) solution to Problem 2.1

Fig. 2.2 Wolfram Alpha solution to Problem 2.1

The third solution is to use the Graphing Calculator to draw the graph of the equation b + d = 9 and find on this graph points with integer coordinates that belong to the angle (including its sides) formed by the vertical line b = 2 and the angle bisector b = d of the first quadrant (Fig. 2.3). The integration of trial and error, computing and graphing provided by the instrumental act motivates mathematical creativity of posing new problems about artificial creatures, something that is collateral to this integration.

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Fig. 2.3 Graphing Calculator solution to Problem 2.1

The creative process of problem solving and posing prompts one to inquire whether it is possible to modify Problem 2.1 by allowing the presence of more than one creature of the same type, or, in decontextualized form needed to support the instrumental act, by considering linear equations with coefficients different from one. With this in mind, consider another problem which was discussed with elementary teacher candidates. Problem 2.2. Three brimps and one drimp have nine legs among them. How many legs does each creature have? Does the problem have more than one correct answer? Does the problem have more than one correct solution? Discussion. This problem (with the first question seeking information) can be solved by assigning legs to brimps, beginning with one leg. Teacher candidates often ask: why do we start with brimps, can we start with a drimp? This is a very good question to which there is no easy answer as it was in Problem 2.1—because brimp has fewer legs then drimp. One has to try both ways in order to understand that by starting with brimps the use of division by three would be avoided. While this difference is not significant for an adult learner, avoiding division could make a difference for a young problem solver. As shown in Fig. 2.4, through this process, two correct answers can be found and described as decomposing the number 9 in two groups formed by three and one addends: 9 = (1 + 1 + 1) + 6 and 9 = (2 + 2 + 2) + 3. If brimp has three legs, then there would be no legs left for drimp. Collaterally to the two decompositions of the number 9, one can pose and solve similar problems. To this end, one can first partition the number 6 as 3 + 3 (two drimps), 2 + 2 + 2 (three drimps) and 1 + 1 + 1 + 1 + 1 + 1 (six drimps). Likewise, one can partition the number 3 as 1 + 1 + 1 (three drimps). This is an example of how creativity in formulating new problems can be collateral to already solved problems. Problem posing requires conceptual

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Fig. 2.4 Two answers to Problem 2.2 as positive integer solutions to the equation 3b + d = 9

understanding of a solution of a problem solved through trial and error. Three more solutions of Problem 2.2 are possible: solving the equation 3b + d = 9 using Wolfram Alpha, solving this equation by noting that just as 3b and 9 are multiples of 3, the term d should be a multiple of 3 (yielding d = 3 and d = 6—see Fig. 2.4) in order to satisfy the equation and, finally, graphing this equation by the Graphing Calculator (setting b = x, d = y) and locating points with positive integer coordinates on the graph. The next problem extends the first two problems to include three artificial creatures. Problem 2.3. Three creatures—brimp, drimp, and grimp—have 10 legs among them. If neither one is a one-legged creature and brimp may not have more legs than drimp while drimp may not have more legs than grimp, how many legs does each creature have? Does the problem have more than one correct answer? Does the problem have more than one correct solution? Discussion. This problem (with the first question, as in the previous two problems, seeking information) can also be solved by using a picture. One can start assigning legs to three creatures keeping in mind that each creature must have at least two legs and that the assignment of legs should start with brimp as it may not have more legs than the other two creatures. The pictorial solution, presented by a teacher candidate and looked like the one shown in Fig. 2.5, displays four correct answers. Once again, the use of a diagram as a psychological tool is an element of the instrumental act allowing one to think with images. Numerically, the picture can be described by the relations 10 = 2 + 2 + 6, 10 = 2 + 3 + 5, 10 = 2 + 4 + 4 and 10 = 3 + 3 + 4. This description can be confirmed by Wolfram Alpha: by entering into its input box the command “solve over the integers a + b + c = 10, 1 < a ≤ b ≤ c” four triples of integers result (Fig. 2.6). The third element of the instrumental act (alternatively, the third solution of Problem 2.3) is the use of the spreadsheet “Creatures” shown in Fig. 2.7 where

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Fig. 2.5 Four answers found through solving Problem 2.3

Fig. 2.6 Using Wolfram Alpha in solving Problem 2.3

possible legs for brimps are on the left, possible legs for drimps are on the top, corresponding legs for grimps are in the middle; the cells A1 and A2 filled, respectively, with numbers 10 (the total number of legs) and 1 (the number of legs not considered) can be controlled by sliders. This new element of the instrumental act allows one not only to check the answer already delivered pictorially and computationally but it enables one to pose and immediately solve a new problem about three creatures as well as to use the spreadsheet in alternative problem-solving and problem-posing contexts. Indeed, computer-supported initial investigation would allow teacher candidates to continue their exploration using the spreadsheet “Creatures” to modify the total

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Fig. 2.7 Four answers to Problem 2.3 generated by the spreadsheet “Creatures”

number of legs or setting up other constrains (e.g., the number of legs that the creatures may not have) eventually asking new questions such as: • How many combinations of the three creatures with the total of 10 legs are there if neither creature may have two legs? • How would the number of combinations of the three creatures change if one modifies the total number of legs in the previous question? • How does the number of combinations of the three creatures change when one removes the constraint of not having a certain number of legs? • How can one explain that, given the number of legs total, with the increase of the number of legs the creatures may not have, the number of combinations of the three creatures either stays the same or increases? For example, why (with 10 legs total) are there more combinations of creatures not having three legs than of creatures not having two legs? Some of the questions could be difficult (like the last question requesting explanation), yet they could emerge from the investigation and can be asked by a student engaged in the explorations. A possible response to the last question is given in Appendix (Question 1). Note that within the instrumental act a challenging question may be asked not only by a teacher but by a student as well. In other words, the manifestation of hidden creativity can be seen as a collateral outcome of explorations structured by the instrumental act and that are not possible without this act. Here one can see a dual nature of the instrumental act and its bi-directional impact on one’s experience. Indeed, the use of the spreadsheet affects both the completion of the task (delivering information sought) and intellectual development of the learner of mathematics (triggering requests for explanation). Put another way, by acting towards the completion of the task, the spreadsheet evinces pragmatic mediation and by acting towards the learner the tool supports their epistemic mediation (Lonchamp, 2012).

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2.3 Iterative Nature of Questions and Investigations Supported by the Instrumental Act The spreadsheet of Fig. 2.7 provides more possibilities for creative observations the explanations of which may be challenging. Using the spreadsheet, one can find out that in the case of 10 legs among the three creatures we have the same number of combinations of creatures not having one leg as not having two legs, the same number of combinations of creatures not having three legs as not having four legs, the same number of combinations of creatures not having five legs as not having six legs, and so on. Also, it appears that even when within the pattern observed with the total of 10 legs and the condition on the ordering the number of restricted legs for the three creatures is removed, the patten does not change: there are 24 ordered combinations of three creatures not having 4 legs and not having 5 legs. The same is true when the total number of legs is 9: there are 19 ordered decompositions of 9 in three addends without 4 and 5. Indeed, three addends can be arranged in one, three and six different orders when, respectively, all addends are the same, two addends are the same, and all addends are different (see Sect. 2.6). However, the case of 11 legs total produces a counterexample: one can check to see that there are 30 and 33 ordered partitions of 11 into addends without 4 and 5, respectively. That is, any statement of that kind depends on the total number of legs among three creatures. These are quite interesting and creative observations that one can either make or be encouraged to make using technology. However, formal explanations of these computationally supported observations are quite challenging. Once again, one can see how technology-driven mathematical observations support one’s epistemic development through problem solving.

2.4 The Joint Use of Tactile and Digital Tools Within the Instrumental Act As was mentioned in Chap. 1, the instrumental act may include both digital and physical tools. With this in mind, another type of problems introduced in this section involves the use of multicolored Unifix (linking) cubes as means of an instrumental activity. This mostly hands-on activity could be further developed by the use of a spreadsheet as the generator of different numeric sequences referred to below as natural number sieves. A simple example of a natural number sieve is the sequence of odd numbers when from the sequence of natural numbers, every other number is eliminated. We may call the set of odd numbers, {1, 3, 5, 7, 9, 11, 13, . . .}, the natural number sieve of order one. In a similar way, every other number from the sequence of odd numbers can be eliminated to create the set of numbers {1, 5, 9, 13, 17, . . .} that may be called the natural number sieve of order two. The next elimination of every other number yields the set {1, 9, 17, . . .}. At that point, the following questions can be asked: Is there a pattern through which numbers in the sieves of order 1,

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2, and 3 develop? What is the difference between two consecutive terms in each sieve? Answering these questions can be enhanced by a spreadsheet that generates natural number sieves of different orders. These sieves are all arithmetic sequences the difference of which is the number 2 raised to the power which is equal to the order of the corresponding sieve. For example, in the sieve {1, 9, 17, . . .} of order three we have 9 = 1 + 23 and 17 = 9 + 23 . Being a part of the instrumental act, the spreadsheet is used to inform the formulation of activities involving linking cubes. Such combination of the digital and the tactile serves to demonstrate how in astute mathematics education what one does by hands has to be informed by argument and computation, two major pillars of formal mathematics. Consider a problem that involves integration of concrete materials and digital tools. Problem 2.4. Using 210 linking cubes, construct a set of towers shown in Fig. 2.8. From this set, eliminate every other tower and from the remaining set of towers eliminate every other tower again. Imagine that the process of elimination continues until the difference in height between two consecutive towers is 32 cubes. How many eliminations have been made? Discussion. In order to answer an information-type question of Problem 2.4, one has to recognize a pattern. This recognition should result in seeing the surviving towers as physical representations of odd numbers each of which differs from the next number by two. Furthermore, the difference between the heights of two consecutive towers surviving the second elimination is four (= 22 ), surviving the third elimination is eight (= 23 ). Recognition of a pattern through visualization allows one to abolish further use of linking cubes (after all, as mentioned above, there are 210 cubes only) by noting that 32 = 25 ; that is, five eliminations have been made. Linking cubes, being a part of the instrumental act, which is placed between the question and the answer, help a student to coordinate the course of mental processes towards concrete thinking about the problem because mathematics, outside of applications (alternatively, without the frames of reference), is too abstract and for many, especially at a young age, the subject matter needs contextualization. For example,

Fig. 2.8 Towers and their consecutive elimination

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as described by Wertheimer (1959), a nine-year-old girl with difficulties in solving simple arithmetical problems, when having been offered contextual problems based on a concrete situation with which she was familiar and the solution of which “was required by the situation, she encountered no unusual difficulty, frequently showing excellent sense” (pp. 273–274). In other words, linking cubes (or other kinds of concrete materials), when used as mediational means for grasping mathematical abstraction, motivate productive thinking and boost on-task behavior. In addition to using linking cubes, it is possible to include any computational tool (e.g., a spreadsheet) into the instrumental act, thus assisting a problem solver in answering another three information-type questions: • After how many eliminations is the height of the second tower equal to 65 cubes? • After how many eliminations is the height of the second tower equal to 1025 cubes? • How many cubes are there in the smallest tower to survive 11 eliminations? Here, one has to recognize another pattern: regardless of the number of consecutive eliminations, the height of the second tower is always one greater than a power of two with the exponent being the number of eliminations. Thus 65 − 1 = 64 = 26 , 1025 − 1 = 1024 = 210 , and 2049 − 1 = 2048 = 211 . Learners of mathematics, elementary teacher candidates included, do appreciate the ease and power of computations provided by commonly available digital tools. For example, entering the command “Table[2^n + 1, {n, 1, 11}]” (or even a simpler one: “2^n + 1, n = 1…11”) into the input box of Wolfram Alpha yields the result shown in Fig. 2.9. As noted in the pre-digital era by Halmos (1975), “The best way to teach teachers is to make them ask and do what they, in turn, will make their students ask and do” (p. 470). Nowadays, this recommendation about the importance of asking and doing, that is, the pedagogy of reflective inquiry (Dewey, 1933), takes on new didactical opportunities when technology is used to support asking questions and posing problems.

Fig. 2.9 Using Wolfram Alpha to ease numeric computations

2.5 Tactile Activities as a Window to the Basic Ideas of Number Theory

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2.5 Tactile Activities as a Window to the Basic Ideas of Number Theory The next set of questions deals with a new type of hands-on activities which can be characterized as reflective abstraction (Piaget, 1973) for “when properly understood, reflective abstraction appears as a description of the mechanism of the development of intellectual thought” (Dubinsky, 1991, p. 98), something that begins with reorganization and reconstruction of the hands-on activities. In agreement with classic perspectives on learning, one can be asked: • Can you make a square by rearranging cubes in the first three/four/five towers survived the first elimination? A reconstruction of the previously fabricated sets of towers into a square provides a tactile affirmation and a visual recognition of a foundational assertion from number theory that the sum of any quantity of the first consecutive odd numbers is always a square number (see Chap. 3, Sects. 3.6 and 3.7). This is a pretty remarkable summation result which can be first discovered by using linking cubes or counters (see Chap. 3, Fig. 3.8), then described symbolically as 1 + 3 + 5 = 32 , 1 + 3 + 5 + 7 = 42 , 1 + 3 + 5 + 7 + 9 = 52 and, finally, generalized to the form 1 + 3 + 5 + · · · + (2n − 1) = n 2 .

(2.1)

Identity (2.1) can be first justified (proved) informally by demonstrating (for small values of n) that the square constructed out of n2 tiles when augmented by 2n + 1 tiles turns into a square constructed out of (n + 1)2 tiles. This visual demonstration can be described in terms of the method of mathematical induction. Indeed, assuming that identity (2.1) is true, one can write 1 + 3 + 5 + · · · + (2n − 1) + (2n + 1) = n 2 + 2n + 1 = (n + 1)2 . Other activities of reconstruction may stem from the following questions. • Consider the set of towers survived the first two eliminations. Can you make a symmetrical cross (a plus sign) using linking cubes from each tower in this set? Can you describe such crosses (plus signs) using numbers connected through addition and multiplication? Is such cross unique for each tower? • Can one reformulate Problem 2.3 (about three artificial creatures) in terms of creating symmetrical crosses? • Can one reformulate Problem 2.3 in terms of creating asymmetrical crosses? • Is the reformulation of Problem 2.3 mentioned in the last two questions possible if the number of legs among three creatures were 11? Once again, the ideas of reconstruction of a tower into a cross, modification of the latter to recognize the non-uniqueness of mathematical behavior, and reformulation of one context in terms of another context as the demonstration of sensitivity of

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2 From Additive Decompositions of Integers to Probability Experiments

Fig. 2.10 Five symmetrical crosses (plus signs) out of 13 linking cubes

mathematical problem posing to numeric data come as a collateral outcome of using the instrumental act in the problem-solving activity leading to “the diversity of the situations and projects the users set for themselves” (Béguin & Rabardel, 2000, p. 186). For example, the fourth tower which survived two eliminations consists of 13 linking cubes and five symmetrical crosses can be constructed out of those cubes (Fig. 2.10). The diversity of situations to explore through the agencies of reflective inquiry (Dewey, 1933), reflective abstraction (Piaget, 1973) and productive thinking (Wertheimer, 1959) may include finding the number of symmetrical crosses that can be constructed from the n-th tower that survived two eliminations, from the n-th tower that survived three eliminations, and, finally, from the n-th tower that survived k eliminations. Here, problem solvers can use their previous explorative experience to develop their own new explorations. That is, once again, the display of creativity is a collateral outcome of being previously engaged in similar explorations guided by the use of technological tools. More information about other types of number sieves and their description through difference equations can be found in (Abramovich & Leonov, 2019). The next section will illustrate close relation between the notions of collateral learning (Dewey, 1933) and collateral creativity mediated by the classroom culture of asking (and answering) questions (Lerman, 1994; Mason, 2000; Harris, 2014; Ulleberg and Solem, 2018) as well as posing (and solving) new problems (Abramovich, 2019; Singer et al., 2015).

2.6 Historical Account Connecting Decomposition of Integers to Challenges of Gambling As is known from the history of the probability theory (e.g., Tijms, 2012; Todhunter, 1949), posing mathematical problems that stem from gambling had often been due to non-mathematicians who wanted to enhance their erudition towards making their practical experience educative and thus becoming more mathematically informed gamblers. That is, creativity in asking explanation-type mathematical questions was often displayed due to experience gained outside of mathematics. Here is an example

2.6 Historical Account Connecting Decomposition of Integers …

31

of such a question (similar to the case of creature discussed in Sect. 2.3) that in the seventeenth century the Grand Duke of Tuscany (Cosimo II de’Medici) had asked Galileo Galilei (the father of all major scientific developments in the seventeenth century Italy): Why when rolling three dice, does the number 11 appear more often than the number 12? Most likely, the question was asked because both 11 and 12 have the same number of unordered partitions in three summands not greater than 6, namely, 11 = 1 + 4 + 6 = 1 + 5 + 5 = 2 + 3 + 6 = 2 + 4 + 5 = 3 + 3 + 5 = 3 + 4 + 4 and 12 = 1 + 5 + 6 = 2 + 4 + 6 = 2 + 5 + 5 = 3 + 3 + 6 = 3 + 4 + 5 = 4 + 4 + 4, yet 11 has 27 ordered partitions and 12 has 25 ordered partitions. The above-mentioned connection of Problem 2.3 to the history of probability theory, made possible by the inclusion of the instrumental act into mathematical problem solving, can be used to consider Problem 2.5. Three dice were rolled. What were the chances to have 10 spots on three faces if it is known that neither die had two spots when rolled? Discission. Using the spreadsheet “Creatures”, one can find theoretical probability of the outcome sought. There are 18 ordered decompositions of 10 into three summands not including the number 2. These decompositions (Fig. 2.11) are generated by the triples (1, 1, 8), (1, 3, 6), (1, 4, 5) and (3, 3, 4); namely, the first and the fourth triples generate three decompositions each while the second and the third triples generate six 18 ∼ decompositions each. Therefore, the probability sought is the ratio 216 = 0.08. Once again, the instrumental act makes it possible to control with a slider the corresponding cells of the spreadsheet. However, an important aspect of the instrumental act deals with the function of contextual understanding of the numbers under one’s control. Whereas none of the three creatures with the total of 10 legs may have more than 8 legs, in the context of rolling three dice, the number 8 has to be replaced by the number 6 even if the number 10 is replaced by 11. That is exactly what it means that “The inclusion of a tool in the behavioral process … sets to work a number of new functions connected with the use and control of the given tool” (Vygotsky, 1930). It is through the need to control the tool that the need for contextual understanding of a problem attains a new functional significance. At the same time, experimental probability of having 10 spots on three faces without having 2 spots on any face can be calculated within a spreadsheet (Fig. 2.12) by simulating 5000 rolls of three dice. The result is shown in cell F1 which displays the number 0.0722. The difference between the theoretical and the experimental probabilities is equal to 0.01. The use of the spreadsheet shown in Fig. 2.12 plays a dual role in the instrumental act. First, the tool pragmatically mediated problem-solving process by verifying the correctness of the theoretical solution obtained through the

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2 From Additive Decompositions of Integers to Probability Experiments

Fig. 2.11 Using the spreadsheet of Fig. 2.7 in the context of probability

Fig. 2.12 Experimental probability is calculated in cell F1

use of a structurally different spreadsheet. As noted by Freudenthal (1978), “It is independency of new experiments that enhances credibility” (p. 193). Second, the tool pragmatically mediated mathematical behavior of a problem solver by taking on the complex coordination of a large quantity of random data. The computational power of the spreadsheet of Fig. 2.12 “immensely extends the possibilities of behavior by making the results of the work of geniuses available to everyone” (Vygotsky, 1930).

References Abramovich, S. (2019). Integrating computers and problem posing in mathematics teacher education. World Scientific. Abramovich, S. (2021). Using Wolfram Alpha with elementary teacher candidates: From more than one correct answer to more than one correct solution. MDPI Mathematics (Special issue: Research on Teaching and Learning Mathematics in Early Years and Teacher Training), 9(17), 2112. Available at https://doi.org/10.3390/math9172112

References

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Abramovich, S., & Leonov, G. A. (2019). Revisiting Fibonacci numbers through a computational experiment. Nova Science Publishers. Béguin, P., & Rabardel, P. (2000). Designing instrument-mediated activity. Scandinavian Journal of Information Systems, 12(1), 173–190. Common Core State Standards. (2010). Common core standards initiative: Preparing America’s students for college and career. Available online at http://www.corestandards.org Department of Basic Education. (2018). Mathematics teaching and learning framework for South Africa: Teaching mathematics for understanding. The Author. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the education process. Heath. Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 95–123). Kluwer. Freudenthal, H. (1978). Weeding and sowing. Kluwer. Halmos, P. R. (1975). The teaching of problem solving. The American Mathematical Monthly, 82(5), 466–470. Hardy, G. H. (1929). An introduction to the theory of numbers. Bulletin of the American Mathematical Society, 35(6), 778–818. Harris, B. (2014). Creating a classroom culture that supports the Common Core: Teaching questioning, conversation techniques, and the essential skills. Routledge. Isaacs, N. (1930). Children’s why questions. In S. Isaacs (Ed.), Intellectual growth in young children (pp. 291–349.). Routledge & Kegan Paul. Lerman, S. (Ed.). (1994). Cultural perspectives on the mathematics classroom. Springer. Lonchamp, J. (2012). An instrumental perspective on CSCL systems. International Journal of Computer-Supported Collaborative Learning, 7(2), 211–237. Mason, J. (2000). Asking mathematical questions mathematically. International Journal of Mathematical Education in Science and Technology, 31(1), 97–111. Ontario Ministry of Education. (2020). The Ontario Curriculum, Grades 1–8, Mathematics (2020). Available online at http://www.edu.gov.on.ca Piaget, J. (1973). In G. Howson (Ed.), Developments in mathematics education: Proceedings of the Second International Congress on Mathematical Education. Cambridge University Press. Singer, F. M., Ellerton, N., & Cai, J. (Eds.) (2015). Mathematical problem posing: From research to effective practice. Springer. Takahashi, A., Watanabe, T., Yoshida, M., & McDougal, T. (2004). Elementary school teaching guide for the Japanese course of study: Arithmetic (Grade 1–6). Global Education Resources. Tijms, H. (2012). Understanding probability. Cambridge University Press. Todhunter, I. (1949). A history of the mathematical theory of probability. Chelsea. Ulleberg, I., & Solem, I. H. (2018). Which questions should be asked in classroom talk in mathematics? Presentation and discussion of a questioning model. Acta Didactica Norge, 12(1), Art. 3. Vavilov, N. A. (2020). Computers as novel mathematical reality: II Waring Problem. Computer Tools in Education, 3, 5–55. (In Russian). Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. [Online materials]. Available at https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm Wertheimer, M. (1959). Productive thinking. Harper & Row.

Chapter 3

From Number Sieves to Difference Equations

3.1 Introduction Once a first grade student asked a question during a lesson on skip counting: Why do we call it skip counting by two when we skip one number only? Indeed, we skip one number only when moving from 2 to 4 to 6 to 8 and so on, which means adding 2 in each case. Apparently, this terminological discrepancy deals with distinction between action (skipping) and operation (adding). With this in mind, consider. Problem 3.1. How can one explain why we add 2 to a number through skip counting by 2 when we skip (i.e., omit) one number only? Discussion. Perhaps using the word skip (meaning omit) might be confusing. Do we skip something when we “skip counting by one”? There is no integer to skip between 0 and 1 (Fig. 3.1). Yet, we skip (meaning move along) an interval between two consecutive points on the number line; in particular, between 0 and 1. And probably it is better to avoid using the word skip with its double meaning such as omit and move along. But collaterally to the question asked by the first grader, the wording “skip counting” may be extended to skip intervals and integers they include while keeping in mind that skipping n integers that follow the integer x on the number line (action) is equivalent to adding n + 1 to x (operation). In other words, skipping the interval (x, x + n + 1) is equivalent to skipping n integers this interval includes (not counting the two endpoints). This distinction between action and operation can bring about and give meaning to the word sieve as a tool of creating different integer sequences; that is, eliminating certain integers from the set of natural numbers based on their location (that is, rank) within the set.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_3

35

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3 From Number Sieves to Difference Equations

Fig. 3.1 Skip counting by ones and twos as skipping intervals not numbers

3.2 On the Notion of a Number Sieve A number sieve, already mentioned in Chap. 2, Sect. 2.4, can be defined as a subsequence of a number sequence developed according to a certain rule. The word sieve is borrowed from the context of prime numbers (Chap. 5) when prime numbers became a subsequence of natural numbers through the process of elimination of composite numbers. Likewise, even and odd numbers can be developed through the process of elimination of certain terms of the natural number sequence. The rules of elimination applied to natural numbers suggest the emergence of recursive definitions of even and odd numbers. Such definitions (alternatively, first order difference equations) should include the first number of the sequence and the rule that every number beginning from the second one is equal to the previous number plus two. So, we have the following recursive formulas xn+1 = xn + 2, x1 = 2,

(3.1)

xn+1 = xn + 2, x1 = 1,

(3.2)

for the even and the odd numbers, respectively. Another definition of the two sequences can be developed by establishing a relationship between a term of a sequence and its rank in the sequence. Such relationship is called a closed formula; closed in the sense of its self-efficiency: in order to find a term of a sequence, one only needs to know its rank within the sequence and does not need to know other terms like in the case of recursive definition when one needs to go back, that is, to recur to what is already known. For even numbers, each term is twice its rank; for odd numbers, each term is twice its rank minus one. Thus, we have the following closed formulas: xn = 2n, n = 1, 2, 3, . . .

(3.3)

xn = 2n − 1, n = 1, 2, 3, . . . .

(3.4)

for the even and the odd numbers, respectively.

3.3 Theoretical Value of Practical Outcome of the Instrumental Act

37

A good question often asked by elementary teacher candidates is as follows: Why do we need to have two types of formulas, recursive and closed, for the same concept? To answer this question, note that the recursive formula stemmed from the introduction of even (or odd) numbers as a sieve developed through the process of elimination of every other natural number starting from two (or from one). But the closed formula reflected more traditional introduction of even and odd numbers dealing with divisibility by two; in other words, reflecting symbolically the physical action of putting objects in pairs. This duality of approaches to symbolic definitions of even and odd numbers raises the question of how their equivalence—one of the big ideas of mathematics—can be demonstrated. This demonstration will be provided below. In a meantime, the use of different definitions in the context of a digitally enhanced instrumental act which integrates pragmatic activities and epistemic development has to be discussed.

3.3 Theoretical Value of Practical Outcome of the Instrumental Act The process of the development of a number sieve using technology has both practical (pragmatic) and theoretical (epistemic) values. To explain, consider the following simple example. Suppose one has to find and record, perhaps as a practice in the use of a calculator or a spreadsheet, the sum of the first two and the first three positive integers, (1 + 2) + (1 + 2 + 3) = 3 + 6 = 9, then the sum of the first three and the first four positive integers, (1 + 2 + 3) + (1 + 2 + 3 + 4) = 6 + 10 = 16, then the sum of the first four and the first five positive integers, (1 + 2 + 3 + 4) + (1 + 2 + 3 + 4 + 5) = 10 + 15 = 25, and so on. As a result of this practical task, the numbers 9, 16, and 25 have been recorded. However, an epistemic value of this task is to recognize that the sums so developed are square numbers, demonstrate this phenomenon by using manipulatives (Fig. 3.2) and then proceed to formal demonstration (proof) for which formula (3.16) mentioned below in Sect. 3.7 can be used. As it was demonstrated through support of various collaterally creative activities in the previous two chapters of the book, the use of a spreadsheet can lead to deeper learning of mathematics. The pragmatic outcome of using a spreadsheet as an element of the instrumental act is in the ease of generating a large number of terms of a sieve, followed, if needed, by developing a sieve within a sieve. The epistemic part of using a spreadsheet deals with the mathematical meaning behind the formulas that define a sieve. Collateral creativity presumes students asking (often unexpected) questions about mathematical ideas they learn by reflecting on what their teaches say and do in the context of the instrumental act. A confident interaction by students with a spreadsheet is indicative of their understanding of mathematics used in computations. If a recursive definition allows one to use a single column (or row) of a spreadsheet by continuously referring to the previous number, a closed definition requires using two columns (or rows) of the tool by continuously referring to the rank of number to be

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3 From Number Sieves to Difference Equations

Fig. 3.2 Making a square number from two sums of consecutive integers

developed by such definition. That is, pragmatic and epistemic uses of a spreadsheet (and any mathematical software for that matter) within the instrumental act go handin-hand. Yet, a mathematical equivalence between two types of definition of an even or an odd number sieve becomes hidden unless a question about their mutual relationship is asked and answered.

3.4 On the Equivalence of Two Approaches to Even and Odd Numbers Equivalence, as one of the big ideas in mathematics, can be recognized, for example, by exploring numerical equivalence of two definitions of the same concept. To this end, one can use a spreadsheet as an instrument which provides a numerical approach to the idea of equivalence followed by a formal demonstration that two definitions, closed and recursive, are equivalent in the sense that the identity of numbers they generate never stops and continues infinitely. The use of a spreadsheet as a tool for conjecturing algebraic formulas through numerical evidence and then proving such formulas by the method of mathematical induction can be found in (Abramovich, 2016). Likewise, the use of the measuring facility of dynamic geometry software should be followed by formal demonstration (proof). Without proof, one can develop a faulty

3.4 On the Equivalence of Two Approaches to Even and Odd Numbers

39

perception of the validity of mathematical conjectures based on the recognition of their empirical evidence. Already in the third century B.C. Archimedes (1912) noted, “Certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said mechanical method did not furnish an actual demonstration. But it is of course easier, when the method has previously given us some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge” (p. 13). Nowadays, this perspective on the study of geometry is reflected in a number of publications by mathematics education researchers (e.g., Carriera et al., 2016; Christou et al., 2004; Trgalová, 2022; Yerushalmy, 1990). Consistent with the ancient idea, enriched by the modern-day tools, the concept of instrumental act can be given new meaning. To this end, students can be asked to use a spreadsheet (Fig. 3.3) as a generator of even and/or odd numbers through closed and recursive definitions. This would be the first step towards making them mathematically motivated and interested to learn whether it is possible to get the closed definitions of even and odd numbers from the corresponding recursive ones and vice versa. It is this interest that may be interpreted as collateral creativity in using technology to appreciate an epistemic value of practical (pragmatic) numeric computations. With this in mind, consider

Fig. 3.3 Using a spreadsheet to recognize numeric equivalence of definitions (3.1) and (3.3)

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3 From Number Sieves to Difference Equations

Problem 3.2. Demonstrate that formulas (3.1) and (3.3) as well as (3.2) and (3.4) are equivalent definitions of even and odd numbers, respectively. Discussion. Moving from (3.1) to (3.3), one can add the following (n − 1) relations x 2 = x1 + 2, x3 = x2 + 2, x4 = x3 + 2, . . . , xn = xn−1 + 2, resulting in x2 + x3 + · · · + xn−1 + xn = x1 + x2 + x3 + · · · + xn−1 + 2 + 2 + · · · + 2 n−1

whence, cancelling common terms from both sides, xn = 2 + 2(n − 1) = 2n, n = 1, 2, 3, . . . . Alternatively, one can use Wolfram Alpha as shown in Fig. 3.4. Transition from (3.3) to (3.1) is pretty straightforward: xn+1 = 2(n + 1) = 2n + 2 = xn + 2, x1 = 2. Similarly, moving from (3.2) to (3.4), one can add the following (n − 1) relations x2 = x1 + 2, x3 = x2 + 2, x4 = x3 + 2, . . . , xn = xn−1 + 2, resulting in x2 + x3 + · · · + xn−1 + xn = x1 + x2 + x3 + · · · + xn−1 + 2 + 2 + · · · + 2 n−1

whence, after the same cancellation, xn = 1 + 2(n − 1) = 2n − 1, n = 1, 2, 3, . . .. Alternatively, one can use Wolfram Alpha as shown in Fig. 3.5. Likewise, moving from (3.4) to the (3.2) yields xn+1 = 2n + 1 = 2n − 1 + 2 = x n + 2, x1 = 1. The use technology as a backdrop of moving from one definition to another and looking at the transition from the closed to the recursive, either a teacher or a student can collaterally ask the following questions as posing problems to think about and opening a window to new mathematical contexts (to be considered in other sections of this chapter): Fig. 3.4 Transition from (3.1) to (3.3) using Wolfram Alpha

3.5 Developing New Sieves from Even and Odd Numbers

41

Fig. 3.5 Transition from (3.2) to (3.4) using Wolfram Alpha

• Is it always the case that a recursive formula immediately follows from the closed formula? • Do there exist more complicated recursive definitions (difference equations) requiring more complicated transition to closed formulas and vice versa? • Can one use technology to obtain a closed formula from a recursive definition of a number sequence? • Can one use technology to obtain either formula from a few first terms of a number sequence? Another method to develop a sieve from a given sequence is through selection of its terms according to a certain rule. For example, even numbers can be selected from natural numbers using the following rule: select numbers, one by one, that are divisible by two. This rule yields the following closed formula: xn = 2n. This formula describes symbolically a physical definition of an even number: if the elements of a set can all be put in pairs, then the cardinality of the set is an even number. Likewise, odd numbers can be selected from natural numbers using the following rule: select numbers, one by one, that are not divisible by two. This rule yields the following closed formula: xn = 2n − 1. In other words, removing a single element from a set of objects put in pairs leaves one element without pair and, therefore, this action changes the cardinality of the set to become an odd number. One can see a difference between two ways of developing a sieve and its definition: the process of elimination yields a recursive definition of the corresponding sieve, and the process of selection yields a closed definition of the corresponding sieve.

3.5 Developing New Sieves from Even and Odd Numbers Interesting instrumental activities with number sequences involving a spreadsheet may deal with developing new sieves from already existing ones. Such activities begin with using pencil and paper to collect numeric data from which one can move to generalization needed for a digital investigation. For example, using the even number sieve—2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, …—the following questions can be explored:

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3 From Number Sieves to Difference Equations

• How can one use the process of elimination to develop multiples of six proceeding from the even number sieve? • How can one use the process of elimination to develop multiples of ten proceeding from the even number sieve? • How can one eliminate multiples of a certain number from the set of natural numbers? • How can one use the process of elimination to develop powers of two proceeding from the even number sieve? One can see that by eliminating two consecutive numbers from the even number sieve, one can get the sequence 6, 12, 18, 24, …. From here, one can develop a recursive definition of the multiples of six in the form xn+1 = xn + 6, x1 = 6. A more complicated task is to develop a formula describing the sequence of numbers that were eliminated from the even number sieve; namely, 2, 4, 8, 10, 14, 16, …. What is the most important in the context of creativity, in general, and collateral creativity, in particular, is one’s ability to see what kind of questions can be asked about what is already known. This requires from a teacher to create conditions conducive to such questions being asked. Following is a scenario describing how questions leading to generalizations can be prompted. Problem 3.3. Develop a closed formula that eliminates multiples of six from the natural number sequence. Discussion. Consider the sequence 1, 2, 4, 5, 7, 8, … representing the elimination of the multiples of three from the natural number sequence. Entering this sequence into the input box of The On-Line Encyclopedia of Integer Sequences (OEIS® , https://oeis.org/), one can find out that it can be expressed through the following closed formula (that is, it is possible to use information available on-line to find a general formula from the first few terms of a sequence—see one of the questions posed at the end of Sect. 3.4)  3n − 1 , n = 1, 2, 3, . . . , xn = I N T 2 

(3.5)

where INT(x) stands for the largest integer smaller than or equal to x. First of all, one can use a spreadsheet to justify formula (3.5) numerically. Collaterally to this justification, one can try to understand why formula (3.5) works in order to develop similar formulas when probing into the nature of numbers eliminated from a sieve. One can note that because INT (2n − 1) = 2n − 1, the sequence xn = I N T (2n − 1), n = 1, 2, 3, . . ., should eliminate all multiples of two. Collaterally to this note, one can make the last notation consistent with formula (3.5).  which can prompt the This results in the formula I N T (2n − 1) = I N T 2n−1 1 question whether the numbers 2 and 1 in the last formula and the numbers 3 and 2 in sequence (3.5) are responsible for the elimination of multiples of 2 and  should exclude the 3, respectively, and if so, the sequence xn = I N T 4n−1 3 multiples of four from the natural number sequence. Numerical verification in the context of a spreadsheet confirms the guess. This motivates the development

3.5 Developing New Sieves from Even and Odd Numbers

43

Fig. 3.6 Eliminating multiples of six from the natural number sequence

of formulas for eliminating multiples of other integers from the set of natural numbers. Once again, by trial and error, it can be confirmed that the sequences  6n−1 7n−1 xn = I N T 5n−1 = I N T = I N T , x , x , n = 1, 2, 3, . . . eliminate, n n 4 5 6 respectively, the multiples of 5, 6 and 7 from the natural number sequence. In particular, one can use Wolfram Alpha and enter into its input box the command “Table [floor((6n-1)/5), {n, 100}] to get the first 100 natural numbers that do not include using empirical induction, one can develop the multiples of six (Fig. 3.6).  In general, which eliminates the multiples of m from the natural sequence xn,m = I N T mn−1 m−1 number sequence. The presence of computational instruments in the teaching and learning process makes it possible to support reasoning by computing and computing by reasoning. In the former case, we see the pragmatic use of the instruments; in the latter case, we see how the instruments affect epistemic development of their users. The activity involving the creation of different number sieves when integrated with the instrumental act creates conditions conducive to fostering collateral creativity. Learners of mathematics are capable of asking a variety of questions resulting from the pragmatic use of technology and fostering their epistemic development. As another demonstration of this pedagogical phenomenon, consider. Problem 3.4. The Zerofree sequence (https://mathworld.wolfram.com/Zerofree. html). 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, . . . .

(3.6)

represents a natural number sieve which excludes multiples of ten. Demonstrate that the sequences  n−1 , n = 1, 2, 3, . . . xn = n + I N T 9 

and

(3.7)

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3 From Number Sieves to Difference Equations

 xn = I N T

 10n − 1 , n = 1, 2, 3, . . . . 9

(3.8)

are equivalent and both generate Zerofree sequence (3.6). Discussion. First note that the side of (3.7) may not be of 10.  right-hand  a multiple = 10k. Then are multiples of 10. Let I N T n−1 If it is, then both n and I N T n−1 9 9 10k = n−1 − ab , 0 < a < b, whence n = 90k + 9a + 1. The last equality shows that 9 b n is not a multiple of 10 unless a = b. It remains to be proved that (3.7) is equivalent to (3.8). Indeed,  I NT

10n − 1 9





   n−1 9n + n − 1 = I NT n + 9 9   n−1 . = n + I NT 9 = I NT

3.6 Polygonal Number Sieves First note that three problems to be discussed in this section are motivated by one of the questions posed at the end of Sect. 3.4 regarding a transition from a recursive formula to a closed formula. This question will be answered in the context of a more complicated situation dealing with creating number sieves leading to polygonal numbers—triangular, square, pentagonal, hexagonal and so on—using both elimination and selection of natural numbers. In order to create a triangular number sieve (1, 3, 6, 10, 15, …) through elimination and selection, one can select the number 1, eliminate the next number, 2, then select 3 (= 1 + 2) and eliminate two natural numbers after 3, i.e., 4 and 5, then select 6 (= 3 + 3) and eliminate three natural numbers after 6 to select 10 (= 6 + 4) and then eliminate the next four natural numbers, i.e., 11, 12, 13, and 14, to select 15 (= 10 + 5), and so on. Here, one can see the relationship between action (elimination) and operation (addition leading to selection) mentioned in Sect. 3.1 above: eliminating (n − 1) consecutive integers that follow the integer x is equivalent to adding n to x thus defining the selection of the number (x + n). The resulting sieve includes the (selected) triangular numbers. The interplay between physical elimination and operational addition (leading to selection of a number to be included into a sieve) suggests that the triangular number tn of rank n is the sum of the triangular number of the previous rank plus n, as to reach tn from tn−1 one has to do (n − 1) eliminations between two consecutive selections. That is, tn = tn−1 + n, t1 = 1.

(3.9)

3.6 Polygonal Number Sieves

45

Problem 3.5. Using formula (3.9), develop a closed formula for triangular numbers. Discussion. Note that the process of elimination and selection resulted in recursive formula (3.9). A transition to a closed formula can be carried out in much the same way as in the case of even (or odd) numbers by adding n equalities to get t2 + t3 + · · · + tn−1 + tn = t1 + t2 + t3 + · · · + tn−1 + 2 + 3 + · · · + n whence tn = 1 + 2 + 3 + · · · + n.

(3.10)

Relation (3.10) is not a closed formula; one has to carry out the summation of the first n natural numbers in order to develop such a formula. But what was found out is that the process of elimination described above is equivalent to the summation of the first n natural numbers and the triangular number of rank n can be defined through this sum. The process of elimination and selection works in the case of developing square numbers (1, 4, 9, 16, 25, …) as follows. Select the number 1 and eliminate the next two numbers; select the number 4 (= 1 + 3) and eliminate the next four numbers; select the number 9 (= 4 + 5) and eliminate the next six numbers; select the number 16 (= 9 + 7) and eliminate the next eight numbers reaching (selecting) the number 25 (= 16 + 9), and so on. The resulting sieve includes the (selected) square numbers. The interplay between elimination (action) and addition (operation leading to selection of a number to be included into a sieve) suggests that the square number sn of rank n is equal to the square number of the previous rank plus 2n − 1 as to reach sn from sn−1 one has to do 2(n − 1) eliminations between two consecutive selections. That is, sn = sn−1 + 2n − 1, s1 = 1.

(3.11)

Problem 3.6. Using formula (3.11), develop a closed formula for square numbers. Discussion. Once again, the process of elimination resulted in recursive formula (3.11). A transition to a closed formula can be carried out in much the same way as in the case of triangular numbers by adding n equalities to get s2 + s3 + · · · + sn = s1 + s2 + s3 + · · · + sn−1 + 3 + 5 + · · · + 2n − 1, whence sn = 1 + 3 + 5 + · · · + 2n − 1.

(3.12)

Relation (3.12) is not a closed formula; one has to carry out the summation of the first n odd numbers in order to develop such a formula. But what was found out is that the process of elimination and selection described in the case of square numbers

46

3 From Number Sieves to Difference Equations

is equivalent to the summation of the first n odd numbers and this sum is the square number of rank n, that is, n 2 . In other words, 1 + 3 + 5 + · · · + 2n − 1 = n 2 . The process of elimination and selection works in the case of developing pentagonal numbers (1, 5, 12, 22, 35, …) as follows. Select the number 1 and eliminate the next three numbers; select the number 5 (= 1 + 4) and eliminate the next six numbers; select the number 12 (= 5 + 7) and eliminate the next nine numbers; select the number 22 (= 12 + 10) and eliminate the next twelve numbers reaching the number 35 (= 22 + 13), and so on. The resulting sieve includes the (selected) pentagonal numbers. The interplay between elimination (action) and addition (operation leading to selection of a number to be included into a sieve) suggests that the pentagonal number pn of rank n is equal to the pentagonal number of the previous rank plus 3n − 2, as to reach pn from pn−1 one has to do 3(n – 1) eliminations between two consecutive selections. That is, pn = pn−1 + 3n − 2, p1 = 1.

(3.13)

Problem 3.7. Using formula (3.13), develop a closed formula for pentagonal numbers. Discussion. Once again, the process of elimination resulted in recursive formula (3.13). A transition to a closed formula can be carried out in much the same way as in the case of triangular and square numbers by adding n equalities to get p2 + p3 + · · · + pn = p1 + p2 + p3 + · · · + pn−1 + 4 + 7 + · · · + 3n − 2, whence pn = 1 + 4 + 7 + · · · + 3n − 2.

(3.14)

Once again, relation (3.14) is not a closed formula; in order to develop such a formula, one has to carry out the summation of n terms of the arithmetic sequence with the first term 1 and difference 3. But what was found out is that the process of elimination described in the case of pentagonal numbers is equivalent to the summation of the n terms of the above-mentioned arithmetic series and this sum is the pentagonal number of rank n. The process of elimination and selection works in the case of developing hexagonal numbers (1, 6, 15, 28, 45, …) as follows. Select the number 1 and eliminate the next four numbers; select the number 6 (= 1 + 5) and eliminate the next eight numbers; select the number 15 (= 6 + 9) and eliminate the next twelve numbers; select the number 28 (= 15 + 13) and eliminate the next sixteen numbers reaching the number 45 (= 28 + 17), and so on. The resulting sieve includes the (selected) hexagonal numbers (the first order sieve of triangular numbers). The interplay between elimination (action) and addition (operation leading to selection of a number to be

3.7 Connecting Arithmetic to Geometry Explains Mathematical Terminology

47

included into a sieve) suggests that the hexagonal number h n of rank n is equal to the hexagonal number of the previous rank plus n − 3, as to reach h n from h n−1 one has to do 4(n − 1) eliminations between two consecutive selections. That is, h n = h n−1 + 4n − 3, h 1 = 1.

(3.15)

Once again, the process of elimination and selection resulted in recursive formula (3.15). Collateral creativity of a student may prompt the following question: • How do we know what kind of elimination and selection leads to which type of a polygonal number? This question asks what comes first—the process (action) or its symbolic description (a name for a sieve). A possible answer to this question is that a mathematician uses different but consistent rules of elimination (action) and selection (through operation of addition) in developing subsequences of natural numbers (sieves) and then assign names to those sieves. The names may come from connecting arithmetic (operation) to geometry (image). It will be shown in the next section how this connection can be done. One can see that the elimination and selection approach to the concept of a number sieve makes it possible to develop different sieves which recursively define many sequences of numbers. This opens a window to different activities, which foster one’s mathematical competence as the crux of recognizing the emergence of collateral creativity in the classroom, such as • • • •

Practice a skill of demonstrating equivalence of different definitions of a concept. Practice a skill of pattern recognition by seeing general in particular. Practice a skill of formulating mathematical conjectures. Practice a skill of proving mathematical conjectures.

3.7 Connecting Arithmetic to Geometry Explains Mathematical Terminology Understanding the meaning of mathematical terminology, which often stems from its imagery, is an important element of teachers’ mathematical competence needed for epistemic control of students’ contextual investigations. With this in mind, consider the set of natural numbers: 1, 2, 3, 4, 5, …. One can develop a new sequence not through the process of elimination (action) but through the process of summation (operation) which, to a certain extent, may be considered as a gradually evolving selection. Consider partial sums of natural numbers: 1, 1 + 2, 1 + 2 + 3, …, 1 + 2 + 3 + … + n, …. One can see these sums through the following lens: select the first n natural numbers and add them to find the sum. Contextualization can be used to develop a method of finding the sum by noting that it can be interpreted as the

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number of handshakes among (n + 1) people who shake each other hand one time only. Indeed, the first person can make n handshakes and leave, the second person can make (n – 1) handshakes and leave, the third person can make (n – 2) handshakes and leave; finally, the penultimate and the last individuals can make one handshake. This gives the sum 1 + 2 + 3 + … + n as the total number of handshakes. In order to find this sum, note that the process of handshaking can be described differently: the first person makes n handshakes and does not leave allowing the second person to make n handshakes and not leave so that all (n + 1) individuals can make n handshakes. Repeating n handshakes (n + 1) times yields the product n(n + 1) which represents twice as many handshakes as the sum 1 + 2 + 3 + … + n. From here, the relation (a possible use of which as an instrument of formal demonstration was mentioned above in Sect. 3.3) 1 + 2 + 3 + ··· + n =

n(n + 1) 2

(3.16)

follows. Contextualization can also be integrated with visualization by representing partial sums of consecutive natural numbers in the form of an evolving equilateral triangle as shown in Fig. 3.7. This image explains the term “triangular numbers” applied to the sums of the first n natural numbers as represented by formula (3.16). Whereas any sum of an arithmetic series (starting from the number 1) can be presented in the form of an isosceles triangle, the term triangular number is reserved for an equilateral triangle only, representing the sequence with the number 1 being both the first term and the difference (i.e., natural number sequence). This note may be an answer to a collaterally creative question as to why other sums of integers differing by the same number are not called triangular numbers (e.g., see Figs. 3.8, 3.9, 3.13). Fig. 3.7 Constructing an image of triangular numbers

3.8 Polygonal Numbers and Collateral Creativity

49

Fig. 3.8 Constructing a square from an isosceles triangle

Fig. 3.9 Relating isosceles and equilateral triangles

Now, consider the set of odd numbers: 1, 3, 5, 7, 9, …. One can develop a new sequence not through the process of elimination but through the process of summation which, to a certain extent, may also be considered as a gradually evolving selection. Consider the partial sums of odd numbers: 1, 1 + 3, 1 + 3 + 5, …, 1 + 3 + 5 + … + (2n − 1), …. One can see these sums through the following lens: select the first n odd numbers and add them to find the sum. Visualization provided by Fig. 3.8 suggests that the partial sums of odd numbers (starting from the number 1) are square numbers. Symbolically, one can show that this statement is true by the method of mathematical induction. Assuming that the equality 1 + 3 + 5 + … + 2n − 1 = n2 is true, by adding 2n + 1 to its both sides one proves the correctness of the inductive transfer consisting in the transition from n to n + 1. Indeed, 1 + 3 + 5 + … + (2n − 1) + (2n + 1) = n2 + 2n + 1 = (n + 1)2 .

3.8 Polygonal Numbers and Collateral Creativity In connection with the diagram of Fig. 3.8, an interesting example of collateral creativity was demonstrated by an elementary teacher candidate (briefly mentioned in Chap. 1, Example 1.1). Responding to the task of transforming a 4 × 4 square made out of 16 counters into an isosceles triangle, the candidate not only created the

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3 From Number Sieves to Difference Equations

triangle shown in the left-hand side of Fig. 3.8 but, as part of the instrumental act supported by concrete materials, re-constructed the isosceles triangle (with lateral sides comprised of four counters) in the form of an equilateral triangle shown in the right-hand side of Fig. 3.9 (all sides comprised of five counters) with one counter left unused. This transition from one type of triangle found through the task to another type of triangle not sought by the task can be seen through the lens of collateral creativity as an unexpected (but favorable) outcome of the instrumental act. This also highlights an important role of students’ interaction with a “more knowledgeable other” (Vygotsky, 1978); in this case, a teacher who is aware of the significance of students’ ideas. In other words, collateral creativity of a student in posing a (new) problem becomes supported by the teacher tapping into their zone of proximal development (Vygotsky, 1987). A hidden mathematical idea behind this transition from an isosceles triangle to an equilateral triangle with one counter left unused can be interpreted in terms of the existence of a pair of square and triangular numbers when the difference between the two numbers is equal to one. Towards revealing this idea to teacher candidates as an outcome of collateral creativity, the following two questions might be asked. Do other such pairs exist? If so, how can one find such pairs? Extending the instrumental act to include a digital tool, a spreadsheet can be used to answer these questions. It turned out that among the first 1000 square numbers only eight can find a pair among triangular numbers. The spreadsheet of Fig. 3.10 shows those pairs of square and triangular numbers that differ by one. The spreadsheet was created by generating square numbers followed by applying the triangular number test (Abramovich, 2016, p. 209) to squares diminished by one. The test states that the number √ N is a triangular number if the sum 8N + 1 is a perfect square and the difference 8N + 1 − 1 is a multiple of two as the last radical is an odd number. Alternatively, one can use Wolfram Alpha and enter into its input box the command “solve over the positive integers n^2–1 = m(m + 1)/2”. The result is shown in Fig. 3.11 that confirms the use of the triangular number set within a spreadsheet. As an aside, note that such pairs of isosceles and equilateral triangles are similar to twin primes (Chap. 5) and can be called twin IE-triangles. To conclude this section, another connection between number theory and geometry can be demonstrated. First, one can clarify why the partial sums p1 , p2 , p3 , …, defined by formula (3.14) are called pentagonal numbers. The process of reconstruction of the sum of four numbers 1 + 4 + 7 + 10 = 22 from an isosceles triangle into a pentagon is shown in Fig. 3.12. Second, once again, 22 counters forming an isosceles triangle can be used to construct an equilateral triangle with one counter left unused (Fig. 3.13). Thus, we have the pair (22, 21) of pentagonal and triangular numbers and, motivated by collateral creativity by the above-mentioned teacher candidate in a similar context, one can try to find other such pairs. The spreadsheet of Fig. 3.14 shows that among the first 1000 pentagonal numbers only five can find a pair among triangular numbers. Alternatively, one can use Wolfram Alpha by entering into its input box the command “solve over the positive integers n(3n-1)/2–1 = m(m + 1)/2”. The result shown in Fig. 3.15 confirms what was found by the spreadsheet of Fig. 3.14. Instrumental activity supported by a spreadsheet and Wolfram Alpha enabled finding

3.8 Polygonal Numbers and Collateral Creativity

51

Fig. 3.10 Locating pairs of square and triangular numbers differing by one Fig. 3.11 Wolfram Alpha confirms the results of spreadsheet modeling

another type of twin IE-triangles. The examples of this section show how students’ collateral creativity supported by the instrumental act and by the teacher’s prompts can lead to nearly unlimited beauty of mathematical discoveries.

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Fig. 3.12 Constructing pentagonal numbers

Fig. 3.13 From the isosceles to the equilateral with one counter unused

Fig. 3.14 Locating pairs of pentagonal and triangular numbers differing by one

References

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Fig. 3.15 Wolfram Alpha confirms the results of spreadsheet modeling

References Abramovich, S. (2016). Exploring mathematics with integrated spreadsheets in teacher education. World Scientific. Archimedes (1912). The method of archimedes. In: T. L. Heath (Eds.). Cambridge University Press. Carreira, S., Jones, K., Amado, N., Jacinto, H., & Nobre, S. (2016). Youngsters solving mathematical problems with technology: The results and implications of the Problem@Web Project. Springer. Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic geometry environments. International Journal of Science and Mathematics Education, 2(3), 339–352. Trgalová, J. (2022). Digital technology and its various uses from the instrumental perspective: The case of dynamic geometry. In P. R. Richard, M. P. Vélez, & S. Van Vaerenbergh (Eds.), Mathematics education in the age of artificial intelligence (pp. 417–429). Springer. Vygotsky, L. S. (1978). Mind in society. MIT Press. Vygotsky, L. S. (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.), The collected works of L. S. Vygotsky (vol. 1, pp. 39–285). Plenum Press. Yerushalmy, M. (1990). Using empirical information in geometry: Student’s and designer’s expectations. Journal of Computers in Mathematics and Science Teaching, 9(3), 23–38.

Chapter 4

Explorations with the Sums of Digits

4.1 Introduction A few years ago, one of the authors asked a group of fifth grade students what they can say about a year number. The following simple problem was posed. Problem 4.1. What is special about the number 2021? Discussion. The goal of the problem was to encourage students to look at numbers from an open-ended perspective. It is open-endedness of mathematical thinking that motivates and provides foundation for collateral creativity in the context of even routine situations before any technology becomes involved. The students provided different answers, among them: • • • • • • • •

2021 is a four-digit number. The second digit is zero. Two digits are the same. 2021 can be split in two consecutive numbers, 20 and 21. 2021 is an odd number. The sum of digits of 2021 is equal to 5. Regardless of the order of the four digits, their sum does not change. The total of nine four-digit numbers can be constructed by permuting digits in the number 2021.

The answers were recorded, and the next stage of the activity was to use students’ observations about the number 2021 as a source of new problems to explore when moving beyond a particular year number. This resulted in the emergence of several collaterally creative questions some of which were formulated by students. That is, once again, collateral creativity was seen as an awoken mathematical problem posing.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_4

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4 Explorations with the Sums of Digits

Problem 4.2. Explore the following questions. • Which digits of the number 2021 do not change in the twenty-first century? • How many years of the twenty-first century have the sum of digits equal to 5? • How many years of the twenty-first century have the sum of digits equal to 6, 7, 8, 9? • How can such years be constructed? • How can one find all such years? • What is the smallest sum of digits of a year in the twenty-first century? • What is the largest sum of digits of a year in the twenty-first century? • Why is the difference between two consecutive years with the same sum of digits always equal to nine? • Is it true that each century has only one palindromic year (i.e., a year which reads forwards and backwards the same)? • How many different four-digit years can be created out of the four digits of 2021? • How many different four-digit years can be created out of the digits 2, 0, and 1 (allowing for the repletion of the digits)? • Are there centuries with more than one palindromic year? • What is the relationship between the number of digits in a year and the number of digits in a century to which the year belongs? This chapter will demonstrate different mathematical developments resulting from investigations prompted by the questions in Problem 4.2 thus highlighting conceptual depth of seemingly ‘simple’ tasks yet bearing potential for collateral creativity through explorations supported by different instruments, both tactile and digital.

4.2 About the Sums of Digits The first question was not very complicated: the first two digits of the number 2021 are the same for any year that belongs to the twenty-first century (see Sect. 4.4 of this chapter). An answer about the number of years in the twenty-first century with the sum of digits being equal to the number 5 can be given by using two major ways of reasoning: procedural and conceptual (Abramovich, 2015; Kadijevich, 2002; RittleJohnson et al., 2001). The former way is to provide an organized list: 2003, 2012, 2021, and 2030. The organized list requires some understanding of how the numbers can be created. This understanding is, in fact, conceptual. More specifically, one has to note that the first two digits of a year in the twenty-first century are 2 and 0; therefore, only the last two digits may vary with the sum of digits 3. How many two-digit numbers have the sum of digits 3? To answer this question conceptually, one can introduce a friendly context of finding the number of ways to put three (identical) finger rings on two fingers or to put three (identical) cookies in two cups. In a classroom setting, base-ten blocks may be used as a substitute for rings or

4.2 About the Sums of Digits

57

cookies. Such a substitute serves a special purpose: it shows how manipulatives can be differently interpreted depending on mathematical context they serve. Indeed, whereas a rod includes ten cubes and a flat includes ten rods, each of them represents a single place value, and therefore in the context of dealing with digits each of them—a cube, a rod, and a flat—represents one thing the numeric value of which is one. Figure 4.1 shows four ways to put three cookies (either three cubes, or three rods, or one rod and two cubes, or two rods and one cube) in two cups. Alternatively, a number of ways to put three (identical) cookies in two cups can be found by solving the equation a + b = 3 in non-negative integers within Wolfram Alpha as shown in Fig. 4.2. The question about the number of ways to put three cookies in two cups can also be resolved by finding the number of permutations of letters in the word YYYS where three letters Y represent three cookies and the letter S represents a separator between two cups, so that the word YYYS describes the case of a cup with three cookies and an empty cup. The number of permutations of letters in the word YYYS is equal to four (YYYS, YYSY, YSYY, SYYY, or, = 4). This formal counting technique involving factorials can be used formally, 4! 3! to count the number of years with the same digits as 2021, that is, with two 2’s, one 0, and one 1. The number of permutations of digits in the string 2021 is equal to 4! = 12. Collaterally creative question is: How many of those permutations (years) 2! are expressed through a three-digit number (that is, begin with zero)? How many years with two 2’s, one 0, and one 1 belong to the twenty-first century? Likewise, one can find the number of years in the twenty-first century with the n! for sums of digits 6, 7, 8, 9, and so on. The result of calculating the expression (n−1)! different values of n is shown in Fig. 4.3. For example, the case n = 5 corresponds to the number of years in the twenty-first century with the sum of digits 6 and it can be described through the word YYYYS (a cup with four cookies and an empty cup), Fig. 4.1 Four ways for a two-digit number having 3 as the sum of digits

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4 Explorations with the Sums of Digits

Fig. 4.2 Alternative representation of the diagram of Fig. 4.1

Fig. 4.3 Using Wolfram Alpha in finding the number of permutations of letters in a word 5! the number of permutations of letters in which is equal to 4! = 5. By making an organized list, one can check to see that the years 2004, 2040, 2013, 2031, and 2022 are the only years in the twenty-first century with the sum of digits 6. How can one explain that as the sum of digits increases by one, the number of years in the twenty-first century with this new sum increases by one as well? Increasing the sum of digits by one (when the last two digits may be changed only) can be interpreted as adding another finger ring or a cookie to the mix (still dealing with two cups or two fingers). Figure 4.4 shows how adding additional square to any number of lined-up squares (top part) creates a new space (bottom part) for a triangle to be placed among

4.3 Years with the Difference Nine

59

Fig. 4.4 Adding additional finger ring/cookie

squares. Formally, the number of permutation of letters in the words with (n + 1) . . . Y S and Y Y  . . . Y S, respectively, and n letters Y and a single letter S, that is, Y Y  differs by one. Symbolically,

(n+1)! n!

n+1

−

n! (n−1)!

n

= n + 1 − n = 1.

4.3 Years with the Difference Nine The difference between two consecutive years (in the twenty-first century) with the sum of digits equal to 5 in the sequence 2003, 2012, 2021, 2030 is equal to 9. The fact that the difference is equal to 9 can be demonstrated conceptually with the use of base-ten blocks (Figs. 4.5 and 4.6). It is enough to deal with the representations of the last two digits of two consecutive years. In Fig. 4.5 the difference between the years 2012 and 2003 is presented as the difference between 12 and 3. In Fig. 4.6 the difference between the years 2021 and 2012 is presented as the difference between 21 and 12. In that way, activities with base-ten blocks on a place value chart are Fig. 4.5 The difference nine between 12 and 3

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Fig. 4.6 The difference nine between 21 and 12

presented in the context of problem solving; something that is quite different from using these tools to do calculations, without connection to any problem. There are six years in the twenty-first century with the sum of digits seven: 2005, 2014, 2023, 2032, 2041, 2050. The difference between two consecutive years in this sequence is also 9. There are seven years in the twenty-first century with the sum of digits eight: 2006, 2015, 2024, 2033, 2042, 2051, 2060. The difference between two consecutive years in this sequence is also 9. These observations raise new questions (to be discussed in the next sections) seeking explanation for the patterns found.

4.4 Calculating the Century Number to Which a Year Belongs One question following from the above observations is pretty routine, yet collateral to the previous task: given a year, which century does it belong to? To this end, note that in the twenty-first century two consecutive years with the sum of digits k, k ≤ 20 (= 2 + 9 + 9), can be written as 2 × 103 + n × 10 + m and 2 × 103 + (n − 1) × 10 + m + 1, where m + n = k − 2. The difference between the two numbers does not depend on the first two digits and can be computed using the last two digits only: (n × 10 + m) − ((n − 1) × 10 + m + 1) = 10 − 1 = 9. For example, when k = 7, n = 4 and m = 1, the corresponding pair of consecutive years is 2041 and 2032 which are 9 years apart. One may be curious whether this difference stays the same for any century. This question is connected to the question about

4.5 Finding the Number of Years with the Given Sum of Digits Throughout …

61

the relationship between the year number of the corresponding century number. For example, which operation transforms the year number 2021 into the century number 21? Let N = 103 a1 + 102 a2 + 10a3 + a4 be a four-digit year number. Regardless of a year number  N of the Common Era, the century  C(N ) to which it belongs is equal to I N T 10N2 + 1. Therefore, C(N ) = I N T 10N2 + 1 = 10a1 + a2 + 1 is the number of the century to which year N belongs.

4.5 Finding the Number of Years with the Given Sum of Digits Throughout Centuries One can use Wolfram Alpha to generate an addition table the entries of which are digits in the range 1 through 9. Using the tool (Fig. 4.7), the following questions can be explored. • How many years are there with 20 as the sum of digits in other Common Era centuries not higher than the twenty-first century? These are 299, 1199, and 2099. Indeed, subtracting 2 from 20 yields 18 which appears in the 9-addition table (Fig. 4.7) only one time: 18 = 9 + 9. • How many years are there in the twenty-first century with the sum of digits 20? There is only one such year in the 21st century: 2099 has 20 as the sum of digits. Indeed, subtracting 2 from 20 yields 18 which appears in the 9-addition table (Fig. 4.7) only one time: 18 = 9 + 9. • How many years are there in the twenty-second century with the sum of digits 20?

Fig. 4.7 Addition table with addends in the range [0, 9]

62

•

•

•

•

• • •

4 Explorations with the Sums of Digits

There are only two such years in the 22nd century: 2189 and 2198 have 20 as the sum of digits. Indeed, subtracting 3 from 20 yields 17 which appears in the 9-addition table (Fig. 4.7) two times: 17 = 9 + 8 = 8 + 9. How many years are there in the twenty-third century with the sum of digits 20? There are only three such years in the 23rd century: 2279, 2297, 2288 have 20 as the sum of digits. Indeed, subtracting 4 from 20 yields 16 which appears in the 9-addition table (Fig. 4.7) three times: 16 = 9 + 7 = 7 +9 = 8 + 8. How many years are there in the twenty-fourth century with the sum of digits 20? There are only four years in the 24th century, 2369, 2396, 2378, 2387, with 20 as the sum of digits. Indeed, subtracting 5 from 20 yields 15 which appears in the 9-addition table (Fig. 4.7) four times: 15 = 9 + 6 = 6 + 9 = 8+ 7 = 7 + 8. How many years are there in the twenty-fifth century with the sum of digits 20? There are only five years in the 25th century, 2459, 2495, 2468, 2486, 2477, with 20 as the sum of digits. Indeed, subtracting 6 from 20 yields 14 which appears in the 9-addition table (Fig. 4.7) five times: 14 = 9 + 5 = 5 + 9 = 8 + 6 = 6 + 8 = 7 + 7. How many years are there in the twenty-sixth century with the sum of digits 20? There are only six years in the 26th century, 2549, 2594, 2558, 2585, 2567, 2576, with 20 as the sum of digits. Indeed, subtracting 7 from 20 yields 13 which appears in the 9-addition table (Fig. 4.7) six times: 13 = 9 + 4 = 4 + 9 = 8 + 5 = 5 + 8 = 7 + 6 = 6 + 7. How can one guess, without making an organized list, how many years with the sum of digits 20 are there in the twenty-seventh, twenty-eighth, twenty-ninth and thirtieth centuries? Is there a pattern? What is the pattern? Does the pattern for the number of years with the sum of digits 20 observed for the 21st through the thirtieth centuries continue for the 31st through the 40th centuries? What about the 41st through 50th centuries? How would responses to the above questions change if the sum of digits changes? How can one use the addition table of Fig. 4.7 to explain that there is no year in the twenty-first century with 21 as the sum of digits and two years with 19 as the sum of digits; no year in the twenty-second century with 22 as the sum of digits and three years with 19 as the sum of digits; and so on? Answers to the questions of that kind will be provided in the next section.

4.6 Partitioning n into Ordered Sums of Two Positive Integers There exist (n − 1) ways to partition an integer n in ordered sums of two positive integers. Only numbers not greater than 18 can be partitioned in the sums of two one-digit numbers. For example, a number n = 10 + k, 0 ≤ k ≤ 8, can be partitioned in the ordered sums of two positive integers one of which is a two-digit number in 2k ways. Therefore, a number n = 10 + k, 0 ≤ k ≤ 8, can be partitioned in ordered sums of one-digit numbers in ((10 + k) − 1) − 2k = 9 − k ways. For example,

4.7 Interpreting the Results of Spreadsheet Modeling

63

Fig. 4.8 Calculating positive values of the expression 9 − k

when k = 8, there is only one way, 9 + 9, something that is confirmed by the above formula (as well as by the table of Fig. 4.7). That is, in the twenty-first century there is only one year, 2099, with the sum of digits 20 and two years, 2089 and 2098, with the sum of digits 19. Likewise, in the twenty-sixth century there are six years with the sum of digits equal 20. The first two digits of a year in the twenty-sixth century are 2 and 5, the sum of which is 7; therefore, the remaining two digits must have the sum 13. Because 13 = 10 + 3, we have 9 − 3 = 6 years in the twenty-sixth century with the sum of digits equal to 20. Figure 4.8 shows the use of Wolfram Alpha in calculating the values of the expression 9 − k for 0 ≤ k ≤ 8.

4.7 Interpreting the Results of Spreadsheet Modeling Using a spreadsheet, one can construct a table which displays the number of years in any given century with the given sum of digits. Figures 4.9 and 4.10 display the results for the twenty-first and twenty-second centuries, respectively. Observing the tables, one can be asked to interpret the results of modeling by answering the following questions. • Why do the largest sums of digits of years in two consecutive centuries differ by one? • Why does the largest number of years in the twenty-first century have the sum of digits equal to 11?

Fig. 4.9 Number of years in the twenty-first century with the given sum of digits

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Fig. 4.10 Number of years in the twenty-second century with the given sum of digits

• Why does the largest number of years in the twenty-second century have the sum of digits equal to 12? • How can one explain monotonic increase and monotonic decrease of the number of years as the function of the sum of digits of a year? To conclude this chapter, note that dealing with some simple questions based on the change of a year creates a fertile context for new problems collaterally arising from deeper investigations supported by physical and digital tools. This wealth of opportunities in the age of technology illustrates true potential for the culture of questioning and investigating that provides quite rich and productive mathematical experience.

References Abramovich, S. (2015). Mathematical problem posing as a link between algorithmic thinking and conceptual knowledge. The Teaching of Mathematics, 18(2), 45–60. Kadijevich, D. (2002). Towards a CAS promoting links between procedural and conceptual mathematical knowledge. The International Journal of Computer Algebra in Mathematics Education, 9(1), 69–74. Rittle-Johnson, B., Siegler, R. S., & Alibali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93(2), 346–362.

Chapter 5

Collateral Creativity and Prime Numbers

5.1 ‘Low-Level’ Questions Require ‘High-Level’ Thinking One of the main pedagogic expectations behind fostering collateral creativity deals with a teacher’s ability to recognize in a student’s unexpected utterance hidden mathematical meaning to allow for revealing and expanding this meaning through a didactically appropriate response. For example, not only schoolchildren but also elementary teacher candidates might sometimes confuse odd and prime numbers, knowing that just as an odd number, a prime number (greater than two) is not divisible by two. But the issue is not divisibility alone, it is the number of divisors (alternatively, the number of multiplicative decompositions) that an integer may have. One of the authors used an exploratory activity with several groups of upper elementary school students (Grades 4–6) and elementary teacher candidates to investigate certain properties of a number representing a calendar year, for example, to determine if it is a prime number. Some (four-digit) years provide very interesting contexts leading to quite deep explorations. To this end, consider Problem 5.1. Explore divisibility of the number representing year 2021. Discussion. The investigation started with an open question by a teacher about what can be said about number 2021 (see Chap. 4, Problem 4.1). Some students saw it being composed of three consecutive digits, 0, 1, 2. Others saw two consecutive two-digit numbers, 20, 21, The conclusion that the number is odd came quickly into their mind. A slight hint from the teacher can help students to realize that it cannot be divided by 3 (using a calculator or the corresponding divisibility rule mentioned in Problem 5.3 below). It is interesting to see how students would handle even numbers (4, 6, 8, etc.) The (non) divisibility by 5 is quite easy to determine visually. Divisor 7 can be checked using a calculator. After a few more searches, students might suggest that no divisor (except, of course, 1 and 2021) exists, so a conjecture that 2021 could be prime comes to students’ mind. How do we check the conjecture?

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_5

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Unfortunately, the sieve of Eratosthenes (see Sect. 5.6.1) is not an effective tool to decide the primality of 2021 and cannot be appropriated as an element of the instrumental act in answering the last question (even by using a sieve, http://www.hbmeyer. de/eratosiv.htm, available on-line with 400 being the largest number). Didactically, two major approaches to decide the primality of 2021 (or 2023, 2027, 2029, for that matter) may be considered. The first approach is a combination of a purely visual/ mental examination (the last digit of 2021 is not an even number, zero included, and not 5; the sum of digits of 2021 is not a multiple of 3) and a computational testing (and the number is not divisible by, say, 7 and 11), perhaps by using a calculator. This can be useful as an initial step pointing at possible, though erroneous, conjecture that 2021 might be a prime number. The second approach which reflects the latest development of digital technology is to take advantage of Wolfram Alpha, already used to support various computations in other chapters of the book. Just entering “2021” into the input box of Wolfram Alpha brings about the following prime factorization 2021 = 43 × 47 implying that 2021 is not a prime number as a product of two prime numbers smaller than 2021. Once again, one can see that control of Wolfram Alpha does not require mathematical skills. Yet being included in the instrumental act of “distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors” (Gauss, 1966, p. 396), the tool supports coordination and course of mental processes (Vygotsky, 1930) responsible for asking further questions about prime numbers. In fact, a pedagogy of asking new questions rather than avoiding them “may lead to the emancipation of pupils’ epistemological position” (de Champlain et al., 2018, p. 500), something that, in turn, can result in the emergence of collateral creativity. Towards this end, the following questions can be posed and explored: • What was the (most recent) year before 2021 that was a product of two different (or even consecutive) prime numbers? The next one? Or, more generally, how frequent are such years? If one tries the products 41×43 (1763—the closest year, before 2021, being a product of two consecutive primes) and 37 × 47 (the candidates for the year before 2021, excluding the product 43 × 43) another interesting question is: • How can one prove without finding the products, that the former is greater than the latter? Likewise, if one tries the products 43 × 59 and 47 × 53 (the candidates for the year after 2021, excluding the product 47 × 47; the latter, year 2491, being the smallest product of two consecutive primes greater than 43 × 47), a question, collateral to the study of prime numbers, is: • How can one prove without finding the products of two-digit numbers that the former is greater than the latter? That is, as Dewey (1938), introducing the concept of collateral learning, put it, “Perhaps the greatest of all pedagogical fallacies is the notion that a person learns only the particular thing he is studying at the time” (p. 49). Indeed, in the process

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of studying prime numbers using such a powerful computational tool as Wolfram Alpha, the learning space of a student can be expanded to include seemingly ‘lowlevel’ questions requiring, nonetheless, ‘high-level’ thinking informed by performing rather basic arithmetic operations. Such hidden complexity of the so-called ‘lowlevel’ questions naturally afforded by use of technology can collaterally expand students’ conventional learning space, leading to another question: • How many divisions of 2021 by consecutive prime numbers one has to make in order to determine that 2021 is not a prime number? Wolfram Alpha can generate a list of all primes in the range 2 through 43 (Fig. 5.1). One can see that there are 14 primes in the range [2, 43]. This leads to the next question: • Is it possible to decide whether 2021 is divisible by some of these primes without actually doing division? This question (which is both pragmatic and epistemic) leads to the discussion of the tests of divisibility by 3 (or by 9) and by 11 which, reduce dealing with the division of multidigit numbers to operating on their digits. Such tests will be considered below in Sect. 5.5. • If 2021 is not a prime, when would we have the closest year represented by a prime number? Using Wolfram Alpha (Fig. 5.2), one can immediately get an answer to this question— the closest to 2021 year represented by a prime number is 2027. A special aspect of this number will be discussed in the next section. In a meantime, note that changing just one digit in a four-digit composite number yielded a prime number. Exploring for primality with the help of Wolfram Alpha another pair, 2037 and 2017, of four-digit numbers that differ by one digit shows that the former and the latter are composite and prime numbers, respectively. A collaterally creative question would be to ask whether any four-digit number is either a prime or one digit short of a prime. One can use Wolfram Alpha to explore this question. In doing so, one may note (Fig. 5.2) that there are no primes between 2039 and 2053 and therefore, any composite number in the range [2040, 2048] would require at least two digits to be modified in order

Fig. 5.1 Wolfram Alpha generates all primes not greater than 43

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Fig. 5.2 Primes greater than 2021

to have a prime. However, one can also see (with the help of Wolfram Alpha) that 2141 is a prime number created by modifying just one digit in the composite number 2041. This raises the following questions: • Does there exist a four-digit composite number which requires at least two digits to be changed in order to have a prime number? • If so, how can such number(s) be found using Wolfram Alpha? The authors leave these questions to be answered by the readers of the book.

5.2 Twin Primes Explorations Motivated by Activities with the Number 2021 Wolfram Alpha, as an instant provider of information, can generate several prime numbers greater than 2021 (Fig. 5.2). The first two among them are 2027 and 2029. One can (or has to be prompted to) note that these two prime numbers are special because they differ by two. Such primes are called twin primes. Figure 5.2 displays the following pairs of twin primes: (2027, 2029), (2081, 2083), (2087, 2089), (2111, 2113), (2129, 2131), (2141, 2143). One may also observe that among 20 prime numbers displayed in Fig. 5.2 more than half are twin primes. Although it is known from the time of Euclid1 that the number of primes is infinite (see Sect. 5.6.2), it is not known, as of the time of writing this book, whether there are infinitely many twin primes. In the spirit of Dewey’s (1933) reflective inquiry as integration of knowing and doing and Piaget’s (1973) reflective abstraction as creative reconstruction of previous experience, the questions about twin primes, with answers easily provided by Wolfram Alpha as the nucleus of Vygotsky’s (1930) instrumental act, may continue as follows. 1

Euclid—the most prominent Greek mathematician of the third century B.C.

5.2 Twin Primes Explorations Motivated by Activities with the Number 2021

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• Can one describe pairs of twin primes based on their last digits? Which numbers may not serve as the last digits of twin primes? • Consider the pair (2027, 2029). What is the next pair of twin primes with the numbers 7 and 9 as the last digits? • Consider the pair (2081, 2083). What is the next pair of twin primes with the numbers 1 and 3 as the last digits? • Consider the pair (2129, 2131). What is the next pair of twin primes with the numbers 9 and 1 as the last digits? • What is the difference between 2027 and the next twin prime with the last digit 7? • What is the difference between 2081 and the next twin prime with the last digit 1? • What is the difference between 2129 and the next twin prime with the last digit 9? The next set of purely conceptual questions is designed to avoid the use of any tool. • Without using Wolfram Alpha, find the smallest pair of twin primes the last digits of which are 7 and 9. What is the next pair of twin primes with the last digits 7 and 9? • Without using Wolfram Alpha, find the smallest pair of twin primes the last digits of which are 1 and 3. What is the next pair of twin primes with the last digits 1 and 3? • Without using Wolfram Alpha, find the smallest pair of twin primes the last digits of which are 9 and 1. What is the next pair of twin primes with the last digits 9 and 1? Now, Wolfram Alpha may be used as an instrument supporting both pragmatic (abolishing time-consuming computations) and epistemic (modifying thinking about patterns) mediation in exploring: • Guess the first five pairs of twin primes with the digits 1 and 3. Check your guess using Wolfram Alpha. Describe what you have found. As a manifestation of collateral creativity, a teacher should be prepared to expect the following questions by students engaged into the above explorations to be placed “within a learning process rather than a teaching process” (de Champlain et al., 2018, p. 499). • Why was my guess about the fifth pair of prime twins with the endings 1 and 3 wrong? I noticed that the difference between the pairs is 30 and thus guessed (131, 133) as the fifth pair, but Wolfram Alpha rejected my guess. Why is it so? • Is there a formula through which prime numbers can be generated like square numbers? • Is there a formula through which a large number of primes can be generated? If so, what is this formula?

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The last three questions are pretty interesting and fundamental for the theory of numbers. Whereas the first question can be answered in terms of the deficiency of reasoning by empirical induction, the second question cannot be answered in affirmative, and the third question will be discussed in Sect. 5.6.2.

5.3 Students’ Confusion as a Teaching Moment and a Source of Collateral Creativity Collateral creativity of a student can be seen as an unexpected manifestation of ‘high-level’ mathematical thinking. Nonetheless, its unexpectedness often reveals immaturity of one’s mathematical experience which is in need of development in order to sustain leaning towards collaterally creative problem-solving outcomes. First, it has to be recognized by a teacher whose role is critical in the development of students’ mathematical knowledge. This knowledge includes the appreciation of the diversity of perspectives on definitions of mathematical concepts which, when presented by a student, may be imprecise, if not erroneous. For example, some elementary teacher candidates think of the term divisibility as an operation and when asked whether 11 is divisible by 2, might answer in affirmative using the equality 11 ÷ 2 = 5.5 as a convincing justification (Abramovich, 2022). Likewise, as was mentioned above, elementary teacher candidates sometimes confuse odd numbers and prime numbers when, for example, call the number 9 a prime number. This confusion of number theory concepts can be considered as a teaching moment to emphasize the importance of definitions of those concepts that open a window to creative discussion of what the definitions describe (see Chap. 3, Sect. 3.4 for distinction between elimination and selection). It is through teachers’ recognizing the presence and sustaining the manifestation of collateral creativity in the classroom that new mathematical knowledge becomes appreciated by their students. In what follows, basic ideas about divisibility and associated concepts will be discussed to motivate a variety of collaterally creative questions about prime numbers eventually being asked.

5.4 Different Definitions of a Prime Number Note that the whole idea behind the theory of numbers is the study of divisibility of integers by prime numbers. In this regard, note that there are several ways a prime number can be defined and investigating the interplay between alternative definitions can foster collateral creativity. With this in mind, consider Definition 5.1. Prime number is a natural number greater than one which is not a product of smaller natural numbers. Collaterally to this definition, the following problem can be posed and discussed.

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Problem 5.2. Can a natural number with exactly two different divisors be a product of smaller natural numbers? Discussion. Consider a natural number n with exactly two different divisors. The first question to be answered is: Can n be equal to one? If n = 1, then n may not have more than one divisor. Therefore, n /= 1. Can a prime number n = k × m, where 1 < k ≤ m < n? For example, the smallest such factors of n are k = m = 2, thus n = 4 − a composite number with three divisors. In fact, the last example was a counterexample to the statement that a natural number n with exactly two different divisors can be a product of smaller natural numbers. In other words, an equivalent alternative to Definition 5.1 is Definition 5.2. Prime number is a natural number with exactly two different divisors. How can one find prime numbers using the above two definitions? The first natural number satisfying Definition 5.1 is the number 2. It is a product of two numbers, 1 and 2, but only 1 is smaller than 2. Likewise, 2 is the first natural number satisfying Definition 5.2: it has exactly two different divisors, 1 and itself. The same can be said about the number 3. It is a product of two numbers, 1 and 3, but only 1 is smaller than 3; likewise, 1 and 3, being different numbers, are the only divisors of 3. Yet the number 4 is not a prime as it can be written as a product of two 2’s. Once, during a classroom discussion of Definition 5.1, an elementary teacher candidate asked as to why the number 9 is not a prime number because just as 3 = 1 × 3 we have 9 = 1 × 9 where only 1 is smaller than 9. This question allowed for justification of Definition 5.2 by pointing out that 9 = 3 × 3 = 1 × 9, that is, 9 has two ways to be multiplicatively decomposed and 3 has only one such way. An unexpected question by the teacher candidate created conditions for the class to appreciate Definition 5.2 provided by a “more knowledgeable other” (Vygotsky, 1978). This definition, because of the word exactly excluded 9 from prime numbers, and because of the word different excluded 1 from prime numbers. Furthermore, the candidate’s example with the number 9, as will be shown in Sect. 5.6.1, provided a springboard into further discussion of prime numbers under the umbrella of collateral creativity.

5.5 Tests of Divisibility and Collateral Creativity Tests of divisibility by 2, 5, and 10 are pretty simple: any number with an even last digit (including zero) is divisible by 2; any number with the last digit either 0 or 5 is divisible by 5; and any number with the last digit 0 is divisible by 10. Formal proofs of the above three statements are based on the fact that any power of 10 is divisible by 2, 5, or 10. The tests of divisibility by 3, 9, and 11 are different. As was mentioned above, those tests are based on the properties of the sums of digits of a tested number. Consequently, they provide many opportunities for collateral creativity.

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Problem 5.3. Prove that an n-digit number N is divisible by 3 (or by 9) if the sum of digits of N is divisible by 3 (or by 9). Discussion. One can represent (a base-ten number) N as a polynomial in the powers of 10 with the digits of N serving as coefficients: N = a1 × 10n−1 + a2 × 10n−2 + · · · + an−1 × 10 + an = a1 × (10n−1 − 1) + a2 × (10n−2 − 1) + · · · + an−1 × (10 − 1) + an = a1 × 99 . . . 9 + a2 × 99 . . . 9     n−1

n−2

+ · · · + an−1 × 9 + (a1 + a1 + · · · + an ) The last expression is divisible by 3 (or by 9) if the sum a1 + a2 + · · · + an is divisible by 3 (or by 9). Indeed, any number, all digits of which are 9’s, is divisible by both 3 and 9. While the last statement seems pretty obvious, representation of such a number as a polynomial in the powers of 10 all coefficients of which are equal to 9 allows one to justify this statement by adding that, due to the distributive property of multiplication over addition, a sum is divisible by 3 or 9 if each term of the sum is divisible by 3 or 9. Problem 5.4. Develop the test of divisibility by 11. Discussion. The test of divisibility by 11 requires one to either subtract from or add to a power of 10 the number 1, depending on whether the power of 10 has an even or an odd number of zeroes. For example, in the former case we have 102k − 1 = 99  . . . 9 = 100 × 99  . . . 9 +99 from where it follows that if 99 . . . 9 is   2k

2k−2

2k−2

divisible by 11, then, by the method of mathematical induction 99  . . . 9 is divisible 2k

by 11 as well (with 99 in the case of k = 1 providing the base clause). Likewise, in the latter case we have 102k+1 + 1 = 1 00  . . . 0 1 = 99 . . . 9 0 + 11 = 909 . . . 09 ×11 × 10 + 11     2k

2k

2k−1

implying that, because 102k+1 + 1 = 909 . . . 09 ×11 × 10 + 11, both sides of the   2k−1

last equality are divisible by 11. Therefore, when in the number N = a1 × 10n−1 + a2 × 10n−2 + · · · + an−1 × 10 + an we have n = 2k (an even-digit number like 3212) so that representing N through the sum of powers of 10 increased or decreased by the number 1, one can show that N is divisible by 11 if divisible by 11 is the sum of its digits taken with alternating signs so that the first and the last digits enter the sum with different signs:

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N = a1 × 102k−1 + a2 × 102k−2 + · · · + a2k−1 × 10 + a2k = a1 × (102k−1 + 1) + a2 × (102k−2 − 1) + · · · + a2k−1 × (10 + 1) + a2k − a2k−1 + a2k−2 + · · · + a2 − a1 . When n = 2 k + 1 (an odd-digit number like 121) we have N = a1 × 102k + a2 × 102k−1 + · · · + a2k × 10 + a2k+1 = a1 × (102k − 1) + a2 × (102k−1 + 1) + · · · + a2k × (10 + 1) + a2k+1 − a2k + a2k−1 + · · · − a2 + a1 . One can see that the first and the last digits enter the sum of digits with the same sign. Students and their teachers alike can be collaterally (co-)creative when using their knowledge of the test of divisibility by 11 by asking someone to write on the board a multidigit number (most likely not divisible by 11) and then modify it (in a variety of ways) to make a modification divisible by 11. For example, if the number written on the board is 123456, the following questions can be asked: • Is it possible by permuting digits in the number 123456 to make it divisible by 11? Why or why not? • Is it possible by permuting digits in the number 1234567 to make it divisible by 11? Why or why not? • Is it possible by just changing a single digit to make a modification of the number 123456 divisible by 11? Can this be done in more than one way? • Can permuting digits in the number 1234567 be done in more than one way to make it divisible by 11? • A palindrome is a number that reads forwards the same as backwards. For example, 1221 is a palindrome which yields another palindrome, 111, when divided by 11. (Same is true for 1331, 1441, 1551). Is every palindrome a multiple of 11? Does every palindrome when divided by 11 yield a palindrome? Why or why not? If not, provide a counterexample.

5.6 Historically Significant Contributions to the Theory of Prime Numbers 5.6.1 The Sieve of Eratosthenes In the third century B.C., a Greek scholar Eratosthenes devised a method (nowadays called the sieve of Eratosthenes; the term sieve used in the previous chapters was borrowed from this context) of separating primes from natural numbers by continuously eliminating all multiples of 2, then all multiples of 3, then all multiples of 5

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(the third prime number), then all multiples of 7 (as 7 is the first number to survive elimination by 2, 3, and 5), and so on. The sieve of Eratosthenes allows one to obtain all the prime numbers less than any given integer N, by crossing out from the set of all natural numbers greater than 1 and smaller than N the multiples of each of the √ primes up to N in turn. All numbers that remain undeleted are the primes sought. Suppose that all multiples of the primes 2, 3, 5, …, pn , where pn is the n-th prime number, were eliminated. Then, only the multiples of primes greater than pn and not divisible by any of the first n primes have survived the elimination process. The smallest multiple of pn+1 (the smallest prime number greater than pn ) that was not 2 . Indeed, the multiples of pn+1 smaller than eliminated by the first n primes is pn+1 2 pn+1 are 2 pn+1 , 3 pn+1 , . . . , pn pn+1 , and each of them was eliminated by one of the first n primes. For example, the smallest composite number that cannot be eliminated by the first three primes—2, 3, and 5—is 49, the square of 7—the smallest prime number greater than 5. In mathematics, prime numbers have long been occupying an important place at the cornerstone of many theoretical developments associated with work of great mathematical minds. Some basic ideas about prime numbers have found their place in school mathematics also. The difficulty in identifying prime numbers and discovering prime factors of composite numbers increases as the magnitude of numbers increases. This is an obstacle for extending the students’ learning about prime numbers beyond the basic ideas. Nonetheless, the study of prime numbers in school mathematics is relevant because “the prime factorization of natural numbers [being unique] is equivalent to factoring polynomials” (Takahashi et al., 2006, p. 213) and, therefore, many topics in high school algebra can be conceptualized in terms of generalized arithmetic. In the age of technology, such conceptualization can be developed even before students reach high school.

5.6.2 Is There a Formula for Prime Numbers? As is well known, unlike the case of natural, odd and even numbers, as well as more complicated numeric sequences such as Fibonacci numbers (see Chaps. 1 and 7), no rule or formula exist to decide whether a number is a prime, or to produce a prime number given its rank. A call for such a formula, whatever its meaning (Matiyasevich, 1999), was probably motivated by the fact, famously proved by Euclid using a proof by contradiction (for the first time in the history of mathematics), that there are infinitely many prime numbers. In his proof, Euclid made an assumption about the existence of the largest prime number pn and then constructed an example of a prime number larger than pn , thus running into contradiction with his own assumption. In this proof, contrary to the statement that the largest prime number does not exist, the finite list of all primes p1 < p2 < · · · < pn was made using which the number N = p1 p2 . . . pn + 1 was constructed. An obvious conclusion is that N is not a prime number as N > pn . This conclusion implied that there exists a prime number P which divides N and does not belong to the above list of n primes; otherwise,

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the difference N − p1 p2 . . . pn = 1 would be divisible by P, something that is not possible. Therefore, an assumption that the list of primes is finite leads to the existence of a new prime number which is not part of the list. In proving that no largest prime number exists, Euclid also formulated the fundamental property of prime numbers: if a prime p divides the product of two integers, then p must divide at least one of those integers. This property of prime numbers can be used to prove the Fundamental Theorem of Arithmetic which states that every integer greater than one can be uniquely represented by a product of one or more prime numbers without regard to their order as factors. In particular this theorem negatively answers a collaterally creative question: Can a number be represented as a product of two prime numbers in more than one way? Whereas the genesis of the very concept of the uniqueness of prime factorization goes back to the time of Euclid, its proof is considered to be given only in the nineteenth century by Gauss.2 Another classic example from the history of mathematics demonstrates the necessity of rigor in conjecturing through empirical induction. It deals with the so-called Fermat primes, the numbers with long and interesting history. In search for a formula producing a prime number given its rank, Fermat3 came across the following five numbers represented through the powers of two increased by one: 0

1

2

F0 = 22 + 1 = 3, F1 = 22 + 1 = 5, F2 = 22 + 1 = 17, 3

4

F3 = 22 + 1 = 257, F4 = 22 + 1 = 65537. All the numbers 3, 5, 17, 257, and 65,537 turned out to be primes. From here, n Fermat conjectured that all numbers of the form Fn = 22 + 1, n = 0, 1, 2, . . . , are prime numbers. Using Wolfram Alpha one can check that all the five numbers are indeed primes. From here one can guess (like Fermat did) that the pattern continues with the growth of the powers of two in the expression for Fn . However, this is not true: the next 5 Fermat number F5 = 22 + 1 is composite. In the digital era, it is pretty easy to come to this conclusion. Indeed, entering the command “factor 2^32 + 1” in the input box of Wolfram Alpha, yields the result shown in Fig. 5.3 and this result refutes the conjecture by Fermat. The prime factorization shown in Fig. 5.3 was known long before the digital era. This is also a truly remarkable gem of number theory. This prime factorization was due to Euler4 who found the two factors theoretically by representing the number F 5 = 4,294,967,297, which is already the sum of two squares, (216 )2 + 12 = 655362 + 12 , as another such sum: 622642 + 204492 . Using Wolfram Alpha as part of the instrumental act makes it possible to find the second representation (Fig. 5.4), something that “immensely extends the possibilities of behavior by making the results of the work of geniuses available to everyone” (Vygotsky, 1930). Introducing a 2

Carl Friedrich Gauss ( 1777–1855)—a German mathematician, commonly regarded as the greatest mathematician of all time. 3 Pierre de Fermat (1607–1655)—a French mathematician. 4 Leonhard Euler (1707–1783)—a Swiss mathematician, the father of all modern mathematics.

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Fig. 5.3 Factoring the 5th Fermat’s “prime” using Wolfram Alpha

Fig. 5.4 Representing 232 + 1 as the sums of two squares

historical component into the modern context of digital technology use, enables one to see the development of mathematical ideas made possible by connecting different concepts of mathematics. To conclude the story about Fermat “primes” note that Gauss linked these numbers to the ancient problem of constructing a regular polygon with compass and straightedge and proved, when he was only 19 years old, that if n is a Fermat “prime”, then a regular polygon with n sides can be so constructed (Conway & Guy, 1996). Another historically famous fact associated with prime numbers is due to Euler who constructed the function f (n) = n 2 − n + 41 (alternatively, f (n) = n 2 + n + 41—a form suggested by Legendre5 (Mollin, 1997)) of an integer variable n that, as shown in Fig. 5.5, for all n ∈ [1, 40] (alternatively, for all n ∈ [0, 39]) produces only prime numbers (although not consecutive ones, except for the first pair, 41 and 43). Furthermore, as n increases, f (n) continues producing mostly prime numbers. For example, in the range n ∈ [41, 120] more than 75% of the values of f (n) are prime numbers. An interesting exploration is to identify twin primes among those numbers 5

Andrien-Marie Legendre (1752–1833)—a French mathematician.

References

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Fig. 5.5 Generating 40 prime numbers through a formula

generated by the function f (n) in the range n ∈ [41, 120]. Once again, collateral creativity by students has potential to integrate historical perspectives as collateral learning of the history of mathematics. The questions may include: • Which palindromes on the list of 40 primes (Fig. 5.5) yield palindromes when multiplied by 11? • Does every palindrome when multiplied by 11 yield a palindrome? • How can one explain that when a palindrome divided by 11 yields a palindrome, the sum of digits of the larger one is twice the sum of the smaller one? (This question is answered in Appendix, Question 2). When reflecting on historical developments of mathematics, in general, and the study of prime numbers, the following quote from David Hilbert’s keynote address to the 1900 International Congress of Mathematicians (Hilbert, 1902), can still inspire future generations of twenty-first century mathematicians to pursue posing and investigating new problems: History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future (p. 437).

References Abramovich, S. (2022). Towards deep understanding of elementary school mathematics: A brief companion for teacher educators and others. Singapore: World Scientific. Conway, J. H., & Guy, R. K. (1996). The book of numbers. Copernicus. de Champlain, Y., DeBlois, L., Robichaud, X., & Freiman, V. (2018). In V. Freiman & J. L. Tassell (Eds.), Creativity and technology in mathematics education (pp. 479–505). Springer. Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the education process. Heath.

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Dewey, J. (1938). Experience and education. MacMillan. Gauss, C. F. (1966) Disquisitiones arithmeticae (translated from Latin by A. A. Clarke). Yale University Press. Hilbert, D. (1902). Mathematical problems (Lecture delivered before the international congress of mathematicians at Paris in 1900). Bulletin of American Mathematical Society, 8(10), 437–479. Matiyasevich, Yu. V. (1999). Formulas for prime numbers. In S. Tabachnikov (Ed.), Kvant Selecta: Algebra and analysis, II (pp. 13–24). The American Mathematical Society. Mollin, R. A. (1997). Prime-producing quadratics. The American Mathematical Monthly, 104(6), 529–544. Piaget, J. (1973). In G. Howson (Ed), Developments in mathematics education: Proceedings of the second international congress on mathematical education. Cambridge University Press. Takahashi, A., Watanabe, T., Yoshida, M., & McDougal, T. (2006) Lower secondary school teaching guide for the Japanese course of study: Mathematics (Grade 7–9). Global Education Resources. Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. [Online materials]. Available at: https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm Vygotsky, L. S. (1978). Mind in society. MIT Press.

Chapter 6

From Square Tiles to Algebraic Inequalities

6.1 Introduction The ideas of this chapter stem from one of the authors’ work with two elementary teacher candidates on the development of a computational learning environment (briefly mentioned in Chap. 1, Example 1.2) for young children to use spreadsheetbased bar graphs as values of fractions virtually representing chances of something to happen (Abramovich et al., 2002). The real-life context included virtual selection of plain and peanut M&M candies and comparison of the likelihood of specific selections expressed via the height of bar graphs. Interactive modifications and reconstructions of bags of M&Ms the quantities of which were controlled by sliders allowed for enriching the learning environment with visual representations of chances. In the course of designing a spreadsheet-based instrument for modeling different real-life scenarios, teacher candidates were asking collaterally creative questions about the behavior of fractions the answers to which not only required transition from numeric inequalities to algebraic ones but, better still, justified the need for this transition. The major conceptual idea that underpins the main topic of this chapter—the need for algebraic inequalities as formal tools of answering a collaterally creative question—can be introduced in a pretty traditional context of using a two-dimensional grid for comparing proper fractions. Similarly to an example of collateral creativity presented in Chap. 1, Sect. 1.4.2, where an accidentally removed square tile led fourth grade students to the recognition of many interesting properties of a modified square, square tiles will be used as a virtual two-dimensional grid allowing for several appropriate modifications not possible with just paper-and-pencil. Once again, in a collaterally creative question by a teacher candidate (Chap. 1. Example 1.2) one can see an unwitting problem posing that happened in the zone of proximal development of the candidate. As will be shown in this chapter, solving the problem posed by the candidate required significant efforts. To begin, consider

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_6

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Problem 6.1. Using square tiles, create a grid to compare fractions 3/4 and 4/5— which one is bigger? (Alternatively, the problem is motivated by comparing chances to pick up a red M&M from the bags of 4 and 5 candies out of which, respectively, 3 and 4 are red). Discussion. This problem was presented to elementary teacher candidates as a reflection on the computational environment introduced in (Abramovich et al., 2002) in which chances were compared by young children through bar graphs representing the values of the corresponding fractions, 3/4 and 4/5: if we have two bags of M&Ms, with 4 and 5 candies, in which, respectively, 3 and 4 are red, for which bag do we have higher chances to pick up a red candy? Similar to the case discussed in Chap. 1, Sect. 1.4.2, the candidates were using square tiles and small stickers with letters X and O (as non-digital technology) to mark fractions. They created a diagram (projected by an overhead to a large white board) shown in Fig. 6.1. The diagram represented a grid comprised of 4 rows and 5 columns in which 3 rows were marked with X’s and 4 columns were marked with O’s. Because there were 15 tiles marked with X’s and 16 tiles marked with O’s, the candidates concluded that 4/5 is greater than 3/4. Accidentally, one of the candidates noted that if the square tiles in the top row and the far-left column were removed, the remaining number of O’s is still bigger than that of X’s. Another candidate noted that the remaining nine O’s and eight X’s represent, respectively, 3/4 and 2/3 of the so reduced grid which means that 3/4 is bigger than 2/3. Furthermore, it was observed that 3/4 resulted from 4/5 and 2/3 resulted from 3/4 through reducing by one both the numerators and the denominators of the original fractions. In general, it was observed that whereas the grid with b rows and d columns can be used to compare proper fractions ab and dc , by removing the top row and the left

Fig. 6.1 Using square tiles and stickers to compare fractions

6.2 Comparing Fractions Using Parts-Within-Whole Scheme

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c−1 column, the remaining grid can be used to compare fractions a−1 and d−1 . It is at b−1 that point that the following collaterally creative question was asked by one of the teacher candidates:

• Is it true that diminishing by one the numerators and the denominators of two fractions, the relationship between the modifications would always be the same as between the original fractions? This question turned out to be similar to the question mentioned in Chap. 1, Example 1.2. That is, once again, collateral creativity can be identified as an unintentional but favorable outcome of the learners’ touching upon hidden ideas of precollege level mathematics through instrumentally supported pedagogical mediation. The authors believe that such outcome is indeed favorable because jointly finding answers to collaterally creative questions leads to epistemic development of both more and less knowledgeable practitioners of mathematics education. Just as in the case of second and fourth grade examples presented in Chap. 1, Sects. 1.4.1 and 1.4.2, the instrumental act that motivated collateral creativity in the mathematics classroom was supported by square tiles, something that elementary teacher candidates have to use in their own classroom. An answer to the above-mentioned question asked by a teacher candidate as a manifestation of collateral creativity will be developed below in the context of pizza sharing.

6.2 Comparing Fractions Using Parts-Within-Whole Scheme In school mathematics, inequalities are used as tools to compare the results of various situations described through different mathematical symbols beginning from integers, then fractions, and, finally, variables. Whereas the order of integers is linear and the magnitude of an integer is defined by its rank within the set of natural numbers, the order of fractions (different from unit fractions) cannot be determined by their rank, although there is one-to-one correspondence between fractions (reduced to the simplest form) and points on the number line. In other words, equivalent fractions, like 2/3 and 4/6, correspond to the same point on the number line. Quantitative comparison of fractions using parts-within-whole scheme (Pitkethly & Hunting, 1996) can be carried out conceptually by using a real-life context. One such context is a fair division of identical circular pizzas to be used throughout this chapter as a “referent unit” (Conference Board of the Mathematical Sciences, 2012, p. 28). Seeing fractions through the lens of dividend-divisor context and using a computer program (its description created within the Geometer’s Sketchpad is included in Appendix) which allows one to construct fraction circles, create conditions for fostering collateral creativity. One of the main arguments of this book is that pedagogy based on the ideas of the instrumental act (Vygotsky, 1930) integrated with a real-life context of productive thinking (Wertheimer, 1959) motivates collateral creativity by the learners

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of mathematics who, quite naturally, as mentioned in other chapters of the book, are capable of asking seemingly simple questions the answers to which are not immediately available even in the presence of a competent teacher. Towards this end, consider Problem 6.2. A pizzeria offers one size pizzas only. A group of nine people ordered five pizzas and a group of seven people ordered four pizzas. In which group would a person get a larger amount of pizza through fair sharing? Discussion. To answer the question quantitatively, one has to decide which fraction, 5/9 or 4/7, is a bigger (or smaller) fraction. This decision can be done qualitatively (i.e., conceptually), without any arithmetical rules of quantitative comparison of fractions. To this end, note that the fraction 5/9, understood as a result of dividing 5 by 9, may represent the result of fair sharing of 5 pizzas among 9 people and the fraction 4/7, understood as a result of dividing 4 by 7, may represent the result of fair sharing of 4 pizzas among 7 people. That is, the fractions 5/9 and 4/7 describe the quantities of pizza each person in the first group and in the second group, respectively, would get through fair sharing. Connecting fractions to division in context, without knowing the result of division as a number, is the only knowledge of fractions that is needed when using parts-within-whole scheme (Pitkethly & Hunting, 1996). This is like comparing the values of the sums 1 + 2 and 2 + 3 informally, without adding numbers, by seeing one and two pizzas in the first case and two and three pizzas in the second case. When all pizzas are identical (being lined up with the same distance between pizzas for the purpose of comparison; alternatively, using one-toone correspondence between the two sets of pizzas), one can recognize more pizza in the second set than in the first set. Within the suggested context, one can begin by finding the largest piece of a pizza that one can get when pizzas are divided in equal parts. Note that numerically, such a piece is typically represented by a unit fraction 1/n, where the numerator represents a pizza as a whole (or “referent unit”) and the denominator represents the number of the pizza’s equal parts. Using virtual fraction circles as tools of the instrumental act, which replaces (often clumsy) drawings by hand with computer-generated images, shows that in both groups one can get half of a pizza as the largest possible piece. Indeed, there are 10 halves in the first group of 9 people (Fig. 6.2) and 8 halves in the second group of 7 people (Fig. 6.3). The use of computer-generated fraction circles, while making hand drawing unnecessary, requires the use of new functions of operating software to allow for the pragmatism of the image construction having a clear epistemic outcome. The remaining halves must be divided among 9 people in the first group and among 7 people in the second group. Numerically, the resulting pieces are 1/18 and 1/14 of a pizza. In the true spirit of collateral creativity, an elementary teacher candidate asked one of the authors if it is still possible to use the whole pizza in doing the comparison. That is, collaterally to an accidental outcome of the division of pizzas when in both groups half of a pizza was left, the candidate surmised that dividing half of a pizza in 9 and 7 pieces and deciding which piece is bigger is equivalent to

6.2 Comparing Fractions Using Parts-Within-Whole Scheme

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Fig. 6.2 Dividing 5 pizzas (and 1 pizza) among 9 people

Fig. 6.3 Dividing 4 pizzas (and 1 pizza) among 7 people

dividing the whole pizza in 9 and 7 pieces, thereby, allowing one to compare larger pieces. Without any knowledge of mathematics, it is quite clear that dividing a pizza among 7 people yields a larger piece than among 9 people. Therefore, each person in the second group would get a larger piece of pizza. Numerically we have 5/9 < 4/7, where 5/9 = 1/2 + 1/18 and 4/7 = 1/2 + 1/14. The last two representations of proper fractions are their Egyptian representations (Chap. 8). The pizzeria context can be extended (modified) further towards nurturing the development of new ideas. This modification is not straightforward and it is based on insight regarding the direction into which the ideas might epistemically develop. Such insight can be provided either by a student or communicated by a teacher who knowingly leads students using guided discovery pedagogy. Put another way, whereas “subject matter has instrumental value as a means of promoting discovery” (McEvan & Bull, 1991, p. 362), the modern-day interplay between mathematics and education calls for teaching strategies that enable students’ discoveries of mathematical ideas under the guidance of their teachers. More specifically, consider Problem 6.3. A pizzeria offers one size pizzas only. A group of nine people ordered five pizzas and a group of seven people ordered four pizzas. Part 1. Suppose a person in the first group took one pizza and left. Likewise, a person in the second group took one pizza and left. Now, in the first group there are 4 pizzas for 8 people; in the second group there are 3 pizzas for 6 people. Compare the amount of pizza an individual in each group would get through fair sharing. Part 2. Suppose that, once again, one of the eight people in the first group took a pizza and left. Likewise, in the second group, one of the six people took a pizza and left. Now, in the first group we have 3 pizzas for 7 people; in the second group we

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have 2 pizzas for 5 people. Compare the amount of pizza an individual in each group would get through fair sharing. Discussion. After one person in each group has taken a pizza and left (Part 1), in each group an individual would get exactly half of a pizza. Numerically, we have 4/ 8 = 3/6 = 1/2. That is, we started (Problem 6.2) with the inequality 5/9 < 4/7 and reducing by one each of the four numbers (5, 9, 4 and 7) we arrived at the equality 4/8 = 3/6. In Part 2, each number in the last equality is reduced by one again and, numerically, the task is to decide which fraction is bigger: 3/7 or 2/5? This time, in the first group, by dividing each of the 3 pizzas in half results in 6 pieces only, not enough to share with 7 people. So, each pizza has to be divided in at least 3 equal parts, thus having 9 pieces of pizza measured by 1/3 of pizza. Each person in the first group can be given 1/3 of a pizza. The remaining 2 pieces have to be divided among 7 people. Likewise, in the second group, each of the 2 pizzas can be divided in 3 equal pieces thus having 6 pieces, so that each of the 5 people can get 1/3 of a pizza. The remaining 1/3 of a pizza has to be divided among five people. Now, one has to compare 2 pieces of a pizza divided among 7 people and 1 piece of a pizza divided among 5 people (each of the three pieces measured 1/3 of a pizza). Once again, collaterally creative idea expressed by students was to make this comparison by using full pizza as a “referent unit” rather than its part. As shown in Fig. 6.4, two pieces and one piece of pizza can be replaced, respectively, by 2 pizzas to be divided into 7 parts and 1 pizza to be divided into 5 parts; therefore, the first division gives a larger quantity of pizza than the second division. Whereas the use of fraction circles is a pragmatic component of the instrumental act supported by the Geometer’s Sketchpad (see the description of the program for construction fraction circles in Appendix), the visual precision of a geometric construction can be used as a springboard into one’s epistemic development. That is, to use the power of the instrumental act in order to embark on a formal proof of what is seen in the diagram of Fig. 6.4. Towards this end, a pretty obvious statement can be made; namely, dividing 2 pizzas among 7 people gives a better deal than dividing 2 pizzas among 8 people. Symbolically, the result of comparison of the deals can be written as 2/7 > 2/8 = 1/4. Furthermore, because dividing a pizza among 4 people is a better deal than dividing it among 5 people, inequality 1/4 > 1/5 holds true. As

Fig. 6.4 Collateral creativity in comparing two fractions by changing the unit

6.3 Collateral Creativity: Calls for Generalization

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Fig. 6.5 Dividing 3 and 2 pizzas among 7 and 5 people, respectively

a result, we have 2/7 > 1/5. That is, we started with the inequality 5/9 < 4/7, then, after decreasing by one each of the four numbers forming the fractions, moved to the equality 4/8 = 3/6 and, after the next such decrease, ended up with the relations 3/7 = 1/3 + x and 2/5 = 1/3 + y, where x > y. Consequently, the inequality 3/7 > 2/5 was corroborated. Put another way, as shown in Fig. 6.5, through the division of 3 and 2 pizzas, the relations 3 = 7 · 13 + 2x and 2 = 5 · 13 + x can be developed, whence, setting x = 1 (a new unit), one can write 2x/7 = 2/7 and x/5 = 1/5 from where it follows that 2/7 > 2/8 = 1/4 > 1/5. That is, dividing 2 pieces among 7 people gives a better deal than dividing 1 piece among 5 people, implying, eventually, that the inequality 3/7 > 2/5 holds true.

6.3 Collateral Creativity: Calls for Generalization In this section, an attempt to generalize from the explorations described in the previous section will be made. It was observed, in the context of creating a computational learning environment for visual comparison of chances of something to happen, that some pairs of equivalent proper fractions can be modified by first decreasing their numerators and denominators by one to get an inequality between so modified fractions and then increase the numerators and denominators of the original fractions by one to get an inequality of the opposite sign. For example, proceeding from the equality 43 = 68 we have (by decreasing and increasing, respectively, each digit by one) the pair of inequalities 23 < 57 and 45 > 79 . The following problem will be used to explore how such fractions can be developed from a pair of equal (alternatively, equivalent) fractions. Problem 6.4. Let

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c a = , 1 < a < b, 1 < c < d. b d

(6.1)

Under which condition do the pairs of fractions 

a−1 c−1 , b−1 d −1

 (6.2)

and 

a+1 c+1 , b+1 d +1

 (6.3)

form inequalities of different signs between their corresponding elements? Discussion. One can check to see that if in addition to relations (6.1) the inequality b+c >a+d holds true, then the inequalities a−1 > b−1 − inequalities (6.1) and (6.4) imply a−1 b−1

c−1 d−1 c−1 d−1

(6.4)

c+1 and a+1 < d+1 hold true. Indeed, b+1 b+c−(a+d) c+1 = (b−1)(d−1) > 0 and d+1 − a+1 = b+1

> 0. In particular, in Problem 6.3 we have 4/8 = 3/6 and 8 + 3 > 4 + 6, so that 3/7 > 2/5 and 5/9 < 4/7. b+c−(a+d) (b+1)(d+1)

Proposition 6.1. Let the integers a, b, c, and d satisfy relations (6.1) and (6.4). Then the elements of the pairs of fractions (6.2) and (6.3) form inequalities of opposite signs.

6.4 Collaterally Creative Question Leads to the Discovery of “Jumping Fractions” The material of this section stems from a question asked by an elementary teacher candidate and formulated at the conclusion of the discussion of Problem 6.1 Same question was raised during one of the authors work with a group of interns on the development of a computational environment aimed at the representation of fractions through bar graphs allowing for their visual comparison (Abramovich et al., 2002). In algebraic form, the question is as follows:   Do there exist positive integers a, b, c, and d such that the pairs of fractions ab , dc  a−1 c−1  and b−1 , d−1 form inequalities of opposite signs between their corresponding elements? Regardless of context, this question demonstrates collateral creativity—an unexpected question asked as a reflection on an activity with fractions—in a pretty mundane (for aspiring teacher candidates) situation because, whenever a fraction

6.4 Collaterally Creative Question Leads to the Discovery of “Jumping …

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was greater than another fraction, the reduction of four numbers by one typically does not change the sign of inequality between the two fractions. For example, 5 4 4 3 > 17 and 11 > 16 , or, for fractions being closer to each other, 45 > 43 and 43 > 23 12 (Fig. 6.1). The reader may try comparing other randomly selected pairs of fractions to see that the pairs sought by teacher candidates are not easy to find right away. Whereas such fractions do exist, collateral creativity by a learner of mathematics, once again, quite unexpectedly, created conditions for the development of new ideas. As a particular illustration, consider Problem 6.5. A pizzeria offers one size large pizzas only. Part 1. A group of 19 people ordered 4 pizzas and a group of 30 people ordered 6 pizzas. In which group would a person get a larger amount of pizza through fair sharing? Part 2. Suppose that one of the 19 people in the first group took a pizza and left. Likewise, in the second group, one of the 30 people took a pizza and left. Now, in the first group there are 3 pizzas for 18 people; in the second group, there are 5 pizzas for 29 people. Compare the amount of pizza the remaining individuals in each group would get through fair sharing. Discussion. In Part 1, common sense (that is, fewer people—better deal) suggests that in the first group, 4 pizzas for 19 people gives a larger piece of pizza for each person than 4 pizzas for 20 people. The latter case can be demonstrated by dividing each pizza in five equal pieces yielding exactly 20 pieces, each measuring 1/5 of a pizza. In the second group, dividing each of the 6 pizzas in 5 equal pieces also yields exactly 30 pieces each measuring 1/5 of a pizza. This division can be concluded in the form of the inequality 4/19 > 6/30. In Part 2, one has to compare dividing 3 pizzas among 18 people and 5 pizzas among 29 people. In the former case, dividing each pizza in 6 equal pieces yields exactly 18 pieces, each measured by 1/6 of a pizza. In the latter case, common sense suggests that dividing 5 pizzas among 29 people gives a better deal than among 30 people, when each individual gets 1/6 of a pizza, exactly the same amount of pizza for each person in the first group. This suggests that in Part 2, a better deal is in the second group. That is, numerically, 3/18 < 5/29. In other words, the transition from Part 1 to Part 2 can be expressed numerically through the pair of inequalities 4/19 > 6/30 and 3/18 < 5/29. The two inequalities may serve as a counterexample to the collaterally creative question raised at the conclusion of the discussion of Problem 6.1 and provide an illustration to Proposition 6.1 formulated at the end of Sect. 6.3. However, finding such pairs of fractions through a system requires transition to algebraic inequalities. Before making this transition, consider Problem 6.6. A pizzeria offers same size large pizzas only. Part 1. A group of 26 people ordered 5 pizzas and a group of 20 people ordered 4 pizzas. In which group would a person get a larger amount of pizza through fair sharing?

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Part 2. Suppose that one of the 26 people in the first group took one pizza and left. Likewise, in the second group, one of the 20 people took one pizza and left. Now, in the first group, we have 4 pizzas for 25 people; in the second group, we have 3 pizzas for 19 people. Compare the amount of pizza the remaining individuals in each group would get through fair sharing. Discussion. In order to show that the inequality 4/20 > 5/26 holds true, first note that its left-hand side indicates that each person in the second group would get 1/ 5 of a pizza after dividing four pizzas in five equal pieces each. In the first group, replacing 26 by 25 yields a better deal, 1/5 of a pizza for each individual (after dividing five pizzas in five equal pieces each), exactly the same amount of pizza as in the second group. Whereas the opposite inequality, 3/19 < 4/25 (after reducing each number by one), is true, the language of pizza sharing requires more involved reasoning, namely, to use common sense twice combined with some understanding of the change of unit. To this end, note that dividing 3 pizzas among 19 people in the first group can be done by dividing each pizza in 7 equal pieces and after giving each individual a piece measured by 1/7 of a pizza, 2 such pieces remain. Similarly, dividing 4 pizzas among 25 people in the second group can be done by dividing each pizza in 7 equal pieces and after giving each individual a piece measured by 1/7 of a pizza, 3 such pieces remain. Now, one has to decide which deal is better: dividing 2 pieces in 19 equal parts or dividing 3 such pieces in 25 equal parts. Because the remaining pieces are all the same size, one can think about them as full pizzas; that is, to compare the fractions 2/19 and 3/25 using the language of pizza sharing. The inequality 2/19 < 2/18 can be justified by reducing the number of participants in the first group from 19 to 18; thus, leading to a better deal. The inequality 3/25 > 3/27 can be justified by extending the number of participants in the second group from 25 to 27; thus, leading to a less good deal. It remains to be shown that 2/18 = 3/27 = 1/9. In other words, the fraction 1/9 is located between the fractions 2/19 and 3/25; that is, 2/19 < 2/18 = 1/9 = 3/27 < 3/25. To summarize Parts 1 and 2, the following pair of inequalities can be presented: 3/20 > 4/26 and 2/19 < 3/25.

(6.5)

In this situation one may seek clarification of the need of reasoning based on common sense twice. First of all, unlike the case of the inequality in the first part of the problem, none of the two fractions, 3/19 and 4/25, is reducible to a unit fraction. Second, an important aspect of this reasoning is the appreciation of the idea of the change of unit. Whereas this example is more involved than the previous examples, it still does not involve large numbers when we indeed need mathematics; only relatively small numbers can be explored qualitatively on the conceptual level, using reasoning based on common sense. Another collaterally creative question seeks explanation of how the two pairs of fractions shown through relations (6.5) can be found. Knowing such fractions can be used in posing contextual problems of the type of Problem 6.6. To this end, let us take advantage of the equivalence of fractions and select the unit fraction 1/5 which

6.5 Algebraic Generalization

89

is equivalent to the fractions 6/30 and 4/20. By common sense, 4/20 < 4/19 as 4 pizzas for 19 people is a better deal than for 20 people (one can come up with this conclusion even prior to any action on pizzas, by noting that 19 < 20). Therefore, 4/ 19 > 6/30. Reducing by one each of the four numbers in the last inequality changes its sign and yields the inequality 3/18 < 5/29. In much the same way, one can start with the unit fraction 1/6 which is equivalent to 4/24 and 5/30. Because 5/30 > 5/31 we have 4/24 > 5/31. The reduction by one of numerators and denominators in the last inequality changes its sign and yields 3/23 < 4/30. The reader may want to prove the last inequality using the context of pizza sharing.

6.5 Algebraic Generalization Similar to Problem 6.6, the unit fraction 1/5 is equivalent to the fractions 4/20 and 5/ 25. By common sense (whatever the context of division), the last fraction is greater than the fraction 5/26. Therefore, 4/20 > 5/26. At the same time, reducing by one each number in the last inequality changes its sign and yields 3/19 < 4/25. This example can be generalized to the following algorithm of developing what may be called “jumping fractions” (Abramovich, 2017, p. 234). Let 1/x be a unit fraction which is equivalent to either xyy or xx2 where x > y > 1. Because xx2 > x 2x+1 , we have y xy

> x 2x+1 . At the same time, one can prove the inequality xy−1 < x−1 by reducing y−1 x2 2 it to the inequality (y − 1)x < (x − 1)(x y − 1) whence x(x − y − 1) + 1 > 0. The last inequality is true as x > y > 1 and, therefore, as integers, x − y ≥ 1, whence x(x − y − 1) + 1 ≥ x(1 − 1) + 1 = 1 > 0. One can see that the context of pizza sharing (or any other division context for that matter) was not used in the (algebraic) proof as context implies concreteness in a sense that it allows one “to probe into the referents for the symbols involved” (Common Core State Standards, 2010, p. 6). Interesting questions to explore with the class are as follows: Can we add a number greater than 1 to x 2 in order to decrease the fractions xx2 and xyy ? Why do we add 1? What if we add 2? As a result of exploring these questions, teacher candidates discovered that whereas 4/20 = 1/5 = 5/25 > 5/27, the inequality sign between 4/20 and 5/27 does not change after decreasing each of the four integers by one as 3/19 > 4/26. The last inequality was again verified in the context of pizza sharing: 3 pizzas were divided in 7 equal pieces each yielding 21 pieces measuring 1/7 of a pizza and 4 pizzas were also divided in 7 pieces each yielding 28 pieces measuring 1/7 of a pizza; in both groups, two identical pieces had to be shared among 19 and 26 people, respectively, indicating a better deal for the first group. This investigation demonstrated two things aimed at the epistemic development of teacher candidates: (i) the number 1 is the only positive integer to be added to x 2 in order to generate “jumping fractions” and (ii) such fractions are indeed very sensitive to numeric modifications.

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6.6 Seeking New Algorithms for the Development of “Jumping Fractions” This section aims at answering the question: Is it possible to find “jumping fractions” by using different strategies? In other words, does the problem of finding “jumping fractions” have more than one correct solution (algorithm)? It turned out that it does. One of the authors learned about a new class of “jumping fractions” from an elementary teacher candidate when, knowing that such fractions are rare and very difficult to find by trial-and-error, proposed to the class that anybody who finds a pair of jumping fractions would not be required to submit the course portfolio for evaluation and receive the highest grade for the course. One student did present several pairs of jumping fractions, (4/14, 2/6), (4/24, 2/10), (4/34, 2/14), and, as a result, received the highest grade for the course. No algorithm explaining the development of those pairs was provided though. What follows is the representation of this type of fractions through an algorithm which can be developed through algebraization. 4 2 To this end, consider two number sequences an = 10n+4 and bn = 4n+2 . One can first prove that an < bn for all n = 1, 2, 3, …. Indeed, the inequality 2 4 < 10n + 4 4n + 2

(6.6)

is equivalent to the inequality 16n + 8 < 20n + 8 which is true for all n > 0. At the same time, the reduction by one each of the two numerators and denominators in the sequences an and bn yields the inequality 1 3 > 10n + 3 4n + 1

(6.7)

Inequality (6.7) is equivalent to the inequality 12n + 3 > 10n + 3, which holds true for all n > 0. Therefore, the pairs of fractions (an , bn ) are “jumping fractions” for all n = 1, 2, 3, …. When n = 1 we have the pair (4/14, 2/6) for which 4/14 < 2/ 6 and 3/13 > 1/5 (Fig. 6.6). At that point, a collaterally creative question was asked by a teacher candidate: Why wasn’t the pair (4/14, 2/6) reduced to the form (2/7, 1/3)? After asking this question, the teacher candidate herself provided an answer: the original pair of “jumping fractions” may not include a unit fraction. That is, a collaterally creative question may be answered by the very inquirer. Another teacher candidate noted that replacing the pair (4/14, 2/6) by the pair (2/7, 2/6), in which none of the fractions is a unit fraction, results in the loss of the “jumping fractions” phenomenon as 2/7 < 2/6 and 1/6 < 1/5. This, once again, shows that “jumping fractions” are sensitive to even equivalent numeric modifications. Finally, the behavior of the “jumping fractions” can be demonstrated by constructing graphs of the pairs (an , bn ) and their modifications. Such graphs are shown in Fig. 6.7—inequality (6.6) at the left-hand side and inequality (6.7) at the right-hand side of the figure. One can see that the graphs are very close to each

6.6 Seeking New Algorithms for the Development of “Jumping Fractions”

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Fig. 6.6 Demonstrating “jumping fractions” 4/14 and 2/6

Fig. 6.7 Graphical demonstration of the pairs (an , bn ) as “jumping fractions”

other, something that can be confirmed numerically—for example, when n = 1, the differences between the two sides of inequalities (6.6) and (6.7) are, respectively, 0.05 and 0.03. 7 5 Similarly, one can prove that the pairs (cn , dn ) where cn = 10n+7 and bn = 7n+5 are “jumping fractions”. When n = 1 we have the pair (7/17, 5/12) for which 7/17 < 5/12 and 6/16 > 4/11. Because 5/12 − 7/17 ∼ = 0.005 and 6/16 − 4/11 ∼ = 0.01, the graphical comparison of this kind of “jumping fractions” would demonstrate that, at least in the first case, the two graphs are visually indistinguishable. By analyzing the pairs (an , bn ) and (cn , dn ), one might conjecture that the pairs 6 4 (en , f n ) where en = 10n+6 and f n = 6n+4 , n = 1, 2, 3, …, represent another example of “jumping fractions”. This conjecture, however, can be refuted through numerical or graphical modeling demonstrating, one more time, that “jumping fractions” are

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6 From Square Tiles to Algebraic Inequalities

indeed quite uncommon; yet they can be discovered due to collateral creativity of students who are eager to ask questions about mathematics they learn and are able to appreciate using technology as a means of continuity of a mathematical conversation.

References Abramovich, S. (2017). Diversifying mathematics teaching: Advanced educational content and methods for prospective elementary teachers. World Scientific. Abramovich, S., Stanton, M., & Baer, E. (2002). What are Billy’s chances? Computer spreadsheet as a learning tool for younger children and their teachers alike. Journal of Computers in Mathematics and Science Teaching, 21(2), 127–145. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. The Mathematical Association of America. Common Core State Standards. (2010). Common core standards initiative: Preparing America’s students for college and career. Available online at: http://www.corestandards.org McEvan, H., & Bull, B. (1991). The pedagogic nature of subject matter knowledge. American Educational Research Journal, 28(2), 316–334. Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30(1), 5–38. Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. [Online materials]. Available at: https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm Wertheimer, M. (1959). Productive thinking. Harper & Row.

Chapter 7

Collateral Creativity and Exploring Unsolved Problems

7.1 Exploring Palindromic Number Conjecture in the Middle Grades The use of famous solved problems as a motivation for the learning of mathematics is a well-known pedagogical approach (Abramovich et al., 2019). Likewise, introducing to learners of mathematics unsolved problems that attracted mathematicians in search for proofs, brings about a taste of real mathematics into the classroom. With the advent of digital technology, such pedagogical opportunities have been significantly enhanced. By trying to interpret the results of modeling mathematical ideas with hidden complexity, students sometimes ask questions the answers to which are unknown not only to their teachers but even to research mathematicians. Questions of that kind might originate in the context of elementary number theory quite a few problems of which, while having mathematically simple formulation, do not have mathematically rigorous solution. Such problems are called conjectures. In the age of technology, simply formulated number theory conjectures can be demonstrated even to young learners of mathematics towards the goal of motivating questions about the properties of numbers they recognize. The following problem was discussed with students by the second author of the article (Abramovich and Strock, 2002), a middle school mathematics teacher (a former student of the first author) in upstate New York, United States. It was also used by the second author of this book with elementary students in schools of New Brunswick, Canada, and their future teachers. Problem 7.1. Take any number with at least two digits. Reverse its digits and add the so modified number to the original one. If the sum does not read the same forwards as backwards, repeat the process with the sum and see whether the new sum reads the same forwards as backwards. Report what you have found.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_7

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Discussion. At the middle school teacher’s suggestion, students started with the number 78. Reversing the digits yielded 87. The sum 78 + 87 = 165. Reversing the digits of 165 yielded 561. The sum 165 + 561 = 726. Reversing the digits of 726 yielded 627. The sum 726 + 627 = 1353. Reversing the digits of 1353 yielded 3531. The sum 1353 + 3531 = 4884—the number reads the same forwards as backwards. Such number is called a palindrome. It took four steps for 78 to reach a palindrome. Students tried other numbers using the spreadsheet shown in Fig. 7.1. They discovered that the number 89 is attracted by a palindrome, 8813200023188, in more than 20 steps. As mentioned in (Guy, 1998), 89 is the smallest such number to be attracted by a palindrome in more than 20 steps. Someone noted that the first iteration yielded 187 the middle digit of which is the sum of other two digits. Another student asked whether this means that the number is divisible by 11 (see Chap. 5, Sect. 5.5, Problem 5.4). Quick computational verification confirmed that other iterations of 89 and that of other numbers are multiples of 11 as well (see asterisks in the spreadsheet of Fig. 7.1 confirming divisibility by 11). Reflecting on this activity, a middle school student noted: “I feel technology helped me because using pen and paper is too time consuming. It would have taken a lifetime to do just what we did in like 10 minutes.” Playing with other numbers by using the spreadsheet of Fig. 7.1 and recognizing their seemingly unavoidable attraction by a palindrome through the reverse and add process which also appears generating multiples of 11 on the way to a palindrome, the students asked collaterally creative questions: • Is the sum of a two-digit number and its reverse always a palindrome? • Is the sum of a two-digit number and its reverse always divisible by 11? • Are the numbers obtained through the reverse and add process always divisible by 11? • Is any natural number attracted by a palindrome? In response to these questions, the middle school students learned from the teacher that the answer to the last question is unknown and asked them to check whether the iterations are always divisible by 11. The teacher said that based on extensive computational experiments it was conjectured that any natural number is attracted by a palindrome. Yet, nobody was able to prove this statement, known as the Palindromic number conjecture (Weisstein, 1999). Its significance for mathematics was recognized by Lehmer (1938), one of the leading computational number theorists of the 20th century, who noted, “the following problem which although it has been

Fig. 7.1 Reaching a 13-digit palindrome in 24 steps

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proposed from time to time has never been solved, as far as I know … [and] it stands apart from the common digital problems and to such a degree that it is a challenge not only to “Digitologists” but to any mathematician” (p. 12). What we know from this quote about the genesis of the problem is that it has been known for at least 90 years. As of the time of writing this book, it still has the status of an unsolved problem. Using the spreadsheet of Fig. 7.1, the following questions can be explored. • Is it possible to find an arithmetic sequence of three-digit numbers that converge to a palindrome in the same number of steps? • Can one find even numbers starting from which a palindrome is reached in a certain number of steps? If so, what is special about such starting values? • The pairs of 4-digit numbers (1069, 1079), (1159, 1169), (1249, 1259), and (1339, 1349) require six steps to reach a palindrome. What can be said about those pairs, both individually and in relation to each other?

7.2 The 196-Problem as a Possible Counterexample to Palindromic Number Conjecture Lehmer (1938) was probably the first to introduce what is now called the 196-problem by mentioning that the number 196 is special because it is unknown if it is attracted by a palindrome through the reverse and add process, and if it is, the number of steps “exceeds 73” (p. 112) and that 1997 is another number “for which the statement may be false” (p. 13). The number 196 is the smallest number for which an attracting palindrome was not found. It belongs to sequence A023108 included in the OEIS® (http://oeis.org/). All the numbers in this sequence have been conjectured to serve as counterexamples to the Palindromic number conjecture. It is interesting to note that even counterexamples to certain statements may be conjectured. Another special property of the number 196, that can be discovered by using the spreadsheet of Fig. 7.1, with the addition of the numbers which Lehmer (1938) found on the 56th and 73rd iterations (apparently, the latter iteration, presented by Lehmer in his paper, 45747660392013285659330918416673654, was published with a misprint), is that the reverse and add process generates the sequence 196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, …, 934217310162393261013712428, …, 45747659181913285659330927106673654 (the correct 73rd iteration), …, the terms of which beginning from the 3rd iteration, 7436, are also multiples of 11 (see Chap. 5, Sect. 5.5, Problem 5.4). Figure 7.2 shows confirmation by Wolfram Alpha that the 73rd iteration, as presented in (Noe and Lee, undated), is a multiple of 11. Other iterations were found by the spreadsheet of Fig. 7.1 to be multiples of 11.

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Fig. 7.2 Verification by Wolfram Alpha of a 35-digit number divisibility by 11

7.3 Formulation of Collatz Conjecture and Its History Another unsolved problem (at the time of writing this book) is the so-called Collatz conjecture named after Lothar Collatz, a German mathematician (1910–1990) who is believed to be the first to come up with it in 1937, although, according to Lagarias (2010), no evidence could be found in Collatz’s publications. In the literature, Collatz conjecture is also known as the 3n + 1 problem for it deals with a sequence defined as follows. Start with any positive integer n and if n is an even number divide it by two to get n/2; otherwise, multiply n by three and add one to get 3n + 1. Whatever the outcome, apply the same rule to either n/2 or 3n + 1. The conjecture states that regardless of the starting number n, the sequence always converges to a three-cycle (4, 2, 1). Put another way, the sequence always reaches the number 1 from which it goes to 4 followed by 2 and 1, thus repeating the cycle (4, 2, 1). For example, as shown in the spreadsheet of Fig. 7.3, starting from the number 7, the cycle (4, 2, 1) is reached on the 14th iteration. A simple formula that does computations is shown in the formula bar of the spreadsheet. Just as the Palindromic number conjecture, the introduction of Collatz conjecture in terms of rules which allow one to build interesting number sequences, is also accessible to young learners of mathematics who can be encouraged to try to guess, after initial explorations, what those sequences have in common. In the sections that follow, it will be shown that the context of this conjecture provided many opportunities for collateral creativity of students at the middle school level. Labelle (2002) explains why the 3n + 1 problem is so interesting by citing its unpredictable behavior in terms of the sequence generated by different numbers which look very random. For example, the number 27 which is relatively small requires 112 steps before it gets into the loop (4, 2, 1) whereas the number 84 requires only 10 steps to reach the number 1 counting 84 as the first step (Fig. 7.4). Labelle (2002) also mentions the number 15733919 which was tried by a computer program resulting in more than 700 iterations before it reaches (4, 2, 1). This randomness in the behavior of the starting values makes the conjecture very difficult to prove. While the investigation of mathematics involved in the search for a proof (or impossibility to produce one) is too complicated for schoolchildren and their future teachers alike, they can still ask deep questions and make investigations, especially

7.3 Formulation of Collatz Conjecture and Its History

Fig. 7.3 From the number 7 to the cycle (4, 2, 1) in 14 steps

Fig. 7.4 The number 10 reaches the number 4 in four steps

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when being supported by such commonly available tool as a spreadsheet. They can also make some small discoveries which could nurture their interest in the study of mathematics. For instance, the length of the sequence is a very interesting aspect to investigate. While it is quite easy to determine the shortest length (yet still worthy of doing), the search for the longest path would be an open-ended problem that can be investigated as well (for example, the number 27 produced a very long sequence, can this one be increased?). Other questions to be investigated are: • Is there something to do with parity (odd vs. even) or divisibility? • What is the biggest number one can get in the sequence? • What is the biggest amplitude (the increase in value between two iterations)? In the next section, some possibilities of investigation of these and other questions using a spreadsheet will be demonstrated under the umbrella of collateral creativity.

7.4 Introducing the Rule: Initial Steps in Conjecturing Using a ‘low entry–high ceiling’ strategy for mathematical enrichment activities accessible for all students, Collatz conjecture was explored by one of the authors with schoolchildren of ages 9–13 in a number of Canadian schools as part of enriching and challenging activities (Freiman, 2006; NRICH Team, 2017/2019). Prior to the use of a spreadsheet, the students were asked to explore all numbers in the range 1 through 10 according to the following rule: choose a number and if it is even, divide it by 2; if it is odd, multiply it by 3 and add 1; step by step apply the same rule to a so developed number and continue until you recognize something special. Very soon, all students reported repetition of the (4, 2, 1) triple. Then they were asked to describe how the process of arriving to this triple was different for each starting number. Already, some observations were made, comparing the number 8 (straightly going down, 4, 2, 1, to stop at 1) with the number 9 which produces quite a long sequence (28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 10, 5, 16, 8, 4, 2, 1, …). Collectively, by checking all numbers from 1 to 10, students saw that the rule always produces the triple (4, 2, 1) at the end of the sequence. They became curious to know whether this is true for larger numbers and were motivated to continue investigating for integers greater than 10. After exploring integers greater than 10 (something that is still possible to do with paper and pencil) and coming to the same conclusion that the sequence seems to always reach the triple (4, 2, 1), students were making observations based on this initial investigation. For instance, some of them shared that an odd number in the sequence is always followed by an even number (e.g., the number 7 produces 3 × 7 + 1 = 22) and once a power of two is reached, it descends along the smaller powers of two to stop at 1 (e.g., the number 16 (= 24 ) goes down through 8, 4, 2, 1). Other questions were discussed prompting further investigation. For example: Does descending to the triple (4, 2, 1) always work? Here, trying to find one number

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to disprove conjecture leads to a need to use technology; but some observations could (and did) eventually narrow the search, as seeing a number being already in the sequence generated by a smaller number, no need to check it. For example, based on the sequence for the number 9 (listed above), the students recognized that the number 52 does not need to be checked). These small yet surprising discoveries seemed triggering students’ interest to go further. At that point, a spreadsheet shown in Fig. 7.3 was used to model the behavior of the sequences for different starting numbers, thus increasing opportunities for new questions and discoveries. Following are examples of such open-ended questions that were investigated by the students. • Is an even number a good candidate for a longer sequence? An odd number? A number divisible by 3? • What would be other ‘good’ candidates for a longer sequence? • How ‘high’ can the sequence go? (This means the largest number in a sequence). • What would be the biggest increase (drop) from one number to the next one? • What are consecutive numbers that produce a ‘summit’ (big increase—big drop— big increase)? Then, the students already familiar with the spreadsheet of Fig. 7.3, were introduced to a more complicated spreadsheet (Fig. 7.4) to refine and deepen their investigation.

7.5 Deepening Investigation: Some Possible Paths Using a Spreadsheet Using the spreadsheet of Fig. 7.4, one can do much more than just generate the sequences converging to the triple (4, 2, 1). The tool can interactively record and, most importantly, save another sequence—the number of steps, S(n), it takes the process to reach the cycle starting from a number n. For example, as shown in the spreadsheet of Fig. 7.4, the number 10 (cell B2) is attracted by the triple after four steps (cell E2). Below is the list of challenging questions that can be explored by students under the guidance of their teacher using the spreadsheet of Fig. 7.4. Although challenging questions are mostly come from teachers, with support of the instrumental act such questions also can be asked by capable and aspiring students, whose “special needs” should not be neglected. Starting from seemingly low-level questions requiring, nonetheless, high-level thinking, one should not consider such questions as coming from nowhere as they are naturally afforded by the students’ appropriate use of technology. In the spirit of Vygotsky (1930), students’ ability to control a digital tool by modifying numeric data involved enables them to reconstruct the entire thinking about a problem and, through this process of reconstruction, to begin asking questions. In particular, classroom teaching, enhanced by the appropriate use of spreadsheets, should be organized along the lines of pedagogy that motivates students to

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ask questions and expects teachers, by providing prompts, to assist them in finding answers. • Does zero appear only once in the sequence S(n)? • Is it possible to have two (three, four, five, and so on) equal numbers as consecutive terms of the sequence S(n)? If so, what are these numbers? • How many steps does the n-th power of 2, n > 2, require reaching the triple (4, 2, 1)? • What is the difference between the number of steps found for the numbers 341 and 85? • What is the smallest five-term sequence of consecutive numbers each of which requires the same number of steps to reach the number 4? What is the first number they all meet in this process? • How can one explain the equality S(13) = S(80)? (Fig. 7.5). • It is known that S(11) = 12. What is the smallest n > 11 such that S(n) = 12. • Without developing the entire path starting from the number 85, how can one determine the number of steps required reaching the triple (4, 2, 1)? • Without developing the entire path starting from the number 341, how can one determine the number of steps required reaching the triple (4, 2, 1)? • Does there exist a monotonically increasing sequence an which has the number of steps S(an ) forming the sequence of consecutive natural numbers starting from the number 1? How can such sequence be described by a formula? • Find a formula for the sequence an when S(an + 1) = S(an ) + 2. Answers to the last two questions are provided in Appendix (Questions 3 and 4).

Fig. 7.5 Coming to 40 from different sides

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7.6 Fibonacci Numbers Emerge Through exploring the 3n + 1 problem, one can see an unexpected context where Fibonacci numbers (briefly mentioned in Chap. 1, Sect. 1.4.3) emerge. As a practicing middle school teacher once inquired, “The wonder of the Fibonacci numbers ... they pop up everywhere ... and what does it mean?” This wonder of mathematical and non-mathematical concepts appearing in seemingly unrelated contexts was noted already by Dewey (1929) who emphasized the educational importance of an “empirical situation in which [familiar] objects are differently related to one another” (p. 86). In the specific context of mathematics, Pólya (1957) argued that “One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else” (p. 15). This unforeseen appearance of Fibonacci numbers means that the notion of mathematical connections is truly the most common thread permeating the entire school mathematics curriculum. Indeed, the starting odd number always generates an even number; the starting even number generates either an even number or an odd number. Consider Fig. 7.6 in which the process is presented in the form of a tree diagram: An even number when divided by two yields either even or odd number; an odd number when multiplied by three and increased by one always yields an even number. Fig. 7.6 The tree diagram develops like Fibonacci numbers

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This observation can be expressed through the diagram E + O → 2E + O → 3E + 2O → 5E + 3O → 8E + 5O → · · · the elements of which develop as Fibonacci-like numbers (Chap. 1, Sect. 1.4.3) starting from O and E. By assigning to the letters O and E the numeric value 1, we have the sequence 2, 3, 5, 8, 13, that is, the first step in the tree diagram is described by the third Fibonacci number F3 = 2 the second step by the fourth Fibonacci number F4 = 3, and so on. Let us assume that step n in the tree diagram is described by the Fibonacci number Fn+2 = Fn+1 + Fn , where Fn+1 and Fn describe the number of possibilities for even and odd numbers, respectively. Then, on step n + 1 there are Fn+1 possibilities for even numbers from evens on step n and Fn possibilities for even numbers from odds on step n. Therefore, there are Fn+2 = Fn+1 + Fn possibilities for even numbers and F #$& possibilities for odd numbers on step n + 1 in the tree diagram. That is, the sum Fn+2 + Fn+1 = Fn+3 represents the total number of possibilities on step n + 1. This testing of “transition from n to n + 1” (Pólya, 1954, p. 111) represents a mathematical induction proof that on step n in the tree diagram of Fig. 7.6 there are Fn+1 possibilities for even numbers out of Fn+2 total possibilities.

7.7 More on the Role of a Teacher in Supporting Problem Posing by Students This chapter described the uses of a spreadsheet in exploring the Palindromic number and Collatz conjectures. Despite the simplicity of the rules, these conjectures still remain unproved. They present examples of open-ended tasks attractive for students even at the elementary level, especially with the use of technology that supports investigations. Using a spreadsheet, one creates modeling data enabling many questions to be raised that would not be possible otherwise (Calder, 2010). Classroom activities associated with explorations of the behavior of sequences generated by different starting numbers, prompt questions leading to a variety of new conjectures. In turn, the verification of conjectures can nurture students’ interest in mathematics. While triggering a positive relationship to mathematics in all students, especially at a younger age, these experiences can lead some of them to more advanced mathematical studies (Appelbaum and Freiman, 2014; Knuth et al., 2019). The role of a teacher in such a classroom is critical for motivating students’ questioning and prompting investigations. But with each new question, facilitated by a teacher when needed, students learn the art of problem posing (Brown and Walter, 1990) by asking mathematical questions and looking for the answers by experimenting and modeling (a process where technology is particularly useful). Therefore, some unexpected (by both students and teachers) observations and questions

References

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can arise. In that way, mathematics learning as the instrumental act becomes a reciprocal process when students learn from teachers and teachers learn from students. The more a teacher learns today from students, the more tomorrow’s students would learn from the teacher and through such learning reciprocity both parties epistemically develop, hopefully bringing mathematics closer to the solution of unsolved problems.

References Abramovich, S., & Strock, T. (2002). Measurement model for division as a tool in computing applications. International Journal of Mathematical Education in Science and Technology, 33(2), 171–185. Abramovich, S., Grinshpan, A. Z., & Milligan, D. L. (2019). Teaching mathematics through concept motivation and action learning. Education Research International, 2019, 13. Article ID 3745406. Available online at https://doi.org/10.1155/2019/3745406 Appelbaum, M., & Freiman, V. (2014). It all starts with the Bachet’s game. Mathematics Teacher, 241, 22–26. Brown, S. I., & Walter, M. I. (1990). The art of problem posing. Lawrence Erlbaum. Calder, N. (2010). Affordances of spreadsheets in mathematical investigation: Potentialities for learning. Spreadsheets in Education, 3(3), Article 4. Dewey, J. (1929). The quest for certainty. Minton, Balch & Co. Freiman, V. (2006). Problems to discover and to boost mathematical talent in early grades: A challenging situations approach. The Montana Mathematics Enthusiast, 3(1), 51–75. Guy, R. K. (1998). What’s left? Math Horizons, 5(4), 5–7. Knuth, E., Zaslavsky, O., & Ellis, A. (2019). The role and use of examples in learning to prove. The Journal of Mathematical Behavior, 53, 256–262. Labelle. J. (2002). Collatz conjecture. Available online at http://online.sfsu.edu/meredith/301/Pap ers/LaBelle,CollatzProblem.pdf Lagarias, J. C. (2010). The 3x + 1 problem: An overview. In J. C. Lagarias (Ed.), The ultimate challenge: The 3x + 1 problem (pp. 3–30). American Mathematical Society. Lehmer, D. H. (1938). Sujets d’étude. Sphinx, 74(8), 12–13. Noe, T. D., & Lee, M. (undated). Table of n, a(n) for n = 0..2390. Online materials (http://oeis.org/ A006960/b006960.txt.). NRICH Team. (2017/2019). Creating a low threshold high ceiling classroom. Available online at https://nrich.maths.org/7701 Pólya, G. (1954). Induction and analogy in mathematics (Vol. 1). Princeton University Press. Pólya, G. (1957). How to solve it. Anchor Books. Vygotsky, L. S. (1930). The instrumental method in psychology (talk given in 1930 at the Krupskaya Academy of Communist Education). Lev Vygotsky Archive. [Online materials]. Available at https://www.marxists.org/archive/vygotsky/works/1930/instrumental.htm Weisstein, E. W. (1999). Palindromic number conjecture. In The CRC concise encyclopedia of mathematics. Chapman & Hall/CRC.

Chapter 8

Egyptian Fractions: From Pragmatic Uses of Technology to Epistemic Development and Collateral Creativity

8.1 Introduction Once, one of the authors met with a small group of high school seniors who wanted to study teaching and become elementary teachers. The goal of the meeting was to do some real-life activities with hidden mathematical meaning. The participants were given paper and pencils and were asked to solve Problem 8.1. Divide fairly among five people three equal size circular pizzas. Discussion. It was expected that the high schoolers would divide each pizza in five equal pieces (sharing the center of a pizza) thus having the total of 15 pieces and the plan was to ask whether a division in a smaller number of pieces is possible. Contrary to these expectations, one participant presented division (Fig. 8.1) in which the number of pieces was 10. Although the intent of the activity was not to talk about fractions but to show different ways of dividing pizzas leading to different number of pieces resulting from each division, this episode can be used to introduce a very special case of representing common fractions, known as Egyptian fraction 1 — representation. Numerically, the diagram of Fig. 8.1 can be described as 35 = 21 + 10 an Egyptian fraction representation of a common fraction as a sum of distinct unit fractions using the so-called Greedy algorithm (see Sect. 8.4) in which 1/2 is the largest unit fraction smaller than 3/5. High school seniors, who divided each pizza in five equal pieces recognized the efficiency of this division and applaud the creativity of their peer. Collaterally, to this comparison in terms of the number of pieces, another participant of the meeting asked if it is possible to have fewer than 10 pieces. Jointly, an answer to this collaterally creative question was found (Fig. 8.2). This final chapter will deal with activities associated with Egyptian fractions and different ways dividing fairly pizzas among people. In support of the activities, Wolfram Alpha, the Geometer’s Sketchpad, and Microsoft Excel will be used.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3_8

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Fig. 8.1 Unexpected division of 3 pizzas among 5 people

Fig. 8.2 Collaterally creative idea of sharing 3 pizzas among 5 people with only 7 slices

8.2 Brief History of Egyptian Fractions Egyptian fractions represent an interesting strand in the history of mathematics. Known for almost four millennia, the term Egyptian fraction means a finite sum of distinct reciprocals of positive integers; alternatively, a finite sum of distinct unit 1 are Egyptian fractions. A unit fraction fractions. For example, 21 + 13 and 21 + 41 + 20 1 itself, like 3 , is an Egyptian fraction, although it can be represented as a sum of 1 1 1 other unit fractions in many different ways; for example, 13 = 14 + 12 + 90 ,3 = 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 + + + , = + + + , = + + + + , and so on. 4 13 156 90 3 6 10 18 90 3 7 10 18 42 90 A remark by a notable French mathematician André Weil (1906–1998) that ancient Egyptians “took a wrong turn” (Graham, 2013, p. 290, italics in the original) when using their notation to represent proper fractions may be revisited because, as will be shown below, Egyptian fractions provide many interesting activities from a mathematics education perspective. Ahmes, an ancient Egyptian scribe, often considered the first mathematician because being the first one saving and presenting mathematical results in a written form, is credited with developing a table, representing fractions of the form 2/n as a sum of distinct unit fractions. For example, 2/5 = 1/ 3 + 1/15 = 1/4 + 1/7 + 1/140 = 1/5 + 1/6 + 1/30 = 1/5 + 1/8 + 1/24 + 1/30. One may observe that whereas, due to the infinitude of unit fractions, the number

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of such representations is infinite, each sum has a finite number of terms. Nonetheless, this table includes only one representation for each fraction of the form 2/n (available online at https://en.wikipedia.org/wiki/Rhind_Mathematical_Papyrus_2/ n_table). The table appeared in the famous Egyptian papyrus roll (ca. 1650 B.C.) found in 1858 by Henry Rhind, a Scottish scholar and collector of antiques (Chace et al., 1927). Although the genesis of using Egyptian fractions is not known, several insightful suggestions of how the representations for that table were found are offered online at the above-mentioned website. A history of ancient Egyptian mathematics provides context for learning fractions and many opportunities of collateral creativity stemming from this context will be demonstrated in the sections of this chapter. Didactical ideas presented below about integrating Egyptian fractions and the context of pizza sharing (see Chap. 6, Sect. 6.2) can forster collateral creativity of teacher candidates and their future students.

8.3 Egyptian Fractions as a Context for Conceptualizations of Fractional Arithmetic Problem 8.2. Find different ways of dividing fairly 5 equal size (circlular) pizzas among 21 people. Compare the number of pieces each way of division yields. Preliminary discussion. Just as in Chap. 6, fair division of pizza means that each person gets the same quantity of pizza. A solution to this problem will be provided at the end of this chapter (see Remark 8.4) to allow for different ideas to be developed. In a mean time, a number of activities involving the use of Wolfram Alpha, the Geometer’s Sketchpad, and Excel spreadsheet dealing with Egyptian fractions will be discussed. The activities will include examples of collateral creativity motivated by the use of technology. To begin, consider the sum 1/2 + 1/4 + 1/20, an Egyptian fraction representation of the fraction 4/5 through the sum of three unit fractions, generated by Wolfram Aplha in response to the command “4/5 as Egyptian”. A competent teacher, leading students to a situation where they may be collaterally creative, can note that the sum 1/2 + 1/ 5 + 1/10 is another Egyptian fraction representation of the fraction 4/5 through the sum of three unit fractions (not provided by Wolfram Alpha). Consequently, students may be asked to decide what is different about the two representations. Students might notice that whereas the denominators of the fractions 1/4 and 1/5 differ by one, the denominators of the fractions 1/20 and 1/10 differ by the factor two. In other words, iterating 1/5 by 1/20 twice yields the same point as iterating 1/4 by 1/20 once. That is, 1/5 + 1/20 + 1/20 = 1/4 + 1/20 = 3/10. As Grade 3 expectations in the United States include conceptualizing “a fraction as a number on the number line” (Common Core State Standars, 2010, p. 24), the idea of iteration can be turned into an investigation on the number line. The following questions, being collateral to the transition to the number line (Fig. 8.3), can be explored:

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Fig. 8.3 Iterating 1/20 on the number line starting from zero

• How many times, starting from zero, does one iterate the fraction 1/20 in order to reach 1/5? • How many times, starting from zero, does one iterate the fraction 1/20 in order to reach 1/4? • How many times, starting from 1/5, does one iterate the fraction 1/20 in order to reach 1/2? • How many times, starting from 1/4, does one iterate the fraction 1/20 in order to reach 1/2? • Which fraction, 1/4 or 1/5, is closer to 1/2? Just as the closeness of integers can be identified on the number line when second graders “relate addition and subtraction to length” (Common Core State Standars, 2010, p. 20), the closeness of fractions can be identified through measuring distances between them as points on the number line. In that way, by counting the iterations of 1/20, one can see that 1/4 is larger than 1/5 and smaller than 1/2 (it will be explained in Sect. 8.4 that the closeness of 1/4 to 1/2, that is, the inequality 1/5 < 1/4 < 1/2, was the reason Wolftam Alpha geneated the representation with 1/4 and not with 1/ 5). At the same time, the equalty 1/4 + 1/20 = 1/5 + 1/10 implies 1/4 − 1/5 = 1/10 − 1/20. The last relation can be interpreted geometrically as follows: on the number line, the distance between the points 1/4 and 1/5 is the same, 1/20, as between the points 1/10 and 1/20. How can one find other pairs of points on the number line with the same distance? The last question leads to the formulation of Problem 8.3. To the left of 1/2 on the number line, find pairs of points, different from (1/4, 1/5) and (1/10, 1/20), with the distance 1/20 between the points. Discussion. One can use Wolfram Alpha to generate the differences 1/2 − n/20 for different values of n by entering into its input box the command “Table [(1/2 − n/ 20), {n, 1, 10}]” (Fig. 8.4). Besides two pairs mentioned above, such pairs are (1/2, 9/20), (9/20, 2/5), (2/5, 7/20), (7/20, 3/10), (3/10, 1/4), (1/5, 3/20), and (3/20, 1/10). One can check to see that 1/2 − 9/20 = 9/20 − 2/5 = 2/5 − 7/20 = 7/20 − 3/10 = 3/10 − 1/4 = 1/5 − 3/20 = 3/20 − 1/10 = 1/20. But what if the difference 1/4 − 1/5 between the unit fractions is represented through fraction circles 1/4 and 1/5 (Fig. 8.5)? Subtracting the two fraction circles yeilds the fraction circle 1/20 the arc of which is equal to 360°/20, that is, 18°. Alternatively, the arcs of the fraction circles 1/4 and 1/5 are, respectively, 360°/4 =

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Fig. 8.4 Using Wolfram Alpha in developing a numeric table Fig. 8.5 Unit fractions as fraction circles

90° and 360°/5 = 72°, so that 90° − 72° = 18°. Likewise, for the pair (1/5, 3/20) we have the difference 1/20 and in terms of the corresponding fraction circles we have 360°/5 − (3 · 360°/20) = 72° − 54° = 18°. This gives an opportunity to connect unit fractions to basic trigonometry and show the interplay between points in the interval (0, 1) and arcs on a circle.

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8.4 The Greedy Algorithm and Some Practice in Proving Fractional Inequalities Apparently, Wolfram Alpha utilizes the so-called Greedy algorithm when generating Egyptian fractions. This algorithm was first used by Fibonacci to convert a given proper (not unit) fraction into an Egyptian fraction. The algorithm is based on finding the largest unit fraction smaller than the fraction to be converted, then finding the difference between the two fractions and continuing (if the difference is not a unit fraction) by applying the algorithm to the difference. For example, the largest unit fraction smaller than 35 is 21 . Indeed, 35 > 36 = 21 . The last inequality can be justified contextually by seeing the fractions 3/5 and 3/6 as dividing 3 pizzas among 5 and 6 people, respectively; obviously, the fewer people—the better deal (Chap. 6, Sect. 6.2). Furthermore, the fraction 1/2 is the largest unit fraction among all unit fractions. Now, 3 6 5 1 1 1 = 21 + x, whence x = 35 − 21 = 10 − 10 = 10 . So, 35 = 21 + 10 , where 21 + 10 is 5 3 an Egyptian fraction representation of 5 developed through the Greedy algorithm. Using a non-Greedy algorithm could be to write 35 = 13 + x, where 13 is not the 9 5 4 − 15 = 15 . That is, largest unit fraction smaller than 35 . We have x = 35 − 13 = 15 3 1 4 4 1 = 3 + 15 . In turn, 15 = 4 + x (here we use the Greedy algorithm because 41 is the 5 4 4 4 largest unit fraction smaller than 15 ; indeed, 15 > 16 = 41 as dividing 4 pizzas among 4 16 15 − 41 = 60 − 60 = 15 people is a better deal than among 16 people). Therefore, x = 15 1 4 1 4 1 4 3 1 . Alternatively, one can write 15 = 5 + x, whence x = 15 − 5 = 15 − 15 = 15 . 60 4 4 8 5 3 1 1 Also, 15 = 16 + x, whence x = 15 − 16 = 30 − 30 = 30 = 10 . That is, 13 + 41 + 60 , 1 1 1 1 1 1 3 + + and + + are three Egyptian fraction representation of the fraction 3 5 15 3 6 10 5 4 13 with 1/3 being the largest fraction. Solving the equation 15 = 17 + x yields x = 105 . In order not to continue subtracting unit fractions from 4/15 to see if the result is a unit fraction, the following problem can be formulated. Problem 8.4. Prove that there exist exactly three Egyptian fraction representations of 3/5 as a sum of three unit fractions with 1/3 being the largest one. 4 Discussion. To begin, note that the equality 53 = 13 + 15 represents 3/5 through the sum of two fractions one of which is 1/3. Therefore, one has to show that 4/15 has only three representations as a sum of two unit fractions. Computationally, this can be shown using Wolfram Alpha (Fig. 8.6). However, confirming computational results by an analytical demonstration can be seen as a way of using technology for epistemic development. Towards this end note that, due to the inequalities 41 < 4 < 13 (justified contextually above), a representation of 4/15 as a sum of two unit 15 fractions may only include unit fractions smaller than 1/3. For the first few such 4 1 4 1 − 14 = 60 , 15 − 15 = 15 , fractions, subtractions result in the following equalities: 15 4 1 1 4 1 13 − = , and − = . The last subtraction resulted in a not unit fraction. 15 6 10 15 7 105 To show that subtracting fractions smaller than 1/7 would also result in not unit 4 − n1 increases monotonically as n fractions, note that the difference f (n) = 15 4 4 1 increases and is approaching 15 < 12 = 3 while producing three unit fractions:

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Fig. 8.6 Computational confirmation of Problem 8.4

f (10) = 16 , f (15) = 15 , f (60) = 14 . To show that no more unit fractions can be 4 produced by f (n), consider the inequality f (n) < 17 , that is, 15 − n1 < 17 whence 105 ∼ n < 13 = 8.07. One can see that f (n) may produce a unit fraction different from 1/4, 1/5 and 1/6 only when 3 < n ≤ 8. Simple computations would show no unit fraction for f (n) when n = 8. This proves that there exist exactly three Egyptian fraction representations of 3/5 with 1/3 being the largest unit fraction (with 1/4, 1/5 and 1/6 being the second largest, respectively, in each representation). The analytic proof can be confirmed computationally by using Wolfram Alpha (Fig. 8.6).

8.5 Pizza as a Context for Introducing Egyptian Fractions Despite all the worldwide research on teaching and learning fractions carried out over the years (e.g., Charalambous & Pitta-Pantazi, 2007; Kor et al., 2019; Lazi´c et al., 2017; Ndalichako, 2013; Pantziara & Philippou, 2012; Steffe & Olive, 2010) and the latest standards-based recommendations in the United States and Canada to consider multiple interpretations of fractions including the dividend-divisor interpretation (Common Core State Standards, 2010; Conference Board of the Mathematical Sciences, 2012; Association of Mathematics Teacher Educators, 2017; Ontario Ministry of Education, 2020), many modern-day elementary teacher candidates are not familiar with this interpretation. At the same time, the dividend-divisor interpretation comes naturally through probing into the properties of division as an operation. In the context of integer arithmetic, when teaching division, a question whether division is a commutative operation arises. Although division is not commutative, this question can open window to fractions because not only 12 pizzas can be divided among 4 people, but, even more realistically, 4 pizzas can be divided among 12 people as well. The latter gives contextual meaning to the operation 4 ÷ 12 without

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providing a numeric outcome of this operation. The diagram of Fig. 8.7 shows how four pizzas can be divided among 12 people fairly—each pizza is divided into 12 equal slices (pieces). As shown in Fig. 8.7, dividing 4 pizzas into 12 pieces each yields 48 pieces. At the same time, dividing 4 pizzas into 3 pieces each (Fig. 8.8) yields 12 pieces— significantly smaller number of pieces than in the former case. This suggests that when a non-unit fraction can be replaced by a unit fraction, due to the equivalence of the two fractions, this replacement means, contextually, that unit fractions can somehow inform a strategy of sharing pizzas among people with the smaller number of pieces than in the former case. With this in mind, consider the case of dividing 5 pizzas among 6 people. A straightforward way is to divide each pizza into 6 pieces as shown in Fig. 8.9 to get 30 pieces, so that each person would get five slices of a pizza

Fig. 8.7 A fair division of four pizzas among 12 people

Fig. 8.8 Measuring four pizzas by 1/3 of pizza

Fig. 8.9 There are 30 pizza pieces to share among 6 people

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Fig. 8.10 There are 12 pizza pieces to share with 6 people

(each slice measured by 1/6 of a pizza). Could the sharing be done more effectively in a sense of having fewer than 30 pieces? Figure 8.10 shows such division—each of the three pizzas is divided in half and the remaining two pizzas are divided in three equal parts each with the total number of pieces equal to 12. That is, each person would get 1/2 of a pizza and 1/3 of a pizza. Decontextualization from pizzas, used, in the words of Vygotsky (1978, p. 115) as “the first-order symbols”, results in the relation 56 = 21 + 13 , representing “the second-order symbolism”. Once again, we see that the appearance of unit fractions is responsible for the smaller number of pieces in comparison with the straightforward division. This time, although a nonunit fraction 5/6 is already in the simplest form, it can be replaced by a sum of two different unit fractions which, alternatively, informs the context of dividing 5 pizzas among 6 people.

8.6 Semi-fair Division of Pizzas It should be noted that whereas both the straightforward division and Egyptian fraction division of pizzas provide each person with identical pieces (either uniquely identical like in the former case or identical in terms of a number of pieces like in the latter case), it is possible to divide pizzas fairly in terms of the quantity of pizza, yet without providing each person with the same number of pieces. Such division may be called a semi-fair division and its enhancement (not always possible) was suggested (Fig. 8.2) by a high school senior, an aspiring elementary teacher candidate. Figure 8.11 shows a semi-fair division of 5 pizzas among 6 people by cutting off an identical piece (measuring 1/6 of a pizza) from each pizza enabling one person to get 5 such pieces and each of the remaining 5 people to get the rest of each pizza (measuring 5/6 of a pizza). One can note that the total number of pieces in which 5 pizzas were divided through this method is 10—smaller than in the case of the straightforward division (30 pieces) and in the case of Egyptian fraction division (12 pieces). It is interesting to note that one of the authors first learned about this method of dividing pizzas from a manager of an eating establishment when the former asked the latter how to divide pizzas in a smaller number of pieces than an obvious straightforward division provides.

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Fig. 8.11 There are 10 pizza pieces to share 5 pizzas among 6 people

The intent of the query was to see the birth of Egyptian fractions in the practice of real life. Instead, the manager demonstrated what may be called practice-fostered mathematics (Abramovich, 2020). This account has important implications for mathematics education, as it signifies that mathematical ideas often emerge as a reflection on common sense and real-life experience. One can only guess if some 36 centuries ago the use of Egyptian fractions was motivated by something purely contextual and then recorded on the papyrus roll. At the same time, it appears that representations by Ahmes of the fractions of the form 2/n through a sum of distinct unit fractions was not motivated by the need of 2 1 = 17 + 91 and the effective sharing. For example, the Greedy algorithm yields 13 2 1 1 1 papyrus roll has 13 = 8 + 52 + 104 . Interpreting each representation as dividing 2 pizzas among 13 people, indicates 26 and 39 pieces, respectively. One can check to see that the semi-fair division gives 24 pieces (see the spreadsheet of Fig. 8.13, cell X2). Finally, Fig. 8.14 created in support of Remark 8.4 shows 14 pieces. Remark 8.1. The semi-fair division shown in Fig. 8.11 (dividing 5 pizzas among 6 people) cannot be enhanced like it was shown in Fig. 8.2 suggested by a high school senior in the case of dividing 3 pizzas among 5 people. If (in Fig. 8.2) each of the 2 pieces measured by 2/5 of a pizza were cut in half, 3 pizzas would be divided in 9 pieces shared among five people: 3 = 3/5 + 3/5 + 3/5 + (1/5 + 1/5 + 1/5) + (1/5 + 1/5 + 1/5). Instead, the aspiring teacher candidate’s solution can be represented through dividing 3 pizzas into 7 pieces forming equal parts for each of the 5 people: 3 = 3/5 + 3/5 + 3/5 + (2/5 + 1/5) + (2/5 + 1/5). Modification of the semi-fair division in the latter form is possible when the difference between the number of people and the number of pizzas is greater than one. For example, one can divide 5 pizzas among 7 people (note 7 − 5 = 2 > 1) by cutting them into 10 pieces forming seven equal quantities: 5 = 5/7 + 5/7 + 5/7 + 5/7 + 5/7 + (2/7 + 2/7 + 1/7) + (2/ 7 + 2/7 + 1/7). Remark 8.2. It should be noted that the Greedy algorithm may produce more unit fractions than a non-Greedy algorithm in converting a common fraction into an 5 1 1 = 15 + 27 + 945 , Egyptian fraction. For example, according to Wolfram Alpha, 21 1 5 5 1 1 where 5 is the largest unit fraction smaller than 21 . At the same time, 21 = 6 + 14 and 1 5 < 15 . That is, the latter conversion of 21 into an Egyptian fraction was developed 6 through a non-Greedy algorithm and it consists of fewer unit fractions than the former conversion. Consequently, when sharing 5 pizzas among 21 people, the number of

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pizza pieces is larger in the case of the former division (63 pieces) than in the case of the latter division (42 pieces). Remark 8.3. Collaterally creative question (motivated by the use of Wolfram Alpha) is to ask for another example of a common fraction the Egyptian form of which obtained through the Greedy algorithm has a larger number of unit fractions than that of obtained through a non-Greedy algorithm. As in many cases associated with the phenomenon of collateral creativity, the question is easy to ask but not easy to answer for it requires one to carry out a number of computations using trial and error. 11 1 1 = 13 + 17 + 1428 (generated by Wolfram Alpha, where 13 is One such fraction is 28 11 the largest unit fraction smaller than 11 because 11 > 11 = 13 and 11 < 22 = 21 ). At 28 28 33 28 11 1 1 1 1 the same time, 28 = 4 + 7 , 4 < 3 . In order to find such a fraction (like 11/28), one has, by trial and error, to find two unit fractions, add them and then, using Wolfram Alpha, to see if the Egyptian form of the sum has more than two unit fractions.

8.7 The Joint Use of Wolfram Alpha and a Spreadsheet This section deals with algebraic generalization needed for computerization of semifair division. Computerization allows for modeling a large number of special cases, observing which is important for the emergence of collateral creativity. In the case of dividing m pizzas among n people, m < n, through the semi-fair method we have each pizza divided into n − m + 1 pieces, n − m of which measured 1/n of a pizza and one piece measured m/n of a pizza. Therefore, the total number of pieces is equal to m(n − m + 1) and when put together make up m pizzas. Indeed,  m (n − m) n1 + mn = m 1 − mn + mn = m. For example, when m = 4 and n = 7 we have 16 pieces in all: for each of the 4 pizzas there are 3 pieces measuring 1/7 of a pizza, that is, the total of 12 such pieces—enough to share fairly among 3 people, and 1 piece measuring 4/7 of a pizza, that is, the total of 4 pieces—enough to share with the remaining 4 people. Furthermore, the difference between the number of pieces obtained through the straightforward division, nm, and the number of pieces obtained through the semi-fair division, m(n − m + 1), does not depend on the number of people, n. Indeed, nm − m(n − m + 1) = m 2 − m. At the same time, such difference in the case of the straightforward and (the Greedy) Egyptian divisions depends on 1 1 1 and 37 = 13 + 11 + 231 , in the number of people. For example, because 35 = 21 + 10 the case of dividing 3 pizzas among 5 people the difference is 15 − 10 = 5 and in the case of dividing 3 pizzas among 7 people the difference is 21 − 21 = 0. One can integrate Wolfram Alpha and a spreadsheet to study numerically the relationship between the number of pieces into which m pizzas can be divided among n people, n > m, assuming that GCD(n, m) = 1. Such a spreadsheet is called an integrated spreadsheet (Abramovich, 2016). By analyzing numerical data, one can make the following observation. The numbers in the columns for semi-fair division

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display a very simple pattern: numbers equidistant from the top and the bottom of a column are the same. For example, when 2 pizzas are divided among 7 people and when 6 pizzas are divided among 7 people, the resulting number of pieces is the same in both cases (Fig. 8.12, cells L2 and L6, respectively). Here 7 − 6 + 1 = 2. Likewise, when 2 pizzas are divided among 11 people and when 10 pizzas are divided among 11 people, the resulting number of pieces is the same in both cases (Fig. 8.13, cells T2 and T10, respectively). Here 11 − 10 + 1 = 2. Also, when 3 pizzas are divided among 13 people and when 11 pizzas are divided among 13 people, the resulting number of pieces is the same in both cases (Fig. 8.13, cells X3 and X11, respectively). Here 13 − 11 + 1 = 3. These special cases can be generalized algebraically. Indeed, when m pizzas are divided among n people and n − m + 1 pizzas are divided among n people in the semi-fair way, the division yields the same number of pizza pieces: in the first case the number is equal to m(n − m + 1) and in the second case—(n − m + 1)[n − (n − m + 1) + 1] = (n − m + 1)m.

Fig. 8.12 An integrated spreadsheet 1

Fig. 8.13 An integrated spreadsheet 2

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Fig. 8.14 Collateral creativity: dividing 2 pizzas among 13 people with 14 pieces

Remark 8.4. In Sect. 8.6, Remark 8.2, the comparison of the number of pizza pieces when 5 pizzas were fairly divided among 21 people demonstrated that the Egyptian 5 1 1 = 15 + 27 + 945 , fraction representations developed through the Greedy algorithm, 21 5 1 1 and not through the Greedy algorithm, 21 = 6 + 14 , yield 63 pizza pieces (cell AO5 of the spreadsheet of Fig. 8.13) and 42 pizza pieces, respectively. That is, the Greedy algorithm not always provides the most efficient solution, a conclusion not possible to obtain without using Wolfram Alpha. This is a role the instrumental act plays in one’s conceptual development. When using the semi-fair division, the number of pizza pieces is 85 (cell AN5, Fig. 8.13). During this discussion, a student demonstrated collateral creativity by offering a more efficient fair division of 5 pizzas among 21 people than the above three methods—the Greedy algorithm, a non-Greedy algorithm, and the semi-fair division. It was suggested to divide each pizza in four pieces measuring 5/21 of a pizza and one piece measuring 1/21 of a pizza. As a result, there are 20 pieces each measuring 5/21 of a pizza and 5 pieces (for a single person) each measuring 1/21 of a pizza; that is, 25 pizza pieces. The unexpectedness of this method of division of pizzas was a manifestation of collateral creativity motivated by the classroom discussion of different methods and the use of drawings as instruments supporting thinking. This method of division makes it possible to divide 2 pizzas among 13 people with 14 pieces as shown in Fig. 8.14. Finally, the content of this remark completes the solution of Problem 8.1. To conclude note that this chapter showed potential of historical developments of mathematics to provide a context for asking collaterally creative questions and deeper investigations using modern technological tools.

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References Abramovich, S. (2016). Exploring mathematics with integrated spreadsheets in teacher education. World Scientific. Abramovich, S. (2020). Pizzas, Egyptian fractions, and practice-fostered mathematics. Open Mathematical Education Notes, 11(1), 1–18. Association of Mathematics Teacher Educators. (2017). Standards for preparing teachers of mathematics. Available online at https://www.amte.net/ Chace, A. B., Manning, H. P., & Archibald, R. C. (1927). The Rhind mathematical papyrus, British museum 10057 and 10058 (Vol. 1). Mathematical Association of America. Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fraction. Educational Studies in Mathematics, 64(3), 293–316. Common Core State Standards. (2010). Common core standards initiative: Preparing America’s students for college and career. Available online at http://www.corestandards.org Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. The Mathematical Association of America. Graham, R. (2013). Paul Erd˝os and Egyptian fractions. In L. Lovász, I. Z. Ruzsa, V. T. Sós, & D. Palvolgyi (Eds.), Erd˝os centennial (pp. 289–309). Springer and János Bolyai Mathematical Society. Kor, L.-K., Teoh, S.-H., Mohamed, S. S. E. B., & Singh, P. (2019). Learning to make sense of fractions: Some insights from the Malaysian primary 4 pupils. International Electronic Journal of Mathematics Education, 14(1), 169–182. Lazi´c, B., Abramovich, S., Mrda, M., & Romano, D. A. (2017). On the teaching and learning of fractions through a conceptual generalization approach. IEJME–Mathematics Education, 12(8), 749–767. Ndalichako, J. L. (2013). Analysis of pupils’ difficulties in solving questions related to fractions: The case of primary school leaving examination in Tanzania. Creative Education, 4(9), 69–73. Ontario Ministry of Education. (2020). The Ontario Curriculum, Grades 1–8, Mathematics (2020). Available online at http://www.edu.gov.on.ca Pantziara, M., & Philippou, G. (2012). Levels of students’ “conception” of fractions. Educational Studies in Mathematics, 79(1), 61–83. Steffe, L., & Olive, J. (2010). Children’s fractional knowledge. Springer. Vygotsky, L. S. (1978). Mind in society. MIT Press.

Appendix

This Appendix is written to provide answers to several questions posed throughout the chapters of the book. Those answers require more than just basic knowledge of school mathematics and may not be immediately obvious. Appendix also includes details of programming spreadsheets used in the book and the description of developing GSP-based environment for the construction of electronic fraction circles used by the authors throughout the book to draw images.

Part 1: Answers to Challenging Questions Introduced in the Chapters of the Book Chapter 2, Section 2.2 Question 1. How can one explain that, given the total number of legs among three creatures, with the increase by one of the number of legs the creatures may not have, the number of combinations of the three creatures either stays the same or increases? For example, why are there more combinations of three creatures not having three legs than creatures not having two legs when the total number of legs is ten? A possible answer. When we exclude 2 from the sum of three numbers equal to 10, we can replace 2 by 1 + 1 only. When we exclude 3 from the sum of three numbers equal to 10, we can replace 3 by 1 + 2 or by 1 + 1 + 1. One can show that in the latter case more modifications for the remaining two integers are possible. For example, let 10 = 2 + 3 + 5 = (1 + 1) + 3 + 5. The only modifications are 10 = 1 + 4 + 5, 10 = 1 + 3 + 6, and 10 = 1 + 1 + 8. Moreover, 2 may be added to 5 to get 7 which then can be represented as 7 = 3 + 4 so that 10 = 3 + 3 + 4. That is, in the case of not using 2 among the addends we have four modifications (see Fig. A.1).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 S. Abramovich and V. Freiman, Fostering Collateral Creativity in School Mathematics, Mathematics Education in the Digital Era 23, https://doi.org/10.1007/978-3-031-40639-3

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Fig. A.1 Four ways not having addend 2

Now, let 10 = 2 + 3 + 5 = 2 + (1 + 2) + 5. It follows that leaving 1 on the second place we have 10 = 4 + 1 + 5 and 10 = 2 + 1 + 7; leaving 2 on the second place we have another two modifications: 10 = 2 + 2 + 6 and 10 = 4 + 2 + 4. Finally, in the case 10 = 1 + 3 + 6 = 1 + (1 + 1 + 1) + 6 we have 10 = 1 + 1 + 8. That is, in the case of not using 3 among the addends five modifications result (see Fig. A.2). A similar situation with the case of not having four and five legs. Let 10 = 1 + 4 + 5 = 1 + (1 + 3) + 5 = 1 + (2 + 2) + 5. One can see that the replacement of 4 by 1 + 3 yields three modifications, 10 = 1 + 3 + 6, 10 = 1 + 1 + 8, 10 = 2 + 3 + 5; when 4 = 2 + 2 we have 10 = 1 + 2 + 7. Moreover, adding 1 to 5 and replacing 4 by 2 + 2 yield one new modification, 10 = 2 + 2 + 6. When 10 = 2 + 4 + 4 = 2 + (1 + 3) + (1 + 3) or 10 = 2 + (2 + 2) + (2 + 2) or 10 = 2 + (2 + 2) + (1 + 3) we have no new modifications. That is, the case of not using 4 among three addends making up 10 (see Fig. A.3) yields five modifications (like in the case of not using 3). But in the case of not using addend 5, we have the replacements 5 = 1 + 4 and 5 = 2 + 3 which, as shown in Fig. A.4, yield six modifications. To conclude, note that although 3 > 2 and 5 > 4, it is due to 2 and 4 being even numbers that given 10 as the total number of legs among three creatures, there are more creatures without 3 and 5 legs than, respectively without 2 and 4 legs. One can check to see that given 9 as the total number of legs among three creatures, the number of creatures without 2 and 3 legs as well as without 4 and 5 legs is, respectively, the same. That is, this arithmetical phenomenon is rather complicated, and the outcome depends both on the parity of the number of legs among three creatures and the number of legs creatures may not have.

Appendix Fig. A.2 Five ways not having addend 3

Fig. A.3 Five ways not having addend 4

Fig. A.4 Six ways not having addend 5

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Chapter 5, Section 5.5 Question 2. How can one explain that when a palindrome divided by 11 yields a palindrome, the sum of digits of the larger one is twice the sum of the smaller one? Alternatively, when a palindrome multiplied by 11 yields a palindrome, the sum of digits of the product is twice the sum of the former palindrome. A possible answer. First note that for any integer N the sum 10N + N = 11N is a multiple of 11. Let N = 10n−1 an + 10n−2 an−1 + 10n−3 an−2 . . . + 102 an−2 + 10an−1 + an be an n-digit palindrome such that ai + an−i+1 < 10, i = I N T

n  2

+ 1, . . . , n.

(A.1)

(An example of such seven-digit number is 2143412). Then 11N = 10N + N = 10n an + 10n−1 an−1 + · · · + 102 an−1 + 10an + 10n−1 an + 10n−2 an−1 + · · · + 102 an−2 + 10an−1 + an = 10n an + 10n−1 (an + an−1 ) + · · · + 102 (an−1 + an−2 ) + 10(an + an−1 ) + an . One can see that the sum of digits of N is equal to an + an−1 + an−2 + · · · + an−2 + an−1 + an and, when inequality (A.1) holds true, the sum of digits of 11N is equal to an + (an + an−1 ) + (an−1 + an−2 ) + · · · + an−2 + (an−1 + an−2 ) + (an + an−1 ) + an = 2(an + an−1 + an−2 + · · · + an−2 + an−1 + an ). That is, when inequality (A.1) holds true, the sum of digits of the number 11N is twice the sum of digits of the number N. For example, the digits of the palindrome 2143412 satisfy inequality (A.1) and 2143412 × 11 = 23577532—a palindrome. At the same time, the digits of the palindrome 189981 do not satisfy inequality (A.1) and 1189981 × 11 = 2089781—not a palindrome. In other words, not all palindromes when multiplied by 11 yield a palindrome.

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Chapter 7, Section 7.5 Question 3. Does there exist a monotonically increasing sequence an which has the number of steps S(an ) forming the sequence of consecutive natural numbers starting from the number 1? How can such sequence be described by a formula? A possible answer. To answer this question, one can use the spreadsheet of Fig. A.5 (changing the value of the slider-controlled cell B2 (values an ) and analyzing entries of cell E2 (values S(an ) recorded in column G) from where the following equalities can be written down: S(1) = 1, S(2) = 2, S(5) = 3, S(10) = 4, S(20) = 5, S(40) = 6, S(80) = 7, S(160) = 8. Thus, the first eight terms of the sequence an sought are: 1, 2, 5, 10, 20, 40, 80, 160. One can recognize a pattern in the development of this sequence: every term beginning from a4 is twice the previous term. Thus, the sequence an can be defined recursively as follows: a1 = 1, a2 = 2, a3 = 5, an = 2 · an−1 , n ≥ 4.

(A.2)

A less obvious way to describe sequence (A.2) which defines an recursively is to use a closed formula involving the greatest integer function (available in the tool kit of spreadsheet formulas, so it can be verified through spreadsheet modeling).   an = I N T 5 · 2n−3 , n = 1, 2, 3, . . . .

(A.3)

Indeed, according to closed formula (A.3)       5 5 5 = 1, a2 = I N T = 2, a3 = I N T = 5, 4 2 1     a4 = I N T (5 · 2) = 10, a5 = I N T 5 · 22 = 20, a6 = I N T 5 · 23 = 40. a1 = I N T

Fig. A.5 Finding the number of steps to reach the triple (4, 2, 1)

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Note that finding formula (A.3) was due to the use of the On-line Encyclopedia of Integer Sequences (OEIS® ) by typing in its input box the sequence 1, 2, 5, 10, 20, 40, 80, 160. Chapter 7, Section 7.5 Question 4. Find a formula for the sequence an provided that S(an+1 ) = S(an ) + 2. A possible answer. To answer this question, one can use the spreadsheet of Fig. 8.4 and develop the following equalities: S(1) = 1, S(5) = 3, S(20) = 5, S(80) = 7, S(320) = 9. From here, the sequence 1, 5, 20, 80, 320, … can be written down and then generalized to the form of the following recurrence a1 = 1, a2 = 5, an = 4 · an−1 , n ≥ 3.

(A.4)

A more sophisticated way to describe sequence (A.4) is to use the closed formula   an = I N T 5 · 4n−2 , n = 1, 2, 3, . . . ,

(A.5)

which, once again, can be easily modeled within a spreadsheet.

Part 2 (for Chap. 8): Constructing Fraction Circles Using the Geometer’s Sketchpad A fraction circle is a physical representation of a unit fraction. How can one construct an image of a unit fraction using The Geometer’s Sketchpad? Recall that a fraction circle represents a sector of a circle defined by the center from which two radii, forming an angle enclosed by an arc, stem. Thereby, three parameters have to be considered: the location of the center, the length of the radius, and the angle formed by the two radii (alternatively, degree measure of the arc). The following seven steps will be presented with reference to Fig. A.6. Step 1 is to construct a segment AB the left endpoint (A) of which is the center of a circle, the fraction, 1/n, of which is the object to be constructed. This segment represents a radius of the circle. The next four steps deal with the construction of the second radius. To this end, one can rotate the right endpoint (B) of the segment about its left endpoint (A) by 360°/n. The rotation requires one to designate the left endpoint (A) as the center of rotation: double click at point A (step 2) and then click (or highlight) the right endpoint (B) (step 3). To rotate, one has to open the Transform menu, choose Rotate and enter into the resulting dialogue box the number 360°/n as the measure of the angle of rotation. Clicking at the button Rotate yields (step 4) a point (C). Connecting the so constructed point (C) with the center (A) completes (step 5) the construction of radius AC.

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Fig. A.6 Six steps required to construct a fraction circle

The next two steps deal with the construction of an arc supported by the angle 360°/n. To this end, one has to highlight two points, (A) and (C), then open the Construct menu and choose Circle by Center + Point. As a result, the whole circle is constructed. Selecting a point on the circle between the points (B) and (C), highlighting three points—(B), the constructed point, (C), in that order—result in the arc connecting the points (B) and (C). Then the circle and the point between (B) and (C) must be made hidden. This completes step (6). To fill the fraction circle with color, after clicking at the arc, one opens the Construct menu and in the line Arc interior selects either Arc segment or Arc sector (step 7). The process of construction of fraction circle 1/2 (when point (B) is rotated about point (A) by 360°/2) is shown in Fig. A.7.

Fig. A.7 Seven steps leading to the construction of the fraction circle one-half

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Part 3: Programming of Spreadsheets 1. Programming of the spreadsheet of Fig. 2.7 (Chap. 2, Sect. 2.2). See also spreadsheets pictured in Figs. A.1, A.2, A.3 and A.4 in Part 1 of Appendix. A slider-controlled cell A1 displays a number to be decomposed in three positive integer addends. A slider-controlled cell A2 displays an addend which does not appear among the three addends. Row 2, beginning from cell C2, displays consecutive natural numbers as one of the possible addends. Column B, beginning from cell B3, displays consecutive natural numbers as another set of possible addends. Cell C3 is entered with the formula =IF(AND(C$2>=$B3, $A$1-C$2-$B3>=C$2,$A$1-C$2$B3>0, ($B3-$A$2)^2>0, (C$2-$A$2)^2>0), IF($A$1-C$2-$B3=$A$2,“ ”, $A$1C$2-$B3),“ ”) which is replicated to the right across rows and columns. This formula is responsible for generating the third addend. 2. Programming of the spreadsheet of Fig. 2.12 (Chap. 2, Sect. 2.6) Cell A2 is entered with the formula =RANDBETWEEN(1,6) which is replicated across rows and columns to cell C5001. This formula randomly generates one of the integers from the range 1 through 6. Cell D2 is entered with the formula =IF(OR(A2=2,B2=2,C2=2),“ ”, IF(A2+B2+C2=10,1,“ ”)) which is copied down to cell D5001. This action marks with the number 1 those triples of addends of 10 that do not include the number 2. Cell E1 is entered with the formula = COUNTIF(D2:D5001,1) counting the number of triples marked with the number 1. Cell F1 is entered with the formula =E1/5000 which calculates experimental probability of not having addend 2 among three addends of 10 over 5000 trials (Fig. A.8).

Fig. A.8 Experimental probability is calculated in cell F1

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3. Programming of the spreadsheet of Fig. 3.10 (Chap. 3, Sect. 3.8) In column A, beginning from cell A3, the first 1000 natural numbers are defined. In column B squares of numbers from column A are displayed. In cell C3 the formula =IF(B3-1=0, “ ”, IF(AND(SQRT(8*(B3-1)+1)=INT(SQRT(8*(B3-1)+1)), MOD(SQRT(8*(B3-1)+1)-1,2)=0),B3-1, “ ”)) is defined and copied down to cell C1002. This formula is responsible for generating triangular numbers smaller than square numbers by one. In cell E3 the formula =IF(A3>COUNT(C$3:C$1002), “ ”, SMALL(C$3:C$1002,A3)) is defined and copied down column E. In cell F3 numbers from column E increased by one are defined. The last two columns display pairs of triangular and square numbers that differ by one. In column G square roots of numbers from column F are displayed (Fig. A.9). 4. Programming of the spreadsheet of Fig. 3.14 (Chap. 3, Sect. 3.8) In column A beginning from cell A3 the first 1000 natural numbers are defined. In cell B3 the formula =0.5*(3*A3^2-A3) is defined and copied down to cell B1002 thus displaying the first 1000 pentagonal numbers. In cell C3 the formula =IF(B3-1=0, “ ”, IF(AND(SQRT(8*(B3-1)+1)=INT(SQRT(8*(B3-1)+1)), MOD(SQRT(8*(B31)+1)-1,2)=0),B3-1, “ ”)) is defined and copied down to cell C1002. The formula is responsible for generating triangular numbers which are smaller than pentagonal numbers by one. In cell E3 the formula =IF(A3>COUNT(C$3:C$1002), “ ”, SMALL(C$3:C$1002, A3)) is defined and copied down column E. In cell F3 numbers from column E increased by one are defined. As a result, pairs of triangular and pentagonal numbers that differ by one are displayed in columns E and F. In

Fig. A.9 Locating pairs of square and triangular numbers differing by one

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Fig. A.10 Locating pairs of pentagonal and triangular numbers differing by one

cell G3 the formula =(1+SQRT(1+24*F3))/6 is defined and copied down column G. This formula generates the ranks of pentagonal numbers displayed in column F (Fig. A.10). 5. Programming of the spreadsheet of Fig. 7.1 (Chap. 7, Sect. 7.1) Cell B2 is controlled by a slider displaying the starting number in the process of reaching a palindrome. The (hidden) range [A6:A20] is filled with the first 15 natural numbers to serve as positional ranks of iterations in the process of approaching a palindrome; the choice of 15 is due to the limitation of a spreadsheet to deal with numbers having at most 15 digits to guarantee precision of computations. In cell B6 the formula =INT(B$2/10^(LEN(B$2)-$A6))-10*INT(B$2/10^(LEN(B$2)$A6+1)) is defined and replicated down column B. This formula enables computational separation of digits in a number displayed in cell B2. When a positional rank (displayed in column A) of a digit in a number is not greater than the number of digits in this number (computed by the spreadsheet function LEN), the formula instructs the spreadsheet to fill a cell with the corresponding digit (including zero); otherwise, the cell is filled with a zero (Fig. A.11). The next step is to recognize a palindrome (seeing it as a string of digits) through analyzing computationally separated digits in terms of their positional ranks. When the digits equidistant from the beginning and the end of the string are paired, one can see that the sums of their positional ranks are one greater than the number of digits in the string. That is, the larger positional rank can be calculated as the difference between the largest positional rank (i.e., the length of the string) increased by one and the smaller positional rank. For example, the second and the fourth digits in the string 12345 have the positional ranks, respectively, 2 and 5 + 1 – 2 = 4. One can use the spreadsheet function INDEX as a tool

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for identifying a palindrome. This function enables a spreadsheet to choose any number within a one-dimensional array [B5:B19] given its position in this array. In cell C6 the formula =IF($A5>LEN(B$2),0,IF(B5=INDEX(B$5:B$19,LEN(B$2)$A5+1,1),1,0)) is defined and replicated down column C to cell C20. This formula instructs the spreadsheet to generate a zero when a positional rank (column A) is greater than the number of digits in a tested integer; otherwise, through the nested conditional function IF, the formula compares digits equidistant from the beginning and the end of the string and generates the number 1 when the digits are identical. Now, the pair of arrays [B6:B20] and [C6:C20] is replicated to the right. When the number of so generated 1’s is equal to the length of the string (i.e., when each digit in the string is marked with the number 1 due to the last formula) we have a palindrome. In cell C2 the formula =IF(SUM(C6:C21)=INT((LEN(B$2) +1)/2),“PALINDROME”,“ ”) is defined. The formula verifies if the number in the cell immediately to the left is a palindrome. In cell D2 the formula = B2+B6+B7*10+B8*10^2+B9*10^3+B10*10^4+B11*10^5+B12*10^6+B13*10^7 +B14*10^8+B15*10^9+B16*10^10+B17*10^11+B18*10^12+B19*10^13 + B20*10^14 is defined. The pair of cells C2 and D2 is replicated to the right across row 2. In cell C3 the formula =IF(C2=“ ”,1,0) is defined and it marks the cell with the number 1 if in cell C2 there is no message “palindrome”; that is, each one indicates step towards a palindrome. The pair of cells B3 and C3 is replicated to the right across row 3. Cell A3 is entered with the formula = COUNTIF(B3:EF3, 1) which counts the number of 1’s in row 2. Cell B2 is controlled by a slider. In cell C2 the formula =IF(SUM(C6:C21)=INT((LEN(B$2)+1)/ 2),“PALINDROME”,“ ”) is defined. In cell D2 the formula = B2+B6+B7*10+B8*10^2+B9*10^3+B10*10^4+B11*10^5+B12*10^6+B13*10^7 +B14*10^8+B15*10^9+B16*10^10+B17*10^11+B18*10^12+B19*10^13 + B20*10^14 is defined. The pair of cells C2 and D2 is replicated to the right across row 2. In cell C3 the formula =IF(C2=“ ”,1,0) is defined. Cell D3 is empty. The pair of cells C3 and D3 is replicated to the right across row 3. In cell D4 the formula = IF(MOD(D2, 11)=0,“*”,“ ”) is defined. Cell E4 is empty. The pair of cells D4 and E4 is replicated to the right across row 4.

Fig. A.11 Reaching a 13-digit palindrome in 24 steps

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6. Programming of the spreadsheet of Fig. 7.4 (Chap. 7, Sect. 7.5). See also the spreadsheet pictured in Fig. A.5 in Appendix. Column A beginning from cell A2 displays consecutive natural numbers. A slidercontrolled cell B2 displays the tested number. Cell B3 is entered with the formula = IF(MOD(B2,2)=0, B2/2, 3*B2+1) which is replicated down column B. The formula either divides a number by 2 or multiplies it by 3 and increases the product by one. Cell E2 is entered with the formula =IF(B2=0,“ ”, IF(B2=4, 0, XLOOKUP(4, B$2:B$1001,A$2:A$1001)-1)). The formula looks for the number 4 in column B and displays its neighbor in column A decreased by one (which is the number of steps it takes the tested number to reach the number 4). Cell G2 is entered with the formula =IF(B$2=0,“ ”, IF(B$2=A3, E$2, G3)) which is replicated down column G. This formula included the so-called circular reference, that is, reference to a cell in which such formula is defined. It that way, all computations from the previous steps are preserved. For example, both numbers 15 (currently displayed in cell B2) and 14 (previously displayed in cell B2) reach the number 4 in 15 steps (cells G16 and G15, respectively) (Fig. A.5). 7. Programming of the spreadsheets of Figs. 8.12 and 8.13 (Chap. 8, Sect. 8.7) In column A consecutive natural numbers starting from the number 2 (cell A2) are displayed. In cell B1 the number 2 is defined. Cell C1 is empty. In cell D1 the formula =1+B1 is defined. In cell E1 the letter E (Egyptian) is entered. The pair of cells D1 and E1 is replicated to the right across row 1. In cell B2 the formula =IF(AND(B$1>$A2, GCD(B$1, $A2)=1), $A2*(B$1-$A2+1),“ ”) is defined and replicated down column B. Column C is empty. The pair of columns B and C beginning from row 2 is replicated to the right to columns R and S (in Fig. 8.12 and to cells AN and AO in Fig. 8.13). Numbers in columns marked by the letter E are entered one-by-one and are obtained from Wolfram Alpha. For example, the number 6 in cell E2 means that because 2/3=1/2+1/6, dividing 2 pizzas among 3 people using the Egyptian fraction division yields the total of 6 pieces.

Fig. A.12 An integrated spreadsheet